Atomic Spectra Atomic Structure - Socrates

-

Atomic Spectra

Atomic StructureB Y

Gerhard HerzbergResearch Professor of Physics

University of Saskatchewan

TRANSLATED

‘WITH THE CO-OPERATION OF THE AUTHOR

B Y

J. W. T. SpinksProfessor of Physical Chemsitry

Univeristy of Saskatchewan

N E W Y O R K

DOVER PWBLICATIONS

1944

COPSRIGHT, 1937, BY

PRENTICE-HALL, INC. COPYRIGHT, 1844, BY

DOVER PUBLICATIONS

FIRST EDITION, 1937SECOND EDITION, 1944

Preface to Second Edition

THE present edition of this work contains a number of cor-rections and additions; Birge's new set of fundamental

constants has been adopted throughout, and various tables,especially the table of ionization potentials, have been broughtup to date.

The author is indebted to Professor J. W. Ellis of the Univer-sity of California at Los Angeles for a list of errors and correc-tions; several of the author’s students have also been helpful inpointing out certain mistakes.

The author is grateful to Dover Publications for their initiativeand interest in making this book available again in a revisedphoto-offset edition in spite of war-time difficulties.

G. H.SASKATOON, SASK., August, 1944.

Preface .

HET present work is the translation of a volume published inGerman by Theodor Steinkopff about a year ago.1 Atomic

Spectra and Atomic Structure constitutes the first part of a morecomprehensive course on atomic and molecular spectra whichthe author has prepared and given recently.

Though in the past few years several excellent accounts havebeen written on the subject of atomic spectra (cf. bibliography),there is still a need for an elementary introduction that is espe-cially adapted to the beginner in this field and also to those whorequire a certain knowledge of the subject because of its appli-cations in other fields.

For these two groups of readers the discussion of too manydetails and special cases does not seem desirable, since it is likelyto obscure the fundamentally important points. Consequently,in thii book the main stress is laid on the basic principles of thesubject. Great pains have been taken to explain them as clearlyas possible. To this end numerous diagrams and spectrogramsare given as illustrations. Always the experimental results serveas the starting point of the theoretical considerations. Compli-cated mathematical developments have been avoided. Instead,the results of such calculations have been accepted without proof,reference being given to sources where proof can be found.Throughout the work an effort has been made to emphasize thephysical significance of the theoretical deductions.

vi Preface

Rather liberal use has been made of small type in the printingof certain portions of the text. These, together with the foot-notes, contain theoretical explanations and details that may verywell be omitted in a first reading without interfering with anunderstanding of the fundamental points. Throughout the book,in making this distinction between small and ordinary type, theauthor has kept in mind the needs of those readers who wish toobtain a thorough knowledge of only the more important prin-ciples. The part printed in ordinary type is self-sufficient and,adequate for that purpose.

In view of the applications, particularly to the study of molec-ular spectra and molecular structure, some points have been moreextensively treated than others that might appear more impor-tant from the point of view of atomic spectra alone. In general,completeness has not, been attempted except in Tables 17 and18, which give, respectively, nuclear spin values and ionizationpotentials. In these tables, results published up to the begin-ning of the present year have been considered.

A discussion of X-ray spectra has been omitted, as one can befound in almost any advanced physics text.

Naturally in the course of the translation the author has usedevery opportunity to improve the original German presentation.It is believed that in many instances the explanations have beenclarified. Also, certain recent findings have been added.

The author is greatly indebted to Dr. J. W. T. Spinks for hiswillingness to undertake the translation and for his prompt andcareful work in carrying it out. He also owes many thanks toDr. R. N. H. Haslam, who was kind enough to read the entireproof and made numerous and valuable suggestions for improvingthe presentation. Finally, the author wishes to express his appre-ciation to Dr. E. U. Condon, Editor of ‘the PRENTICE-HALLPHYSICS SERIES, and the staff of Prentice-Hall, Inc., for theirhelpful co-operation during the publication of this volume.

G. H.

’ G. Herzberg, Atomspektren und Atomstruktur (Dresden, 1936).

Table of ContentsCBA-RP R E F A C E . . . . . . . . . . . . . . . . . . . . . . .

INTRODUCTION . . . . . . . . . . . . . . . . . . . .Observation of spectra. Light sources. Emission and ab-sorption. Examples. Spectral analysis. Units.

I. THE SIMPLEST LINE SPECTRA AND THE ELEMENTS OF

ATOMIC THEORY . . . . . . . . . . . . . . . .

1. The empirical hydrogen terms . . . . . . . . . .The Balmer series and the Balmer formula.series.

Other hydrogenRepresentation of spectral lines by terms.

2. The Bohr theory of Balmer terms . . . . . . . .Basic assumptions. Electron orbits in the field of a nucleuswith charge Ze. Energy of Bohr’s orbits (Balmer terms).Spectra of hydrogen-like ions. Continuuni at the series limit.

3. Graphical representation by energy level diagrams .Energy level diiam and spectrum. Consideration of the

quantum number k, and the fine structure of the H lines.Selection rule for k.

4. Wave mechanics or quantum mechanics . . . . .Fundamental principles of wave mechanics. Mathematicalformulation. .Physical interpretation of the * function. TheHelsenberg uncertainty principle. Wave mechanics of the Hatom. Momentum and angular momentum of an atom accord-i n g to wave mechanics. Transition. probabilities and selectionrules according to wave mechanics.magnetic dipo e radiation.

Quadrupole radiation and

5. Alkali spectra . . . . . . . . . . . . . . . . .The principal series. Other series. Theoretical interpreta-tion of the alkali series. Alkali-like spark spectra. TheMoseley lines.

6. Spectrum of helium and the alkaline earths . . . .Helium. Heisenberg's resonance for helium. The alkaline

II. MULTIPLET STRUCTURE OF LINE SPECTRA AND ELEC-TRON SPIN . . . . . . . . . . . . . . . . . .

1. Empirical facts and their formal explanation . . .Doublet structure of the alkali spectra.Selection rule for J; compound doublets.

Quantum number J.Triplets and singlets

of the alkaline earths and helium. Prohibition of interom-binations; intercombination lines. Higher multiplicities; termsymbols. Alternation of multiplicities.

vii

PAGE

V

1

1 11 1

1 3

23

28

54

64

7171

. . .viii Contents

CHAPTER P A G E

2. Physical interpretation of the quantum numbers . . 82Meaning of L for several emission electrons. Physical inter-pretation of J: cause of multiplet splitting. Selection rule forJ. Physical interpretation of S.

3. Space quantization: Zeeman effect and Stark effect 96General remarks on Zeeman effect and space quantizationNormal Zeeman effect. Anomalous Zeeman effect. Pasche-Back effect. Stark effect. Statistical weight.

III. THE BUILDING-UP PRINCIPLE AND THE PERIODIC SYS-

1 .

2.

3.

TEM OF THE ELEMENTS. . . . . . . . . . . . . 120

The Pauli principle and the building-up principle . . 120Quantum numbers of the electrons in an atom. Pauli prin-ciple. Prohibition of intercombinations. Application of thePauli principle.

Determination of the term type from the electronconfiguration . . . . . . . . . . . . . . . . 128

Russell-Saunders coupling. Terms of non-equivalent electrons.Terms of equivalent electrons. Electron distribution with anumber of electrons present.

The periodic system of the elements . . . . . . . 138H (hydrogen). He (helium). Li (lithium). Be (beryllium).B (boron). C (carbon). N (nitrogen). 0 (oxygen). F (flu-’orine). Ne (neon). Succeeding periods of the periodic system.

IV. FINER DETAILS OF ATOMIC SPECTRA . . . . . . . . 1521 .

2.

3.

4.

Intensities of spectral lines . . . . . . . . . . . 152General selection rules (dipole radiation). Special selectionrules (dipole radiation). Forbidden transitions. General re-marks on the intensity ratios of allowed lines. Sum rule.

Series limits for several outer electrons, anomalousPage 10. The last number in the first line of the table

should be:terms, and related topics . . . . . . . . . . . 162

Series by excitation of only, one outer electron. Series byexoitation of two electrons; anomalous terms. Excitation ofinner electrons. Term perturbations. Pre-ionization (auto-ionization).

1.23954 x 10-c.Page 35. The clause in the second line of the last paragraph

should be in brackets. thus:

Other types of coupling . . . . . . . . . . . . . 173 Page 194.(j,j) coupling. Transition cases. add :

The interval rule; analysis of multiplets. . . . . . 177General remarks concerning the analysis of atomic spectra.Land6 interval rule. Example of a multiplet analysis. Page 219.

16.50, respeetively, should be changed to:0.749 and 17.27V. HYPERFINE STRUCTURE OF SPECTRAL LINES . . . . 182

1. Isotope effect . . . . . . . . . . . . . . . . . 182Isotope effect for the H atom. Isotope effect for more com-plicated atoms.

Contents ix

CHAPTER2 . Nuclear spin . . . . . . . . . . . . . . . . . 185

Magnitude of the nuclear spin and its associated magneticmoment. Vector diagram allowing for nuclear spin. Selectionrule for F; appearance of a hypermultiplet. Determination of

I and g from hyperfine structure. Zeeman effect o f hyperfinestructure. Statistical weight. Determination of nuclear spin

by the Stem-Gerlach experiment. Results. Importance ofnuclear spin in the theory of nuclear structure.

VI. SOME EXPERIMENTAL RESULTS AND APPLICATIONS . . 197Energy level diagrams and ionization potentials . . 197Magnetic moment and magnetic susceptibility. . . 202Magnetic moment of an atom. Paramagnetii. Paramag-netio saturation. Diamagn etism.solutions and in solids.

Paramagnetiim of ions inMagnetocalorio effect; production of

extremely low temperatures.

Chemical applications . . . . . . . . . . . . . 213Periodicity of chemical properties.(valence). The ionization potential.

Types of chemical bindingElectron affinity. Ionic

compounds. Atomic compounds (homopolar valence). Acti-vated states and collisions of the second kind; elementarychemical processes.

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 237

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . 245

ERRATA

(where d 7 is ‘an element of volume)Between the 8th and 9th lines from the bottom,

Results. The nuclear spins obtained by the abovebrieflyIn the first line of Table 20 the numbers .7157 and

Illustrations xi

IllustrationsFIGURE PAGE

1. Emission spectrum of the hydrogen atom in the visibleand near ultraviolet region [ Balmer series, Herzberg(41)] . . . . . . . . . . . . . . . . . . . . . 5

2. Absorption spectrum of the Na atom [Kuhn (42)] . . 53. Emission spectrum of the Na atom (arc with one Na

electrode) . . . . . . . . . . . . . . . . . . . 54. Arc spectrum of magnesium . . . . . . . . . . . . 65. Spectrum of a mercury-vapor lamp . . . . . . . . . 66. Part of the Fe arc spectrum with large dispersion

(2.7 A/mm.) . . . . . . . . . . . . . . . . . . 77. Band spectrum of the PN molecule [Curry, Hersberg,

and Herzberg (43)] . . . . . . . . . . . . . . . 78. Schematic representation of the H-atom spectrum . . . 1 29. Circular Bohr orbits for the H atom (n = 1 to n = 4) 1 6

10. Elliptical Bohr-Sommerfeld orbits for the H atom withk = 1, 2, and 3 [from Grotrian (8)] . . . . . . . . 18

11. Higher members of the Balmer series of the H atom(in emission) starting from the seventh line and show-ing the continuum [Herzberg (41)] . . . . . . . . 23

12. Energy level diagram of the H atom [Grotrian (8)] . . 2 413. Energy level diagram of the H atom, including fine

structure [Grotrian (8)] . . . . . . . . . . . . . 2614. Photograph of electron diffraction by a silver foil . . . 3015. De Broglie waves for the circular orbits of an electron

about the nucleus of an atom (qualitative) . . . . . 3116. Vibrations of a string: fundamental and overtones . . 3217. Diffraction of De Broglie waves at a slit (uncertainty

principle) . . . . . . . . . . . . . . . . . . . 3718. Radial part of hydrogen eigenfunctions for n = 1, 2, 3 4 019. Nodal surfaces of the part of the hydrogen eigenfunction

independent of r (for l = 3, m = 1) . . . . . . . . 4120. Probability density distribution of the electron for

n = 1, 2, 3 for the H atom plotted as a function ofthe distance r . . . . . . . . . . . . . . . . . . 43

x

FIGURE PAGE

21. Electron clouds (probability density distribution) of theH atom or hydrogen-like ions in different states [afterWhite (51)] . . . . . . . . . . . . . . . . . , 44

22. Probability distribution of momentum and velocity inthe states n = 1, l = 0, and n = 2, l = 0, of the Hatom . . . . . . . . . . . . . . . . . . . . . 48

23. Example of a quadrupole. . . . . . . . . . . . . . 5324. Energy. level diagram of the Li atom [after Grotrian

(8)] . . . . . . . . . . . . . . . . . . . . . . 5725. Energy level diagrams of Li and Li-like ions up to 0 VI 6 126. Moseley diagram of the terms of Li-like ions . . . . . 6327. Energy level diagram for helium . . . . . . . . . . 6528. Energy level diagram for potassium [Grotrian (8)] . . 7 229. Examples of line doublets . . . . . . . . . . . . . 7430. Origin of a compound doublet 2P - “0. . . . . . . . 7531. Some calcium triplets (Ca I) . . . . . . . . . . . 7632. Energy level diagram of Ca I [Grotrian (8)] . . . . . . 7733. Origin of a compound triplet . . . . . . . . . . . . 7834. 4P - ‘D transition for C+ at 6806 A . . . . . . . . . 8035. Addition of 11 and 12 to give a resultant orbital angular

momentnm L for II = 2, 12 = 1, and L = 3, 2, 1 . . 8 436. Precession of II and 12 about the resultant L . . . . , 8437. Vector addition of L and S to give a resultant J for

different examples . . . . . . . . . . . . . . . . 8938. Example of an inverted term 4D. . . . . , . . . . . 9239. Examples of line splitting in a magnetic field (Zeeman

effect) [after Back and Lande (6)] . . . . . . . . 9740. Precession of the total angular momentum J in a mag-

netic field H about the field direction as axis . . . 9841. Space quantization of the total angular momentum J

in a magnetic field H for J = 2 and J = K. . . . . 9942. Schematic representation of the Stern-Gerlach experi-

ment . . . . . . . . . . . . . . . . . . . . . 10143. Space quantization for J = % and J = 1. . . . . . . 10244. Normal Zeeman effect for a combination J = 3 + J = 2 10445. Anomalous Zeeman effect of the sodium D lines,

2P112 + 2Slp and 2P~l~ 4 2Si/2 . . . . . . . . . . . 10746. Anomalous Zeeman splitting of a 3S1 -+ 3P1 transition . 107

xii Illustrations

FIGURE PAGE47. Addition of magnetic moments in an atom (in explana-

tion of the Lande g-formula) . . . . . . . . . . . 10948. Origin of the magnetic moment for a 2P1,z state . . . 11149. Paschen-Back effect for a *P -+ TS transition . . . . . 11350. Stark effect splitting of the helium line X4388 [after

Foster (132)] . . . . . . . . . . . . . . . . . . 11451. Production of an electric dipole moment in an electric

field (Stark effect) and precession of J about the fielddirection . . . . . . . . . . . . . . . . . . . . 115

52. Stark effect of the D lines of Na . . . . . . . . . . 11653. Stark effect in a strong electric field for a 3D term (analo-

gous to the Paschen-Back effect) . . . . . . . . . 11754. Radial charge distribution for the ground states of H,

Li+, Na+, K+ . . . . . . . . . . . . . . . . . . 13655. Energy level diagram for C I . . . . . . . . . . . . 14356. Energy level diagram for N I . . . . . . . . . . . . 14457. Rough representation of the energies of the shells for

different nuclear charge Z (to explain the filling upof inner shells) . . . . . . . . . . . . . . . . . 148

58. Origin of the most important nebular lines (transitionsbetween the low terms of 0 II, 0 III, and N II) . . 158

59. Energy level diagram of the 0 atom, with differentseries limits. . . . . . . . . . . . . . . . . . . 163

60. Origin of an anomalous triplet of the alkaline earths . . 16561. Energy level diagram of Be I with anomalous. term

series [Paschen and Kruger (78)] . . . . . . . . . 16662. Energy level diagram of Zn Ib [Beutler and Guggen-

heimer (80)] . . . . . . . . . . . . . . . . . . . 16863. Perturbed 3F terms of Al II compared with the hydro-

gen-like terms 4Rln2 and with the IF terms . . . . . 17064. Pm-ionization of the terms lying above the first ioniza-

tion potential of an atom or an’ion . . . . . . . . 17265. Relative positions of the terms of a ps configuration . . 17566. Observed relative positions of the first excited 3P and

‘P terms of elements of the carbon group. . . . . . 17667. Hyperfine-structure of three spectral lines . . . . . . 18368. Isotope effect for the 6215 A “line” of Zn II (schematic) 18469. Vector diagram and corresponding energy level diagram

allowing for nuclear spin . . . . . . . . . . . . . 187

Illustrations. . .

x i i i

FIGURE PAGE

70. Precession of the angular momentum vectors about thetotal angular momentum F for the component F = 5of a 5Fl term with I = 2 . . . . . . . . . . . . . 188

71. Energy level diagram showing the hyperfine structurefor the three spectral lines reproduced in Fig. 67 . . 189

72. Zeeman splitting of the hyperfine structure componentsF = 1 and F = 2 of a term with J = $5, I = 35. . . 191

73. Energy level diagram of Al I [Grotrian (8)] . . . . . 19874. Energy level diagram of Cl I [Kiess and de Bruin (103)] 19975. Energy level diagram of Hg I [Grotrian (8)] . . . . . 20276. Energy level diagram of Ni I [Russell (104)] . . . . . 20377. Dependence of magnetization P on field strength and

absolute temperature, H/T (Langevin) . . . . . . . 20678. Dependence of the ionization potential of the neutral

atom on atomic number . . . . . . . . . . . . . 21779. Hydrogen molecule . . . . . . . . . . . . . . . . . 22480. Dependence of the potential energy of two H atoms on

nuclear separation . . . . . . . . . . . . . . . . 225

Tables IntroductionTABLE PAGE

1. Conversion factors for energy units . . . . . . . . . 102. Rydberg constants and first members of Lyman series

for hydrogen-like ions . . . . . . . . . . . . . . 213. J values for doublet terms . . . . . . . . . . . . . 7 34. J values for triplet terms. . . . . . . . . . . . . . 7 85. L values and term symbols for terms with different

electron configurations . . . . . . . . . . . . . . 8 76. Multiplicities for different values of S . . . . . . . . 917. Possible multiplicities for various numbers of electrons 958. Symmetry of the eigenfunctions of helium . . . . . . 1259. Possible states of an electron . . . . . . . . . . . . 127

10. Terms of non-equivalent electrons . . . . . . . . . . 13211. Terms of equivalent electrons. . . . . . . . . . . . 13212. Derivation of terms for two equivalent p-electrons , . 13413. Electron configurations and term types for the ground

states of the elements . . . . . . . . . . . . . . 14014. Intensities for a *P - 2D transition . . . . . . . . . 16115. 4P - 4D transition for Cf [Fowler and Selwyn (59)]. . 17816. Fe I multiplet [Laporte (82)]. . . . . . . . . . . . 18017. Observed values for nuclear spin . . . . . . . . . . 19518. Ionization potentials of the elements. . . . . . . . . 20019. Calculated and observed values for the magnetic mo-

ments of the rare earth ions . . . . . . . . . . . . 20920. Electron affinities . . . . . . . . . . . . . . . . . 21921. Homopolar valency . . . . . . . . . . . . . . . . . 2 2 6

DURING the last few decades the investigation of atomicand molecular spectra has had a decisive influence on

the development of our present ideas of atomic and molec-ular structure. This investigation has shown above allthat only certain discrete energy states are possible for anatom or molecule. The investigation of atomic spectra inparticular, with which we shall occupy ourselves in thisbook, has given us information about the arrangement andmotion (angular momenta) of the electrons in an atom.Furthermore, it has led to the discovery of electron spin andto a theoretical understanding of the periodic system of theelements. The data on the fundamental properties ofdifferent atoms obtained by means of spectra form a basisfor an understanding of molecule formation and the chem-ical apd physical properties of the elements.

In this book we shall be concerned exclusively withoptical line spectra in the, restricted sense of the term-thatis, with atomic spectra in the region from 40A to the farinfrared, and not with X-ray spectra, which extend fromapproximately lOOA to lower wave lengths. The essentialdifference between optical line spectra and X-ray spectra isthat the former correspond to energy changes of the outerelectrons of an atom, and the latter to energy changes of theinner electrons.

Observation of spectra. The separation of light into itsspectral components can be accomplished either by refrac-tion or diffraction. Both phenomena depend upon the waveiength, but in opposite ways: the greater the wave length,the greater is the diffraction of light; but the greater thewave length, the smaller is the refraction of light. For the separation of light by diffraction, gratings are used; forseparation by refraction, prisms. Both methods may be

1

2 Introduction - -

employed except in the region below 1250& where a gratingis necessary. The prism method has the advantage oigreater light intensity, whereas the grating method gener-ally affords greater resolving power.’ The construction anduse of spectroscopes and spectrographs will not be dealtwith here. Information on these topics is given in bibli-ography references at the end of this book: (la), (2a), (3),(4), (11), (14).

Spectra in the far infrared can be investigated only withthermopiles or bolometers; however, below 13,OOOA photo-graphic plates are generally used. By using a photographicplate a large region of the spectrum may be obtained at onetime.

Lenses, prisms, and windows of glass can be used only inthe region from 3~ to 36OOA. At lower wave lengths, glassabsorbs light almost completely and this necessitates theuse of quartz or fluorite. Quartz begins to absorb ap-preciably at 18OOA, and therefore fluorite must be usedbelow this wave length. Fluorite itself begins to absorbstrongly at 1250A, so that below this wave length onlyreflection gratings can be used, with complete exclusion oflenses and windows.2 Since air absorbs strongly at 19OOA,the whole spectrograph must be evacuated for photographsbelow this wave length. Also, in this region the gelatin onthe photographic plates absorbs, and makes necessary theuse of specially prepared plates.3

Light sources. There are many possibilities for the pro-duction of light for spectroscopic investigations. Theprincipal ones are temperature radiation and all kinds ofluminescence-electroluminescence, chemiluminescence, andfluorescence.

In temperature radiation of gases, the atoms or moleculesare excited to light emission by collision with other atoms or

1 Shortly before the short wave-length limit of transmission, a prism canin some cases provide a greater resolving power than a grating.

z Melvin (40) has recently found that LiF transmits down to 1080 A.a These difficulties disappear for the very penetrating X-rays below 4 A.

Introduction 3

molecules, the necessary energy being derived from thekinetic energy of the colliding particles. Therefore a hightemperature is required. Such emission occurs, for ex-ample, in flames, although it is then often mixed withchemiluminescence. Excitation of gases by high tempera-ture alone is obtained, however, in any electric furnace ofsufficiently high temperature-for example, in the Kingfurnace.

Luminescence includes all forms of light emission in whichkinetic heat energy is not essential for the mechanism ofexcitation. Electroluminescence includes luminescence fromall kinds of electrical discharges-such as sparks, arcs, orGeissler tubes of different kinds operating on direct or alter-nating current of low or high frequency. Excitation inthese cases results mostly from electron or ion collision; thatis, the kinetic energy of electrons or ions accelerated in anelectric field is given up to the atoms or molecules of the gaspresent and causes light emission. Chemiluminescence re-sults when energy set free in a chemical reaction is convertedto light energy (see Chapter VI). The light from manychemical reactions (for example, Na + Cl,) and from manyflames is of this type. Photoluminescence, or fluorescence,results from excitation by absorption of light (for example,in fluorescein, iodine vapor, sodium vapor, and so on). Theterm phosphorescence is usually applied to luminescencewhich continues after excitation by one of the abovemethods has ceased.

Emission and absorption. By any of the foregoingmethods, characteristic emission spectra can be obtained foreach substance. They usually vary for a given substanceaccording to the mode of e x

To obtain the absorption spectrum of a substance, lightwith a continuous spectrum (as that from a filament lamp)is passed through an absorbing layer of the substance being

4 Conversely, conclusions as to the mode of excitation may be drawn fromthe kind of spectrum observed.

4 Introduction

investigated and is then analyzed with a spectrograph. Weobtain light lines (absorption lines) or bands on a darkbackground on the photographic plate.5 (See Fig. 2.)The intensity of the absorption can be altered by varyingthe thickness of the ‘absorbing layer., or, in the case of gases,by changing the pressure.

Examples. Examples of simple and complicated opticalline spectra are given in Figs. 1, 2, 3, 4, 5, 6. In the spectraof H, Na, and Mg (Figs. 1-4), regularities are immediatelyapparent, whereas with Hg and Fe such regularities are noteasily recognizable. Actually, both the complicated andthe simple spectra consist of series of lines, or series of linegroups (cf. the figures), whose separation and intensity de-crease regularly toward shorter wave lengths. When thenumber of these series is large, a complic ated spectrumresults. Two such series are indicated in the Hg spectrum(Fig. 5). Fig. 7 shows a typical example of a band spectrum(PN) for comparison with the line spectra. It obviouslyshows a completely different type of regularity. Thisdifference led quite early to the assumption that line spectraare emitted or absorbed by atoms, band spectra by molecules.This assumption has in the course of time been completelyjustified, notably by the fact that with it all the details ofa spectrum can be explained satisfactorily. It has also beenindependently verified by the experiments of W. Wien oncanal rays, and by the determination of line width, which,as a result of the Doppler effect, depends on mass.

Spectral analysis. As already stated, each chemicalelement gives rise to a characteristic line spectrum bysuitable excitation (flame, arc, spark, electric discharge).Conversely, the appearance of a line spectrum can be usedas an analytical test for the presence of an element-a testwhich has the advantage that extraordinarily small amountsof an element can be detected. This method of analysis,

6 Obviously, the reverse holds for visual observation-dark lines appear ona light background.

Introduction 5

I /IHB H, Ha

Fig. 1. Emission Spectrum of the Hydrogen Atom in the Visible and NearUltraviolet Region [Balmer series, Herzberg (41)].position of the series limit.

I-1, gives the theoretical

Fig. 2. Absorption Spectrum of the Na Atom [Kuhn (42)]. The spectro-gram gives only the short wave-length part, starting with the f i f t h line of theprincipal series. The lines appear as bright lines on a dark continuous back-ground, just as on the photographic plate.

f i t tI ,. /,j .

zzizhcucv=? SHARP SERIES (short leaders)%$%2ss%z

DIFFUSE SERIES (long leaders)

Fig. 3. Emission Spectrum of the Na Atom (Arc with One Na Electrode).Three series can be clearly recognized; one of them, the principal series, coin-cides with the absorption series of Fig. 2.

SHARP SERIES

DIFUSE

SERIES

SingletsPRINCIPAL

SERIES

233

22cl

Fig. 4.A

rc Spectrum of M

agnesium.

The different series of the neutral M

g atom are indicated above

and below

. See

Chapter

I, section

6. The

spectral lines

indicated w

ith dotted

lines do

not belong

tothe norm

al series.T

he few w

eaker unm

arked lines in the spectrogram

are lines of Mg+ and im

purities.

II I

*I1*-I 1 jI I 11 ~~~~.~~~~~ ‘/ii: 11 1 1 / ”

:I

II

IIII..-..

II

I III.

-Fig.

5. Spectrum

of

a M

ercury-Vapor

Lam

p.T

wo

spectrograms

with

different exposure

times are given, together w

ith the ware

lengths of m

ost of the lines.T

his spectrum is often used

as a comparison spectrum

.II

If

!I

.1li~

.l.l.--~

..---...-

-.-

--

nlr

,:rl

(’ /’ y

11, i

k 1

,,I

Fig. 6.Part of the Fe A

rc Spectrum w

ith Large D

ispersion (2.7 &m

m.).

The

multiplet indicated

below is described fully

in Chapter

IV, section

4.

Fig. 7.B

and Spectrum

of

the PN

M

olecule [C

urry, H

erzberg, and

Herzberg (43)].

8 Introduction

called spectral analysis, has recently been considerably de-veloped [see bibliography: (15), (16), (17), (18), (19)], butthe results will not be discussed here. Rather, we shallconcern ourselves with the structure of atomic spectra and

the conclusions which can be drawn regarding atomicstructure. However, a knowledge of the structure of thespectrum is of some importance to the spectra-analyst,particularly in the choice of suitable lines for spectro-

analytical tests.

Units. In the infrared, wave lengths are usually meas-ured in terms of 1-1: 1~ = lo-+ mm. In the ordinary opticalregion, wave lengths are measured in Angstrom units:IA = lo+ cm. For wave lengths above 2OOOA, the valuein air under standard conditions, Xr:r 6 is generally used,while X,,, is usually employed for wave lengths below 2000Asince these wave lengths are almost always measured witha vacuum spectrograph.

For the purpose of investigating regularities in spectraand their connection with atomic structure, it is very helpfulto use, instead of the wave length of a given line, the fre-quency or a value that is proportional to the frequency.The frequency (number of vibrations per second) is:

where c is the velocity of light. That is, Y’ is usually a verylarge number. (For X,,, = 1OOOA; Y’ = 3 X lo’“.) Be-cause of this and also because the accuracy of X sometimesis markedly greater than that of c, wave numbers are gener-ally used in spectroscopy:

V’ 1 1’ = c = X,,, = nn,i*Xair

6 When n is the refractive index of air for the wave length concerned,

Therefore Xlir is somewhat smaller than &.c.

Introduction 9

where nair is the refractive index for the wave lengthconsidered. The value v is simply the reciprocal of thewave length in vacuo - that is, the number of waves in1 cm. in vacuo. (Dimensions, cm-l; for X,, = lOOOA,hw = 100,000 cm-l.) In order to obtain the vacuum wavenumber, we must first convert the wave length in air to thewave length in vacuo by multiplying by n,i,, and thentake the reciprocal value. This computation is muchsimplified by using such tables as the Kayser Tabelle derSchwingungszahlen (21).

As will be further explained in Chapter I, the frequencyV’ and the energy E of a light quantum are related by thefundamental equation E = hv’, where h is Planck’s constant(h = 0,624 X lo-*’ erg sec.). The frequency or the wavenumber can therefore serve as a measure of the energy.When a single atom or molecule emits light of wave numberV, the emitted light quantum has an energy E = hv’ = hcv.Therefore 1 cm-l is equivalent to 1.9858 x lo-r6 ergs permolecule. If we consider the elementary act for one molinstead of a single atom or molecule, we must multiply bythe number of molecules in one mol, N = 6.0228 X 1023.Then 1 cm-l is equivalent to 11.960 X 10’ ergs per mol, or2.8575 cal. per mol. using the chemical atomic weight scale.

Finally we must mention the electron-volt, which is verywidely used in atomic physics. One electron-volt is theenergy of an electron which has been accelerated through apotential of. 1 volt.’ The kinetic energy acquired by anelectron of charge e falling through a potential V (inelectrostatic units) is eV ergs. With e = 4.8025 X lo-loelectrostatic units and one volt = 1/299.776 electrostaticunits, it follows that one electron-volt is equivalent to1.6020 X 1O-*2 ergs per molecule, which corresponds to8067.5 cm-’ or 23,053 cal. per mol. All these conversionfactors are collected together in Table I.

1 The volt used here is the absolute volt, which differs slightly from the in-ternational: 1 VOlti*rt = 1 . 0 0 3 4 VO1t.b..

10 Introduction

TABLE 1CONVERSION FACTORS OF ESERGY UNITS

-1 IUnit cm-* ergs/molecule c a l . / m electron-volts

-1 cm-‘. . . 1 1.9858 X lO+ 2.8575.I erg/molecule 5.0358 X 1015

1.239*J) WY1 1.4390 X 10” 6.2421 X 10”

1 fd./mol,.~,.,,, 0.34996 6.9494 x 10-l’ 1 4.3379 x 10-s1 electron-volt , 8067.5 1.60203 X lo-‘* 23053

I l

The values for e, h, N and c are taken from Birge (141).These values differ rather considerably from those used in theoriginal printing of this book; but they are only insignificantlydifferent. from those used in the author’s Molecular Spectra andMolecular Structure I: Diatomic Molecules.

CHAPTER I

The Simplest Line Spectra and the Elementsof Atomic Theory

1. The Empirical Hydrogen Terms

The Balmer series and the Balmer formula. Thesimplest line spectrum is that of the H atom, which is itselfthe simplest atom (see Fig. 1). This spectrum consists, inthe visible and near ultraviolet, of a series of lines whoseseparation and intensity decrease in a perfectly regularmanner toward shorter wave lengths. Similar series areemitted by the alkali atoms, though in greater number andoverlapping one another (see Fig. 3). The spectra of all theother elements likewise consist of such series, which, how-ever, on account of much overlapping, are not always soeasily recognizable.

The apparent regularity of the so-called hydrogen serieswas first mathematically formulated by Balmer. He foundthat the wave lengths of the lines could be representedaccurately by the formula:

x=2!!&n12 - 4

where n1 = 3, 4, 5, . . . , and G is a constant. The equationis now generally written in the form:

where Y is the wave number of the line (see Introduction,p. 8). In this equation a single constant R, the Rydbergconstant, appears and has the value 109,677.581 cm-’( = 13.595 volts).’ In spite of the simplicity of the formula,extraordinarily close agreement is obtained between experi-

l Cf. Birge (145).

Simplest Line Spectra and Atomic Theory [I, 1 I, 2] Bohr Theory of Balmer Terms 13

! :450

al

0 10

0

O.-

Q

Q

Q--

x).-

ml

Do.-

00

Qo-

mental and calculated values, theagreement being within the limitsof spectroscopic accuracy (1 : 10’).

Other hydrogen series. Whenthe number 2 in the Balmer formulais replaced by n2 = 1, 3, 4, 5, . . .and nl is allowed to take the values2, 3, . . . ; 4, 5, . . . ; 5, 6, . . . ; 6,7 , . . .respectively, other series ofwave numbers or wave lengths areobtained. The spectral lines of Hcorresponding to these series haveactually been observed and are foundto have exactly the predicted wavelengths. The first series (n2 = 1)was discovered by Lyman in the farultraviolet; the others, in the infra-red, by Paschen (n2 = 3), Brackett(n2 = 4), and Pfund (n2 = 5).

All these line series of the H atomcan be represented by one formula:

R Rv=2-9.122 nl

(I, 1)

where n2 and nl > n2 are integers,and n2 is constant for a given series.With increasing values of the ordernumber nl, v approaches a limitVC+J = R/n22. That is, the separationof consecutive. members of a givenseries decreases so that v cannotexceed a fixed limit, the series limit.In principle, an infinite number oflines lie at the series limit.

Fig. 8 gives a schematic represen-tation of the complete H spectrum.

Representation of spectral linesby terms. According to formula

(I, 1) the wave number of any line of the H-atom spec-trum is the difference between two members of the series,T(n) = R/n?, having different values of n. These mem-bers are called terms. The lines of other elements alsocan be represented as the difference between two such terms.This conclusion follows empirically from the fact that they,likewise form series. Therefore, quite generally the for-mula for the wave number of a line is:

v = Te - T1 (I, 2)However, the term T usually has a somewhat more com-plicated form than that for the H spectrum. In addition,the first, and second members of the formula are obtainedfrom diferent term series (see below).

The converse of the fact that each spectral line can berepresented as the difference between two terms is embodiedin the Rydberg-Ritz combination principle, which states that,with certain limitations, the difference between any twoterms of an atom gives the wave number of a spectral lineof the atom. For example, the difference between T(4)and T(lO) for hydrogen gives the sixth line of the Brackettseries.

2. The Bohr Theory of Balmer TermsThe fundamental relation between the terms of an atom

and its structure was first recognized by Bohr. Eventhough the Bohr theory is now extended and altered in someessential respects by the new wave or quantum mechanics,we must deal with it briefly at this point, since a knowledgeof this theory considerably simplifies an understanding ofmodern theories. In fact, a number of phenomena in spec-troscopy can be dealt with by using the Bohr theory alone.

Basic assumptions. According to the Rutherford-Bohrtheory, the atom consists of a heavy nucleus with a chargeZe, about which Z electrons rotate. (Z = the ordinalnumber in the periodic system of the elements-that is, theatomic number.) In order to explain the characteristiclight emission by atoms, Bohr proposed two basic assump-tions. (1) Of the infinite number of orbits of an electronabout an atomic nucleus, which are possible according to

Simplest Line Spectra and Atomic Theory [I, 2

classical mechanics, only certain discrete orbits actuallyoccur. These fulfill certain quantum conditions. Further-more, in contradiction to the classical Maxwell theory, theelectron, in spite of accelerated motion, emits no electro-magnetic waves (light) while in one of these discrete orbits.(2) Radiation is emitted or absorbed by a transition of theelectron from one quantum state to another-by a quantumjump-the energy difference between the two states beingemitted or absorbed as a light quantum of energy hv’(h = Planck’s constant, v’ = frequency). The light quan-tum is emitted when the atom goes from a state of higherenergy to one of lower energy, and is absorbed in theconverse case (conservation of energy). The relationhv’ = E,, - E,, therefore holds, E,, and E,, being theenergies of the upper and lower states, respectively. Thisrelation is the Bohr frequency condition. The index n of Edistinguishes the different orbits and their energy valuesfrom one another.

The wave number of the emitted or absorbed light isobtained from the frequency condition:

v’ E.., Es,y=-=h-c 7iG (I, 3)

From the similarity betwe equations (I, 2) and (I, 3)-in both cases v is the difference between two quantitieswhich can take only discrete values, that is, which can benumbered by integers-we see that, apart from a factor, theterms of equations (I, 1) and (I, 2) are equal to the energiesof the quantum states. The E values contain an arbitraryadditive constant. If we take the additive constant so thatE = 0 when the electron is completely removed from thenucleus, the energy values of the different quantum stateswill be negative, since, by the return of an electron to sucha state, energy will be liberated. (A positively chargednucleus attracts electrons.) The terms in (I, 1) and (I, 2)are positive quantities (for hydrogen, T = El+). Therefore

T1 = .- 2,. T, = - 2

I, 2] Bohr Theory of Balmer Terms 1 5

Here - E = W is the work that must be done in order toremove an electron from a given orbit to infinity (separationenergy). Apart from the factor hc, the terms are thereforeequal to the separation energies of the electron in the givenstates. For the lowest state of the atom, the ground state,the separation energy is called the ionization energy, or theionization potential, which accordingly is equal to the largestterm value of the atom. Similarly, apart from the factorhc, the term differences are equal to the energy differencesof the given atomic states.

This connection between term values and energies isshown’experimentally in the work of Franck and Hertz.They observed that, when collisions between electrons andatoms take place, an inelastic collision-that is, an energytransfer from the electron to the atom-can occur when, andonly when, the kinetic energy of the electron is greater thanthat calculated from the term difference for the transition ofthe atom from the ground state into an excited state. Theamount of energy lost by the electron is exactly equal to theexcitation energy of the atom as calculated from the spec-trum. Furthermore, after such a collision, there can beobserved the emission of a spectral line corresponding to thetransition from the excited state to the ground state. [Cf.Geiger-Scheel (lc) .]

Electron orbits in the, field of a nucleus with charge Ze.Taking first the simplest case, in which the orbits are circlesof radius r, we apply Newton’s fundamental law: force= mass X acceleration. Here the force is Coulomb’s at-traction Ze*/+; the acceleration is the centripetal accelera-tion v2/r. Hence

28 m3 28-=-rz r Or

r=-mv2 (I,4)

where m and v are, respectively, the mass and the velocityof the electron. Thus far we have applied only classicalmechanics, which leads to the conclusion that every valueof r is possible, depending on the value of v.

16 Simplest Line Spectra and Atomic Theory [I, 2

According to Bohr (see earlier text), only certain orbitsactually do occur and these are selected by the postulate thatthe angular momentum mvr is an integral multiple of h/2~;that is,

hmvr = 12-j2?r where 78 = 1, 2, 3, . . .

This is an assumption which cannot be further justified.Here n is called the principal quantum number. For agiven value of n, the values of r and v are now unambigu-ously fixed by equations (I, 4) and (I, 5). For r, we obtain:

It is apparent that the radii of the possible orbits are pro-portional to n2.

In Fig. 9, for the case of hydrogen (Z = 1), the first feworbits from n = 1 to n = 4 are drawn to scale. For the

smallest possible orbit ;that is, with n = 1:

h*r = - = alI = 0.529A4u2rne2n=l n=2 a=30 ?b=J0k-2

This radius is of thesame order of magnitudeas the radius of the atomgiven by kinetic theory.

There are three refine-

6 10Xldscm.

ments of this simplified0 2 4 6

Fig. 9. Circular Bohr Orbits for the Htheory.

Atom (n = 1 to R = 4). (1) In real i ty theelectron revolves, not

about the nucleus itself, but about the common center ofgravity; also, the nucleus revolves about that center.Therefore the mass of the nucleus enters into the equations.It may be shown [cf. Sommerfeld (5a)] that equation (I, 6)still holds if m is replaced by the so-called reduced mass:

mM- -p-?R+l%I

I, 2] Bohr Theory of Balmer Terms 1 7

where M is the mass of the nucleus. Here cc is approxi-mately equal to m because M/(m + M) is very nearlyequal to 1. (m = 9.1066 X 1O-28 gm. and, for hydrogen,M = 1.6725 X 1O-24 gm.)

(2) In general, not only circular orbits but also ellipticalorbits are possible (compare above). Evidently the onecondition mentioned above is not sufficient to fix unam-biguously both axes of the ellipse. Therefore Sommerfeldintroduced a new and more general postulate than theoriginal one of Bohr-namely, for the stationary statesthe so-called action integral .$ pi dqi extended over oneperiod of the motion must be an integral multiple of h.

fpi dqi = nib (4 7)

Here ni is a whole number, pi any generalized 2 momentumwhich depends on the corresponding co-ordinate qi. Thispostulate implies the previous one: If dqi = dp where bp isthe angle of rotation, then pi = p,, the angular momentumof the system. According to classical mechanics, theangular momentum of any isolated system is a constant.Therefore

f

2r

p, da = p,s

dp = 2up, = n,h; n, = 1,2,3, - - - (I, 8)

that is, as before the angular momentum is an integralmultiple of h/2?r. However, for an ellipse, r is not constantand therefore we have from (I, 7) an additional condition:

$p,dr = n,.h; r&r = 0,1,2, -** (I, 9)

where p, is the linear. momentum in the direction of r.Here n; is called the radial quantum number; n,, which. willhenceforth be replaced by k, is called the azimuthal quantumnumber. Just as previously by (I, 5) the continuous rangeof r values was reduced to (I, 6), now, by conditions (I, 8)and (I, 9), the possible values of the major and minor axes

* This term is not defined here because it is not particularly essential for thefollowing considerations. For a complete explanation, the reader is referred tothe texts on advanced dynamics.

P

..I1 8 Simplest Line Spectra and Atomic Theory [I, 2

of the elliptical orbits are reduced to the following [cf.ox, Pm

hz n2 aaa=w.Z=T.n2

(I, 10)where the principal quantum number n is now defined asn = k + n,. Here k may take the values 1, 2, . . . n (k = 0was considered impossible in this theory since for zeroangular momentum the electron would have to traverse thenucleus). Consequently n z k. For n = k, a = b; inother words, we have the circular orbits discussed in con-nection with Fig. 9 (with the same meaning for n). Fromrelation (I, 10) it follows that a/b = n/k. The principalquantum number n is thus a measure of the major axis of theelliptical orbit, whereas the azimuthal quantum number is a

Fig. 10. Elliptical Bohr-Sommerfeld Orbits for the H Atom with k = 1,2, and 3 [from Grotrian (8)]. The positive nucleus is at the focus 0 of theellipse. The energy difference between orbits with equal n but different k isv e small . The smallest value of n for a given k is n = k. Same scale asin F ig . 9.

I, 2] Bohr Theory of Balmer Terms 19

measure of the minor axis. On the other hand, according to(I,8), k(= n,) gives the angular momentum of the atom inthe specified state in units h/21. Fig, 10 shows the ellipticalorbits (drawn to scale) for hydrogen, with various n values,for k = 1, 2, and 3.

(3) Sommerfeld also applied relativistic mechanics to themotion of the electron. He found that the orbit is anellipse, the axis of which rotates uniformly and slowly aboutthe center of gravity (rosette motion) instead of remainingstationary.

Energy of Bohr’s orbits (Balmer terms). For circularorbits, the total energy is:

E = potential energy + kinetic energy = - F + am&JUsing formula (I, 4), we obtain:

E= -T+?& B3;

This equation holds also when the motion of the nucleus isconsidered. Substituting from (I, 6) the value for T andusing P instead of m, we obtain:

27?pe* 24E,, = -T.3

The same expression is obtained for the energy of theelliptical orbits [cf. Sommerfeld (5a)]. Thus the energydoes not depend on the azimuthal quantum number k-thatis, on the minor axis of the ellipse.

However, if relativity is also considered, a very slightdependence on k results-namely (as found by Sommerfeld),

E n. I = 2y.$[l+fq;-;)] (I,12)

where a = 2ne2/hc = 7.2977 X 10” is the so-called Sommer-feld fine structure constanL3

The second term in brackets is very small ‘because of theterm or9; hence, for most purposes the simplified formula(I, 11) may be used. The state of lowest energy evidently

8 Further terms with c@, etc., are included in the exact formula, but areusually negligibly small.

f

20 Simplest Line Spectra and Atomic Theory [I, 2 I, 2] Bohr Theory of Balmer Terms 21

has n = 1. This state, according to Bohr’s theory, is thestable ground state of the hydrogen atom (smallest orbitin Fig. 9).

From equation (I, 11) and Bohr’s frequency condition(I, 3), it follows that the wave numbers of the emitted spectrallines are given by:

v=&(“,-E.J =$h($--$) (I, 13)

where nl and n2 are the principal quantum numbers of theupper and lower states.

The formal agreement of this formula with the empiricalBalmer formula (I, 1) for the hydrogen series is obvious.By substituting the known values of P, e, c, h, and Z in thenumerical factor of equation (I, 13), we obtain the Rydbergconstant R, which formerly had been obtained purelyempirically from the Balmer spectrum. For hydrogenR = 2?r2pe4/ch3, and the agreement between the calculatedand observed values is as close as can be expected from theaccuracy with which the above constants are known. Theformula for the Balmer series is obtained from (I, 13) byusing n2 = 2. This Series thus results from the transitionsof the hydrogen atom from different higher energy stateswith nl = 3, 4, . . ., to the state n2 = 2. In the remaininghydrogen series the lower state has a different principalquantum number. (See also Fig. 12 and discussion insection 3 of this chapter.)

Spectra of hydrogen-like ions. Taking Z = 2 in (I, 13)gives the spectrum which Would be emitted by an electronmoving about a nucleus with charge 2; that is, the spec-trum of He+. Analogously,, for Z = 3 and Z = 4, weobtain the spectra of Li++ and Be+++. The generalformula is :

Y = RP($--$), whereR =y (I,14)

The mass of the nucleus enters into R because of the de-

pendence of R on k. Substituting EL, we obtain:

R=Rc++&-)where R, is the value of R obtained for an infinitely heavynucleus-that is, when m is used instead of p in the formulafor R (I, 14). It follows that R varies slightly for He, Li,and Be. The values for R, calculated from RH by usingaccurate values for the masses [see Bethe (48)], are givenin the second column of Table 2.

TABLE 2

RYDBERG CONSTANTS AKD FIRST MEMBERS OF LYMANSERIES FOR HYDROGEN-LIKE IONS

Hydrogen-l ike Ion R (cm-l) ~2. I (cm-9 A?, I YILP (.A)

H 109,677.581He+ 109,722.263

* Li++ 109,728.723Be+++ 109.730.624

* Referring to the isotope of mass 7.

82.259.56 1215.664329,188.7 303.777740,779.a 134.993

1,317,118.1 75.924

Apart, from this small correction and apart from thefactor Z2, corresponding to a strong displacement to shorterwave lengths, the spectra of these ions are identical in alldetails with the hydrogen-atom spectrum. The third andfourth columns, respectively, of Table 2 give for these ionsthe calculated wave numbers and wave lengths of the linescorresponding to the first line of the Lyman series (n = 2 ---f

n = l).” These and other lines indicated by (I, 14) havebeen found at exactly the calculated positions. Fromformula (I, 11) it follows that for He+ the separation energyW1 of the electron from the lowest level (the ionization po-tential) will be very nearly four times that for the hydrogenatom, where it is equal to RH = 13.595 volts. For Li++it will be nine times as great, and for Be+++ sixteentimes.

4 The relativity correction of (I, 12) has been allowed for in the calculationof the table, taking k = 2 for the upper state (k = 1 would give a slightlydifferent wave number).


Continuum at the series limit. As already stated, theenergy of an atomic state is known apart from an additiveconstant. The latter is chosen so that E = 0 when theelectron is completely removed from the atom; therefore allstable atomic states will have negative E values. A positivevalue of E would, accordingly, indicate more energy thanthat for the system with its parts infinitely separated and at.rest; that is, the two parts possess relative kinetic energy.They approach or separate with a velocity (kinetic energy)that does not disappear-even at infinity.

According to classical mechanics (disregarding radiation)the electron in this case moves, not in an ellipse, but in ahyperbola. This behavior is similar to that of heavenlybodies that come from space with a great velocity anddescribe a hyperbolic orbit about the sun as focus (forexample, the orbit of a comet). Since, according to thequantum theory, only the periodic motions in the atom arequantized, these hyperbolic orbits can occur without anylimitation; in other words, all positive values of E are possible.Hence, extending from the limit of the discrete energylevels, there is a continuous region of possible energy values :the discrete term spectrum is followed by a continuous one.Just as in elliptical orbits,’ according to Bohr (but in con-trast to classical theory) electrons will not radiate in hyper-bolic orbits. Radiation results only through a quantumjump from such a state of positive energy to a lower state ofpositive or negative energy. When the relative kineticenergy is Al?, for a transition to the discrete state n2,formula (I, 13) changes to:

Y= $+A; (I, 15)

As AE can take any positive value, the series of discrete lineswhose limit is at R/n12 is followed by a continuous spectrum,a so-called continuum. Such a continuum actually occurswith the Balmer series in absorption in the spectra of manyfixed stars, and is also observed in emission spectra fromartificial light sources [Herzberg (41)]. In absorption. it

I, 3] Energy Level Diagrams 23

corresponds to the separation of an electron from the atom(photoeffect) with more or less kinet i c energy (dependingon the distance from the limit); in emission, it correspondsto the capture of an electron by a proton, the electron goinginto the orbit with principal quantum number n2.

The beginning of the continuum, the series limit, cor-responds to the separation or the capture of an electron withzero velocity (AE = 0). If the transition takes place fromthe ground state to the ionized state (absorption in coldgas), the wave number of the series limit gives directly theseparation energy (ionization potential).

The intensity of the continuum falls off more or lessrapidly from the limit. Fig. 11 gives as an illustration thecontinuum for the Balmer series in emission.

I IH, H, CONTINUUM

Fig. 11. Higher Members of the Balmer Series of the H Atom (in Emis-sion) Starting from the Seventh Line and Showing the Continuum [Herzberg(41)]. H, gives the theoretical position of the series limit. The photographwas more strongly exposed than Fig. 1, and consequently some weak molecularlines not belonging to the Balmer series are also present-for example, one inthe neighborhood of the position of H,.

In Fig. 2 the continuum can be seen beyond the serieslimit for Na in absorption. The ionization potential forNa may be obtained directly from this limit (5.138 volts).

3. Graphical Representation by Energy Level Diagrams

Energy level diagram and spectrum. Consideration ofthe hydrogen spectrum and of hydrogen-like spectra hasalready shown that in a discussion of the spectrum the termsare of far greater importance than the spectral lines them-selves, since the latter can always be derived easily from theformer. In addition, the representation by terms is much

A descriptive picture of the terms and possible spectrallines is obtained by graphical representation in a Grotrianenergy level diagram. Fig. 12 shows the energy leveldiagram for the H atom. The ordinates give the energy,

-1cni

c 4-* “1 F11 -

lo- 2 I

P

30,000

9-

8-

7-

6- ~ 60.000

t6- / 70,000

/

rl- '


simpler since the number of terms is much smaller than thenumber of spectral lines. For example, there is only oneseries of terms for H, but there is an infinite number of seriesof lines.

Fig. 12. Energy Level Diagram of the H Atom [Grotrian (8)].


and the energy levels or terms R&z2 which occur are drawnas horizontal lines. The separation of the levels decreasestoward the top of the diagram and converges to a value 0 forn + co. Theoretically there is an infinite number of linesin the neighborhood of this point. A continuous termspectrum joins the term series here ‘(indicated by cross-hatching). At the right, the energy scale is given in cm-‘,increasing from top to bottom (term values are positive).As previously explained, the value 0 corresponds to the com-plete separation of proton and electron (n = a). To theleft is a scale in volts beginning with the ground state aszero. This volt scale can be used directly to obtain theexcitation potential of a given level by electron collision-that is, the potential through which electrons must beaccelerated in order to excite H atoms to a given level oncollision (see Franck-Hertz experiment, p. 15).

A spectral line results from the transition of the atomfrom one energy level to another. Accordingly, this line isrepresented in the energy level diagram of Fig. 12 by avertical line joining the two levels. The length of the lineconnecting the two levels is directly proportional to thewave number of the spectral line (right-hand scale). Thethickness of the line gives a rough measure of the intensityof the spectral line. The graphical representation of thedifferent series is readily understood from the figure, as isalso the fact that the lines approach a series limit.

The absorption spectrum of an atom at not too high atemperature consists of those transitions which are possible from the lowest to higher states. Fig. 12 shows that for Hatoms this spectrum is the Lyman series with a continuumstart ing at the series limit (see above). In general, there-fore, H atoms will not absorb at longer wave lengths than1215.7A.”

6The appearance, in absorption in some stellar spectra, of the Balmerseries whose lower state is not the ground state of the atom, is due to the factthat, on account of the high temperature of the stellar atmosphere, a con-siderable portion of the atoms are in the first excited state.

26 Simplest Line Spectra and Atomic Theory” [I, 3

Since the terms for hydrogen-like ions differ from those ofthe H atom only by the factor Z2 (apart from the very smalldifference in Rydberg constant and relativity correctionterm), quite analogous energy level diagrams may be drawnfor them. Practically the only difference is a correspondingchange in the energy scale.

40,000

60,000

60*000

70,000

80,000

90,000

Fig. 13. Energy Level Diagram of the H Atom, Including Fine Structure[Grotrian (8) J.and so on.

The Balmer series is indicated, as usual, with II,, Hg, H7,The broken lines refer to forbidden transitions.


When an atom reaches an excited state by the absorptionof light, it can return to a lower state or to the ground statewith the emission of light. This is called fluorescence. Thelongest wave length capable of exciting fluorescence isknown as the resonance line for the atom concerned. Fig.12 indicates that, for H, this line is the first line of theLyman series. The resulting fluorescence is called reso- .nance fluorescence, or resonance radiation.

Consideration of the quantum number k, and the finestructure of the H lines. Each of the simple levels in Fig.12 with a given value of n actually consists, according toequation (I, 12), of a number of levels lying very close to oneanother. In the Bohr theory these levels differ in the lengthof the minor axis of the ellipse-that is, in the azimuthalquantum ‘number k. For a given value of n, n such sub-levels are present. Because of the small value of the factorcy2, the levels lie so close together that their splitting cannotbe shown in the figure.

In Fig. 13, therefore, the levels with different k are drawnside by side at the same height, whereas states with equal kand different n are drawn above one another.s The numberof sub-levels increases with increasing n. According to theRydberg-Ritz combination principle, each sub-level shouldbe able to combine with any other sub-level; in other words,their energy difference should correspond to a spectral line.Consequently each hydrogen line should consist of a numberof components corresponding to different possible originsfrom the various term components.

Selection rule for k. Using spectral apparatus of greatresolving power, it has indeed been possible to resolve theBalmer lines and also several He+ lines into a number ofcomponents; however, the number of components is muchsmaller than might be expected on the basis of the com-bination principle. This discrepancy is due to the fact that

6 Such a group of levels, drawn over one another, corresponds to the groupof ellipses with the same value of k in Fig. 10 (p. 18).


the number of possible combinations is limited by certainso-called selection rules. Such rules play an equally im-portant role in all other spectra. Most of the selection rulesare not absolutely rigid, since so-called forbidden lines oftenappear, though very weakly. (See Chapter IV.)

These selection rules can be derived if we take into con-sideration that, for large quantum numbers, the quantumtheory must coincide with the classical theory, and thenassume that the rules so derived for large quantum numbersalso hold for small quantum numbers (Bohr’s correspondenceprinciple). The details of this derivation will not be givenhere. The result, however, is that, in the present case, kmay alter only by + 1 or - 1. According to this selectionrule, in Fig. 13 only those transitions indicated by solidlines between neighboring term series can occur. Thecombinations indicated by broken lines are forbidden. Foreach line of the Lyman series, there is consequently only onepossible origin; for each line of the Balmer series, there arethree possible origins; for the Paschen series, five; and so on.However, observations show certain deviations from thistheory; for example, there appear certain components whichare forbidden according to the k selection rule. This dis-crepancy was first explained by several new assumptions,which will be discussed in Chapter II, section 2.’

In Fig. 13 and similar illustrations that appear later inthis book, the wave number of a transition obviously isgiven, not by the length of the oblique line representing it,but by the vertical distance between the two levels.

4. Wave Mechanics or Quantum Mechanics

The Bohr theory of the atom gave a surprisingly accuratequantitative explanation of the spectra of atoms and ionswith a single electron. But, for atoms with two electrons(He), serious discrepancies with experiment were en-countered. Quite apartfrom these and other discrepancies

7 Extended discussions of the hydrogen fine structure are given by Sommer-feld (5b); Grotrian (8); White (12).

I, 4] Wave Mechanics 29

there was the difficulty of understanding the quantumconditions themselves. The attempt to solve this problemfound expression in wave mechanics (De Broglie, Schro-dinger) and quantum mechanics (Heisenberg, Born, Jordan,Dirac), which were put forward almost simultaneously andproved to be different mathematical formulations of thesame physical theory. In the following discussion the wavemechanical formulation will be principally used whereverthe Bohr theory proves inadequate.

Only a brief and necessarily incomplete account of theelements of wave mechanics will be given here. For furtherdetails one of the numerous texts in the bibliography shouldbe consulted.

Fundamental principles of wave mechanics. Accordingto the fundamental idea of De Broglie, the motion of anelectron or of any other corpuscle is associated with a wavemotion of wave length:

where h = Planck’s constant, m. = mass, and v = velocityof the corpuscle. For an electron, replacing these symbolswith numerical values, we obtain:

(I, 17)

where V = electron energy in voltsV

300 = i $ v2 . For

example, for electrons of 100-volt energy the De Brogliewave length is 1.226-X.

In order to calculate the motion of an electron, we mustinvestigate the accompanying wave motion instead of usingclassical point mechanics. However; classical mechanicscan be applied to the motion of larger corpuscles for thesame reason that, problems in geometric optics can ‘becalculated on the basis of rays, whereas actually the prob-lems deal with waves. Wave mechanics corresponds towave optics. Accordingly, if we use appropriate wave


lengths, we should expect diffraction phenomena also forcorpuscular rays. From formula (I, 17), electrons with nottoo great energy should have a wave length of the sameorder as X-rays. The above prediction by De Broglie was

Fig. 14. Photograph of ElectronDiffraction by a Silver Foil. Elec-trons with a velocity of 36,000 volts,corresponding to a wave length of0.0645 A, were used in the expe riment[after Mark and Wierl (49)].

confirmed in experimentsfirst carried out by Davissonand Germer. The experi-ments show the correctnessof De Broglie’s fundamentalprinciples. Fig. 14 is an ex-ample of diffraction ringsproduced by the passage ofa beam of electrons througha silver foil. Diffractiontakes place at the individualsilver crystals. The figureagrees in all details with aDebye-Scherrer X-ray photo-graph. Stern and his co-workers also have shownthat analogous diffraction

phenomena are exhibited by atomic and molecular rays.According to De Broglie, the frequency Y’ of the vibra-

tions may be calculated from the Planck relationE = hv’

where E is the energy of the corpuscle.(4 18)

For a given mode of motion it is necessary to decidewhether we are dealing with progressive or standing waves.Progressive waves correspond to a simple translational mo-tion of the corpuscles considered (potential energy V = 0).In this case, just as for waves propagated in a very longstring, any frequency is possible for the wave motion-thatis; any energy values are possible for the corpuscle. How-ever, when the corpuscle takes up a periodic motion as aresult of the action of a field of force (potential energyV < 0) and has not sufficient energy to escape from this

I, 4] Wave Mechanics 3 1

field (for example, circular motion or oscillatory motionabout an equilibrium point), the wave returns to its formerpath after a certain number of wave lengths.

Fig. 15 shows this behavior diagrammatically for acircular motion. The waves which have gone around 0, 1,2, . . . . times overlap and will, in general, destroy one anotherby interference (dotted waves in Fig. 15). Only in the

Nucleus of an Atom (Qualitative). Solid 1’Fig. 15. De Broglie Waves for the Circular Orbits of an Electron about the

me represents a stationary state(standing wave); dotted line, a quantum-theoretically impossible state (wavesdestroyed by interference).

special case where the frequency of the wave and, therefore,the energy of the corpuscle are such that an integral numberof waves just circumscribe the circle (solid-line wave) do thewaves which have gone around 0, 1, 2, . . . times reinforceone another so that a standing wave results. This standingwave has fixed nodes, and is analogous to the standingwaves in a vibrating string which are possible only for cer-tain definite frequencies, the fundamental frequency and itsovertones (cf. Fig. 16). It follows, therefore, that a station-ary mode of vibration, together with a corresponding state ofmotion (orbit) of the corpuscle, is possible only for certain


energy values (frequencies). For all other energy values (fre-quencies), the waves destroy one another by interference,and consequently, if we assume the relation between waveand corpuscle indicated by the observed diffraction phenom-ena, there is no motion of the corpuscle corresponding tosuch energy values. Even quantitatively the results arethe same as in Bohr’s theory; namely, the interferencecondition in Fig. 15 is evidently

nX = 2ar, where n = 1, 2, 3, - - -With (I, 16) this result leads directly to Bohr’s originalquantum condition (I, 5), from which the Balmer termswere derived. However, here this condition and, with it,Bohr’s discrete stationary states result quite naturally fromthe interference conditions.

Mathematical formulation. In order to determine morerigorously the stationary energy states or stationary wave

states, we must set up thewave equation Schrodinger_just as in the case of thevibration of a string. Let \kbe the wave function whichis analogous to the displace-

n 3110ment y of a vibrating string

w from its equilibrium position.(In a later paragraph we shalldeal with the physical mean-ing of \k.) Since we are

I ’Fig. 16. Vibrations of a String:

dealing with a wave motion,Fundamental and Overtones. * varies periodically with

t i m e a t e v e r y p o i n t i nspace. We can therefore write:

f = $ cos (2&t) or J/ sin (27ru’t)These expressions are combined in the usual convenientform :

I@ = #. e-z*iv’t (I, 19)Here # depends only upon the position (x, y, z) and gives the

I , 4 ] W a v e M e c h a n i c s 33

amplitude of the standing wave at this point. For thevibrating string the corresponding amplitude functions areshown in Fig. 16.

Schrodinger's differential equation for the amplitude # ofthe atomic wave function is:

a** a** a**z2+G+s+8$(E-V#=0 (I, 20)

In this equation, m is the mass of the particle, E the totalenergy, and V the potential energy. This Schrodinger equa-tion replaces the fundamental equations of classical mechanicsfor atomic systems. The frequency Y’ of the vibrations in(I, 19) is obtained from the fundamental assumption (I, 18) :

E = hv’

and hence we can also write:* = ~e-22rimt’ (I,21)

When it is assumed, similar to the case of the vibratingstring, that 1c, is everywhere single valued, finite, and con-tinuous, and vanishes at infinity, then the Schrddinger equa-tion (I, 20) is soluble, not for unrestricted values of E, but onlyfor specified values of E, the so-called eigenvalues. The cor-responding wave functions are called the eigenfunctions ofthe problem. They represent the stationary states forwhich the wave motion is not everywhere destroyed byinterference. The discrete energy values of an atom whichare experimentally observed in the spectrum appear here asthe eigenvalues of the atomic wave equation.

Without the above boundary conditions the wave equa-tion could be solved for any value of E (that is, any fre-quency), but the solution would not be unique. For ex-ample, we would obtain different values of 9 for a pointaccording to whether the inclination to a fixed axis weregiven by the angle (o, or 360’ + (c, and so on. The differentJI values at each point would destroy one another by inter-ference (cf., also, Fig. 15 and accompanying discussion).

The amplitude curves (eigenfunctions) for the vibratingstring, whose differential equation is much simpler, are


represented in Fig. 16. The eigenvalues are the frequenciesof the fundamental vibration and its overtones: yo, 2ro,3v0, *a*. Other frequencies are impossible.

The eigenfunctions for the wave equation of the H atomare given graphically in Fig. 18 and discussed on page 38.

Equation (I, 20) is, as stated above, the differential equationfor the amplitude # of the wave function 9. The wave equationfor \E itself, which contains both the spatial co-ordinates andthe time, is:

-&(g+a$+g) + v!F =$$f (I,22)

In all the cases with which we have to deal in the following discus-sion, this equation can be solved by substituting 9 from equation(I, 21), which immediately leads to (I, 20) for the amplitudefunction 9. Therefore, in the following considerations equation(I, 20) may always be taken as the starting point.

It should be noted that the imaginary quantity i occurs in(I, 22).complex.

Hence it is essential, according to (I, 21), for Q to beThe function $ sin 27r(E/h)t would not solve the time-

dependent Schrodinger equation (I, 22).Equations (I, 20) and (I, 22) refer only to the one-body prob-

lem. If the system contains several particles, these equationsmust be replaced, respectively, by:8

and-&$-($+$+g) + vq =$$ (I,23)

+$$+-g)+g& v)$ = 0 (I, 24)

where mk is the mass of the kth particle whose co-ordinates arexk, yk, zk. Therefore \k and # are functions of 3N co-ordinates -that is, they are functions in 3N dimensional space (configurationspace) if N is the number of particles.

Physical interpretation of the q function. According toBorn, the value of 4 for a given value of the co-ordinates isrelated to the probability that the particle under considerationwill be found at the position given by the co-ordinates; in otherwords, the probability is given by 1 \k I* or \k. \k* where \k* isthe complex conjugate of \k. The corresponding relationfor light-namely, that the number of light quanta at a

* For the derivation of these equations, see Sommerfeld (5b).


given point is proportional to the square of the amplitudeof the light wave at that point-is readily understood whenit is remembered that, according to elementary wave theory,the light intensity is proportional to the square of the ampli-tude of the light wave and, on the other hand, is naturallyproportional to the number of light quanta, since each lightquantum contributes hv’ to the intensity.

When qI, \k2, q3, . . . are eigenfunctions of a vibrationproblem, \k = Zci\ki is also a solution of the differentialequation. With a vibrating string this means that a num-ber of overtones, and possibly also the fundamental, can beexcited at the same time, as is usually the case. On theother hand, when we have * = ZciQi for an atomic system,this does not mean that the different characteristic vibra-tions *I, e2, . . . are excited in one and the same atom withamplitudes cl, c2, . . . , but it corresponds to the followingstate of knowledge concerning the system: The relativeprobabilities of being in the states given by *I or \k2 or!I!* **a are in the ratioslcll*: ]c21* : 1~312 es-. A givenatom can be found in only one state. London (50) ex-presses this result by saying that the “as well as” ofclassical physics has become “either . . . or” in quantummechanics.

From the probability interpretation of WP* it follows that.$ w?* & = 1(where dr is an element of volume), since the prob-ability that a given particle will be found somewhere in space is 1.The condition previously stated, that \k must vanish at infinityand be everywhere finite, also follows from this9 Eigenfunctions*i for which f qi*i* dr + 1 must be divided by a factor sochosen that $Wi\Iri* d7 = 1 (normalization). Likewise, it canreadily be shown mathematically that

. sq,,f&,,,* dz’ = 0, for n + rn (I, 25)

That is, eigenfunctions belonging to different eigenvalues areorthogonal to one a n o t h e r . The system of eigenfunctions istherefore a normalized orthogonal system.

0 In fact it follows that 9 must vanish more rapidly at infmity than 1 /r.


The Heisenberg uncertainty principle. The Heisenberguncertainty principle is very closely related to wave theory.In order to determine, as accurately as possible in wavemechanics the velocity or momentum of a particle, the DeBroglie wave length must be defined as accurately aspossible, since

p=mv=;

This equation is the converse of (I, 16). In order to meas-ure X accurately, the wave train must be greatly extended,and in the limiting case must be infinitely extended if wewish to give X or p an absolutely exact value. Then, ac-cording to wave mechanics, the position at which the par-ticle under consideration will be found is completely uncer-tain, since the probability of finding it at a definite point is*** and, when the wave is infinitely extended, this quantityhas everywhere the same value $#*. If then p is exactlymeasured, the corresponding position will be completelyuncertain.

Conversely, when we wish to define the position of aparticle as accurately as possible, the wave function mustbe so chosen that it differs from zero only at one given point.According to Fourier’s theorem, there can be produced afunction limited to a small region by the overlapping of sinewaves, but only by the overlapping of many waves ofdifferent wave lengths. In the limiting case (completelydefined position), the wave lengths must take all valuesfrom 0 to 00 ; this makes the wave length and, therefore,the momentum completely uncertain. We arrive then atthe law: Position and momentum cannot be simultaneouslymeasured exactly. Heisenberg has formulated this relation-ship somewhat more precisely: When Aq and Ap are theuncertainties with which q and p can be measured simul-taneously, the product Aq.Ap cannot be of a smaller orderof magnitude than Planck’s quantum of action.

Aq- Ap Z- h (I, 26)


This holds for any co-ordinate and the correspondingmomentum.

The Heisenberg principle will now be verified for a simple case.Consider the diffraction of a matter wave at a slit of width Aq(Fig. 17). Through this slit the position of the particle is knownwith an accuracy Aq. The point within the slit through whichthe particle passes is completely un-certain. The particles are deflectedbv the slit and will form a diffraction I a Ipattern on a screen. How a single par-ticle behaves behind the slit is, in prin-ciple. indeterminate within certainlimits. For example, if the particle Iappears at A, it has acquired an addi-tional momentum Ap in the verticaldirection above the original momen- Fig. 17. Diffraction of Deturn where Ap = p sin a. Accordingto the ordinary diffraction theory, the

~~$~tYw~ti$e~.‘ht cun-

diffraction angle CY is of the order X/Aq (the smaller the slit andthe greater the wave length, the greater the diffraction). Substi-tuting, we obtain Ap N pX/Aq. But, according to De Broglie,PA = h, and therefore Ap. Aq - h. Thus, when the position islimited by the slit to a region Aq, the momentum in the samedirection is uncertain to at least an extent Ap = h/Aq since, foreach of the points in the diffraction pattern, we can give only theprobability of the particle’s hitting the screen at that point.

Wave mechanics of the H atom. In order to deal inwave mechanics with the H atom or hydrogen-like ions, theCoulomb potential - Ze2/r must be substituted for V in thewave equation (I, 20). Z is the number of charges on thenucleus (for H, Z = 1). The differential equation

z+g+g+qn (E+T),=O (I,W

(m = mass of the electron) must’then be solved under theconditions that 4 is everywhere single valued, continuous,and finite.

The calculation, which is simple in principle, will not bedealt with here.‘O It gives the result that the differentialequation can be solved for all positive values of E but notfor all negative values of E. More particularly, it is found

*O For textbooks on wave mechanics, see bibliography: (5b) and (23) to (32).

38 Simplest Line Spectra and Atomic Theory [I, 4 I, 4] Wave Mechanics 39

that only those negative values of E for which2dmel22 RhcZZ

Es= -h22=na, with n = whole number,

lead to a solution. For all values other than these, the DeBroglie waves in a Coulomb field completely destroy oneanother by interfere&e. Thus the possible energy valuesfor a hydrogen atom and a hydrogen-like ion as given bywave mechanics are exactly the same as those given bythe Bohr theory [cf. equation (I, 11)]. It will be remem-bered that the latter were in quantitative agreement withthe experimentally observed spectra of the hydrogen atomand hydrogen-like ions. Making allowance for the factthat the nucleus also moves has the same effect as in theBohr theory: in the energy formula the reduced mass p =mMi(m + M) must be used instead of the electron mass m,where M is the mass of the nucleus. The influence of rela-tivity has been disregarded in (I, 27).

It should perhaps be stated here that, while wave me-chanics agrees with the old Bohr theory in this case, it reallyhas made a distinct advance beyond that theory: first, it isin agreement with many experiments which the Bohr theorycontradicts; and second, in contrast to the Bohr theory, allthe results can be derived from one fundamental assumption(the Schrodinger equation).

To each eigenvalue of the Schrodinger equation-that is,to each stationary energy state-there belongs, in general,more than one eigenfunction. These eigenfunctions aredistinguished by two additional quantum numbers 1 and m,which are always integers. One of them, l, correspondsto the Bohr quantum number k, which was a measure ofthe minor axis of the elliptical orbit: The quantumnumber l is called the azimuthal quantum number, or thereduced azimuthal quantum number. If the relativitytheory is considered, there is also a very small difference inenergy for states with different l but equal n. The value ofl, together with k, is indicated in the energy level diagramfor hydrogen (Fig. 13). For a given value of n, l takes thevalues 0, 1, 2, . . ., n i 1; that is, l = k - 1. The quantum

number m, called the magnetic quantum number, takes thevalues -l, -l + 1, -l + 2, ..., +l for a given pair of nand Z values. This gives the following scheme:

I :

: D ‘si P . 0 FEach zva ue in the last line corresponds to one eigen-P

function different from the others. For each value of n,there are as many different eigenfunctions as there are num-bers in the last line below the n value under consideration.

The mathematical form of the eigenfunction is the following: .

ti nlm = Ce--p(2p)* L”,‘++:(2p) P/ml (cos O)dmq (I, 28)referred to a system of polar co-ordinates T (distance from theorigin)? 0 (angle between radius and z-axis), and cp (azimuth of T-Zplane, inclination to a fixed plane). Here p is an abbreviation forZr/nu~; that is, for the lowest state of the H atom (Z = 1, n = 1),p is equal to the distance from the origin measured in termsof uH as unit (radius of lowest Bohr orbit = 0.528 A). L2,%‘(2p)is a function (Laguerre poiynomial) of 2p; its form depends on nand 1. Pi”’ (cos ~9) is a function of the angle 6 (the so-calledassociated Legendre polynomial), and has a different form ac-cording to the values of m and 1.

The eigenfunctions can be split into two factors, one ofwhich depends only upon the distance r from the origin,and the other only upon the direction in space. For thevalues n = 1, 2, 3, the dependence on r is shown in Fig. 18(see p. 40). For a given value of n, the function shows adifferent form for different values of I; similarly, it shows adifferent form for a given value of 1 and different values of n.The form of the function is, however, independent of m.In Fig. 18 the radius of the corresponding Bohr orbit isrepresented by a vertical line on the abscissa axis. In allcases, # finally decreases exponentially toward the outsideand is already very small at a distance which is, on theaverage, about twice the radius of the corresponding Bohrorbit. For n > 1, # goes once, or more than once, through


42 I& junction corresponding to the nodes of a vibrating string.5 $23

(see Fig. 16). For E = 0, the number ofnodal spheres isv2m n- 1, as is shown in Fig. 18. Since in these eases the23a$

eigenfunctions $ are also spherically s y m m e they are

$%Irepresented completely by Fig. 18.

$2 For I >.O, the number of nodal spheres is smaller (see

3 Fig. 18) and equals n - 1 - 1. However, new nodal sur-.!? 2-79 faces appear since $ then depends on direction also. Ince.d different directions from the originbe*sk the variation with T is the same asi’t;.B81

in Fig. 18 but the function must be

2-g multiplied by a constant factor

42depending on the direction. For

N- 6 some directions this factor is zero.G-sa The resulting nodal surfaces are

11 cc.2 partly planes through the z-axis,a-B“08

and partly conical surfaces withaiz the z-axis as the axis of the cones.*SB

[;

For 1 = 3 and m = 1, these nodalsurfaces are given in Fig. 19.

,,f Etiz O~*$r~$$~~~~T h e Eigenfunction. Independent of

variation of the + function- withI (for t = 3, m = 1). Thethree nodal surfaces are: the

-- *wzaa direction depends on m and Z but ~‘~~$~~U~~~;. ““od,~~is=.-%3

not on n. Since the number of iz; 8~h~~;;;~~B~!~~-

3;nodal surfaces caused by this de-

a.3 pendence on direction is I, in all cases the total number of

21 nodal surfaces is n - 1.

7: Thus in quantum mechanics the principal quantum num-

1.g ber is given-a meaning that is easily visualized-namely, thetotal number of nodal surfaces + 1. The azimuthal quan-turn number l gives the number of nodal surfaces goingthrough the mid-point. It is clear that the number of nodalsurfaces can only be integral. Thus while integral quantum

l1 In formula (I 28) P\“’ (co8 @)eimo = 1.t ,

.2.2

the value zero before the exponential decrease sets in; thatis, on certain spherical surfaces about the nucleus, the $function is always zero . These are the nodal surfaces of the


numbers are introduced into the Bohr theory as assumptionsquite incomprehensible in themselves, they appear in wavemechanics as something quite natural.

As we have seen above, q itself has no immediatelyapparent physical meaning, but *!I!* = ##* has. Theprobability of finding the electron in a volume element dr isgiven by ##* d7. The variation of #ICI* is naturally similarto that of 9 (Fig. 18). The dotted curves in Fig. 20 repre-sent, for the same n and I values as in Fig. 18, that part of$#* which depends on r (all drawings made to same scale) ;they represent simply the squares of the correspondingfunctions of Fig. 18. The zero positions thus lie at thesame T values as for $. However, since #JI* = I+ 1 2’ isalways positive, the zero positions are, at the same time,also the positions of the minima of ##*.

The solid lines in Fig. 20 represent #$* multiplied by r2(again all drawn to same scale). This has the followingmeaning: The dotted curves of Fig. 20 show the variationof $#* along a definite radius vector. If we now wish to de-termine how often a given r value occurs independent of thedirection of the radius vector, we must integrate ##* overthe whole surface of the sphere for that value of T. Thisgives a factor proportional to r2, since the surface of a sphereequals 47~~. This is shown by the solid curves of Fig. 20.It is seen from the dotted curves with 1 = 0 that the prob-ability of finding the electron near the mid-point of theatom is greater than at some distance from the mid-point.In spite of this, however, the electron is, on the whole, moreoften at a point which is some distance from the mid-point,since there are many more possibilities for such a point (allpoints of the spherical shell of radius r). Therefore thelargest maximum in the solid curves of Fig. 20 lies at anoticeable distance from the zero point (origin). The elec-trons are found most frequently at this distance, the dis-tance of greatest electron probability density, which hasapproximately the same magnitude as the major semi-axis of the corresponding Bohr orbit (also indicated in Fig.20). However, according to. wave mechanics, any other

`

4 4 Simplest Line Spectra and Atomic Theory [I, 4

distances T (even those that are considerably greater) havea probability different from zero. The electron is, so tospeak, smeared out over the whole of space, However, becauseof the exponential decrease toward the outside, the prob-ability of finding the electron at any great distance outsidethe region of the Bohr orbit is very small, although notexactly equal to zero.

Since we no longer have distinct electron orbits, it isperhaps better to speak of electron clouds about the nucleus.Fig. 21 is an aid in visualizing these electron clouds andgives, for different values of n, I, and m, an approximate

n=3,m=f2 n=-3,m=*1 n==a.m=o n=l,m-0132

Fig. 21. Electron Clouds (Probability Density Distribution) of the H Atomor Hydrogen-like Ions in Different States [after White (51)].not uniform for all the figures but decreases with increasing n.

The scale is

differ only in the sign of m have the same electron cloud.States which


picture of what one might expect to see when one is lookingat an H atom with an imaginary microscope with enormousmagnification.

In the figures, the brightness indicates roughly the densityof the electron clouds. These clouds have a rotational sym-metry about a vertical axis in the plane of the figure l2 (thez-axis mentioned earlier). The states with 1 = 0 andn = 1, 2, 3 are spherically symmetrical. For n > 1, alter-nate light and dark rings appear, corresponding to theabove-mentioned nodal spheres of the $ function. Thecloud is subdivided into spherical she&l3 For 2 > 0, onecan see in Fig. 21 the nodal cones which, for the particularvalue 1 = 1, degenerate into a plane perpendicular to thez-axis. Different pictures are obtained for different valuesof m and equal n and 1. With increasing n and 1, the pic-tures become more and more complicated (cf. n = 4, 1 = 2,m = 0).

However, these are the pictures of the atoms (in par-ticular, the H atom) which, according to our present-dayknowledge, we have to use. The term electron cloud, whichis customarily given to the pictures, must not be interpretedas meaning that, in the case of H for example, one electronoccupies at one time the whole of the space occupied by thiscloud. On the contrary, in wave mechanics the electron isconsidered as a point charge, and the density of the cloud ata specified point gives only the probability of finding the

If This results from the fact that the dependence of the wave function fion the azimuthal angle (p is given by e imp [equation (I, 28)], which by multi-plication with the complex conjugate gives a constant-namely, 1. When itis stated that $ has nodal planes through the z-axis, thii statement holds forthe real and imaginary parts of $ individually, since cos rnr or sin m(p has just2m zero positions in the region 0 to 2~. There are consequently m nodalplanes. However, the more accurate theory shows that, in forming $$*,the complex ti function must be introduced-not the real or the imaginarypart alone (cf. p. 34).

I* We must emphasize again that, in spite of the greater density at themiddle of the cloud (indicated by the greater brightness in Fig. 21), theelectron is most often in the outermost spherical shell, since this has a muchgreater extent. If the electron has A = 3, it is, therefore, mostly at a greaterdistance from the nucleus than it is for n = 2 or n = 1.


electron at that point. In order to observe the picture oneshould, strictly speaking, observe a large number of H atomsin the same state. Since, however, the orbit of the electroncannot be definitely determined according to wave me-chanics, we can in many cases make calculations as thoughthe electron were smeared out he whole space.

According to ordinary wave mechanics, just as in Bohr’stheory, the energy of a stationary state for the H atom andhydrogen-like ions depends solely upon n. States of differ-ent Z (having different minor axes of the elliptical orbit in theBohr theory) but equal n have the same energy.degenerate.

They areThis degeneracy is, however, removed when we

allow for the relativity theory. A small difference in energythen occurs between states with different Z and equal n; thisdifference, as also in Bohr’s theory, gives rise to the Jinestructure of the Balmer lines. However, a complete agree-ment of calculated and observed fine structure is obtainedonly by allowing also for electron spinJ14 which will be dis-cussed in Chapter II. The degeneracy between states withequal n’ but different 1, which results if the relativity theoryor electron spin is not considered, occurs only in the case of apoint charge in a pure Coulomb field of force (H atom).However, in the general case, for example with the alkalis(see section 5 of this chapter), such degeneracy disappears.States with different Z can then have noticeably differentenergies for the same value of n. The type of eigenfunctionor electron cloud remains the same as in Figs. 18, 20, and 21.When more electrons are present, to a first very rough ap-proximation’ the electron cloud is simply a superpositionof the probability density distributions of the individualelectrons (Fig. 21).

The fact that, for a given n and 1, there are still a numberof different eigenfunctions according to the value of m(namely, m = - I, - 1 + 1, - Z + 2, . . .

J+ 1, making

I4 The spin also follows as a necessary consequence of Dirac’s relativisticwave mechanics.


21 + 1 different eigenfunctions) also holds in the generalcase. Even then these states have exactly equal energies.This is connected with the fact that, when no outer field ispresent, states with different spatial orientations of thesystem have equal energy and are degenerate with respectto one another. This is called space degeneracy, which weshall discuss in greater detail when dealing with the Zeemaneffect (Chapter II).

Neither of these degeneracies occurs for n = 1 , since thenthe only possible value for Z is 1 = 0, and the only possiblevalue for m is likewise 0. (See scheme, p. 39.)

Momentum and angular momentum of an atom accordingto wave mechanics. The representations in Fig. 21 give apicture of the probability of finding an electron at a givenposition in space, or, in other words, the shape of the elec-tron cloud about the nucleus. They tell nothing, however,about the motion of the electron or its velocity at differentpoints in space. The Heisenberg uncertainty principle in-forms us that the simultaneous position and velocity of anelectron cannot be given with any desired accuracy-thatis, the velocity of the electron cannot be given for eachpoint. However, we can reach at least some conclusionsabout the velocity or the momentum of the electrons in anatom; for example, we can calculate the velocity distributionover the various possible values just as we calculated theprobability distribution of the various positions of the elec-tron in the atom (Figs. 20 and 21).

To illustrate, Fig. 22 gives the probability that the elec-tron will have the velocity or the momentum given by theabscissae for the ground state of the H atom (n = 1, Z = 0)and for an excited state (n = 2, Z = 0) [Eisasser (52)].The curves correspond to the solid curves of Fig. 20. Ac-cording to Fig. 22, the most frequently occurring velocity inthe ground state is 1.2 X lo* cm./sec.; in the first excitedstate, 0.4 X lOa cm./sec. For the latter state, a velocity of


1.1 X lo8 cm./sec. does not occur, although greater andsmaller values are both present.

Fig. 22. Probability Distribution of Momentum and Velocity in the StatesR = 1, 2 = 0, and R = 2, 2 = 0, of the H Atom. The curves give the squareof the momentum wave function given by Elsasser (52). The value of themomentum in units of h/2raH = 1.96 X lo-19velocity in cm./sec. is shown as abscissae. The

gr. cm./sec. or the value of theordinate is proportional to

the probability of finding the electron in the H atom with the given value ofmomentum or velocity.

Quite definite statements may be made regarding theangular momentum of an atom. The co-ordinate associatedwith the angular momentum is the angle of rotation. Thelatter, in contrast to a Cartesian co-ordinate, is completelyuncertain-a result which follows from the rotational sym-metry of the charge distribution. It is, therefore, not incontradiction to the uncertainty principle that the angularmomentum corresponding to a given stationary state has anabsolutely definite value. Calculation shows that thenumerical value of the angular momentum is m h/2u,or approximately Z(h/2?r). (The approximate value will beused in most of our subsequent considerations.) This factgives at the same time a descriptive meaning to the quantumnumber 1: it is the angular momentum of the atom in units ofh/2?r. For 2 = 0, the angular momentum of the atom iszero. That is the reason for introducing 1 instead of k - 1,


In the Bohr-theory, k represented the angular momentum,& . . .the electron in a given orbit; we now represent the angularmomentum by 1 = k - 1. While the value k = 0 did notoccur in the Bohr theory, in wave mechanics the valueI = 0 (angular momentum = 0) does occur and correspondsto k = 1. In the Bohr theory, angular momentum = 0meant the so-called pendulum oscillation orbit in which theelectron would have had to go through the nucleus, andthis was excluded as impossible. Now, an angular mo-mentum equal to zero means simply that the electron clouddoes not rotate; the electron does not need to fall into thenucleus because of this. The value I = 0 does not meanthat absolutely no motion takes place, but only that themotion is not such that an angular momentum results (cf.Fig. 22).

The fact that even in wave mechanics each stationarystate of the atom has a perfectly definite angular momentumshows that the atom can still be regarded as consisting ofelectrons rotating about a nucleus, as in the original Bohrtheory. (We must not, however, speak of definite orbits.‘7Consequently the Bohr theory is adequate in many cases.In particular, we can in many instances use the angularmomentum 2 and the other angular momenta in the sameway as ‘in the Bohr theory, the results being confirmed byexact wave mechanical calculations.

In the following discussion the angular momentum vectorswill be indicated by heavy (boldface) type to distinguishthem from the corresponding quantum numbers, printedin regular type. Thus 2 means a vector of magnitude-h/2= = Z(h/2?r).

That I is connected with the angular momentum can furtherbe understood from the following. According to De Broglie[equation (I, 16)]:

x h h or p=tZ-E-nw p

19 Just because of the fact that (p is quite uncertain, an absolutely definitevalue can be given to the angular momentum.


where m, is the electron mass, p the momentum, and X the cor-responding De Broglie wave length. If we wish to introduce theangular momentum p, into the De Broglie equation, X must bemeasured in the corresponding co-ordinates, that is, the angleof rotation cp. From the expression previously given for the Hatom (I, 28), it follows that 9 = $e-?+” contains the factoreihv-2rdt) (m = quantum number). According to the usual wavetheory, this factor represents a wave propagated in the directionof increasing or decreasing (a according to the sign of m-that isa wave which travels around the z-axis with angular velocity2rv Im. Its wave length is 2*/m, since, when p increases by 2*/m,

ei(m~--)rv’~) acquires its original value once more. Substitutingthis wave lengthI in the expression p = h/X gives as the angularmomentum about the z-axis:

hP, =m-

27rFor a given n and l, m can have the values -l -1+1 .q.,+l.These are, therefore, the angular momenta about the) z-axis inunits of h/2*. All of these states have the same energy. Thisevidently means that the angular momentum itself is Z(h/2*)and has components equal to m(h/2?r) along the z-axis, dependingon its orientation to this axis.replaces 2. and M replaces m.)

(Cf. Fig. 41, p. 99, in which JMore accurate wave mechanical

treatment shows that the angular momentum is dm h/2uand not Z(h/2*). [Cf. Condon and Shortley (13). 3 In ChapterII the above relations till be discussed in greater detail.

Transition probabilities and selection rules according towave mechanics.. In wave mechanics, as in the Bohrtheory, the transition of an atomic system from one sta-tionary state to another is associated with the emission oflight according to the Bohr frequency condition. However,this occurrence can be treated from a far more unifiedviewpoint by wave mechanics (Dirac) than by the Bohrtheory. If an atom is in an excited state, the probabilityof its transition to a lower state can be calculated. Theatom remains for a certain time in the upper state (meanZije). The transition to the lower state follows after a timewhich is in inverse proportion t o the transition probability:the greater the probability, the shorter the time. The

16 The magnitude of this wave length can also be immediately obtainedwhen we consider that, the real part of the $ function of the H atom has mnodal planes.


life in a given excited state for the individual atoms variesexactly as do the lives of individual atoms of a givenradioactive substance. The mean life is usually of theorder of lo-* sec. The intensity of the emission or theabsorption of light by a large number of atoms dependson the magnitude of the transition probability. Definitepredictions about the polarization of the emitted light canalso be made in certain cases [see Condon and Shortleyu3n

Detailed calculations show that, for the H atom and alsofor more general cases, an important selection rule operates-namely, that the intensity is extremely small except when

Al=+1 o r - 1 (I,29)

That is to say, practically only those states can combinewith one another whose I values differ by only one unit.This selection rule corresponds exactly to the earlier selec-tion rule for k. Thus Fig. 13 can be used also for a wavemechanical representation of the transitions for an H atom.There is no selection rule for the quantum number n. Anyvalue of An is possible for a transition:

An = 0, 1, 2, 3, . . The different values of An correspond to the differentmembers of a series.”

The simplest classical model capable of radiating electromag-netic waves is an oscillating electric dipole (Hertz oscillator).Electromagnetic waves are radiated with the same period withwhich the electric charge flows back and forth in such a dipole(for example, in a linear antenna). The-intensity of the radiationdepends upon the magnitude of the alteration of the dipolem o m e n t . The dipole moment is a vector whose components aregiven, in the case of a system of point charges, by the followingexpression : I* CWi, Cwii, ceizi. According to wave mechanics,

I7 For H, a transition with An = 0 would correspond to a transition betweenfine structure terms with equal n; for n - 2, the transition has a wave numberof only 0.3 cm-l, or a wave length of about 3 cm. Observations of absorptionof this wave length in activated hydrogen are still doubtful [see Betz (53);Haase (140)].

18 As is well known, the magnitude of the vector is cd for two charges (+ Eand - t) separated by a distance d.


the probability density of the electron may be given for any pointin the atom. This may, for most practical purposes, be regardedas though, on the average, a certain fraction of the total charge e(given by the probability density) is at the point under considera-tion. That is, we can treat the atom as though the electricdensity at a point is: p = cq@*. Therefore, in the case of oneelectron (hydrogen atom) the components of the electric momentfor the whole atom for a stationary state n are:

P, =s

tP,ik,,*x dr

P, = Se\k,!k,*y dr

P, = f&.Q.*z dr (I, 30)where dr is again an element of volume. Since the nucleus itselfis taken as the origin of co-ordinates, its contribution need notbe taken into consideration. The integrals are independent oftime because the time factors for q,, and q,,* cancel; on accountof the symmetry of the charge distribution, the integrals areactually zero. There is consequently neither a static dipolemoment nor one altering with time. This means, in agreementwith experiment, that even according to the classical theory theatom does not radiate while in a stationary state; whereas inBohr orbits it should radiate (if we had not made the a lditionalad hoc assumption that it does not radiate), since the atom withthe electron in these orbits has a dipole moment varying withtime.

Dirac has shown that the radiation emitted by an atom in thetransition from state n to state m may be obtained by replacing*,,9,,* in equation (I, 30) by \k,,!~,,,*, regarding the resulting Pas an electric moment (transition moment) and then completingthe calculation in the classical manner. Since \k, contains thetime factor e-*ri(&n’*)l (I, 21), and q,,,* the time factor e+2ri(E=‘h)t,\k,*‘,* and Pznm = E $ ?P,,~,,,*x dr (and correspondingly, Pyn”,Pznm) are fro longer constant in time but have the time factore--friI(Bn-Bn)/hlt; that is, they vary with just the frequencythat would be obtained from the Bohr frequency conditionv’ = ; (E, - I#,,,). The result is an emission or absorption ofthis frequency in a purely classical way. An analogous state ofaffairs holds for a system containing a number of particles. I tis necessary only to sum the integral over the different particles;for example,

(I, 31)


According to what has been said, we can put, for the variableelectric moment associated with the transition from n to m:

prim = h (I, 32)where Rnm is a vector with components

R,“m =s

~&,,*s dr, R,“m = - - . , R,fm = - . - (I, 33)

The vector R”” gives the amplitude of the vibration of the transi-tion moment Pnln associated with the transition from n to m.

Remembering that the intensity of light radiated from an atomis equal to the number of transitions per second (that is, thetransition probability) multiplied by hv’,,, we obtain (usingthe classical formula for the intensity of electromagnetic wavesradiated by a vibrating electric dipole) the expression:

Anna - 643T4 ,,JRnmRmn.?h.“: .for the probability of the transition from n to m where v = v’/c

is ‘the wave number. The transition probability therefore de-pends upon the quantities RUM’. RratN itself is determined bythe eigenfunctions of the two states involved [see (I, 33)]. Thuswe see that a knowledge of the eigenfunctions is particularlyimportant for the calculation of ‘transition probabilities. Thequantities Ran can be arranged in a square array (verticalcolumns n, horizontal columns m), which is called a matrix.Ry%+ire the matrix elements. When RpaBn = 0 for a given pairof values of n and m, the transition from n to m is forbidden.Detailed calculation shows that combinations between all statesfor which 1 does not differ by f 1 have Rnwt = 0; that is, theselection rule Al = f 1 holds. Other selection rules can besimilarly derived. Such selection rules always depend upon thesymmetry properties of the atomic system under considerationand of the corresponding eigenfunctions. In Chapter II, sec-tion 3, the derivation of the selection rules for the magneticquantum number M will be given as an illustration. (Cf. alsop. 68 and p. 154.)

Quadrupole radiation and magnetic dipole radiation. A sys-tern of electric charges such as that illustrated in Fig. 23 has nodipole moment (Ceixi = 0). In spiteof this, the system gives an external a 5.electric field, which, however, falls off

w-c +2e -C

more rapidly with increasing distancethan that of the dipole, which itself

Fig. 23. Example of a Quad-

falls off more rapidly than that of therupole.

monopole (the potentials are proportional to l/13, l/rP, llr, respec-tively). An assemblage of charges such as that in Fig. 23 is calleda quadrupole. Its action is characterized by a quadrupole moment,

.

54 Simplest Line Spectra and Atomic Theory, [I, 5

which in the above case is given simply by CE~Z~~, where 2 is theaxis along which the charges are located. It is immediately seenthat this expression is not zero. In general a quadrupole is morecomplicated than the one given in Fig. 23, and likewise the quad-rupole moment is usually more complex. The general case willnot, however, be discussed here.

Just as a variable dipole moment leads to radiation (dipoleradiation), so also does a variable quadrupole moment lead toradiation (quadrupole radiation). The latter is, however, con-siderably weaker. The transition probability, similarly to theabove, is obtained by substituting ei$,$m* dr for ti in CEizi2,and integrating. Therefore quadrupole radiation depends uponthe integral ~x?$+,#,* dr, whereas dipole radiation depends upon$ xJ’nJ/m* dr. Because of this difference, transitions which arestrictly forbidden for dipole radiation may occur - though quiteweakly-due to quadrupole radiation. The ratio of the transitionprobabilities of ordinary dipole radiation to ordinary quadrupoleradiation is found to be about 1 : lo-*.

Finally, it may happen that, for a transition, the variation ofthe electric dipole moment will disappear, whereas that of themagnetic dipole moment does not (cf. Chapter II, p. 111). Ac-cording to classical theory, a variable magnetic dipole momentsuch as that produced, for instance, by an alternating current ina coil gives rise to electromagnetic radiation. Correspondingly,in wave mechanics, it gives rise to a transition probability whichmay be different from zero even if the ordinary dipole transitionprobability is zero. Again, the transition probability due tomagnetic dipole radiation is small compared with that due toelectric dipole radiation (1 : 10b5).

Actually, cases have been observed in which transitions thatare strictly forbidden by the electric dipole selection rules takeplace due to quadrupole or magnetic dipole radiation. (SeeChapter IV.)

5. Alkali Spectra

The principal series. The absorption spectra of alkalivapors (Fig. 2) appear quite similar in many respects to theabsorption spectrum of the H atom (Lyman series). Theyare only displaced, to a considerable extent, toward longerwave lengths.19 These spectra also consist of a series oflines with regularly decreasing separation and decreasingintensity; This series is called the principal series. It

19 We disregard for the moment the splitting of the lines of the heavieralkalis, with which we shall deal in Chapter II. This splitting is still so smallfor Li that it cannot be noticed with the usual spectroscopic apparatus.

I, 5] Alkali Spectra 55

cannot, however, be represented by a formula completelyanalogous to the Balmer formula. On the other hand, sincethe lines converge to a limit, we must be able to representthem as differences between two terms. One of theseterms is a constant T~s (known as the fixed term), and hasthe frequency of the series limit. The other (known as therunning term) must depend on a running number (ordernumber) m in such a way that the term disappears asm -900.

It has been found that the series can be satisfactorilyrepresented with R/(m + p)” as the running term. R is theRydberg constant, and p is a constant number < 1; p iscalled the Rydberg correction.. It is the correction that, forthe alkalis, must be applied to the Balmer term (p = 0 givesthe Balmer term). The running number m takes valuesfrom 2 to 00. The quantity n* = m + p is called the effec-tive principal quantum number. Thus the formula for theabsorption series (principal series) for the alkalis 20 is:

Y= Tps - (m : p)”A continuous spectrum follows at the series limit, as shownin Fig. 2.

Other series. In emission, other series in addition to theprincipal series may be observed for the alkalis. These

p one another. Fig. 3 (p. 5) shows theurn. The three most intense of the addi-

tional series have been given the names diffuse, sharp, andBergmann series. The last is also sometimes called thefundamental series. The lines of the diffuse and the sharpseries frequently appear diffuse and sharp, as their namesindicate. The Bergmann series lies further in the infraredand consequently does not appear in the spectrogram i nFig. 3. The limits of these series and, therefore, theirlimiting terms differ from the limiting term of the principal

SQ This formula does not hold so exactly as that for the H spectrum. Moreexact agreement with experiment can be obtained by adding to the denomi-nator an additional term which depends on m.


series, but the sharp and the diffuse series have a commonlimit (see Fig. 3). Tss is the common limiting term for thesharp and the diffuse series; TBS, for the Bergmann series.The running terms are quite analogous to those of the prin-cipal series, but the Rydberg correction has a different valuefor each series. Thus we have:

PRINCIPAL SERIES:R

” = Tps - (,n + p)” (m = 2, 3, . . . )

SHARP SERIES:R

Y = TM - -on + 4)

(m = 2, 3, * * *).

DIFFUSE SERIES:R

Y = TSS - -(m + d)*

(m = 3, 4, . . . )

BERGMANN SERIES:R

v= Tes- (m+f)* (m = 4, 5, . . . )

The values found empirically show that Tps = R/( 1 + s)“,Tss = R/(2 + PI’, Tss = R/(3 + d)2; that is, the limitingterms belong to one of the series of running terms. If weput mP as a symbol for R/(m + P)~, mS for R/(m + s)~, andso on, the series may be written:

PRINCIPAL SERIES: v=lS-mP ’ * .) (I, 35)SHARP SERIES: Y = 2P - mS i::$;; . . . ) (I, 36)DIFFUSE SERIES: v=2P-mD (m = 3, 4, . . . ) (I, 37)BERGMANN SERIES: v=3D-mF (m = 4, 5, . . . ) (I, 38)

Theoretical interpretation of the alkali series. Fromthe four series of the alkalis it is evident that four differentterm series or four sets of energy levels exist, and these canbe designated by S, P, D, F. In Fig. 24 these series aregiven for Li in the manner explained in an earlier section.The ground state of the alkali atom is 1S, since in absorptiononly the principal series appears and this has 1S as thelower level. The S terms 2S, 3S, . . . follow after it. Thelowest P state occurring is 2P, and it lies above the 1S termby an amount equal to the wave number of the first line ofthe principal series 1S - 2P. The series of P terms followafter it. The principal series in absorption corresponds totransitions from the ground state to the various P states;


the converse holds for emission. The sharp series cor-responds to transitions from the higher S terms to thelowest P state. The lowest D term lies still higher than thelowest P term (namely, by 2P - 3D), and, analogously,the 4F term is higher than 3D. All term series go to thesame limit, whereas of course the line series have differentlimits (cf. above).

The similarity of this energy level diagram (Fig. 24) tothe generalized energy level diagram of H (Fig. 13, p. 26) isobvious. The main difference is that the members of the

=s ‘P ‘0 ‘F‘=L 0 1 2 9

~~~.~~z-x- - ----_ cm-6S 7P- 706s \‘X’I-

7)z’ZII6DF6F-----

-5.00’

10.001

15,oot

20,OOf

26.00(

so.ooa

35,000

40.000

----.

45.000

,I

o-

D-

I-

)-

I-

I-

Fig. 24. Energy Level Diagram of the Li Atom- after Grotrian (8)]. Thewave lengths of the spectral lines arc written on the connecting lines repre-senting the transitions. Doublet structure (see Chapter II) is not includedSome unobserved levels are indicated by dotted lines. The true principalquantum numbers for the S terms are one greater than the empirical runningnumbers given (see p. 61); for the remaining terms, they are the same.

58 Simplest Line Spectra and Atomic Theory [I, 5 I, 5] Alkali Spectra 59

different adjacent term series no longer have almost exactlythe same height. This is to be expected, theoretically, forthe terms of a single electron moving in a field which is notthe Coulomb field of a point charge. The structure of thealkali spectrum therefore leads to the conclusion that, forthe alkali atoms, a single outer electron moves about an atomiccore 21 whose field shows marked deviations from the Cou-lomb field of a point charge, which are due to the finite extentof the core. Furthermore, it follows that the S, P, D, Fterms are distinguished from one another by the value of thequantum number 1 (I = 0, 1, 2, 3, . . .); that is, by the orbitalangular momentum of the outer electron. On the basis ofthe old Bohr theory, each term series would correspond to aseries of elliptical orbits, as in Fig. 10 (p. 18). The factthat the series of P terms begins with m = 2, the D termswith m = 3, and the F terms with m = 4 is also in agree-ment with this assumption, since, if the order number m isidentified with the principal quantum number n, m must be~Z+l(seep.38). The selection’ rule Al = f 1 is alsofulfilled; only neighboring term series combine with oneanother.

The Rydberg correction (the deviation from the hydrogenterms) is greater, the nearer the electron comes to the corein its orbit according to the old Bohr idea. The correctionis greater still if the orbit penetrates the core (so-calledpenetrating orbits), as then the effective nuclear charge Zeftacting on the electron is appreciably altered. In the im-mediate neighborhood of the nucleus the whole nuclearcharge acts, but at a great distance it is shielded by the coreelectrons down to Zdf = 1. Accordingly, the Rydberg cor-rection should be greatest for S terms, smaller for P terms,still smaller for D terms, and so on (see Fig. 10). This isactually the case. The Rydberg correction is extremelysmall for F terms; that is, they are practically Balmer terms.In contrast, the Rydberg correction for S terms is so large

*I The stable electron group obtained by removal of the outermost electronor electrons is called the core or kernel.

(for Li, 0.59) that we are not certain what the true principalquantum number is-that is, whether the ground term for Lihas n = 1 or 2. The numbers in Fig. 24 are not the trueprincipal quantum numbers of the emission electron. W eshall find out later,what these are.

The common limit of all term series (Fig. 24) correspondsto the removal of the outer electron (the emission electron),which is moving about the atomic core. Beyond thislimit, as in the case of hydrogen, extends a continuous termspectrum which corresponds to the removal of the electronwith more or less kinetic energy. The existence of this con-tinuous term spectrum is proved from observation of con-tinuous spectra extending beyond the limit of the line series(cf. Figs. 2 and 3). The height of the limit of the termseries above the ground state 1S gives the energy of ionization(ionization potential) of the alkali atom. From Fig. 24 wecan see directly that this is equal to the wave number of thelimit of the principal series (see also p. 23); for Li, theionization potential is 43,486 cm+ or 5.363 volts.

Alkali-like spark spectra. Just as the spectra of He+,Li++, and Be+++ are similar to that of hydrogen, the spectraof the alkali-like ions (ions with the same number of elec-trons) are very similar to the alkali spectra (Paschen,Fowler, Bowen, Millikan, Edlen, and others). The spectraof ions are usually called spark spectra and those of neutralatoms arc spectra, since the former are generally producedin an electric spark (or condensed discharge), and the latterin arcs. This corresponds to the fact that the excitation po-tential of the spectra of ions is much greater than for thespectra of neutral atoms, on account of the necessity ofproducing ionization or multiple ionization of the atom inthe former case. The spectra of singly, doubly, etc.,charged ions are called spark spectra of the first, second, etc.,order. The arc spectrum is indicated by the Roman nu-meral I placed after the symbol for the element; the firstspark spectrum is indicated by the Roman numeral II; and


so on. The following groups analogous to the alkalis havebeen investigated :

Li I, Be II, B III, C IV, N V, 0 VI, F VII, Na IXNa I, Mg II, Al III, Si IV, P V, S VI, Cl VIIK I, Ca II, SC III, Ti IV, V V, Cr VI, Mn VIIRb I, Sr II, Y III, Zr IVCs I, Ba II, La III, Ce IV, Pr V

In section 2 it was shown that, in the series H I, He II,Li III, the spectra and the corresponding term valuesdiffered by a factor Z2. If the spectra of the above series ofatoms and ions were completely similar to H, the wavenumbers of the lines or of the term values should similarlydiffer only by a constant factor (Z - p)“, where Z - p is theeffective nuclear charge acting on the outer electron(Z = atomic number or order number of element, andp = number of core electrons).

In each of the above series, Z - p goes through theintegral number values 1, 2, 3, . . . . Therefore, if thesealkali-like spectra were also hydrogen-like, division of allterm values by the factor (Z - p)” should result in the samevalues for each member. Actually, though the spectra arecompletely analogous in all details (the same number andtype of terms), the individual term schemes do not coincideexactly after division throughout by (Z - p)“. Fig. 25shows this for the series from Li I to 0 VI. As previouslyexplained, this result is due to the fact that the field in whichthe outer electron moves is not exactly the Coulomb field ofa point charge and, therefore, the term values are not simplyproportional to (Z - p)‘, as in formula (I, 11). However,because of their close similarity to hydrogen, the D and Fterms of all members of the above groups and some of the Pterms do coincide approximately after division by (Z - p)’(cf. Fig. 25).

To the right of Fig. 25 is given the position of the H termswith n = 2, 3, 4, . . . , and a’hypothetical term with n = 1.5.The effective principal quantum numbers of the terms cantherefore be read from this scale. For the first P term of


the Li series, this number is nearly 2; for the first D term, 3;and so on. Thus 2, 3, . . . are also the true principalquantum numbers of the terms-that is, they are the prin-cipal quantum numbers which the electron would have if thecore of the atom were very small so that the terms wereidentical with Balmer terms.

In contrast to the P, D, and F terms, the S terms are farfrom being hydrogen-like; for the various members of one ofthe above series of elements, these terms have a noticeablydifferent position after division by (Z - p)“. (Cf. Fig. 25for the Li row.) However, even for these S terms the trueprincipal quantum numbers can be determined from Fig. 25and from similar figures for the other series. With increas-

TGq

cni’0

6,000

16.000

26,000

26,OOC

46,001

60.001

1

.$

6.

4.

8,

I-

I-

I-

I-

D-l

D--

Li IS P D F

;,sT---’;6-6’---;6--s--s---

-.,-I- (-

--a=+----

_-------.z-

! -

_ - - - - - - .z-p=1

Be IIS P D F

z-,x,-,-e 6-6-e-;6e-6-6-

l - 4 ” - ‘ -

? = - - - -

- - - - - - -2 -

_ - - - - - -z - p = 2

B III CIV N VS P D F S P D F S P D

_------- __T----__ - - - - - -.------6- .-------- - - - - - -:-5s6-6- ~6-4 6-6-- - - - - - -

-,-=.I- 4 - ; - , , 4 - 4 - - -----.

-F--s---~

___---__z-

_-------z-p=3

?----

_--------2 -

_ _ - - - - - -z-p=4

_ - - - - -2 -

.----- - - - - - - 1 . 5

Z-p=6 Z - p - 6

OVIS P D

------..------.-s-6-. - 4 - 4 -

--2-=-

Fig. 25. Energy Level Diagrams of Li and Li-like Ions up to 0 VI.

2


ing nuclear charge Z, the core is pulled strongly together andthe external field becomes more and more like a Coulombfield with nuclear charge Z - p. The terms in the aboveseries must therefore become more and more hydrogen-likewith increasing Z - p. In Fig. 25 this effect is seen for Pterms as well as for S terms; it is particularly marked for thelatter. The effective principal quantum number for the 1Sterm is 1.86 for N V and 1.88 for 0 VI, as compared to 1.6for Li; that is, it approaches the value 2, which is, therefore,the true principal quantum number for the emission electronof Li in the ground state and also for Be+, etc. In ananalogous manner the true principal quantum numbers forthe emission electrons in the ground states of Na, K, Rb,and Cs and the corresponding ions are found to be 3, 4, 5,and 6, respectively.22

The Moseley lines. Another representation of the relationbetween the spectra of the alkalis and the alkali-like ions is oftenused. For the terms of hydrogen-like ions,

T RZ8 or=-n2

Plotting m against the nuclear charge should therefore give astraight line going through the origin. The same is true for thehydrogen-like terms of the alkali-like ions when they are plottedagainst Z - p. In Fig. 26 the @@ values for some terms inthe Li group are plotted in this way. We see that the hydrogen-like D and F terms coincide (within the limits of accuracy of thedrawing) with the broken lines which represent the Balmer terms.P terms and S terms also lie on straight lines, but are displacedparallel to the corresponding lines for the Balmer terms (S termsbeing displaced more than P terms). These lines are named afterMoseley, who first discovered the corresponding relation forX-ray spectra. The extent of the parallel displacement is ameasure of the incompleteness of the shielding of the nuclearcharge by the core electrons. The slope of the line equals l/n;hence the slope can be used to derive the true principal quantumnumber. It is evident from Fig. 26, as well as from Fig. 25, thatthe true principal quantum number for the lowest S term (groundstate) of ions of the Li group is 2. A similar state of affairs holds

a For more extensive treatment, see: (7), (8), (9), (11), (12), (13).

I,5] Alkali Spectra 63

for the Na, K, Rb, and Cs series, but the Moseley lines becomeincreasingly curved.

In the Moseley diagram, terms of equal principal quantumnumber (for example, the lowest S and P terms of alkali-likeions, as in Fig. 26) give parallel lines-that is, m - @%@ isa constant. It is easily seen from this that T1 - Tt is a linearfunction of Z - p. This is called the law of irregular doubletsor screening doublets. It is of importance since, when T1 - Tzis known for two members of a series of ions (such as Li I andBe II), the value T1 - Ts can be calculated for other membersof the series. For n = 2, T1 - Te is the frequency of the firstmember of the principal series. Thus, the wave length of thisline may be predicted for higher spark spectra of a series-a factthat is, of course, important in the analysis of these spectra. Anextended discussion of Moseley diagrams and the irregular doubletlaw is given in Grotrian (8) and White (12).

3.0

2.5

l.5

Li I 1 B III CIV N V OVI

Fig. 26. Moseley Diagram of the Terms of Li-like Ions.


6. Spectrum of Helium and the Alkaline EarthsHelium. The emission spectrum of helium consists of a

number of series in the visible region of the spectrum, as wellas in the near and far ultraviolet regions. The number ofthese series is essentially the same as in the spectrogram forMg given in Fig. 4, which will be further treated at the endof this section. There are twice as many line series as forthe alkalis (cf. Fig. 3) : two principal series in the visible andnear ultraviolet (which have different limits), as well as twodiffuse, two sharp, and two fundamental series. Theseseries can again be represented by transitions in an energylevel diagram, but the necessary terms are twice as numer-ous as for the alkalis. There are two series of S terms, twoseries of P terms, and so on.

In the energy level diagram of Fig. 27 the terms are dis-tinguished by ‘S, 3S; ‘P, 3P; and so on.these symbols, see Chapter II.)

(For the meaning ofCorresponding terms of

the two systems with the same order number differ in theireffective principal quantum numbers-that is, in the magni-tude of their Rydberg corrections. The terms of one sys-tem generally lie noticeably deeper than the correspondingterms of the other if the same limit is assumed for all theterm series. This state of affairs was described by earlierinvestigators as due to two different kinds of helium; par-helium (indicated by the left upper index 1) and o r t h o h e l i u m(indicated by the left upper index 3). Parhelium differsfrom orthohelium in having, besides the states with n = 2,3 , ..., an additional deep-lying S state with principalquantum number 1. This is the normal state of the Heatom. Transitions from higher P terms of parhelium (‘P)to the normal state give rise to the far ultraviolet principalseries at 584-504A; this series also appears in absorption[Collins and Price (4 ) ] . Besides this principal series,there exists in the visible and near ultraviolet regions an-other principal series of parhelium corresponding to thetransition from higher ‘P terms to the 2 *S state (cf. Fig. 27).

I, 6] Helium and Alkaline-Earth Spectra 65

Combinations of terms of the para system with those ofthe ortho system have not been observed.23 The term sys-tem of He thus splits essentially into two partial systems,which do not combine with each other (right and left parts ofFig. 27). In particular, the lowest state of orthohelium,2 %S, which lies 19.72 volts above the ground state 1 IS, doesnot combine with the ground state. Those terms whichcannot go to a lower state with the emission of radiationand, correspondingly, cannot be reached from a lower stateby absorption are called metastable. The 2 ‘S state is alsometastable, since the selection rule AZ = f 1 does not allowany transition to 1 ‘S. The metastability of the 2 % state

Fig. 27. Energy Level Diagram for Helium. The running numbers andtrue principal quantum numbers of the emission electron are here identical.The series in the visible and near ultraviolet regions correspond to the indicatedtransitions between terms with R Z 2.

*a The weak intercombination line reported by Lyman at 591.6 A is anNe line according to Dorgelo (55).


is, however, stronger than that of the 2 IS state, since thetransition 2 3S + 1 IS would contradict the prohibition ofan ortho-para transition as well as ∆l = f 1. Transitionswith AI = 0 can occur in an electric field (for example,2 IS --+ 1 ‘S), but not ortho-para transitions (cf. Chapter IV).

The ionization potential of helium as obtained from thelimit of the series 1 IS - m ‘I’ (see Fig. 27) is 24.46 volts.As previously stated, it was in no way possible to derive thisvalue from the Bohr theory, but quantum mechanics givesthe spectroscopic value within the limits of accuracy ofcalculation [Kellner (56); Hylleraas (57)]. The same istrue of the ionization potentials of the helium-like ions, Li+and Be++, whose spectra stand in the same relation to theHe spectrum as those of the Li-like ions to Li. The spectro-scopic values for the ionization potentials of Li+ and Be++are 75.28 and 153.1 volts, respectively.

An explanation of the splitting of the He term schemeinto two practically non-combining systems could be ob-tained from the old Bohr theory only in a very arbitrarymanner. This splitting, however, follows necessarily fromwave mechanics. A complete understanding of it is possibleonly by inclusion of the electron spin, which will be discussedlater.

Heisenberg’s resonance for helium. The theoretical basis forthe explanation of the splitting of the He term scheme was givenby Heisenberg (58) when he applied wave mechanics to a systemwith two electrons. The wave equation for a system such as He,consisting of two electrons moving in the field of a fixed charge 2e(nucleus), is obtained from (I, 24) by substituting

if TI and rt are the distances of the two electrons from the nucleus,and ~12 is the distance of the two electrons from each other.Hence, we obtain:

+F(E+g+g-;)$=o (I,40)

I, 6] Helium and Alkaline-Earth Spectra 67

To zero approximation the repulsion of the electronsbe disregarded. Then, equation (I, 40) is just the sum of twohydrogen wave equations with Z = 2. Each electron may there-fore take any of the ordinary hydrogen energy values with Z = 2,and the eigenfunctions are:

~(ZlY& 22YsZ2) = Qn,(~lylzl)~~,(~zy22)

where the q’s are ordinary hydrogen eigenfunctions [equation(I, 28)]. This result may easily be verified. n1 and n2 are theprincipal quantum numbers of the two electrons. When electron1 is in its lowest energy state (nl = 1) and electron 2 in the staten2* n, the eigenfunction can be written in an abbreviated form:

lb = oiWd2)where the numbers 1 and 2 in parentheses stand for the co-ordinates of electrons 1 and 2. Evidently the state in whichelectron 1 is excited to n1 = h, and electron 2 is is the lowest state,with eigenfunction (~,,(1)~1(2), has exactly the same energy as thestate ~l(l)‘pn(2). This resonance degeneracy is removed if e2/r12,

the electrostatic repulsion of the two electrons in (I, 40), is con-sidered. Because of the coupling between the two electrons,the system will periodically switch over from the state ~01(1)~J2)(electron 2 excited) to the state ~,,(1)~~(2) (electron l-excited),and back again. This is quite similar to the case of two equalcoupled pendulums or two equal coupled electric osoillatingcircuits. If at first only one pendulum (or circuit) is excitedafter a time only the other will be excited, and so on.

Mathematically, the eigenfunction of the perturbed system(including the electrostatic repulsion) to a first approximation isAp1(l)p,,(2) + Bp,,(l)~l(2). Calcula t ion shows that eitherA = B, or A = - B; hence we have (omitting the constantfactor) : .

*a = odhd2) + dlh@) or

90 = 4)d2) - o*wd2) (I,41)These two eigenfunctions correspond to two different eigenvalues,E. and E,, into which the originally twofold degenerate level issplit by introducing the interaction. The first function issymmetric-that is, it remains unaltered by an exchange of theelectrons (exchange of numbers 1 and 2 in parentheses); whereasthe second is antisymmetric-that is, it changes sign for thisoperation.

In the mechanical example, the two eigenfunctions #a and #,,correspond to the two stationary vibrations by superposition ofwhich the observed exchange of energy between the two resonat-ing pendulums (circuits) may be represented. These vibrationsare: the symmetric vibration, in which the two pendulums (or cir-cuits) are always in phase (t t ) ; and the antisymmetric vibration

68 Simplest Line Spectra and Atomic Theory [I, 6 I, 6] Helium and Alkaline-Earth Spectra

in which they are in opposite phases (t +).the two vibrations are evidently different.

The frequencies ofSuperposition of the

two vibrations results in the periodic transfer of all the vibra-tional energy from one pendulum (circuit) to the other.

Similarly, by superimposing*‘. = $,s&2ri(Edh) and I& = J/~e--2~iWalh)t

we obtain a continuous switching over from (al(l)p,,(2) to(~~(1)~01(2). Namely, for t = 0,

*‘. + *a - $8 + Pa = 2Pl(l)Van(2)whereas after a certain interval when e-2ri(E*‘h)t = + 1 ande-?ri(Eo/h) 1 =

E, =!= E’.)- 1 at the same time (which is possible since

q-d + 9’0 = $8 - #a = 24%(l)(P1(2)After a further equal interval of time, 9’. + 9, will again equal2pl(lj(p,(2); that is, the second electron will be excited once more,and so on.

Actually, however, according to the statistical interpretationof wave mechanics, this superposition of \k, and \k, cannot occurin one and the same atom. Either C, (with energy E,) or *a(with energy E,) is excited in the atom. As shown by the func-tions. (I, 41), in each of these stationary states P, and *a, both(~dlM2) and (od%O) are contained; or, in other words, ineach of these stationary states partly electron 1 is excited andpartly electron 2.

The above considerations show that, to every one excited stateof the hydrogen atom with certain n and 1 values, there correspond,in the system with two electrons (He), two excited states withsomewhat different energies, due to Heisenberg’s resonance.One of these states is always symmetric; the other, antisym-metric.exist; the

For the ground state, the resonance degeneracy does noteigenfunction is (pl(l)(p1(2), and there is only one state,

which is symmetric. These theoretical results agree exactly with‘the observed energy level diagram of Fig. 27. The parheliumlevels are the symmetric levels; the orthohelium, the antisym-metric. Even quantitatively, the calculated energy levels andparticularly the energy differences of the two term systems agreeclosely with the observed values.

There is, however, one important difficulty which cannot besolved at this stage; namely, it is found theoretically that thetransition probability between symmetric and antisymmetric termsis exactly equal to zero. This may easily be seen in the followingway: For electric dipole radiation, the transition probability(p. 53) is proportional to the square of

where 5) is one of the three co-ordinates of the Ph particle. Inthe present case, for two electrons this will bc:

B S!Pn!F.m*(xl + x2) cl7

If we now consider the transition between a symmetric and anantisymmetric state, we have to substitute \k,, = %,, and*m = Q’.. However, then the integrand and, therefore, theintegral change sign when the two electrons are exchanged(exchange of index numbers 1 and 2), because 4, then changessign (cf. above), whereas \k, and (~1 + 22) do not. Since thevalue of the integral cannot depend upon the designation of theelectrons, it follows that the integral must equal zero. Thisresult holds, not only for the transition probability produced byordinary dipole radiation, but also for any other type of radiation(p. 53), since the term replacing (Q + 22) would also be un-altered by changing the index numbers. Even the transitionprobability induced by collisions with other particles (electroncollision., and so on) will be exactly equal to zero, because theinteraction term, necessarily, is always symmetric in the two elec-trons of He. There is, consequently, no way of bringing about atransition between symmetric and antisynimetric energy levels.If all the atoms are at one time in a state of one system, as is thecase for normal He (symmetric state)! they should never go overto the other system, and hence the latter system should be un-observable. This conclusion flatly contradicts the fact that both

systems are actually observed; As will be seen later (cf.. ChapterIII, section 1), this is due to the presence of electron spin.

The alkaline earths. A s in the case of He, the alkalineearths and the other elements in the second column of theperiodic system have twice as many series and, correspond-

ingly, twice as many terms as the alkalis. This fact may beclearly seen by comparing the spectrograms of Na (Fig. 3)and Mg (Fig. 4). The Mg spectrogram, it is true, showsmainly diffuse and sharp series and only one line of oneprincipal series. The other lines of this principal series andthe other principal series lie in another region of thespectrum.

The alkaline earths thus have two partial systems ofterms which practically do not combine with each other andlie at different heights. As for He, only one of them, thepara system, has a low-lying state, the ground state 9.


The lowest term of the ortho system, however, is a 3P term-not a 3S term. (Cf. the Ca energy level diagram, Fig;32, p. 77.) Just as with He, the two term systems con-verge to the same limit. From the splitting of the energylevel diagram into two partial systems, we may concludethat, as for He, there are only two electrons outside the atomiccore of the alkaline earths. The same conclusion holds forthe alkaline-earth-like ions. The energy level diagram andthe difference between the two term systems will bc con-sidered in greater detail in Chapter II.

.

. I

. .

? ?

,

CHAPTER II

Multiplet Structure of Line Spectra andElectron Spin

1. Empirical Facts and their Formal Explanation

Doublet, structure of the alkali spectra. As shown inChapter I, the quantum numbers n and 1 just suffice.tocharacterize the different term series of the alkalis (Fig. 24,p. 57). However, they no longer are adequate for He andthe alkaline earths, since for these there are twice as many-term series as for the alkalis-that is, there are two com-plete term systems, which are distinguished by a left upper index 1 or 3 on the term symbol. The physical meaning ofthis method of distinction will be made clear in the ‘subse-quent discussion. Even if ‘we provide an explanation .byassuming that the atom under consideration exists in twodifferent forms (for example, orthohelium and parhelium),the insufficiency of the quantum numbers thus far introducedbecomes still more obvious when we examine the alkalispectra with spectral apparatus of greater dispersion. It isthen found that each of their lines is double, as is generallyknown for the D line of Na. The line splitting increasesrapidly in the series Li, Na, K, Rb, and Cs. It can bedetected for Li only by using spectral apparatus of veryhigh dispersion. However, for the D line of Na, thesplitting is 6 A. Fig. 29(a), page 74, shows this andsome other Na doublets. The line splitting can naturallybe traced back to a term splitting. Either the upper or thelower, or both of the terms involved are double, that is,split into two levels of slightly different energy.

To illustrate, Fig. 28 gives the energy level diagram ofpotassium. The scale used in the diagram is just sufficient

71

72 Multiplet Structure and Electron Spin [II, 1

to show the splitting. The ground state and other S termsare single; the P terms are split, the splitting decreasingwith increasing order number. The components are drawnside by side. If the ground state were split and the P stateswere single, all the lines in the principal series (1S - mP)would have the same splitting (in cm-l) ; but this is not thecase. On the other hand, all lines of the sharp series(2P - mS) have the same splitting, since the commonlower state 2P is split while the upper states mS are notsplit. The lines of the diffuse series (2P - mD) have thesame splitting, for the same reason. The D terms them-selves are split, but the splitting is so much smaller that itmakes scarcely any difference in the case of potassium

, (see below).

25.000

Fig. 28. Energy Level Diagram for Potassium [Grotrian (8)]. Here m isthe empirical order number of the terms (see p. 55). For S terms, the true

principal quantum number of the emission electron (p. 62) is 3 greater than m; ,for P terms, it is 2 greater; for D and F terms, it is equal to m.

II, 1] Empirical Facts 73

Quantum number J. Since the quantum numbers thusfar introduced do not suffice, we distinguish, at first for-mally, the components of the doublets by an index number-that is, a new quantum number. We could write : PI andP2. But, instead, we use as indices: for the P terms, 3 and3; for the D terms, i and 4 ; and so on. , The reason for thisnomenclature will become apparent later. In Fig. 28 thesesymbols are written. over the corresponding term series.In addition, a left upper index 2 (doublet) is given to all theterm symbols (see below). The S terms are given a sub-script 3, although they are actually single. This new quan-tum number (subscript) is designated as J, a.nd was calledthe inner quantum number by Sommerfeld. The differentvalues of J occurring are summarized in Table 3.

TABLE 3

J VALUES FOR DOUBLET TERMS

Term

S

P

D

F

G

L

0

1

2

3

4

J

Each individual term of the alkalis is now characterizedby the three quantum numbers n, l and J. In the futurewe shall write L instead of 2 when we wish to characterize thewhole atom and not a single electron. The selection ruleis the same as for 1 (Chapter I, section 4):

AL,=fl

Selection rule for J; compound doublets. The splittingof the D terms for potassium is so small that, for mostpurposes, they can be treated as if they were single. Ac-cordingly, the D terms are not drawn separately in Fig. 28.Thus there will be practically no difference in the splitting

7 4 Multiplet Structure and Electron Spin [II, 1

of the sharp (“P - ‘8) and the diffuse (“P - 20) series.This also holds for Na, of which a few of the diffuse andthe sharp doublets are shown in Fig. 29(u). The splittingof the D terms becomes noticeable for Rb and Cs, as wellas for the alkali-like ions Ca+, Sr+, and so on. If the indi-vidual doublet term components could combine with oneanother without restriction, four components would beexpected for each of the lines of the diffuse series (since each

w (6)Fig. 29. Examples of Line Doublets. (a) Some Na doublets (part of

the Na emission spectrum reproduced in Fig. 3, taken with larger dispersion).(Z4 Compound doublet of Ca+.designate the terms.

True principal quantum

component of the upper D term should combine with eachof the two components of the lower P term). Actually,only three components are observed, as is shown in thespectrogram for a 2P - 2D transition of Ca+, in Fig. 29(b).Using the J values given above, we obtain agreement withexperiment if we assume for the new quantum number J theselection rule: l

AJ = 0 or + 1 or - 1 (II, 1)Fig. 30 shows the energy level diagram (not drawn to

scale) corresponding to the Ca+ doublet reproduced in Fig.29(b). Transitions allowed by the selection rule are givenas solid vertical lines, the horizontal distance between thelines corresponding to their frequency difference. The

1 If we had distinguished the comwnents of the P and D terms simply bythe indices 1 and 2, a representation of the observed transitions would not havebeen possible with such a simple selection rule.


spectrum produced in this way is drawn schematically inthe lower part of Fig. 30. For the transition, 2D512 - 2P112,AJ = 2. This transition is forbidden by the selection rule,and actually does not appear in Fig. 29(b); however, it isshown by a dotted line in Fig. 30. Asalready stated, ‘the splitting of the upper

J

‘DD(

4

%D term is relatively small, and thus, s/z

using low dispersion, we obtain doubletsonly, as for the sharp series, since2P3i2 - 2D3/2 and *Pa,2 - 2D5,2 practic-ally coincide. Using greater dispersion, %

as in Fig. 29(b), we find that one com- “P

ponent of the doublet, and only one, is I

double. However, this group of lines iscalled, not a triplet, but a compound -J-p+doublet, since it results from the com- Y-bination of doublet terms. The lines of Fig. 30. Origin of

the Bergmann series (2D - *F) similarly ~PC~~~~~f.r$?$;~consist of such compound doublets, (b1.1 Intensities a r ewhich are incompletely resolved still ~e~s~~$~K~~.t~ck-more often than those of the diffuseseries. Allowed combinations for the different series are also indicated in Table 3.

Triplets and singlets of the alkaline earths and helium.A more accurate investigation of the two systems of linesof the alkaline earths, using high dispersion, shows that thepara system consists of single lines (singlets), whereas theortho system consists of threefold lines (triplets). Thesplitting of the latter increases rapidly with increasingatomic number of the element in the second column of theperiodic system. For Hg, the splitting is so great thatdifferent lines of one and the same multiplet lie in differentregions of the spectrum.

Similar to the spectrum of the alkaline earths, even underlarge dispersion, the lines of the para system of He appearsingle, whereas those of the ortho system appear as very


close triplets.2 The symbols already used for He and thealkaline earths (left upper indices 1 and 3) are now under-standable (see Fig. 27, p. 65).of some of the calcium triplets.

Fig. 31 shows spectrogramsAs in the case of the alkalis,

I I I I

(4 (b) (4 6-f)Fig. 31. Some Calcium Triplets (Ca I).

(d) Anomalous triplet (see p. 165).(a), (b), and (c) Normal triplets.

large dispersion (2 A/mm.). These photographs were taken with fairly

the line splitting can be traced back to a splitting of theterms-this time into three components. Fig. 32 shows theenergy level diagram for calcium, with this splitting takeninto consideration.

As in the case of doublet terms, the components of thetriplet terms can be distinguished by indices J, whichmust now be assumed to be integers and to have the valuesgiven in Table 4 (p. 78). The reason for this choice willbe made clear-later. For the alkalis, we found that the Sterms of the doublet system are single. Similarly, here theS terms of the triplet system are single. In spite of thatfact, they are given a J index which, in this case, is equal to 1.These S terms must be clearly distinguished from the Sterms of the singlet system (IS) of the same element, whichlie somewhat higher (cf. Fig. 32). The former combineonly with triplet terms, although they themselves are single;the latter, only with singlet terms.

* For He, two of the components lie so close together that for a long time thelines were thought to be doublets.


Use of the J selection rule (II, 1) gives the possible com-binations indicated in Table 4. For small resolution, allthe resulting lines of the triplet systein are threefold,since then only the splitting of the lower term (whichis the greater) is effective. Even under greatest reso-lution the lines of the principal series (3S - “P) andof the sharp series (“P - “S) are only threefold, sincethe 3S terms are single. However, each line of the dif-fuse series (“P - 3D) and of the Bergmann series (3D - 3F)

VOltS

“F

5-

4-

3

235,ooc

40,ooc

1

45,ooc

cb-4

Singlets Tripletsls, ‘p, ‘0, ‘fi “s, ‘PO “P, “P, ‘Da “0, “DI “&z=n,n nvnsn n n ?a=>’-- ?‘gZ- g - 1 - -

*= .,- I~., - ,-7-l?--n--6r

.n

9 :H.

I.Z&“?~ ~~-6~-6-;~~,~~~, ___- , ---- ‘I----

5 __-- 6 6---- 5- 5- 5-----6.

Fig. 32. Energy Level Diagram of Ca I [Grotrian (8) ]. The diagram showsonly the normal terms. (For the anomalous terms, see p. 164.) The n valuesare true principal quantum numbers. The transitions corresponding tospectrograms (a) to (c) in Fig. 31 are included, among others, in this figure.


Term

S

TABLE 4

J V A L U E S FOR TRIPLET TERMS

L J1

then consists of six components. The spectrogram ofFig. 31(b) shows this for the second member of the dif-fuse series.in Fig. 31(a)

The two lines of the Bergmann series, shownand (c), under the same dispersion are still

simple triplets, since the splitting of the 3F terms is con-siderably less than that of the 3ti terms, which is, in turn,considerably less than that of the 3P terms.

Fig. 33 shows, in greater detail, the origin of a compoundI triplet (“P - “0) in an energy

abc

*

Y *

level diagram analogous to Fig.30. Each of the components ofthe line triplet would be a narrowtriplet if three of the lines (dottedlines in the diagram) were notforbidden by the selection rule.The group of lines obtained inthis way agrees exactly with theobserved spectrogram in Fig.31(b).

It follows from Fig. 33 thatthe separations of the pairs of

Fig. 33. Origin of a Com-&Td Triplet. [Cf. Fig. 31

lines a and b, and d and e, mustbe equal. From the -fact thatthis relation is satisfied by an

observed group of six lines in an unknown spectrum, we canconclude, conversely, that the lines actually belong together


and form such a compound triplet. Apart from this thereare other checks (intensity and interval rules, Chapter IV,section 4).

Prohibition of intercombinations; intercombination lines.As already stated, terms of the triplet system of He prac-tically do not combine with the terms of the singlet system,and conversely. That is, a prohibition of intercombinationsis observed. This also holds for the alkaline earths. How-ever, for them, some intercombination lines (combinationsbetween singlet and triplet terms) actually do appear, al-though they are very weak compared to the allowed transi-tions. The number and intensity of forbidden lines whichdo appear increase with increasing atomic number. Someof these intercombination lines are included in the energylevel diagram for Ca I (Fig. 32). The best-known exampleof such an intercombination line is the Hg resonance lineX2537, corresponding to the transition 3P1 + 5!5’~ (Fig. 74,p. 202). (The Ca line X6573 is analogous.) This is one ofthe strongest Hg lines, but it is considerably weaker than thecorresponding non-intercombination line ‘PI ---) 5!fJ0 at X1849.It should be noted that the selection rule AJ = 0 or f 1holds also for these intercombination lines, with the addi-tional restriction that

J = 0 does not combine with J = 0 (II, 2)

Thus, for Ca or Hg, the lines 3Po --f %b and 3P2 --) ‘So eitherdo not appear at all or appear extremely weakly.3

Higher multiplicities; term symbols. For many elementsother than those dealt with thus far, not only singlet,doublet, or triplet terms, but also terms of higher multi-plicity occur; and, correspondingly, higher multiplets oflines are observed-such as, quartets, quintets, and so on.Also for the higher multiplicities, it is found experimentallythat terms of different multiplet systems usually do not

* The selection rules do not hold quite rigorously. See also Chapter IV.


combine with one another or combine only very weakly(prohibition of intercombinations).

In accordance with the suggestion of Russell and Saun-ders, terms are now generally distinguished by using the mul-tiplicity as the left upper index of the letter giving the L

,ivalue (S, P, D, . . . ) ; this prac-

- tice is analogous to the methodalready used for singlets and

f

s triplets. The J value is given% as the right lower index. Thus‘4 each individual component of a

multiplet term can be charac-terized. For even multiplicities,

*.:: J takes half-integral values;::I*:: for odd multiplicities, integral:: values. (The reason for this

difference will be explained later.)Hence we have symbols such as

% 2p1,2 (read “doublet P onehalf "), 3D2, 3S1, ‘F,,,, and so on.These symbols are also used forsinglet terms where the J value

a. iqd: q !$c”rstE g ggff is equal to the L value; for ex-

- ~~~~rn~th~;;ri~~~a~~u~~~a b c d sfghPi 34. 4P 7 JD Transition turn number, or even the whole

for C at 6800 A. The relative.separations of term and line com-

electron configuration, precedes ponents are drawn to scale from this symbol, as we shall see later.

data given by Fowler andSelwyn (59). If higher multiplicities occur,

the spectra appear more andmore complex. In principle, however, there are the same.regularities as described earlier in this chapter-similarseries of line multiplets (principal series, and so on), andthe same selection rules; hence we need not go into furtherdetail here (see Chapter IV).

As an illustration of a somewhat more complicated multi-plet, a ‘P - 4D transition of the C!+ spectrum is given in


Fig. 34, similar to the compound triplet of Fig. 33. Itshould be noted that a 4P term has only three components.This fact and the given values of J are explained in thefollowing section.

Alternation of multiplicities. The atoms of the elementsB, Al, and the other earths, which follow the alkaline-earthcolumn in the periodic system, have doublet terms like thealkalis, as their spectra show (see the energy level diagramof Fig. 73, p. 198). However, quartet terms also have beenobserved for them, and consequently their energy leveldiagram splits into two partial systems (doublets andquartets), just as in the case of the alkaline earths (singletsand triplets).

All the elements of the carbon group have singlets andtriplets, and sometimes quintets; those of the nitrogen grouphave doublets and quartets, and sometimes sextets; thoseof the oxygen group have singlets, triplets, and quintets;the halogens have doublets and quartets; and the inert gaseshave singlets and triplets, as we have already seen for He.Even and odd multiplicities, therefore, alternate in successivecolumns in the periodic system.

Quite analogous to the alkalis and alkaline earths, thereare, for the other elements, series of arc and spark spectra;for example, C I, N II, 0 III, whose spectra, apart from ashift to the ultraviolet, are completely similar to one another.The Sommerfeld-Kossel displacement law thus holds: The

first spark spectrum of an element is similar in all details tothe arc spectrum of the element preceding it in the periodicsystem; similarly, the second spark spectrum is similar to thefirst spark spectrum of the element preceding it, or to the arcspectrum of the element with atomic number two less, andso forth. On the other hand, arc and spark spectra of thesame element are fundamentally different. The multi-plicity and type of the terms of an atom or ion are thus de-termined solely by the number of electrons. The nuclearcharge affects only the position of the spectrum. The


alternation of multiplicities may therefore be expressed inthe following generalized form:

The terms of atoms or ions with an even number of electronshave odd multiplicities; the terms of atoms or ions with anodd number of electrons have even multiplicities. This ruleholds also for elements not fitting into one of the eightcolumns of the periodic system; for example, the rare earths.

2. Physical Interpretation of the Quantum Numbers

Meaning of L for several emission electrons. For the Hatom and the alkalis (which hape one emission electron), Lis the same as 1, which is itself proportional to the orbitalangular momentum of the electron. For elements with alarger number of emission electrons, such as the earths orthe elements of the oxygen group, the quantum number Lwas at first introduced purely empirically to distinguish thedifferent term series (S, P, D, . . .) of a term system. Itsnumerical value and, from this, the symbol for the corre-sponding term were obtained from the combination proper-ties, the same selection rule being assumed for L as for 1,that is, AL = f 1. Further information was obtained fromthe investigation of multiplet structure and of the Zeemaneffect. In more general cases, transitions with AL = 0 arealso observed (see Chapter IV). The question is: In themore general cases what meaning does L have in our modelof the atom?

If we recall that a definite, constant orbital angular mo-mentum 2 is ascribed to the emission electron of the H atomor of the alkali atoms, it appears very plausible, even in acomplicated atom, to ascribe to each individual electron adefinite, constant orbital angular momentum li, where 1; is avector of magnitude 0, 1, 2, . . . in units h/2*.

That this assumption is true to a fist approximationfollows from the consideration that in complicated atomseach electron may be thought of as moving in the smeared-out field of the other electrons. This smeared-out field isapproximately spherically symmetrical, and an electron

II, 2] Interpretation of Quantum Numbers 83

moving in a spherically symmetric field has, according towave mechanics, quantum numbers n and 1, where 1 isproportional to the angular momentum (see p. 46f.).

The individual angular momenta produce, when addedvectorially, a resultant which depends on the number, mag-nitude, and direction of the respective vectors. Classically,since these can take all possible directions, the resultantmomentum can, in general, take all values up to C:/ Zi I,the last when all Zi are in the same direction.4 Quantummechanics, however, shows that for atomic systems theresultant orbital angular momentum, as well as the indi-vidual angular momenta Z;, can be only an integral multipleof h/2?r.6 The resultant orbital angular momentum is thusL’(h/2?r), or more accurately dL’(L’ + 1) h/2r where L’ istaken temporarily as the corresponding quantum number.The individual li can therefore be oriented only in certaindiscrete directions to one another. For the case of two elec-trons with orbital angular momenta I1 and tz, the possibleresultant L’ values are given by:L’ = (Zl + 121, Vl + Is - 0, 01 + In - 3, . . ., 111 - /*I

Fig. 35 (p. 84) shows the possible resultants for lI = 2,12 = 1. Thus we obtain as many different states of theatom as there are different L’ values. They are distin-guished by the orientation of the orbital planes to oneanother (to use the old Bohr mode of expression).

However, the individual electrons do not move even ap-proximately independent of one another, as do, for example,the planets in the solar system; rather, they exert strongforces on one another (interactions), due partly to theirelectric repulsion and partly to the magnetic moments re-sulting from their angular momenta (see section 3). Theseinteractions have magnitudes which depend on the par-ticular circumstances. For example, if the two electrons

4 In general, the smallest possible value for the resultant is 0. But it will begreater than 0 if one k is larger than the sum of the magnitudes of all the others.

K The basis for this conclusion is quite analogous to the basis for the integralvalue of 1, given on page 41.


have very different principal quantum numbers, the inter-actions are relatively small on account of the large mean

I,=1 L=32.8 -1

separation; whereas they will, in gen-eral, be rather large when the prin-cipal quantum numbers are equal.

This interaction now has the effect1,=2

1,=2 Il*=l

1,=2that the direction of the individual

L=2

t

angular momenta Is no longer con-L=l stant with time (as in the case of the

one-electron problem) but carries outFigS 35S Addition Of a precessional movement (just as the21 and 2~ to Give a Result-

ant Orbital &Wu MO- direction of the earth’s axis carriesmentum L f o r 1, = 2,18 = 1, and L = 3, 2, I. out a very slow precession due to the

interaction with the gravitationalfield of the sun, which seeks, on account of the flatteningof the earth at the poles, to set the earth’s axis perpendic-ular to the orbital plane). In classical as well as in wavemechanics, the resultant angular mo-mentum L’ remains, however, constant 18

c------- - - in magnitude and direction during thisprecession of the individual momenta.The precession for the case of two elec- Ltrons is shown in Fig. 36. The greaterthe interaction, the greater will be thevelocity of p r e If this veloc-ity is of the same order as the angular ~

1,

velocity corresponding to the individ- of ~&$*~,~~~~~ual angular momenta themselves, the ~;~~~t~bY~P~~~~latter lose their meaning completely, VgLes B pict& df thesince then the electron does not de- ~~~t~~” tkLpm&s. . .scribe, even to a first approximation, arotational motion about the individual angular momentumvector as axis, but rather a much more complicated motion.For very strong coupling (very high velocity of precession),this motion reduces, in a first approximation, to a simple

8 In the case of the precession of the earth’s axis, the interaction is so smallthat the period of precession is 25,000 years.


rotation about the precessional axis (the axis of the resultantangular momentum). In this case only the resultant L’has an exactly defined meaning.

If the selection rules for the quantum number L’ of theresultant orbital angular momentum are derived in the wayoutlined in Chapter I, section 4, it is found that theselection rule AL’ = f 1 usually holds, although AL’ = 0can also occur. L’ therefore has just the properties ob-served for the empirically introduced L. Therefore L’ mustbe identified with L. Thus the different term series S, P,D, . . . of an atom with more than one emission electron aredistinguished by different values (0, 1, 2, . . .) for the resultantorbital angular momentum L of the electrons. Hence theselection rule

A L = 0 , *l (II,3)holds. In addition, there is the rule that, so long as theinteraction of the electrons is not very large, only thosequantum transitions take place for which only one of theemission electrons makes a jump - that is, only one altersits I value, the alteration being in accordance with theselection rule (I, 29): AZ = f 1. For example, a state ofan atom with two emission electrons with L = 1, 2, = 1,& = 0 cannot combine with a state L = 2, ZI = 3, & = 3,although this combination would be allowed according to(II, 3) alone.

For strong coupling of the angular momentum vectors,the energy of the entire system will obviously differ accord-ing to the orientation of the individual angular momenta toone another. Thus in the case of two electrons (consideredabove), the energies of the statesL = (II + z2>, (I1 + 12 - 11, (I1 + 12 - 3,

a.-, 111 - 121 (II,4)differ--the difference being greater, the stronger the cou-pling (interaction). The observed magnitude of the energydifference is a direct measure of the strength of the. coupling.

As we have shown above, when there is strong interactionin an atom, the individual angular momenta Zi no longer


have any exact meaning as angular momenta; only theirresultant L has an exact meaning. The momenta li are,however, still of importance in determining the number andtype of the terms. Both in the Bohr’theory and in quan-tum mechanics, Ehrenfest's adiabatic law holds: For avirtual, infinitely slow alteration of the coupling conditions,the quantum numbers of the system do not change 7 and, inparticular, the number of terms does not vary. Hence, ifwe “uncouple " the individual orbital angular momenta byassuming their interactions removed, we come, in the limit-ing case, to the state in which each individual Zi actuallyhas the meaning of an angular momentum and in which wecan carry out the above vector addition. Thus we obtainthe correct number and type of the resulting terms.

Consequently, for the case of a number of electrons in anatom, we ascribe to each electron an I value that would cor-respond to the angular momentum of this electron for in-finitely small or vanishing coupling. Electrons with 1 = 0are called s-electrons; those with 1 = 1, p-electrons; thosewith 1 = 2, d-electrons; and so on (small letters being usedin contrast to capital letters, which represent terms of acomplete atom or ion). The principal quantum number ofthe electron is added to this, and we have such symbols as

1s, 2p, 4d and so on. At all events, even in the actualatom, the quantum numbers Zi still retain their importancefor deriving the number and type of terms, but do notalways correspond to angular momenta-at least not in thestrict sense of the word.

Table 5 shows the term types given by various electron configurations (cf. Table 10, p. 132).

If all but one of the Zi are zero, the resulting L value willnaturally be that of the single Zi. This single 2 then retainsliterally its physical meaning of an angular momentum.

7 The converse of this law is: Only such magnitudes can be quantized asremain constant (invariant) for adiabatic changes. According to Ehrenfest,this converse may be considered the fundamental law of the old quantumt h e o r y .


Such, for example, is the case for most (normal) terms of thealkaline earths and He. The term type S, P, D, . . . thendepends only upon the 1 value of this one emission electron,just as for the alkalis. However, even for the alkalineearths there are terms-the so-called anomalous terms (seeChapter IV, section 2)-for which two electrons have1 =l= 0. For elements of the carbon group and beyond, theoccurrence of such terms with more than one electronhaving I + 0 is quite general.

TABLE 5

L VALUES AND TERM SYMBOLS FOR TERMS WITHDIFFERENT ELECTRON CONFIGURATIONS

Electron Configuration L Term Symbol

s p 1 Ppp 0 1 2 S P Dp d 1 2 3 P D Fdd 0 1 2 3 4 S P D F G

ppp 0 1..1 2 223SPPjPDDF

When there are three electrons for which 1 + 0, the vectoraddition may be carried out simply by combining the l values of two electrons and then combining each of theresulting L values with the I of the third electron.

Physical interpretation of J: cause of multiplet splitting.On the basis of the foregoing, a term with a given L is single.How can we then explain the observed splitting into multi-plets of the terms with a given value of L? As we shallanticipate here (cf. the following section), investigationsof the anomalous Zeeman effect have shown that the indi-vidual components of a multiplet are distinguished fromone another by the total angular momentum of the atom. I nfact, using the above nomenclature for distinguishing thesub-levels (empirical quantum number J), the total angularmomentum is found to be equal to J X h/2s, where, as willbe remembered, J is the quantum number at first intro-

88 Structure and Electron Spin [II, 2

duced purely formally in order to distinguish the sub-levels.According to quantum mechanics, the exact value of thetotal angular momentum is, not J(h/2r), but dJ(J + 1)x h/2r, as in the case of 1. As before, the difference canin many cases be disregarded.

The total angular momentum of the atom is thus not! equal to the resultant (integral) orbital angular momentum

L, which is the same for all components of a multiplet term,but can take as many different values as the multiplicity ofthe term. Thus, to obtain the total angular momentum J,one has to add vectorially to L an additional integral or half-integral angular momentum vector S, whose exact meaningwe shall leave undefined for the moment. According to thequantum theory, L and S cannot be oriented to each otherin any arbitrary direction but only in certain directions(similar to the case of the individual ti), and therefore onlycertain discrete values of the resultant J are possible. Thelargest and the smallest values of J for a given pair of valuesL and S are obtained by a simple addition and subtractionof the corresponding quantum numbers L and S.* In thiscalculation only the magnitude of the resultant vector is ofimportance, since J, naturally, can only be positive. Inter-mediate values of J are also possible, and these differ fromthe extreme values (sum and difference) by integral amounts,just as in the addition of the fi to form a resultant L. Inthis case we have therefore:J = (L + S), (L + S - 1), ( L + S - 2),

. . . , |L- S| (II, 5) In other words, the rule is: the vector addition of L andS is such that the different possible values of their vectorsum have integral differences. Fig. 37 illustrates this rulefor the cases L = 1, S = 4; L = 1, S = 1; L = 2, S = 1;L = 1, S = 3; L = 2, S = 9. When L > S, it is easily

‘This S, naturally, has nothing to do with the S of the S terms. Theformer is a quantum number; the latter, a symbol for L = 0. This nomencla-ture is internationally used and must therefore be used here, although it maylead to confusion.


seen from (II, 5) that the number of possible J values for agiven value of L is --

2s + 1On the other hand, if L < S, the number of possible Jvalues for a given value of L is

2L + 1In particular, for S terms (L = 0, 2L + 1 = 1) there isonly one value of J; namely, J = S.

L=

L = l s=s/,

‘P Term

S=l

I=1

2

L=2 _ S=%

‘D TermFig. 37.

Examples.Vector Addition of L and S to Give a Resultant J for Different

For a given combination of L and S, all the possible orientations ofL and S with respect to one another and the corresponding total angular mo-menta are illustrated.tion is fised in space.

The vector J is indicated by a heavy line. Its direc-The magnitude of the vector J (and, correspondingly,

of L and S) is taken as J(h/2~), and not dJ(J + 1) h/2+, as it should bestrictlyspeaking.

Note that, in drawing such figures, the direction of the first vector is quitearbitrary. It is only for simplicity that all these have been drawn vertically.


, The number of possible J values-that is, possible valuesof the total angular momentum-is equal to the numberof components into which a term of given L is split. Evi-dently when the angular momentum is different for twostates, the energy will, in general, also be different, as wehave already seen when dealing with the vector addition ofthe Zi. Now, we had previously associated some terms with

a system of higher multiplicity, though they themselvesactually had a smaller number of components; for example,we had “S terms, although the S terms are always single.The reason for this apparent inconsistency is nbw clear.The important thing for the behavior of a term is not thenumber of its components but the magnitude of its additionalangular momentum vector S. For 3S terms, the quantumnumber of the additional angular momentum S equals 1,as for 3P and 3O terms. This value of S gives three com-ponents for P, D, . . . terms (cf. Fig. 37), but only one com-ponent for S terms since L = 0. In spite of this fact, the“S term behaves like 3P, 30, . . . terms since for all of themS = 1. The value 2S + 1 is generally called the multi-plicity of a term, which gives the number of possible Jvalues or the number of components only when L > S.

According to the above discussion, the vector additionsin Fig. 37 represent the cases of 2P, .3P, 30, ‘P, 4O terms.The 4P term has only three components (since L < S), butin spite of that is called a quartet term. Table 6 gives themultiplicities (2S + 1) for different values of S.

As Table 6 shows, the multiplicity 2S + 1 is even whenS is half integral (for example, for the alkalis, S = 3, anddoublets result), but is odd when S is integral (for example,for the alkaline earths, S = 1 or 0, and triplets and singletsresult). Conversely, in order to explain an observed evennumber of components (for example, doublets), we mustnecessarily assume that S is half integral: whereas, for anodd number of components, S must be integral.

Just as in the combination of the ti to give L, a precessionof the components Z and S takes place about the resultant J


TABLE 6

MULTIPLICITIES FOR DIFFERENT VALUES OF S

S Multiplicity of the Terms

0 Singlets?4 Doublets

1 Triplets% Quartets2 Quintets

95 Sextets. . . . . .

(cf. Fig. 36, p. 84). The greater the interaction of L andS, the faster will be the precession and the greater will bethe difference in energies of the states with different J;that is, the greater will be the multiplet splitting. Further-more, according to Dirac’s wave mechanical theory of theelectron spin, Sommerfeld’s fine structure formula (I, 12)still holds if k is replaced by j + 3, where j corresponds toJ for the case of one electron. It thus follows from (I, 12)that, for the case of one emission electron, the doubletsplitting is proportional to Z4. Strictly speaking, this con-clusion should hold only for hydrogen-like ions, but qualita-tively this rapid increase in multiplet splitting with increas-ing Z will also hold for all other cases. This result is inagreement with experiment. For example, for Li (Z = 3),the splitting of the lowest 2P level is 0.34 cm-l, for Cs(Z = 55), it is 5540 cm-l; for Be (Z = 4), the total splittingof the first 3F level is 3.03 cm-l, for Hg (Z = 80), 6397.9cm-l. On the other hand, according to (I, 12), the splittingshould decrease with increasing n and L. This effect isalso observed. For a not too high atomic number, themultiplet splitting is, in general, relatively small; that is,the velocity of the precession of L and S about J is small.L and S therefore retain, to a good approximation, theirmeaning as angular momenta, even when the interaction isallowed for. However, for heavy elements, sometimes only

, .

92 Multiplet Structure and Electron Spin [II, 2 II, 2] Interpretation of Quantum Numbers 93

J retains its meaning as an angular momentum (seeChapter IV, section 3).

If the components in a multiplet term lie energetically inthe same order as their J values (smallest J value lowest)the term is called regular or normal and, in the conversecase, inverted. For example, most of the P and D terms ofthe alkalis and t h e alkaline earths are regular doublets or

triplets (Figs. 30 and 33). Similarly, theJ‘4

quartet terms of C+ in Fig. 34 (p. 80) areJ/z also regular. Fig. 38 gives a 4D term as an612 example of an inverted term. The reason

for the appearance of the inverted order of1/2 the terms will not be discussed here [cf.

L=Z, s=s

Fig. 3 8 . Ex-White (12) and Condon and Shortley (13)].

ample of an In-verted Term ‘D.

Selection rule for J. Wave mechanicalCorresponding to calculation of the transition probability (cf.t h e interval rule(see p. 178), the Chapter I, section 4) shows that the selec-separation of the

components in-tion rule AJ = 0, f. 1 holds for the quan-

creases from top turn number J of the total angular momen-to bottom, con-trary to a normal turn of an atomic system. In addition, itterm. is found that a level with J = 0 does not

combine with another level with J = 0.These results agree exactly with the selection rules (II, 1)and (II, 2), which were derived purely empirically fromspectroscopic observations (see p. 74 and p. 79).

Physical interpretation of S. What meaning can we nowgive to the additional angular momentum S in our atomicmodel? Historically, the, first assumption held that thisangular momentum was the angular momentum of theatomic core. The assumption has proved untenable, sincefor the alkalis, for example, the atomic core is formed bythe ground state of the corresponding alkali ion and, ac-cording to the spark spectrum, this is a GSO state (just as forthe inert gases, according to the Sommerfeld-Kossel dis-placement law) and thus has J = 0, L = 0, and S = 0.That is to say, a n angular momentum of the atomic core

cannot be present in the case of a neutral alkali atom.Apart from this, it is difficult to see why the angular mo-mentum of fhe atomic core should be half integral. Butthat J and therefore the additional angular momentum Sreally must be half integral for even multiplicities is alsoconfirmed definitely by the investigation of the anomalousZeeman effect, as will be seem later. Coudsmit and Uhlen-beck were thus led to the assumption that the additionalangular momentum S is due to the electron or electronsthemselves. According to this assumption, each electronperforms a rotation about its own axis as well as a motionabout the nucleus. This rotation is such that the angularmomentum s has the same magnitude for each electron,3 (h/2a) ; the rotation is usually called the spin, or the electronspin.g The assumption of electron spin has been verifiedby an extraordinarily large amount of experimental materialand must. be regarded today as entirely correct.1°

When several electrons arc present, the individual spinvectors si combine with one another just as in the case ofthe Zi previously discussed. It is the resultant spin vectorwhich, according to Goudsmit and Uhlenbeck, is identicalwith the above empirically derived, additional angularmomentum vector S. Analogous to Z, the resultant spinvector S can take only certain discrete values according tothe quantum theory, the maximum value being obtainedwhen all the Si are parallel. In that case, if N is the numberof electrons, the corresponding quantum number S is equalto N/2, since each electron contributes 4. For otherorientations of the si, the following S values arc possible :

N--9 12

N-2 1-- 92

‘-*9 z or 0

The smallest value is 3 or 0 according as N/2 is half integral

8 The rigorous quantum mechanical formula for the magnitude of thevector s would be Js(s + 1) h/2r; that is, with S = $; 1l-43 h/2ir.

*a The assumption appears as a necessary result of Dirac's relativistic wavemechanics. However, this theory has thus far been completely worked outonly for the one-electron problem.


or integral. It follows that S is half integral or integral ac-cording as the number of electrons is odd or even. The em-pirically obtained alternation law of multiplicities followsdirectly from this result, since the multiplicity is equal to2S + 1 and will therefore be even or odd, according as thenumber of electrons is odd or even (see p. 81).

The same conclusions that we have derived for the spinfrom analogy and consideration of the old quantum theorymay also be reached by an accurate wave mechanical treat-ment [Condon and Shortley (13)]. Also, we can obtainthese conclusions rather more simply and schematically (butless rigorously) by assuming that the spins of the individualelectrons in an atom will be either parallel or antiparallel toone another. It is then obvious that the resultant will behalf integral or integral according to whether the number ofelectrons is odd or even.

The exact theoretical derivation shows that, to a first ap-proximation, states with different S (different multiplicities)do not combine with one another. This prohibition ofintercombinations has also been observed empirically (p.79). We therefore have the selection rule for S:

AS = 0 (II, 6)Both theory and experiment show that this selection rule isadhered to less and less strictly as the atomic number in-creases.

The alkalis have one electron outside the atomic core (seeChapter III). Consequently S = 3, and doublet termsresult, in agreement with experiment.

The alkaline earths and He have two electrons outside the atomic core.

!Their spins can be either parallel t t or anti-

parallel t I to one another; that is, S = 1 or 0, and there result triplet as well as singlet states. Each state with a

given L can, in general, occur as a triplet state as well as asinglet state. The ground state, which occurs only as asinglet state, forms an exception (cf. the energy level di-agrams of Figs. 27 and 32), which will be explained in thenext chapter.


With three electrons outside the core, S can have thevalues # and 3, corresponding to t t t and It t. This givesquartets and doublets.

Table 7 lists the possible term multiplicities for variousnumbers of electrons.

TABLE 7

POSSIBLE MULTIPLICITIES FOR VARIOUS NUMBERSOF ELECTRONS

Number ofElectrons I

Possible Multiplicities

DoubletsSinglets, tripletsDoublets, quartetsSinglets, triplets, quintetsDoublets, quartets, sextetsSinglets, triplets, quintets, septetsDoublets, quartets, sextets, o c t e t sSinglets, triplets, quintets, septets, nonets

I

According to the preceding discussion, the spectrum of the Hatom should also be a doublet spectrum. Actually, it has beenshown that the hydrogen fine structure can only be explainedquantitatively by taking account of this fact. The relationsare, however, complicated in this case by the fact that the separa-tion of terms with different l (and equal n) is of the same orderas the doublet splitting. We shall not go into these complications[see White 12)] but merely note that, according to this interpre-tation, the lines of the Lyman series are not single, as assumed inFig. 13, but consist of two components like the lines of theprincipal series of the alkalis. (The experimental investigationof the fine structure of the Lyman lines offers many difficultiesbecause the lines lie in the vacuum region, and has therefore notyet been carried out.)

The fact that multiplet splitting does occur shows that an interaction between Z and S exists. It is, in general,small for a not too great atomic number. This interactionis due to the fact that a magnetic moment is associated withthe electron spin, just as with any rotation of charges.The magnetic moment of the spin is influenced by themagnetic moment associated with the orbital angular mo-


mentum L, the magnitude of this interaction dependingupon their orientation to each other. Therefore, as alreadymentioned, a precession of L and S about the direction ofthe total angular momentum J takes place (cf. Fig. 36,p. 84).

For the magnitude of the magnetic moment of the elec-tron, Goudsmit and Uhlenbeck made the assumption thatit is twice as great as follows from the classical connectionbetween magnetic moment and angular momentum. Themeaning of this assumption will be amplified in section 3of this chapter.

As already noted, states with different S (different multiplicitiesbut with other quantum numbers the same) have appreciablydifferent energies (cf. Figs. 27 and 32). For a not too highatomic number, the energy difference is appreciably greater thanthe energy difference between the individual components of amultiplet. Although it might appear that this energy differenceof terms with different multiplicities is due to the different inter-actions of the spins resulting from their different orientations, theinteraction of the spin vectors Si due to their magnetic momentscannot possibly be very much greater than that of L and S. Infact, theoretically it should be appreciably smaller. The energydifference of the various multiplet systems must, therefore, haveanother origin, which will be dealt with in Chapter III.

3. Space Quantization : Zeeman Effect and Stark Effect

General remarks on Zeeman effect and space quantiza-tion. The necessity for the assumption of an angularmomentum or spin of the electron itself and, in particular,its double magnetism is made especially clear in the explana-tion of the Zeeman effect of spectral lines.be described as follows.

This effect mayWhen a light source is brought

into a magnetic field, each emitted spectral line is split intoa number of components. To a first approximation, thesplitting is proportional to the strength of the magneticfield. Fig. 39 shows three examples of such splitting.

The splitting of the lines is evidently due to a splitting ofthe terms in the magnetic field. The influence of a mag-netic field on energy levels is, perhaps, most clearly under-

II, 3] Zeeman and Stark Effects 97

stood by considering how a magnetic needle behaves in amagnetic field. The potent ia l energy of the magneticneedle depends upon its direction with respect to the mag-netic field. Therefore, if the needle is displaced from the

(4 (b) (c)Fig. 39. Examples of Line Splitting in a Magnetic Field (Zeeman Effect)

[after Back and Lande (6)].(a) Normal Zeeman triplet of the Cd line GUS.47 A (‘I’ - *lI transition).

Above the exposure was so made that only light polarized parallel to the fielddirection could reach the plate (single component at the position of the originalline). Below, the components were polarized perpendicular to the field; theylie symmetrical to the original line.

(b) Anomalous Zeeman splitting of the two D lines of Na, 5889.96 Is and5895.93 A (2,s - *P transition). Above, with magnetic field. Below, without magnetic field.

(c) Anomalous spl i t t ing of the Zn line 4722.16 A (“PI - %I transition).

direction of the field and then released, it will vibrate backand forth about its equilibrium position (the position ofminimum potential energy - that is, when the needle is inthe direction of the field) and can be brought to rest onlyby the dissipation of its energy. Like the magnetic needle,the atom generally has a magnetic moment ~1. The rotationof electric charges which, even according to wave mechanics,takes place in the atom always leads to the production of amagnetic moment in the direction of the axis of rotation.This effect follows the same laws that operate when a cur-rent flows through a wire ring (circular electric current.).The greater the angular momentum of thc atom, the greaterwill be the magnetic moment 1. Because of the inherentconnection between magnetic moment. and angular momen-tum, we have to take into account the gyroscopic forceswhen we discuss the behavior of an atom in a magnetic field.


The effect of these gyroscopic forces is that the rotationalaxis of the atom (direction of p) does not vibrate back andforth about the position of minimum energy, but executes aprecessional motion with uniform velocity about the direc-

Ii/ ------ --I -- --4 ’

+iA0

N’ JII

tion of the field- (so-called Larmorprecession). This precession isshown diagrammatically in Fig. 40.

Just as for the combination ofthe li vectors, the velocity of pre-cession depends upon the strengthof the coupling; that is, in thiscase the velocity depends upon thefield strength H of the magnetic

Fig. 40. Precession of field. It is directly proportional~~~~~~~~1~~~~~~ to the latter. (This holds also forabout the Field Direction asAxis.

the vibration frequency of a mag-net.) So long as no energy is dis-

sipated, the precession continues at a constant angle to thefield direction; that is, J has a constant component M inthe direction of the field. The energy in a magnetic field(as for the magnetic needle) is:

W = W o - HUH (II, 7)where PH is the component of the magnetic moment in thefield direction and Wo is the energy in the field-free case.When v or J is perpendicular to the field direction, W = Wo.

Just as two angular momentum vectors in an atomicsystem cannot, according to quantum mechanics, take anyarbitrary direction with respect to one another but only. certain discrete directions, so an angular momentum vectorcan take only certain discrete directions in a magnetic field.This means that J (and therefore p) is space quantized in amagnetic field. Just as the resultant in the afore-mentionedcase of two angular momentum vectors can take only inte-gral or half-integral values, so in this case the component M

‘, of the angular momentum J in the direction of the field can beonly an integral or half-integral multiple of h/2n. It will be


integral when J is integral, or half integral when J is halfintegral. Thus the following relation holds:

M=J, J - l , J - 2 , . . ..*-J (II, 8) This gives 2J + 1 different values.

The left half of Fig. 41 illustrates the possible orientationsof J to the direction of the magnetic field H for J = 2 andJ = $-. The precession which J carries out about the fielddirection, as in Fig. 40, can take place only at one of thegiven angles to the field direction. For J = 3, only thedirections parallel or antiparallel to the field are possible.

Corresponding to this space quantization, the energy ofthe system in a magnetic field cannot take any value

A H1M ’

N+ -----‘G

0 T 3@\

43

c-_---e4I

I Without Field With FieldI

; ,-+Y( J - 2 <-+;

Fig. 41. Space Quantization of the Total Angular Momentum J in aMagnetic Field H for J = 2 and J = x.tions to the magnetic field.corresponding energy values


between Wo + H 1~ 1 and W o - H 1~ |; but it can take only2J + 1 discrete values. The right half of Fig. 41 shows this

term splitting in a magnetic field. According to (II, 7),the splitting is proportional to the field strength. It is, infirst approximation, symmetrical about the undisplacedterm. All the energy differences between the individualcomponents are the same, since HpH is proportional to M(cf. below) and the possible M values have whole-numberdifferences.

The space quantization itself is independent of the fieldstrength. It remains even when the field strength de-creases to zero, although then all the 2J + 1 different states,differing in orientation, have equal energy: they are de-generate.

This degeneracy in the field-free case is the same as thatalready mentioned for the H atom (p. 47). There we had a2 l + 1 fold space degeneracy. Now, in the general case,J replaces l . Without field, there are consequently 2J + 1different eigenfunctions which belong to the same eigen-value; with field, there are 2J + 1 slightly different eigen-values or energy values belonging to these 2 J + 1 differenteigenfunctions.”

The existence of space quantization is shown most strik-ingly by the Stern-Gerlach experiment in which a beam ofatoms is sent through an inhomogeneous magnetic field. Insuch a field, a body with a magnetic moment is subject, notonly to a force moment tending to turn the direction of themagnetic moment into the field direction, but also to a de-flecting force due to the difference in field strength at thetwo poles of the body. Depending on its orientation, thebody will therefore be driven in the direction of increasingor decreasing field strength. Suppose we now send through

11 In the field-free case, any linear combination of the 2J + 1 eigenfunctionsis an eigenfunction of the same energy value. The eigenfunctions ?tr’lh fieldwill be approximately equal to those without field only when one has chosen the“correct” linear combinations of the field-free eigenfunctions that. are appropri-ate for the problem; that is, when one has placed the z-axis of formula (I, 28)in the direction of the field.


such an inhomogeneaus field atoms possessing a magneticmoment (Fig. 42). If atoms with all possible orientationsto the field are present, a sharp beam should be drawn outinto a band. Actually, a splitting of the beam into 2J + 1different beams takes place. In Fig. 42, J is assumed to be3 and a splitting into two beams results. This experimentshows unambiguously that in a magnetic field not all orien-tations to the field, but only 2J + 1 discrete directions, arepossible.

A

Fig. 42. Schematic Representation of the Stem-Gerlach Experiment. Abeam of atoms possessing a magnetic moment (J = h comes from the left.,passes through an inhomogeneous magnetic field between the poles N and S,and falls on the receiving plate A. The directions of the angular momenta ofthe atoms are indicated by the small arrows.

It must be noted that, whereas the rigorous quantum theoret-ical value for the magnitude of J is dm h/27r (cf. p. 88),the rigorous value for the component M of J is M(h/2r), not&(M + 1) h/27r. Therefore the maximum component of thevector J in the direction of the field is J(h/2*) and not dmX h/2s.. This may at first seem rather puzzling because in classicalmechanics the maximum component of a vector in a given direc-tion is equal to the magnitude of the vector. In fact the magni-tude of a vector in classical mechanics may be defined either (a)by the usual square root of the sum of the squares of the threecomponents, or (b) as the largest value its component can have onsome fixed axis, In classical mechanics, the two definitions areequivalent and hence the distinction between (a) and (b) isnever made explicit. In quantum mechanics, the two are notequivalent-the magnitude being dm h/2* in the sense of(a), and J(h/2a) in the sense of (b), as stated above.

Thus in quantum mechanics the component of J is alwayssmaller than its magnitude, which means that the angular momen-tum vector cannot point exactly in the direction of the field (fixedaxis). This is illustrated on the right in Fig. 43 for J = 3 and


J = 1, whereas on the left tbe more naive representation of Fig. ,41 is given. For larger values of J the difference between the twoways of representation - that is, between definitions (a) and (b)-

becomes smaller and smaller in accordance with the correspondence principle (see p. 28). For the cases where only the corn-

ponents of the angular mo-menta matter, definition (b)is sufficient even in quantummechanics. For some cal-culations, however, it isnecessary to use definition(a). (See below.)

, h/W ,

Fig. 43. Space Quantization forJ = s and J = 1. To the left is thenaive representation (see Fig. 41) takingthe magnitude of the total angular mo-mentum equal to J(h/2r). To the rightis the exact representation taking themagnitude equal to dJ(J + 1) h/2r.

The difference between (a)and (b) in quantum me-chanics is intimately con-nected with Heisenberg’suncertainty relation. If theangular momentum couldpoint exactly in the direc-tion of the field, it would ofcourse mean that the othertwo components were equalto zero. As then all thethree components of theangular momentum wouldhave exact values, it followsfrom Heisenberg’s uncer-tainty relation that the threecorresponding co-ordinates(the angles about the CG, y-,and z-axes) are completelyundetermined. This is onlypossible if the probabilitydistribution is sphericallysymmetrical-that is, if theangular momentum is zero(‘8 state; cf. p. 135). Assoon as the angular momen-tum J is different from zero,only one of the three com-

ponents p,, p,, p, can have an exact value, p, = M(h/27r) ; whereas,for the other two, only the sum of the squares is known, p2 + py’=P-p: = J(J + l)(h/27r)* - M*(h/27r)*, which can never beequal to zero.

With increasing magnetic field strength and therefore in-creasing velocity of precession, Jloses its meaning of angularmomentum. This is similar to the previously considered


case of the vectors lie For strong fields, only M retainsa strict physical meaning, since there results what is essen-tially a rotation of the system about the direction of thefield.

Normal Zeeman effect. The magnetic moment resultingfrom the revolution of a negative electric point charge isgiven classically by:

r= -g&P (II, 9)where fi = angular momentum and m = mass of thecharged particle. Because of the negative sign of thecharge, the magnetic moment has the opposite directionto the angular momentum. For atoms, the angular mo-mentum is J(h/27r) [or more accurately, dJ(J + 1) h/2~].The magnitude of the magnetic moment is thus:

cc= - GC& J or more exactly, *-

For J = 1, the magnetic moment is accordingly:e h

p”=anrcZ;;which is known as the Bohr magneton and has the value0.9273 X 10-m erg/oersted. The component of r in thefield direction is:

pH= e hM- - -2mc2if (II, 10)

Substituting this value of pH in (II, 7), we find that theenergy in a magnetic field is:

W = Wo + hoM, 1 eHwhere o =--2r 2mc (II, 11)

Here o is the so-called Larmor frequency, which may beshown to be the frequency of precession. From (II, 11)we see that the state with smallest energy has its angularmomentum antiparallel to the field direction (M < 0).Because of the negative sign in (II, 9), the magnetic mo-ment is then in the field direction’.

From (II, 11) it follows that terms with different J valueswill have different numbers of components (2J + 1) in a


magnetic field, but that the separation of consecutive com-ponents must be the same for all terms of an atom for agiven field strength. This separation is ho. If two termscombine, it may be shown theoretically (cf. below) that theselection rule for M is:

∆M = 0 , fl (II, 12) with the addition that the combination

M = 0 + M = 0 is forbidden for ∆J = 0 (II, 13) Because there is equal splitting for all terms, the number

of line components is always 3 since all lines with equal ∆M

Without Field With Field M

Fig. 44. Normal ZeemanEffect for a Combination J = 3 + J = 2.’ The arrows repre-senting the transitions form threegroups (indicated by brackets).The arrows in each group haveequal length andfore, to one and

give rise, there-the same line in

the splitting pattern (lower partof figure).

coincide (see Fig. 44). Lineswith ∆M = 0 fall om the posi-tion of the original field-freeline; lines with ∆M = f 1 lie tothe right and left at a distance

0Avnorm = - =C

4.6699 X 1O-5 X H

This kind of splitting is calledthe normal Zeeman effect. Itis observed only for singlet lines(S = 0). [Cf. Fig. 39(a),.. p.97.)

It should perhaps be addedthat, for observations madein a direction perpendicularto the field, the lines with ∆M = 0 are polarized parallel to the field (π -components) the lines with ∆M = f 1, perpendicular to the field

(u-components). [Cf. Fig. 39(a).] These results are inagreement with more detailed calculations, as given below.

The selection rule for the magnetic quantum number M will nowbe briefly derived, according to the methods previously men-tioned (p. 5 1 f.), as a simple example of the wave mechanicalderivation of selection rules. Let #’ = x'eiM'Q and $” = xfreiW“Qbe the eigenfunctions of the upper and lower states, respectively.

=s s

x’x*“zp & dps

eilM’-M”)v d9

This integral is different from zero only when

s

2rei(M’-itf”) Q &,,

does not vanish. Such ii the case only when M’ = M'' T h u sthe z component of the transition moment will always vanishunless M’ = M'' or, in other words, light polarized in the zdirection (direction of the field) will be emitted only when theselection rule AM = M' - M'' = 0 is obeyed.

The x component of the matrix element becomes:

S$‘**“x dr =sss

x’x*‘reifM’--M”)Qp cos pp &7 d.2 dp

=ss

xfx*ffp2 & dp.SeiW'--M") 9 ~0s 9 a9


,

With the field direction taken as the z-axis, the dependence ofthe eigenfunction on the azimuthal angle (p is completely allowedfor in the factor eiMp. [ T he form of the dependence on cp givenpreviously for the H atom is generally true (p. 39).] Thus (Pdoes not occur in x. The matrix element R, associated withthe transition, has components

syQb*“x dT =

sX’XW’ei(M’-d4”)px &

and similarly for y and z.We introduce co-ordinates z, p (distance from z-axis), and (o.

Then dr = p do dz dp; x = p cos ~0; y = p sin 9. Consideringfirst the z component of the matrix element, it is:

~~‘I,$*“z C-IT =SSSx’x*“ei(~.--M”)Qlp dp dz dp

which is different from zero only if the second integral does notvanish. By using cos cp = f(e+ + e+‘), the second integralbecomes :

1 Sei(hf'--M"+I)Qdp

5 .

+ fSei(M'-M"-l)~d~

This vanishes unless the exponent in at least one of the twointegrals = 0; that is, we have the selection rule ∆M = + 1 or- 1. The same result is obtained for the y component. Forboth components of the transition moment perpendicular tothe field direction, we therefore have the selection rule ∆M = f 1.In this way we obtain not only the selection rules but also thepolarization rules. The components of the splitting pattern with


AM = 0 are polarized parallel to the field direction; those withAM = f 1, perpendicular to the field direction. These resultsare in agreement with experiment. [Cf. Fig. 39(a).] A m o r edetailed treatment, which we shall not discuss here, leads to theadditional rule that the transition M = 0 * M = 0 is forbiddenfor transitions with AJ = 0.

Anomalous Zeeman effect. For all lines that are notsinglets, the so-called anomalous Zeeman effect is observed.[See Fig. 39(b) and (c), p. 97.] It consists of a splittinginto many more than three components with separationsthat are rational multiples of the normal splitting Av,(Runge's rule). This effect can be explained only by as-suming that the magnitude of the term splittings for a givenfield strength is not the same for all terms but differs ac-cording to the values of L and J. We may account forthis formally by replacing equation (II, 11) by:

W= W,+hoMg (II, 14)

where g, which is called Lande's g-factor, is a rational numberwhich depends upon J and L. It is quite obvious that, evenif we retain the above selection rule AM = 0, f 1, thenumber of line components obtained in a magnetic field willnow depend upon the number of term components (2J + 1).

Consider, for example, the D lines of sodium, which cor-respond to the transitions *PIi --) 2S1,2 and *Pal2 -+ *Sli2.Since M has only two values for each of the terms 2P112 and2S1,2, and has four values for .*Pa12, it is clear that with adifferent g value for each of the three terms the number ofcomponents of the splitting pattern for one D line of Na willbe different from that for the other. As Fig. 45 shows, weobtain four and six components, respectively, in agreementwith experiment. [Cf. Fig. 39(b) .] Conversely, the ob-

served difference in the splitting patterns of D1 and D,shows that the two P levels, 2Pt,2 and 2Pa,2, differ from eachother in the magnitude of their total angular momentum(the orbital momentum being the same), since it is thiswhich is space quantized. Thus J is actually to be identi-fied with the total angular momentum, as we have already


assumed in the foregoing discussion. Finally, the fact that

Ul D2

Fig. 45. Anomalous Zeeman Effect of the Sodium D Lines, *PIIs + Y&pand SPs 2 + 2S,,2. [Cf. Fig. 39 (b), p. 97.] The components designated by Qhave & = f 1. those designated by z have AM = 0. It should be notedthat, contrary to Fig. 44, arrows indicating transitions with equai AM no longerhave the same length, because of the difference in the splitting in the upperand lower states.

i

actually, J must be taken equal to + in both cases. Withno other choice of J values isit possible to obtain a splittingof each of the terms into two *scomponents. For example, if Jwere equal to 1 for both terms, ,. ,I, IIthe terms would split into threecomponents each, and the lineinto six components (cf. Fig. 46).In a similar manner, the cor-

*p,

rectness of the other J values inTables 3 and 4 can be shown (p.73 and p. 78). Fig. 46 givesthe explanation of the splitting Fig. 46.for a %S1 ---) 3P1 transition, a Splitting o f

t i o n . [ C f .spectrogram of which is repro- According t o

duced in Fig. 39(c).(II, 13), t h e+ M = 0 is ,

We saw above (II, 14) that the anomalous Zeeman effect line.

Anomalous Zeemana *S, -c aP1 Transi-Fig. 39 (c), p. 97.] the selection rule transition M = 0forbidden. since at

the same time ∆J = 0. This

t rans i t ion i s ind ica ted by a do t ted


can be explained formally by introducing the factor y + 1into formula (II, 11) for the splitting of a term in a magneticfield, where g depends on L and J. The original formula(II, 11) with’g = 1 for all terms was based on the assump-tion that the magnetic moment was given by the classicalformula (II, 9). This assumption must, therefore, be in-correct for atoms showing an anomalous Zeeman effect (forwhich g + 1). So long as we consider only a revolution ofpoint-like electrons about the atomic nucleus, it is difficultto understand any deviation from formula (II, 9). Buteven on the basis of the classical theory, the rotation of anon-point electron about its own axis would lead to a ratioof mechanical angular momentum to magnetic momentdifferent from that given by (II, 9). Thus we can wellimagine that the magnetic behavior of the spin of the elec-tron is not the same as that arising from orbital motion.

The extent of the departure from the normal orbital typecan be obtained, for example, when the behavior of theground state *S of the alkalis in a magnetic field is consid-ered, since in this state J results wholly from the spin of oneelectron. It is found empirically that, for this *S state,

∆W = f ho1that is, g = 2; whereas, if J = s =behavior, we ought to have:

3 had a normal magnetic

∆W = f hoior g = 1 . However, the empirical splitting ∆W = ho1 is ob-tained for the 3 term if we assume that the magnetic momentof the electron due to its spin is one whole Bohr magneton,

e kl-2mc2r

and notk 1

-&ma s would be the case if the electron behaved normally. Theassumption that the electron has a magnetic moment of onewhole Bohr magneton (whose direction is opposite to thatof the spin), in spite of the fact that its spin is only *h/21,


was first put forward by Goudsmit and Uhlenbeck, simul-taneously with the hypothesis of elect.ron spin, and leads to

!

a complete explanation of the splitting in all other cases aswell as in the special case considered above. It is clear thatin the general case g depends on the values of L, S, and J,and will differ from the limiting values g = 1 for S = 0, andg = 2 for L = 0. Theoretically it is found (cf. below)that the following formula holds (Lande) :

s=1+ J(J + 1) + S (S + 1) - L(L + 1)cr 2J(J + 1)

(II,,

15)

Suppose an atom has the values of L,S, and J given in Fig. 47. The length

‘of these vectors is proportional tom, m, and m,respectively, if we take the rigorousformula. The magnetic moments ~1,and ~8 associated with L and S arc in-cluded in the same figure. The resultingmagnetic momrnt would lie in the direc-tion of J if the magnetic moment psconnected with the spin were normal,since then &S would equal ~LIL.Actually, the magnetic moment of thespin is twice as large as if it were normal;that is,

J L

The resultant p therefore falls, not in thedirection of J, but in the direction shown,which is different from J and precesseswith L and S about the direction of thetotal angular momentum. Since thisprecession is, in general, much fasterthan the Larmor precession, usually onlythe component of 0 in the J direction,VJ, need be considered in calculating themagnetic effect. This is (see Fig. 47):

Fig. 47. Addition ofMagnetic Moments in anAtom (in Explanation ofthe Lande g-formula).The length of the v&or

tc is taken equal to L.herefore IS is double the

length of S. It should henoted that the directionof the angular momentumvectors is opposite to thatof the corresponding mag-netic moments.

bJ = FL cos 6% J> + vs cos (8, J) (II, 16)In this,

e -?!- m andOr. = - 2mc 2?r

(II, 17)

110 Multiplet Structure and Electron [II, 3 II, 3] Zeeman and Stark Effects 111

Here the minus sign indicates., as before, that the magneticmoment has the opposite direction to the corresponding angularmomentum vector.

In the calculation of the magnetic splitting, it is PR, the com-ponent of pJ in the direction of the field, which matters. Inorder to obtain (II, 14) instead of (II, 11) for the energy in amagnetic field, we have to replace formula (II, 10) for go for-mally by

ehJfg‘!H = - 2mc 2~

This substitution means we have to take as definition for g

-!- h lo(z-qq g IJ = - 2mc 2s (II, 18)

The factor g can be calculated from (II, 16) when both cos termsare known. From Fig. 47, using the obtuse-angled triangleformed by L, S, and J, we obtain, with the help of the cosine law:

cos (L9

J; = J(J + 1) + L(L + 1) - S(S + 1)2-m

cos (S, J) = J(J + 1) + S(S + 1) - L(L + 1)2mm

(II, 19)

Substituting (II, 17), (II, 18), and (II, 19) in (II, 16), and omit-ting the factor kc &, we find:

= J(J + 1) + L(L + 1) - S(S + 1)2J(J + 1)

+ 2 [ J ( J + 1 ) + S ( S + 1 ) - L ( L + 1 ) ] 2J(J + 1)

(II, 20)

This last expression shows clearly the meaning of the factor 2 inthe second term. If the factor 2 were not present, that is, if theelectron spin were magnetically normal, g would obviously equal 1.But, by including the factor 2, we obtain from (II, 20) the Landeformula already given i n (II, 15). Thus g is a rational numberwhich is generally different from 1, in agreement with experiment.For J = S and L = 0, g = 2, a value that we have already used.

If, in the derivation of the g-formula, we had used simplyJ(@%r), L(hj2r), S(h/2*) for the magnitudes of the vectorsmstead of the accurate quantum mechanical values, obviously a different formula would have been obtained. The fact that theLande g-formula gives extraordinarily close agreement with

experiment is further evidence of the correctness of the quantummechanical formula for the magnitudes of the angular momentumvectors.

From (II, 15) the g values for 2P1,2 and 2Ps12 arc 3 and j.Using these values, the energy level diagram for the Zeemansplitting of the sodium D lines has been drawn in Fig. 45, and isin quantitative agreement with experiment. [Cf. Fig. 39(b).]

It might at first appear remarkable that the term ?Pl12 showsany splitting at all. According to our earlier discussion, for2P1,2 the vectors L and S are in opposite directions [cf. Fig. 37,p. 89] and, since L = 1 and S = 3, we would expect zero mag-netic moment because of the double magnetism of S; corre-spondingly, no splitting should result. The above formula gives,however, g =I= 0. A magnetic moment will therefore bc present.When the accurate wave mechan-ical values for the angular momen-tum vectors are taken, J, L, and Sfor 2Pl,e do not fall in a straight linebut produce the diagram shown in

iFig. 48. It is seen that the. twomagnetic moments of L = 1 andS = 3 do not compensate eachother. When the length of L repre-sents at the same time the magni-tude of PL, PS is twice as long asS and p has the indicated directionand magnitude. The whole systemof vectors precesses about J. Themagnetic behavior depends only upon pJ, the component of p in the L-l S=%, i=ndirection of J. We can easily see

i from Fig. 48 that pJ is not zeroFig. 48. Origin of the Mag-

and, correspondingly, the splittingnetic Moment for a *P,,~ State.(Cf. Fig. 47.)

adiffers from zero. The differencebetween the old quantum theory and the new quantummechanics is particularly striking in this case.

The foregoing considerations have shown that when L and Sin. a stationary state differ from zero, the magnetic moment pis not constant, but continually changes its direction (precessesabout J). The possibility of magnetic dipole radiation, mentionedpreviously, depends on this fact.

i

In addition to the term splitting, the relative intensitiesof the individual components in the transitions can also bepredicted theoretically. [Cf. Hund (7); Condon andShortley (13).]


The line splitting will vary according to the values of J,L, and S in the upper and lower states--that is, accordingto the term type in the upper and lower states. Converseley,the investigation of the Zeeman effect forms a very effectivemeans of establishing the type of’ term taking part in atransition. This is particularly useful for complicated linespectra. For example, it enables us to find which linesbelong to a Rydberg series since they must all show thesame Zeeman effect. [Further details may be found inBack and Lande (6).]

From the above discussion it is clear that the double mag-netism of the electron is fundamental to the explanation of theanomalous Zeeman effect and phenomena related to it..Actually, the double magnetism of the electron, as well asthe spin of the electron itself, may be derived from Dirac'srelativistic wave mechanics without ‘the use of any addi-tional assumptions. The fact that such a large body ofcomplicate4 phenomena (Fig. 39, p. 97, shows only thcsimplest examples) can be dealt with completely andquantitatively must be regarded as one of the remarkableachievements of wave mechanics.

Paschen-Back effect. With increasing field strength,when the magnetic splitting becomes greater than the multi-plet splitting, Paschen and Back found that the anomalousZeeman effect changes over to the normal. This has the follow-ing reason: When the magnetic splitting becomes greaterthan the multiplet splitting, the precessional velocity o ofJ in the magnetic field about the field direction becomesgreater than the precessional velocity of S and L about J(see above). The resultant motion is, therefore, betterdescribed as an independent precession of S about the fielddirection and a similar precession of L about the field direc-tion, the motion being somewhat disturbed by the couplingof L and S. Hence we say that L and S are uncoupled bythe magnetic field. To a first approximation, each of thesevectors is therefore space quantized in the magnetic field


independently of the other ivith components b and MS,respectively. For each value of l!!L = L, L - 1, L - 2,. . . - L, MS can take each of the values S, S - 1, . . . ,- S. The magnitude of the term splidting is then, to afirst approximation :

∆W = hoML + 2hoMs ((II, 21) (S has double magnetism) and is, therefore, again an integralmultiple of the normal splitting,as in (II, 11). MS

x,-+hFor MI,, we haave the same Mu ‘-

selection rules as for M, and for+1--+----

the same reasons as those given‘p o-+~

earlier :ml---< -

\:: .IAML = 0, f 1 (II, 22)

For MS, the following selectionrule is obtained from theory:

AMs=O (II, 23)Taking into account these selec-tion rules and using (II, 21), a

I/’normal Zeeman triplet is ob- *s O--<:,tained for a transition between \1

two multiplet terms in a strongmagnetic field. Fig. 49 showsthis for a 2P -+ 2S transition (forexample, the D lines of Na). It

ll ll llY *

should be compared with Fig. 45 Fig. 49 . Paschen-Back Ef-

(p. 107), which applies to the ~$~~&i$$?;%?~~$~~same transition in a weak field. in the normal Zeeman effect theyIn a more rigorous treatment a s& y!Fz J$\h2ggg;tf;correction term of the form shown here, each component ofahMsML must be added to ∆W the “normal ” triplet has two

in (II, 21), because of the inter-components.

action of L and S which is ‘naturally still present. As aresult of this, each component of the normal line triplet willgenerally give a narrow doublet, triplet, etc., according asthe original field-free transition was a doublet, triplet, etc.,


transition. The reason for this line splitting is apparent fromFig. 49 for the ease of a doublet transition.“* M o r e com-

Fig. 50. Stark EffectSplit t ing of the HeliumLine X4388 [after Foster(132)]. Above, the ex-

posure was so made thatonly light polarized par-allel to the field directioncould reach the plate.Below, only light polar-ized perpendicular to thefield could reach theplate. In each patternthe field increases fromtop to bottom.

plicated splitting patterns are obtainedby using intermediate fields. [Incom-plete Paschen-Back effect. See Backand Lande (6); White (12).]

It is readily seen that the totalnumber of term components is thesame in both strong and weak fields:(2L + 1) X (2S + l), in agreementwith the Ehrenfest adiabatic law (men-tioned previously).

Stark effect. As Stark first dis-covered, a splitting of the spectral linesalso takes ‘place in an electric field.Fig. 50 illustrates the splitting of theHe line X4388 in the two directionsof polarization (parallel and perpen-dicular to the field). In each patternthe strength of the field increases fromtop to bottom. [For experimentaldetails, see Foster (132).] As will beseen, the patterns are not symmetricalabout the original line, in contrast tothe Zeeman patterns. The splitting ofthe lines in an electric field can natu-rally be traced to a splitting of theterms. The relationships are, how-ever, not quite so simple as for theZeeman effect, and therefore the Starkeffect is of no particular value as ahelp in the analysis of a spectrum.On the other hand, apart from its in-

trinsic interest and as an application of quantum theorythe Stark effect plays a very important part in the theoriesof molecule formation from atoms, of the broadening ofspectral lines, and of dielectric constants.

y The two components of the central “line” in this case coincide almostexactl since nh;lJ~Jf L is zero for both upper states; but in higher approxima-tion they would not coincide.


The components of the angular momentum J can takeonly the values + J, J - 1, J - 2, . . . , - J with respectto any preferred direction. This rule holds also for thedirection of an electric field. Thus space quantizationtakes place also in an electric field. If, and how, the stateswith different M differ from one another energetically de-pends upon the kind of field acting.

An electric field does not act on the magnetic momentassociated with J. The result of the action of an electricfield is, rather, that the atom becomes electrically polarized,as shown schematically in Fig. 51.The positively charged nucleus K be-comes separated from the center ofgravity S of the negative charges byan electric field E. There results anelectric dipole moment, proportional tothe field, whose magnitude dependsupon the orientation of the orbit, thatis, of the angular momentum J, to thefield. The atom seeks to set itself inthe direction of smallest energy, justas in the case of a magnetic field.Because of the gyroscopic forces, thisproduces a precession of J about thefield direction such that the com- Fig. 51, Production ofponent M of J is constant (see Fig. ~e$~“,“E~~$~ FEFi51). The stronger the field, the morerapid will be the velocity of preces-

$t;~fE~;~~U~~~~~;~Direction. The shaded

sion. The energy shift is given by ~$Ce,,~~$~~et$o~-the product of the field strength and The angular moment&the dipole moment-a result anal- ~~~~~~~n~r*‘end’cul~r toogous to that of the magnetic case.However, since the dipole moment is itself proportional tothe field strength, the term shift in the Stark effect is propor-tional to the square of the field strength. Closely connectedwith this relation is the fact that, in an electric field, theterm components, which differ only in the sign of M, have


the same energy. Obviously the dipole moment producedby the field will not bc altered by reversing the direction ofrotation (change from + M t o - M), and consequentlyt h e e n e r g y shift for + M and - M is the same. Thus thereis qualitatively an essential difference between the Starkeffect and the’ Zeeman effect. The number of term com-ponents in an electric field is therefore J + 3 or J + 1, ac-cording as J is half integral or integral.

If the behavior of an atom (other than hydrogen) in anelectric field is calculated according to quantum mechanics,

as first done by Foster (61) for the

{

7---------“pqa 11.

?? M case of helium, it is found that the2 a/a

I ! magnitude of the shift of the terms by

j; t+ ‘/a an electric field depends, in a rather

“Ph T - - - - - - - -r j

I jcomplicated manner, on the quantum

II ! ?‘/a numbers of the given atomic state andji ; ! :ii :

: ;; :

i ts distance from neighboring terms.::; it:: : In general, the component with small-ii ,

Iest | M | lies lowest (that is, M = 0, or

*svJ<- - - , -‘!i !M = f g,

Fig. 52 illustrates the Stark effect of2 % the D lines of Na. This case has been

I I I thoroughly investigated experimen-

I tally and agrees closely with the accu-

Fig. 52. Stark Effectrate theoretical formulae [see Condon

of the D Lines of Na.Field-fret terms and

and Shortley (13) ; Ladenburg (GO)].

transitions arc indicated From the illustration it may be seenby broken lines. that the splitting of the lines and terms

is, in contrast to the Zeeman effect,not symmetrical about the field-free positions. In the caseof the D lines of Na, a shift to longer wave lengths takesplace. In the case of He (cf. Fig. 50), some components areshifted to longer and some to shorter wave lengths.12

If the field becomes so great that the velocity of precession

I* That the shift in thii case is in both directions (though in general it is inonly one direction) is due to the fact that, for He, a number of terms are fairlyclose to one another (cf. below).


about the field direction is greater than the velocity of pre-cession of L and S about J, an uncoupling takes place, as inthe Paschen-Back effect. L and S are then independentlyspace quantized with respect to the field direction in sucha way that the components are ML and Ms. States withpf,l = L,L - 1, ..., 0 have noticeably different energies.For each of these states, MS can take the values + S, S - 1,. . . , - S. When ML = 0, states with different MS do not

have different energies, since no electric dipole moment canbe induced in the electron itself. When ML + 0, on theaverage a magnetic field in the direction of the electric fieldresults from the precession,because of the magnetic mo-ment associated with L. Thisproduces, as a secondaryeffect , an energy difference forthe states with different Ms.

With Field

Fig. 53 shows these relationsfor a 3D term in an energylevel diagram; they are espe-cially important in the theory

I’I

of the electron structure ofmolecules.

Although the Stark effect was

0I’ -*Ip-I 0 21I --1I III +l-0 0

first discovered for hydrogen and 0- 1

although it is particularly laige Fig. 53. Stark Effect in ain this case, theoretically it is Strong Electric Field for a aD Term

more complicated because of the(Analogous to the Paschen-Back

fact that states with different LEffect

and equal n are degenerate with one another, except for relativityand spin effects. Disregarding this last influence, Bohr’s theoryhad already given the observed splittings both qualitatively andquantitatively. In fact, this application was one of the strikingsuccesses of Bohr’s theory. Wave mechanics gives exactly thesame formulae. Both theories show that a level with given nsplits, in an electric field, into 2n - 1 equidistant levels. Thissplitting increases linearly with field strength (linear Stark effect)and takes place symmetrically with respect to the field-freeposition of the terms. For the upper state of H,, the total split-


ting amounts to 7.8 cm-l for a field of 10,000 volt/cm. In thecase of non-hydrogen-like atoms, the splitting is in general verymuch smaller.

In order to obtain theoretically the Stark effect splitting of Hand H-like ions for low field strengths, relativity and spin effectshave to be considered, and these then give an unsymmetricalsplitting of the individual fine structure components.

For hydrogen, at very high field strengths a quadratic effect issuperimposed on the linear effect, and results essentially in a one-sided shift of the whole splitting pattern. The experimentallyobserved magnitude of this effect is in agreement with wavemechanical calculations but not with the old Bohr theory [Rauschvon Traubenberg (133)].

For atoms with several electrons, the linear Stark effect be-comes important if the splitting due to the quadratic effect iscomparable to the energy difference between states with differentL and equal n, as is the case for H and H-like ions even for veryweak fields. For atoms other than hydrogen, the linear effectmay easily occur for the higher series numbersillustrates that result.

In fact, Fig. 50This spectrogram also shows another im-

portant fact which is connected with the above. With increasingfield, theory shows that the selection rule AL, = f 1 for the termsof one emission electron no longer holds exactly because L losesmore and more its meaning as angular momentum. Thus in anelectric field transitions may take place which would be forbiddenin the absence of a field; for example, S - S, S - D, P - P,P - F, and so on. The He line λ4388 (Fig. 50) corresponds tothe transition 2 ‘P - 5 ‘D.boring transitions 2 ‘P -

But in an electric field the neigh-5 ‘P, 5 IF, 5 ‘G likewise take place.

These give rise to the lines to the right in Fig. 50. It is seen thatthese lines gradually vanish toward weaker fields, and also ap-proach. positions different from the non-forbidden lines to the leftin the figure. For strong fields, the whole pattern tends to becomesymmetrical, as for the hydrogen-like spectra, because then thesplitting is large compared with the separation between 5 lP,‘D, ‘F, ‘G.

In certain cases the fields due to ions present in a discharge, oreyen interatomic fields, are sufficient to cause the appearance ofsuch forbidden transitions (cf. also Chapter IV).

Statistical weight. There is no means of furthersplittingthe individual term components present in a magneticfield.ls They must therefore be regarded a s actually simplethat is, no longer degenerate. These different states are

I8 Here it is assumed that the degeneracy between states with different I hutequal 7~ for H atoms and H-like ions has already been removed-for instance,by an electric field.


ascribed the same a priori probability, or the same statisticalweight; that is, it is assumed that they will appear equallyoften under the same conditions. This assumption hasalways been found to be correct.

As the magnetic field grows weaker and weaker, naturallythe equality of the statistical weights of the individual termcomponents does not change. For the limiting case I-I ---f 0,groups of such term components coincide. The resultantterm which is thus formed has a statistical weight that is asmany times larger than the weight of a simple term as theterm itself has components in a magnetic field. Hence, ifwe take the statistical weight of a simple term equal to 1,then the term with angular m o m e n t u m J has a statistical weight2J + 1, since this is the number of simple term componentsof which it consists in a magnetic field-and, therefore, alsoin the absence of a magnetic field-corresponding to thedifferent possible orientations of J. More generally ex-pressed, the statistical weight of a term is equal to its degreeof degeneracy .I4

For two states with different J values, Jt and Jz, theprobability that the atom will be found in one of these states.is given by the ratio (2J1 + 1) : (2J2 + 1). This is true ifthe states in question have approximately equal energies sothat their Boltzmann factors emslkT are equal.

However, not all degeneracies are removed by an electricfield. Terms with equal positive and negative M have,equal energy. Thus, with the exception of M = 0, eachterm in an electric field is doubly degenerate-that is, it stillhas a statistical weight 2.

An important alteration in the absolute value of thestatistical weight results from nuclear spin, and will bediscussed further in Chapter V.

Among other applications, statistical weights are of im-portance in the calculation of the intensities of spectral lines(see Chapter IV), of the specific heat of gases, and of chem-ical constants.

14 In wave mechanical terms, the statistical weight is, accordingly, equal tothe number of independent eigenfunctions belonging to a given energy value.

CHAPTER III

The Building-Up Principle and the PeriodicSystem of the Elements

1. The Pauli Principle and the Building-Up Principle

We have previously considered the terms of atoms withseveral electrons-in particular, those with several emissionelectrons. We shall now treat this topic in greater detailand investigate how the energy level diagram and theground state of any atom can be theoretically derived.

Quantum numbers of the electrons in an atom. A singleelectron moving in a spherically symmetrical but non-Coulomb field of force (for example, the emission electronof an alkali) can always be characterized by two quantumnumbers, the principal quantum number n and the azimuthalquantum number 1. According to quantum mechanics, ncan be considered as an approximate measure of the extentof the region in which the electron preferably remains.Different values of n (1, 2, 3, . . .) correspond to widelydifferent energy values. 1 gives the angular momentum ofthe electron in its orbit and, for a given value of n, can takeall integral values from 0 to n - 1. The energy differencebetween states with different I and equal n is, in general,not so great as that between states with different n. Thepossible states of an electron can thus be divided intoprincipal groups or levels which differ from one another intheir n values, and into sub-groups or sub-levels which havea given n but different I values. This division is clearlyindicated in the energy level diagram of lithium (Fig. 24, p.57). Even for given n and 1 values, several different statesof an electron are possible: first, due to the various possibleorientations of the vector 2 (for example, in a magneticfield), and, second, due to the electron spin s = 3 which can

120

III, 1] Pauli Principle 121

set parallel or antiparallel to a magnetic field (for example,that of 2).

In an atom with several electrons, the motion of each indi-vidual electron can also be regarded, to a first approxima-tion, as a motion in a centrally symmetric but. non-Coulombfield of force. This field results from the overlapping ofthe Coulomb field of the nucleus and the mean field of theother electrons. Therefore, to this approximation a definitevalue of n and 1 can also be ascribed to each electron in acomplicated atom. The approximation will be particularlygood when WC are considering a single electron with large n,as is usually the case for most of the higher terms of anatom (terms of an emission electron). Then the action ofthe remaining electrons may really be described, to a closeapproximation, as due to their mean field. In contrast tothis, if we are dealing with a number of electrons which haveequal n and are thus approximately equidistant from thenucleus, taking a mean field gives a relatively poor ap-proximation, since the action of the other electrons on agiven electron is strongly dependent on their momentarypositions. In this case, the field in which the given electronmoves can, in general, no longer be considered (evenapproximately) to be centrally symmetric and the quantumnumbers n and I have no longer any exactly definablemeaning. In spite of this, they can still be used to obtainthe number and type of the terms, on account of theadiabatic law (see Chapter II, section 2).

The normal state of an atom is that state in which all theelectrons are in the lowest possible orbits. The lowestpossible orbit of a single electron in a centrally symmetricfield is the 1s orbit. (n = 1, 1 = 0), which is also called theK shell. Accordingly, one might perhaps think that, forthe normal state of uranium, all the 92 electrons are in thisK shell, and analogously for all other atoms. However,such a conclusion can easily be shown to be incorrect,; for,if the electron configuration of the ground state of an atomaltered regularly with the atomic number, it would ob-

122 Building-Up Principle; Periodic System [III, 1

viously be quite impossible to explain the observed periodicityof the chemical and spectroscopic properties of the elements.Furthermore, according to this assumption, the ground stateof any atom would have to be an S state, which is not thecase according to the analysis of the different spectra. F o rexample, B and Al have a 2P state for the ground state.We have already noted that, in the case of Li, all threeelectrons are not in the K shell; one (the emission electron)is in a 2s orbit (L1 shell, n = 2, I = 0), as can be concludedfrom a comparison of the Li spectrum with those of Li-likeions (cf. Chapter I, section 5).

Pauli principle. In order to understand the building-upof the periodic system and the periodicity in the propertiesof the atoms and in their energy level diagrams, we mustintroduce a new assumption. This is the Pauli exclusionprinciple, which prevents the filling of the various shellswith an arbitrary number of electrons. To formulate theprinciple conveniently, let us imagine the atom to bebrought into a very strong magnetic field, which is so strongthat not only is the normal Paschen-Back effect operative(uncoupling of L and S), but also the different Zi and thedifferent si are uncoupled from one another in such a waythat all the Zi and all the si are space quantized independ-ently of one another in the direction of the field. That’ is,for each single electron, the components of 2 in the directionof the field can take one of the values ml = I, I - 1, I - 2,. . . , - 1, whereas the components of s can take one of thevalues m, = f 3. The number of possible states, withwhich we are concerned here, will not be altered by theassumption of a strong magnetic field.

Pauli's principle now states: In one and the same atom, notwo electrons can have the same set of values for the four quan-tum numbers n, l , ml, and m,.’

1 The same conclusions will result if, instead of assuming an extremely strongfield, we assume that the interactions between the individual electrons are soreduced that even a weak field produces the independent space quantizationof the k and the Si. Moreover, instead of the four quantum numbers n, 2,


It follows that only a limited number of electrons canhave the same set of values for the quantum numbers nand 1. The detailed meaning of this fact will become clearin the following discussion.

The Pauli principle does not result from the fundamentalsof quantum mechanics, but is an assumption which, al-though it fits very well into quantum mechanics, cannot forthe time being be theoretically justified.

If #(Z1y&, * * * ) s,y,,z,,) is the eigenfunction of a system con-taining n electrons, generalizing our previous considerations, weobtain

for the probability of finding the system in a configuration withthe coordinates of the individual electrons within the limits zl andx1 + &, yl and y1 + dyl, . . . , z,, and zn + dz,. Since there isno way of distinguishing individual clcctrons, ##* must be inde-pendent of the numbering of the electrons. Therefore, if anytwo electrons are exchanged (exchange of the correspondingindices in $#*), +J* = I#l2 must remain unaltered. Such is thecase either if # itself is unaltered or if it simply changes sign; thatis, $ must be symmetric or antisymmetric with respect to anexchange of any two electrons. [For a more rigorous proof, seeCondon and Shortley (13); cf. also p. 67.)

The quantum mechanical formulation of the Pauli principle is:The total eigenfunction of an atom with several electrons must beantisymmetric in all its electrons. That is, of the two systems ofstates mentioned above, only the one which is antisymmetricactually occurs. It may be shown that this formulation isidentical with the statement of the principle given above.

At first it would appear to follow from this quantum mechanicalformulation of the Pauli principle that, of the two term systems ofHe previously considered, only the antisymmetric, to which theground state does not belong, could occur (Chapter I, section 6).

ml, and tn.., we can also use the quantum numbers n, I, j, and ?nj, where j isthe total angular momentum of a single electron (that is, j = I f $) and mj isthe component of j in the direction of a field (mj = j, j - 1, . . ., - j). SeeChapter IV. We can easily show that this gives the same number of possiblestates for an electron; with n = 2 and 2 = 1, it gives the following states:

73 -t,*+t I -3, -f,,,i,There are thus six states (see p. 127).


The assumption of electron spin is necessary to explain the factthat actually both systems occur. If the spin is included, thetotal eigenfunction is obtained by combining the hitherto con-sidered total eigenfunction # (co-ordinate function) with a furthereigenfunction 8, the spin. eigenfunction.the new total eigenfunction #’ = #s/3.

To a first approximation,Let @+ be the spin func-

tion of a single electron with the spin directed upward, and p-,correspondingly, the spin function of an electron with the spindirected downward-that is, if the spin was originally dircctrdupward, I/3+ 1 2 g ives the probability of finding the values + 3 or- 3 for the spin of the electron in any given direction; similarly,

be I/3-1.if the s in was originally directed downward the probability will

Consequently there are the following four possibilitiesfor the total spin function of the two electrons 1 and 2:

t t /31+/32+

t 1 i31+/32-

1 t h-82+

1 + B1-s2-

In a magnetic field such as that associated with 1, the statesrepresented by the first and fourth eigenfunctions have differentenergies, which are also different from those of the second andthird, although the latter are degenerate with each other. If themutual interaction of the two spins is taken into account, asplitting of the originally degenerate states into two differentstates with eigenfunctions B1+B2- + /?I-&+ and pl+fi2- - p1-p2+occurs. This effect is quite analogous to the Heisenberg res-onance phenomenon (Chapter I, section 6). We therefore havethe following four spin functions of the two electrons:

Br = /31+/32+

BII = L&+/32- + 81-/32+

&II = f31+/32- - h-82+

BIV 5 /31-/?a-

Only one of these eigenfunctions is antisymmetric in the elec-trons-namely, @III; the others are symmetric. However, thetotal eigenfunction ,#’ = #e/3 can now (and this is the importantpoint) be antisymmetric for both term systems-that with sym-metric as well as that with antisymmetric #. We have only torecall that the product of a symmetric and an antisymmetriceigenfunction is antisymmetric, whereas the product of two sym-metric or of two antisymmetnc functions is symmetric. Then ,if the one antisymmetric B is combined with the symmetric 4,or if one of the three symmetric /3’s is combined with the antisym-metric #, the total eigenfunction $’ will be antisymmetric; that is,

Pauli Principle 125

according to the Pauli principle, both term systems can actuallyappear. (See Table 8.)

TABLE 8

SYMMETRY OF THE EIGENFUNCTIONS OF HELIUM

fi I B I #,’ ICo-ordinate Function Spin Function Total Eigenfunction Term System

Symmetric Symmetric Symmetric (Does not occur)Antisymmetric Antisymmetric Singlet system

Antisymmetric SymmetricAntisymmetric

AntisymmetricSymmetric

Triplet system(Does not occur)

The term system whose co-ordinate function is symmetric hasa statistical weight of 1 (only one spin function &I! belongs to it,singlets); the system with antisymmetric co-ordinate functionhas a weight of 3 (three spin functions, triplets).responds to S = 0; the latter, to S = 1.

The former cor-The three symmetric

spin functions correspond to the three possible orientationsM.y = + l,O,- 1, of the spin vector S = 1 in an external mag-netic field or in the field due to the orbital motion when L j= 0.

The occurrence of both term systems is thus entirely due to theexistence of spin. However! the latter has nothing to do withtheir energy difference. T h i already exists for the co-ordinatefunctions, without considering spin, and results from the electro-static interaction of the electrons (see Chapter I, section 6).

The different multiplet systems of atoms with more than twoelectrons may be obtained by analogous, though more complex,considerations. [Hund in (1d) ; Condon and Shortley (13).]

Prohibition of intercombinations. A total prohibition of inter-combinations exists between states with antisymmetric totaleigenfunction I+%’ and states with symmetric total eigenfunction.This can be shown in the same manner as in Chapter I, section 6.If an atom is once in an antisymmetric state, it must remain inthat state for all time. The fact that Pauli's exclusion principleholds for all atoms proves that they are all in the states withantisymmetric total eigenfunction. Transitions to states withsymmetric total eigenfunction never take place; hence thesestates do not occur.

The prohibition of transitions between states with symmetricand antisymmetric co-ordinate functions +. and $J,, (the prohibitionof transitions between states of different multiplicity) holds onlyapproximately-that is, so long as it is possible to use the separa-tion I,V = #s@, since then the matrix element lpnm (p. 53) can be


separated into two factors, one of which depends only upon #.The latter factor, as we have seen, is zero for a transition’betweenstates with symmetric and antisymmetric 9, and thus the com-plete transition probability is zero. As soon as the coupling be-tween spin and orbital angular momentum becomes appreciable,such a separation $’ = #a/3 is no longer strictly possible, andtherefore pm no longer splits into two factors, one of which be-comes zero for a combination between two terms of differentmultiplicities. The smallness of the multiplet splitting showsthat the coupling between spin and orbital angular momentum isweak for elements with low atomic number. Consequently therule prohibiting intercombinations holds almost absolutely forthem (AS =rigorously.

0). With increasing atomic number, it holds less

Application of the Pauli principle. Table 9 shows, forthe possible states of an electron in an atom, the divisionsinto groups and sub-groups up to n = 4 (cf. scheme, p. 39).The order given is, in general, the energy order of the states.Each of the before-mentioned sub-groups with given n and lis once more subdivided according to the value of ml. Allof the latter sub sub-groups of states have the same energyfor a given n and given I value in the absence of a magneticfield. Actually, each of these values (given n, I, ml) shouldbe once more subdivided into two sub-groups with m. = + 3andm, = -3. For the sake of simplicity, this subdivisionis not carried out in Table 9. Instead, the presence of anelectron in a cell (n, l, ml) is represented by an arrow whosedirection (up or down) indicates whether m, is + 3 or - 6.On the basis of the Pauli principle, only two electrons canbe in each such cell (n, I, ml), and then only when they haveantiparallel spin directions, since otherwise these two elec-trons would have the same four quantum numbers n, I,ml, m..

The number of electrons in a subgroup (n, Z) is given bythe exponent of the symbol representing the subgroup.For example, (2~)~ or 2p2 represents two 2p electrons. Themaximum number of electrons which can have the same nand Z and yet not violate the Pauli principle is given by thenumber of arrows in Table 9 between the corresponding


TABLE 9

POSSIBLE STATES OF AN ELECTRON

vertical lines. This maximum number is obviously equalto 2(2Z + 1), since 21 + 1 is the number of possible mlvalues for a given 1. The entrance of any additional elec-tron into such a sub-group (n, l ) is forbidden by the Pauliprinciple, since the additional electron would then neces-sarily have the same ml and m, as one of the electrons al-ready present. When a sub-group or shell contains themaximum number of electrons, it is called a closed shell.

The ground state of an atom is the one in which all electronsare in the lowest possible energy states. On the basis ofthe Pauli principle, this is not (ls)N, where N is the totalnumber of electrons, but it is the state in which all thelower shells are filled only so far as the Pauli principle allows.Thus in the ground state not all of the electrons are equiv-alent; rather, they can be differentiated as inner and outerelectrons. Excited states of an atom result, in general,when the outermost and least firmly bound electron (emis-sion electron) is raised to any of the higher orbits orlevels. There are also excited states in which two or moreof the outer electrons are simultaneously in higher orbits,or in which one of the inner electrons is raised to an outerorbit (see Chapter IV). However, such states are observedrelatively seldom in the optical line spectra of the lighteratoms, but are found more often in the spectra of theheavier elements.

The terms of an atom of nuclear charge Z + 1 may beobtained in the following way: the nuclear charge Z of the

128 Building-Up Principle; Periodic Sysem [III, 2

preceding element in the periodic table in its ground state(or perhaps also in an excited state) is increased by 1, andthen an additional electron is added to one of the shells not.yet filled. Beginning with hydrogen, the terms of all theelements in the periodic system can be derived in this way(Bohr, Mainsmith-Stoner). The principle underlying thisprocedure is called the building-up principle. It is clearthat the electron configurations of the ground states willshow a periodicity, since, after a certain number of electronshave been added,. the outermost electron will be once more,for example, an s-electron. Correspondingly, the otherconfigurations of the outer electrons recur periodically.

2. Determination of the Term Type from the ElectronConfiguration .

The above method gives us only the electron configura-tion (n and l values of the individual electrons) ; thus far itdoes not tell us the term type of the ground state and theexcited states. The term type is obtained by addingtogether the respective angular momentum vectors 2 and sof the individual electrons. For this purpose it is necessaryto make some definite assumptions about the mutual cou-pling or interaction of the individual angular momentumvectors.

Russell-Saunders coupling. The assumption that seemsto apply to most cases is the Russell-Saunders coupling,which we have already used implicitly in the precedingchapter. In this coupling it is assumed that, when severalelectrons are present in an atom, each with a definite Ii andeach with si = 3, the individual ti vectors are so stronglycoupled with one another that states with different resultantL have very different energies. Further, it is assumed thatthe individual si vectors are so strongly coupled with oneanother that states with different resultant S have a con-siderable energy difference. As explained in Chapter II,for strongly coupled vectors, only the resultant (in this case

III, 2] Term Type from Electron Configuration 129

L or S) has an exact meaning as angular momentum. Thevectors that are strongly coupled with one another mustalways be added together first. The resultants L and S,according to the Russell-Saunders coupling, are then lessstrongly coupled with one another and their resultant is J.Each allowable value of L can be combined with eachpossible S; that is, the spins can take all orientations whichare possible on the quantum theory for each state character-ized by a definite value of L. The interaction between Land S gives the multiplet splitting of each term, whichwithout this interaction would be simple.

The Russell-Saunders coupling can be written symbolically:(Sl, s2, * * - M, 12, *--) = (S,L) = J (III, 1)

It should be noted here that the considerable difference betweenthe energy levels of corresponding terms of different multiplicity(different S) is actually due, not to a strong magnetic interactionof the respective si, but to the Coulomb interaction of the elec-trons and the Heisenberg resonance phenomenon (p. 66), whichis entirely independent of spin. The spin merely makes possiblethe actual appearance of the different term systems (p. 124). Inspite of this we can proceed in practice, as t h o u g h the energydifference were due to the magnetic interaction of the spins.

In Chapter IV we shall consider still another kind of couplingof the individual angular momentum vectors which occurs forheavy elements.

Terms of non-equivalent electrons. Non-equivalentelectrons are electrons belonging to different (n, 1) sub-groups (cf. Table 9). In order to determine the terms re-sulting from two non-equivalent electrons, we must, firstof all, find the possible values of the resultant L. In thecase of a p-electron and a d-electron, L = 3, 2 , 1 (cf. p. 87);that is, the two electrons form F, D, and P terms. Thespins of the two electrons can be either parallel or anti-parallel. S is therefore 1 or 0, which means that triplet aswell as singlet terms result. In all, there will be six terms:‘P, ‘D, ‘F, 3P 3D, 3F. Similarly, two non-equivalentp-electrons give’the terms: ‘S, ‘P, ‘D, 3S, 3P, 3D.

If a third non-equivalent electron is added, its 1 must be addedvectorially to the previously calculated L; and s must similarly


be added to S. If, for example, an s-electron is added to pd,the L values remain the same. The possible S values are now 8,), $; that is, the possible terms are: 2P, ZD, zF, 2P, 2D, “F, “P, 4D, 4F.Two different doublet terms for each L will be formed, sincep-electrons and d-electrons with parallel as well as antiparallelspins can give S = 3 with the addition of an s-electron. On theother hand, S = 3 can be obtained in only one way-namely,w h e n all three spins are parallel to one another (quartet). Thethree spin configurations for s p d are t 1 t, 1 t t , t t t . If the thirdelectron that is added is an f-electron (I = 3), the possible Lvalues are 3 + 1, 3 * 2, 3 + 3 (where the sign r indicates vectoraddition). This gives the following L values: 2, 3, 4; 1, 2, 3, 4, 5;0, 1, 2, 3, 4, 5 6. As before, the possible S values are 3, $, $.Thus we obtain: *S(2), *P(4), 5D(6), sF(6), ‘G(6), 2H(4), “I(2),‘S(l), 4P(2), 4D(3), ‘F(3), ‘G(3), ‘H(2), ‘1(l), where the numbersin parentheses indicate the number of the corresponding terms.For example, 4P(2) means two 4P terms. Even in the abovecomparatively simple case the total number of terms belonging tothe same electron configuration (pdf) is considerable.

These and other examples are given in Table 10 (p. 132).

Terms of equivalent electrons. When we are dealingwith equivalent electrons (having the same n and the sameE), some of -the terms derived for mm-equivalent eletronsare no longer possible. For example, for two equivalentp-electrons (p”), not all the terms (‘S, ‘P, ID, %S, JP, “0)previously derived for two non-equivalent p-electrons arepossible. In considering non-equivalent electrons, weassumed that no account need be taken of the Pauli prin-ciple in adding together the individual I and s values; itwas supposed that each orientation allowed by the quantumtheory did actually occur. This assumption is in factjustified, since, when the electrons have different n or 1values, the Pauli principle is already satisfied. When,however, the two electrons have equal n and equal I, that is,when they are equivalent, they must at least differ in theirvalues of ml or m,. When, for example, the two p-electrons(p2) both have 2 with the same direction, which gives a Dterm, ml is the same for both (+ 1, 0, or - 1) and therefore,according to Pauli's principle, the two electrons cannot bothhave m, = + 4, or both m, = - 4. That is, their spins


can only be antiparallel for a D term, giving lD only,and 30 is not possible, although it would be with non-equivalent elcc trons. Further consideration shows thattwo equivalent p-electrons give only the terms IS, sP, 1D(see below). * similarly, three equivalent p-electrons give9, ?P, 20. These and additional examples are given inTable 11 (p. 132).

Particular mention should be made of closed shells-thatis, shells in which the maximum number of equivalent elec-trons is present. In order to fulfill the Pauli principle, allthe electrons must be in antiparallel pairs (S = 0). Inaddition, L = 0, since the state can be realized in only oneway in a magnetic field-namely, with ML = Cm, = 0.Therefore, a closed shell always forms a 4!& state.

In deriving the terms of an electron configuration thatconsists of one or more closed shells and a few additionalelectrons (for example, 29 3s2 3~9, the closed shells can beleft entirely out of consideration. The terms are the sameas for the electrons that are not in closed shells; for, in deter-mining the resultant L and S, the respectiveti can be addedtogether in any desired groups to form partial resultants,and these can be added together to give the total resultant.Since the partial resultants for closed shells are zero, theyhave no influence on the total resultant.

From the latter statement the following may be derived:the term 5% for a closed shell must result when the shell isdivided into two parts, the term types for each part derived,and the resulting angular momentum vectors added to-gether. For example, adding the angular momenta of theterms of p2 vectorially to the corresponding quantities for p4must give those for a p6 %SO state, that is, zero. From thisit follows that the quantum numbers S and L must be thesame for these two electron configurations; that is, theterms of the configuration p4 are the same as those of p2.This result can also be obtained directly (see Table 11).Generalizing, we can say that the terms of a configuration xq

are the same as the terms of a configuration X~Q where r is


the maximum number of r-electrons, that is, 2t.Y + 1).(See p. 127.) For example, the terms of l>b are the same asthose of p; they give only one 2P term; or, the terms of d7are the same as those of d3 given in Table 11.

TABLE 10

TERMS OF NON-EQUIVALENT ELECTRONS

ElectronConfiguration

s.sSPsdPPpdd d585s s ps s dSIJPspdp p pppd

pdf

ElectronConfiguration

Terms

‘As, %‘P, ap‘D, SD‘S, ‘P. ‘D, %, sP, =D‘P, ‘D, ‘F, aP, SD, V’‘S, ‘P, ‘D, ‘F, ‘G, 3.5, “P, aD, aF, T.%s, %s, ‘S‘P, fP, ‘P‘D, ‘D, ‘Dw, ‘P, ‘D, ‘S, ‘P, ‘D, ‘S, ‘P, ‘D‘P, %D, ‘F, tP, tD, ZF, ‘P, ‘D, ‘F*S(2), *P(6), *D(4), *F(2), ‘Ss(l), ‘p(3), ‘WA ‘F(1)*S(2), *P(4), *D(6), *F(4), *G(2), ‘SW, ‘P(2), ‘D(3), ‘F(2),

‘C(1)*s(2), *P(4), “D(6), *F(s), *G(6), ‘H(4), w!)‘SO), ‘P(2), ‘D(3), V(3), ‘G(3), ‘H(2), ‘W

TABLE 11

TERMS OF EQUIVALENT ELECTRONS

Terms

1s‘S, ‘D, ‘P‘P, ‘D, ‘S%S, ‘0, “P‘P1s’‘S, ‘D, ‘6, “P, =F‘P, ‘D(2), fF, ‘G, Vf, ‘P, ‘F‘S(2), 1D(2), ‘F, ‘G(2), ‘I, sP(2), aDD, aF(2), W, “if, SD%, =P, 2D(3), ‘F(2), 2G(2), aff, V, ‘P, ‘D, V”, ‘G, “S


We shall now derive, in greater detail, the terms of equivalentelectrons for a special case. According to the adiabatic law al-ready used, we should be able to derive all possible terms by usingany desired alteration of the coupling conditions. In thePaschen-Back effect, L and S are space quantized with respectto the field direction, independently of each other, with com-ponentsML=L,L-I,..*, -L,andMs=S,S-l,...,- S. ML and MS represent the components of the total orbitaland spin momenta in the field direction. On the other hand, forcomplete uncoupling of the individual electrons from one another,the individual 1; and s; are space quantized with componentsml=& l - l , 1 -2 , s-s, -I, andm,= =t&. Under thesecoupling conditions Crnl and Cm, represent, respectively, thecomponents of the total orbital and spin angular momenta in thefield direction. Therefore, according to the adiabatic law, Crnlmust equal ML, and Cm, must equal MS, for all the configurationswhich the electrons under consideration are allowed, according tothe Pauli principle, to assume in the cells of Table 9. Exactly thesame MI, and MS must be obtained as from the L and S values ofthe resulting terms, and, conversely, these may be derived fromthe calculated Cm, and Cm, values.

Table 12 gives the possible configurations for the case of twoequiyalent pelectrons,2 as well as the corresponding valuesCm = Mt and Cm, = MS (see p. 134).

In order to determine the resulting terms, it is useful to beginwith the highest value of Cm, = Mb, which must be equal to thehighest occurring value of L. In the present case, the highestvalue of ML is 2 and thus a D term results. Since this ML occursonly with MS = 0, the term is ‘D. Apart from ML = 2,ML = + 1, 0, - 1, - 2 also belong to this term, each havingMS = 0.. They are indicated by A in the last column of Table12. There are two terms each with ML = f 1, MS = 0, andthree with ML = 0, MS = 0. Which of them is selected for thecomponents of the ‘D term is of no special importance for -thisderivation. Of the remaining ML and MS values, the maximumML is + 1 and the largest MS is + 1. These values must belongto a 3P term, since only for such a term can the largest values ofML and MS be + 1. ML = 0 and - 1 also belong to this 3Pterm, each of the ML values having MS = + 1, 0, - 1. Alto-gether, for sP we obtain nine configurations. In Table 12, theseare marked +. One configuration remains: ML = 0, -MS = 0 ;this is indicated by X. It can give only a ‘S term. Thus, Iwo

f It must be noted that, because of the identity of electrons, the configura-tion t 1 in a single cell is not different from 4 t ; whereas, if the two electrons arein different cells, the configuration t 1 is different from 4 t . Only the variousstates-not the electrons themselves-can be identified. This assumption maybe considered a supplement to Pauli's principle [cf. Slater (134)].


equivalent p-electrons give the terms ‘9, “P, IS and no others.The terms of the other configurations hsted in Table 11 may bederived in a similar manner.

TABLE 12

DERIVATION OF TERMS FOR TWO EQUIVALENT p-ELECTRONS

+1t+

t

iIc

0

+2 0 A

t +1 +1 ++1 0 A

1 ::0 +

+:+

0 +0 0 A0 00 - 1 :

‘t 0 0- 1 +1 :? - 1 0

ii t - 1 0 tj + - 1 - 1 +i t+ - 2 0 A

- 1

ZTTII = ML Zm a = MS

A procedure similar to the one outlined here for equivalentelectrons can also be followed in the case of non-equivalent elec-trons.simpler.

However, the method previously used (p. 129) is much

If we have a configuration containing equivalent as well as non-equivalent electrons (for example, p%d) the corresponding partialresultants must be taken from Tables 10 and 11. These are thenadded together to give a total resultant in which each term of onepartial configuration (P” in the example) is combined with eachterm of the other (sd in the example).of terms result.

In general a large numberIn the above relatively simple example, there

are 28 terms: 2&‘(2), 2P(1), 2D(5), 2F(4), 2G(2), ‘S, 4f72), ‘D(4),‘J(2), ‘G, SD.

The angular momenta 2 and s of the individual electronshave a well-defined meaning only when they influence oneanother but slightly (in principle, not at all). If this wereactually the case, all terms with the same electron configu-


ration would have the same energy (cf. p. 83). Actuallythey have not. In many cases, considerable energy differ-ences occur; these are larger, the larger the interaction.

In this connection two rules operate. The first wasformulated by Hund: Of the terms given by equivalent elec-trons, those with greatest multiplicity lie deepest, and of thesethe lowest is that with the greatest L. For the cases given inTable 11, this term is the last one in each line. Thesecond rule states: Multiplets formed from equivalent elec-trons are regular when less than half the shell is occupied,but inverted when more than half the shell is occupied. For aproof of this rule, the reader is referred to Condon andShortley (13).

Electron distribution with a number of electrons present.It has already been stated that, when a number of electronsare present in an atom, the wave mechanical charge distri-bution is given, to a first approximation, by a superpositionof the charge distributions of the individual electrons (cf.Fig. 21, p. 44). It is clear that superposition of the elec-tron distributions of s-electrons, each having a sphericallysymmetrical charge distribution, must give a sphericallysymmetrical total charge distribution for the resulting Sterm. A more detailed wave mechanical calculation showsthat S terms resulting from somewhat more complicatedelectron configurations (possibly containing p-electrons andd-electrons) also have a spherically symmetrical chargedistribution. This holds in particular for closed shells,which always give YYO terms; the spherical symmetry holdsrigorously for these shells, even when the interaction of theelectrons is taken into exact account.

The more accurate wave mechanical calculation of thecharge distribution for atoms with a number of electrons istoo complicated to be further considered here. The Har-tree method of self-consistent fields has shown itself to beparticularly useful, but likewise will not be dealt with here.


In Fig. 54 are given the results of such calculations forthe radial charge distribution curves for the ground states ofthe Li+, Na+, and K+ ions. Since these ions have closedshells only, the charge distribution will be completelydescribed by the curves. The distribution for the groundstate of the hydrogen atom is given (drawn to the samescale) for comparison. Fig. 54 corresponds in all details tothe solid curves of Fig. 20 (p. 43). The summation ofr2$,* over all electrons, Cr2#,2, is indicated here; whereas,in Fig. 20, r*#,.* was drawn, since only one electron waspresent. The curves in Fig. 54 thus give the mean chargedistribution (that is, the sum of the probability densities of

Fig. 54. Radial Charge Distribution for the Ground States of H, Li+,Na+, K+. The curves for Na+ and K+ are drawn according to the work ofHartree and his co-workers (62) and (63); the curve for Li+, according toPauling and Goudsmit (9). All curves are drawn to the same scale.


the individual electrons) for the given ions, referred to thewhole spherical shell of radius r. The charge distributionhas pronounced maxima at certain distances from thenucleus. For Li+, there is only one maximum, correspond-ing to the one closed shell (n = 1, K shell). The meandistance of the electrons from the nucleus is 0.28 A, whichis considerably smaller than for the H atom (0.79 A) be-cause of the greater nuclear charge. At the same time theheight of the maximum is greater. This difference is dueto the fact that the eigenfunctions have naturally been sonormalized that the probability of finding an electron some-where in the atom is equal to the number of electrons; thatis, the area under the curves (Fig. 54) must be equal to thenumber of electrons. In the case of one electron, theprobability = 1.3 Thus, owing to the contraction in thedirection of the abscissae and the increase in the number ofelectrons, the height of the curves must increase withincreasing Z. In the case of Na+ there are two maxima, thefirst of which is mainly due to the electrons in the K shell,and the second to the closed L shell (n = 2). Because ofmuch higher nuclear charge, the mean distance of the Kelectrons from the nucleus is only 0.07 A; that of the Lelectrons, 0.41 A. K+ shows still another maximum, due t othe additional M shell, For the electrons of the M shell,the mean distance is 0.82 A. The maxima for the K and Lshells are again pushed farther inward. Correspondingresults are obtained when still more shells are added (Rb+and Cs+).

It is seen that we can also speak of a shell structure of theatom from the wave mechanical viewpoint. Such a state ofaffairs holds for all atoms and ions. However, the distribu-

J The normalization is generally so carried out that Jt’$,* dr = 1, and not4~J7+: dr = 1 (where 4& dr = volume of spherical shell). The factor 4~is introduced into the normalization of the directional part of the eigenfunction.Therefore, for one electron the function #$* averaged over the different

directions in space is equal to & #I’. Thus, we can also say that the curves

in Fig. 54 represent 4&Z++*.


tion of most of the electrons that are not in closed shells isnot spherically symmetrical, but is similar to the electrondistributions for a single electron given in Fig. 21, p. 44.

3. The Periodic System of the ElementsThe electron configurations and the term types of the

ground s ta tes for all the elements of the periodic system aregiven in Table 13, pages 140-141 (cf. Table 9).

H (hydrogen). The lowest orbit of the one electron ofthe H atom is a 1s orbit. The ground state is therefore aV112 state. The higher states correspond to the variousother pairs of n and I values. They are, according to thevalue of 1, %, *I>, ?L), . . . states. However, for equal nthe states nearly coincide (see Chapter I).is a doublet spectrum, since S

The spectrum

for He+, Li++,=s=$. The same is true

times larger.and so on, except that the terms are 4, 9, . . .

He (helium). This element has a nuclear charge 2 andcan have 2 electrons. On the basis of the Pauli principle,both electrons can go into the K shell (n = 1) only whenthey have antiparallel spin directions (t I), since, if thespins were parallel, all four quantum numbers would be thesame for the two electrons (n = 1, I = 0, ml = 0,m, = + a). Therefore, in the ground state S = 0, and,since both electrons are 1s electrons, L = 0. Thus theground state is a IS state (closed shell).not given by this electron configuration.

A triplet state isAn excited state

results when an electron (the emission electron) goes to ahigher orbit. Then both electrons can have, in additionthe same spin direction; that is, we can have S = 1 as wellas S = 0. Excited triplet and singlet states are possible(orthohelium and parhelium). The lowest triplet statehas the electron configuration 1929; it is a 2 3S1 state. Itis the metastable state already referred to (p. 65). Thecorresponding singlet state is 2 ‘So and lies somewhat higher.

III, 2] Periodic System of the Elements 139

We can easily see how the two term systems of helium areobtained in this way (cf. Fig. 27, p. 65).

Li (lithium). If an electron with n = 1 is added to thehelium-like Li+ ion, this electron would have the same fourquantum numbers as one of the electrons already present.This is forbidden by the Pauli principle. The K shell isthus complete with two electrons, and with it also the firstperiod of the periodic system. The third electron can onlygo into the next shell (n = 2, l = 0), or to a still higher shell(see Table 9, p. 127). The Li terms are doublet terms, likethose of hydrogen, since S = s = 4. The ground state(19*29) is a *S1i2 term. In the first excited state the emissionelectron has n = 2, l = 1 (Table 13); a 2P state results.The combination of this state with the ground state givesthe red resonance line of Li (a doublet like the D lines of Na) .The principal quantum number n does not alter in thistransition, which according to page 51 is allowed.

Sometimes the electron configuration is written in frontof the term symbol in order to give a more accurate repre-sentation of the term of an atom containing a number ofelectrons; for example, for the term just discussed, ls22p 2P.

Be (beryllium). In the lowest state of the next element,beryllium, the additional electron can have the same quan-tum numbers n = 2, 1 = 0, ml = 0, as the electron pre-viously added in the case of Li, but must then have oppositespin. The ground state of Be is thus ‘S, since all the elec-trons have 2 = 0 and form pairs (closed shells only). Ac-cording to the Pauli principle, a corresponding qS state (twoelectrons with parallel spins) does not exist. However,just as soon as one of the outer electrons has quantumnumbers n, 2 different from those of the other, its spin can beeither parallel or antiparallel to that of the other. Thusthere is a triplet term corresponding to each higher singletterm (for example, 1s22s3d 3D, lo), analogous to He. TheL value of the term is equal to the E value of the outermostelectron. The singlet and triplet line series ordinarily

TABLE 13

ELECTRON CONFIGURATIONS AND TERM TYPES FOR THEGROUND STATES OF THE ELEMENTS

(Numbers and symbols in parentheses are uncertain.)

TABLE 13 (Continued)

ELECTRON CONFIGURATIONS AND TERM TYPES FOR THEGROUND STATES OF THE ELEMENTS

(Numbers and symbols in parentheses are uncertain.)

T222 :2 :2 12 !2 I

2-t2 I2 I2 42 I2 e2 f2 e

762 0. . ...<2 e2 02 02 02 02 82 62 6t.....2 0a 02 0a 01 0z 01 01 0

Ki1 0..‘...I 0! 0! 0! 0! 0I 0I 0I 0. . . . . .I 0I 0

000000-8-

i.-:=

8

I

3s 3p 2d- I

N 0 I

4s4p 401 :/ I 9s .ip 51 5f .5Q=8 2 0 10 2 0sl. . . . 2 0 10 12. 0 -. . . . . . . . . . . ..*........................812 0 10 120 1ii,.i..6..ib".iii.;..;...(.ij . . . . . . 1

S 2 6 10 (2) 2 0 (1)8 2 0 10 (3) 2 0. (1)S 2 6 10 (4) i 6 (1)8 2 0 10 0 2 0.S 2 0 1 0 7 2 0I S 2 0 1 0 7 2 0 118 2 0 10 (8) 2 0 (l!18 2 6 10 (9) 2 0 (1)18 2 0 10 (10) 2 0 (1)IS 2 0 10 (111 2 0 (1)LS ,2 6 10 13 2 0

0

58 ip Lid Sfh-

:=<

I,...,

, , . .

I

::

. . . .

. . . .

- -

. . . .

-

--Z=

1

. . . .

.I.;111111I111

*...

. . . .I

I33I333-.33.*..3333-

I. H2 . Hc

3 .4. Be5.-B0. 07. N8. 09. F

EIcmerlt

55. cs50. Ba. . . . . . . . . . .57. L a. . . . . . . . ...*58. C e

1. . . . . . . . . .1. . . . . . .!: .

!) ;!) :!) iP :, :

1 :1) :1) ;

0 i2) :2 i2 i. . . . . . .2

2222221. . . . . . . . . .122 12 22 32 42 52 0

18. . . .!..S

S888

59. PI00. N d01. I102. sm03. E u04. G d05. lx00. DY

07. H oOS. Er69. Tm

IO. Ne

11. Na 122 12 22 32 42 52 0

2 02 0. . . . . . . . . . a20 120 :20 120 f20 L20 e20 720 eI . . . . . . . . . .a 0 Ia2 0 ia2 0 ia2 0 io1 6 102 0 101 0 101 0 101 0 101 0 10. . . . . . . . . .1 0 10! 0 10! 0 10I 0 10! 0 10! 0 10I 0 10I 0 10. . . . . . . ...*I 0 100 100 10

0 100 100 100 100 10

18

12. MI13. A l14. Si15. P10. s17. CI

1 8 1 2 0 10 14 2 6. . . . . . . . . . . . . . . . . . . . . 1. .I

. . . . . . . . . . . . . . . . . . .1818

I 2 0 10 14 2 0 12 0 10 14 2 0 2

1 8 2 0 10 14 2 6 31s 2 0 10 14 2 0 418 2 0 10 14 2 0 518 2 0 10 14 2 0 018 I 2 0 10 14 2 0 7

70. Yb. . . . . . . . . .71 . Lu18. A

19. K30. ca. . . . . . . . . . . ,21. 8c

72 . Hf73. Ta7 4 . w75 . Re70 . OS77. Ir78 . et* ,........7 9 . A uB3. H381. Tl82. Pb83. Bi84. PO85 . -80. Rn

87. -88. Ra. . . . . . . . . .89. AC

90. Th91. Ps

12. . . . . . . .22 ;

:2 ;1 i2 i

; j

22. Ti23. 1’2 4 . C r3 5 . M n20. Fe37. co2 8 . N i. . . . . . . . . . . .2q. c u30 . 21131 . Da32 . Ga33 . As34. 8.335 . Br

18 12 0 10 2 0 9. . . . .18 I

. . . . . . . . . . . . . 14 1. . . . .14 I

. . . . . . . . . . . . . . . . . . .2 0 10 2 0 10

1% 2 0 10 14 2 0 101s 2 0 10 14 2 0 1018 3 0 10 14 2 0 1018 2 0 10 14 2 B 1018 2 0 10 14 2 0 1018 2 0 10 14 2 0 1018 2 0 10 14 2 0 10

18 2 0 10 14 2 0 10

2 i,......:122 12 22 32 42 s1 0

1--,

1. . . .1

1

,

1

2 0 12,.....

(2)(2)(2)(2)-

18 1 2 0 10 14 1 2 0 10. . . . I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 2 0 10 14 I 2 0 1018 2 0 10 14 2 0 1018 2 0 10 14 2 0 1018 2 0 10 14 2 0 10

2 0. . . . . . . . . . . . . . . . . . .2 0 (1)30 . Kr

37. Rb38 . 8r. . . . . . . . . . . .30. Y40. Zr41. Cb42. MO

43. Ma4 4 . R u45. Rh40. Pd. . . . . . . . . . . . ,47 . AK48. Cd49. In50. Sn

2 0 (2)2 0 (3)2 0 (1)

1 01 0. . . . . . . . . . . . . . . . . . .10 1

92 . 11

51. Sb52. Te

!O 2!O 4!O 5! 0 (5)!O 7IO 8

observed (see Chapter I) result from transitions of thisemission electron. There can also occur terms for whichboth outer electrons are in orbits other than the lowest(n = 2, l = 0). Under these circumstances several termswill, in general, result from a given electron configuration(see section 2 of this chapter). These are the so-calledanomalous terms of the alkaline earths, which cannot be ar-ranged in the normal term sequences. (Cf. also Chapter IVparticularly Fig. 61, in which the complete energy leveldiagram for Be is reproduced.)

1 4 1


B (boron). Again on the basis of the Pauli principle, thefifth electron cannot be added to the so-called L1 shell(n = 2, I = 0), but must be added at least to the Ls shellwith n = 2, I = 1 (cf. Table 9, p. 127). Since all the elec-trons of boron (except that in the Lz shell) form pairs(closed shells), a doublet spectrum is normally produced.The ground state is 2P (not 25, as for Li), since now L = I= 1. Otherwise the energy level diagram is similar tothat for Li. (Cf. the energy level diagram of aluminum inFig. 73, p. 198, which is quite similar to that of boron.)However, boron can also have terms in which only oneelectron is in the 2s shell, and the other two are in the 2pshell or higher orbits. Then all three outer electrons canhave parallel spins; that is, S can be 3. Quartet terms andalso anomalous doublet terms result. Up to the presenttime, these quartet terms have not been observed for B,although they are known for C+,, which has the samenumber of electrons as B, and for Al [Paschen (64)].

C (carbon). In the lowest state of carbon, two electronsare in the 2p shell (I = 1). According to the precedingsection, this gives three terms: 8J’, ID, ‘S; of these 3P is thelowest and is therefore the ground state of the C atom.The ID and lS terms do not lie very far above the groundstate, since they belong to the same electron configuration.

When one of the two emission electrons of the C atomgoes from the 2p orbit to a higher orbit, normal series ofsinglet and triplet terms result. The number of term seriesis, however, much greater than for boron and the precedingelements in the periodic system, since now several terms canresult for each electron configuration (Table 10). Fig. 55gives the energy level diagram for C I, so far as it is known.It is drawn in a manner which differs from that of thepreceding energy level diagrams because of the presence ofdifferent terms belonging to the same electron configuration.Terms belonging to the configurations ls22s22p np (n = .2,3 , . ..). 2pns (n = 3, 4, .‘..), and 2pnd(n = 3, 4, s-e)

4


are placed under one another, with the singlet terms indi-cated to the left and the triplet terms to the right. Whennecessary, terms of the same electron configuration arebracketed together. We see from the figure that, forexample, six terms result for 2p np if n > 2 (cf. Table 10),and these draw closer together as n increases (cf. p. 84).Apart from these normal terms, additional relatively low-lying terms are possible, which result when an electron isbrought from the 2s shell (which is complete in the groundstate) to the 2p shell, for example, ls22s2pJ. In the ener-getically lowest term of this configuration all four outerelectrons have parallel spins and the result is a %‘2 term,

VOli11.21:

IO

6

6

4

2

(

r--

- b

SP

I_

a

b--

p-1 10.099'm

-‘P

‘ P -b / +“---=:s- a

s ‘P-I -‘P

II

40,ooa

I 50~00(1I

I eopoa

II

'S-

i

70*00(

.I

60,Ool

I----'P ____ -__-- (__--9JM~

Fii. 55. Energy Level Diagram for C I. The unobserved 3p IS term is indi-cated with a dotted line.


which has thus far not been observed. Other terms of thesame configuration are represented to the right in Fig. 55.

N (nitrogen). The succeeding element, nitrogen, in itslowest electron configuration has three electrons in the LI1shell (2~~). According to Table 11, these give the terms45, 2D, 2P; of these, according to the Hund rule (p. 135),

l!

l(

a

6

4

2

0

Itsa - -4-

4p :I

z- ;3P

1

I -

I -

=I

‘1

2P.

- - -

28’29’ np 2*‘2p’ I4 29’2p’ n d cm.---------,

10,001

138 B-

-‘P

I 2*2p’ ‘P-ao*oooof

I

‘-

40.00c

6o.ooa

60,000

70,000

80,000

90.000

I100,000

1 110.000I

----ds _ _ _ _ -_-__i_-_-_-___I

Fig. 56. Energy Level Diagram for N I.

-1

3-

3-

)-

I-

l-

,-


‘5’ lies lowest and is therefore the ground state of the Natom, in agreement with experiment. In this state all thethree 2p electrons have parallel spins. Higher excitedstates result when one electron goes from the 21, orbit to ahigher orbit. Series of quartet and doublet terms arethereby produced; and the number of term series is again,a s with carbon, much larger than for Li.

Fig. 56 represents the N I energy level diagram in a man-ner similar to that of the C I diagram, Fig. 55. A few terms,drawn to the right of the dotted line, do not go t o the normalseries limit (cf. Chapter IV). In addition, a term “P forwhich one electron is brought from the 2s shell to the 2pshell is indicated. If an electron goes from the ‘k-shell tohigher orbits, sextet terms can result, since the five electronscan then have parallel spins. Such terms have not yet beenobserved for N I.

O (oxygen). The lowest possible orbit for the added elec-tron in the case of oxygen is 2p; hence four equivalent 2pelectrons are present, and produce the terms 3P, ‘D, ‘S,just as for carbon (Table 11, p. 132). Again, 3P is thelowest and is the ground state for oxygen. According tothe Pauli principle, the spins of the four outer electronscan never all be parallel in the 2p shell. They can, how-ever, all be parallel when one electron is brought from the 2pshell into higher shells, and then excited quintet terms, aswell as singlet and triplet terms, are formed. The oxygenlines lying in the visible region are combinations of thesequintet terms. The energy level diagram of the oxygenatom will be treated in detail in the next chapter (cf. Fig.59, p. 163).

F (fluorine). For this element, one electron is lacking tocomplete the L2 shell. The ground term is therefore thesame as for boron with its one electron in the L2 shell,namely 2P. The difference now is only that the “Pground term of F is an inverted term (?P3,* lies lower than2P1,2), whereas the ground term for B is regular. This *P


term is the sole low-lying term of fluorine. For all otherterms, the principal quantum number of at least one elec-tron is raised. Excited doublet and quartet terms thenresult (cf. the energy level diagram of Cl, Fig. 74, p. 199).

Ne (neon). This element has six 2p electrons, and forit the L2 shell and also the whole L shell are filled. As aresult, the ground state is lSo (cf. He and Be). Excitedstates result when one electron goes from the L2 shell to ahigher orbit. As in the case of helium, the energy necessaryfor this transition is very great, since the principal quantumnumber must be altered. In addition, for a single electronin an orbit with n z 3, the nuclear charge is almost com-pletely shielded by the nearly complete inner Lz shell, andhence the excited terms are rather hydrogen-like; whereas,for the electrons in the closed 2p shell (ground state), thenuclear charge is much less completely shielded, and there-fore the ground state lies considerably lower than the cor-responding hydrogen term n = 2 (cf. Chapter VI). Infact, it actually lies lower than the hydrogen term n = 1.The large first excitation potential, together with the termtype of the ground state (IS), is responsible for the character of an inert gas (Chapter VI). The excited states of Ne aresinglet and triplet states, as for He.

Succeeding periods of the periodic system. On the basisof the Pauli principle, after eight electrons have been added(Ne), no more electrons can enter the L shell, since anyadditional electron would necessarily have the same fourquantum numbers as one of the electrons already present(cf. Table 9, p. 127). The eleventh electron must thereforego into the M shell (with n = 3). The lowest possible statehas 2 = 0. The ground state of the element Na (withnuclear charge 11) is therefore ls22s22ps3s 2Sl12. Apartfrom the insertion of the L shell (2~~2~6) and the alterationin principal quantum number, this is exactly the same asfor Li.


We understand from the preceding discussion the funda-mental reason why the second period of the periodic systemis completed with Ne (cf. Table 13, p. 140).

As already stated, the number of terms and the termtypes are not altered by a closed shell. This fact, togetherwith the Pauli principle which first made possible the con-cept of a closed shell, provides the essential basis for thetheoretical explanation of the periodicity of the propertiesof the chemical elements (cf. Chapter VI, section 3).

Apart from the altered principal quantum number andthe built-in closed L shell, the eight elements (Na, Mg, Al,Si, P, S, Cl, A) following Ne have exactly the same electronconfigurations in the ground state as have the preceding eightelements. According to the building-up principle, theexcited states should also be quite analogous, except for aslight difference introduced by the possibility of. excitationto a d level without change in principal quantum number.All this is in full agreement with experiment. For the termsof these elements, we need only refer to Table 13.

At argon, the Ml and M2 shells (n = 3, 1 = 0, 1) arefilled; but it can be seen from Table 9 that the whole Mshell is not filled at this stage, since I can also be 2. ForI = 2 (MI shell), ml = + 2, + 1, 0, - 1, - 2, and hencethere can be ten electrons in the MS shell. However, asthe argon spectrum shows, the energy necessary to bringan electron from the Mz shell to the MS shell is very great -even somewhat greater than that required to bring anelectron into the N1 shell (4s orbit). The latter, that is, thefirst excitation potential of argon, is also considerable (11.5volts), and this, together with the fact that the ground stateis a 1s state, makes argon an inert gas (see above). Ifanother electron is added with a corresponding increase innuclear charge, it goes into a 4s orbit, since according to theevidence of the argon spectrum the 4s orbit lies lower than3d. This explains the early occurrence in the periodicsystem of another alkali metal, namely K, with a ground


state %S. Thus the third period, as well as the second,contains only eight elements.

For Ca, two electrons are in the 4s orbit. Ca correspondsto Mg, which has two 3s electrons. If after Ca the buildingup process should go on as after Mg, we would expect thenext electron to enter the 4p shell. But the spectra show

i is


that, for the succeeding elements (SC to Ni), the 3d shellis first filled (cf. Table 13). The reason for this is explainedin Fig. 57. The energy of the different shells (n z 2) isgiven here very roughly for small (~ 20) and large (~ 90)nuclear charge Z, with a correspondingly altered scale.For large Z, the shells with the same principal quantumnumber lie relatively close to one another; whereas, withdecreasing Z, the field becomes less and less hydrogen-like(particularly for the outer shells) and consequently shellswith the same n separate from one another (indicated by theconnecting lines), until in some cases they are no longergrouped together.

For example, with small Z, the 3d shell lies somewhatabove the 4s shell (see above). With increasing Z, the 3dshell tends to be lower; eventually it is lower than the 4sshell. This happens first when Z = 21, and therefore the3d shell (2Mz) is first filled for the elements following Ca (cf.Table 13). At the same time the 4s shell does not alwaysremain filled with two electrons; for example, the Cr I spec-trum shows that the lowest term (‘&), formed from 3d64s,is lower than the lowest term from 3d44s2, because of thenot very different heights of the 3d and 4s shells. In thesame way the 3dl”4s 25 term for Cu lies lower than the3d04s2 2D term. Thus the 3d shell is completely filled forthe ground state 25 of Cu. In this state Cu, having ones-electron outside the closed shells, is similar to the alkalis.This similarity is in agreement with the common form ofthe periodic system in which Cu is placed in the alkalicolumn. The succeeding elements (Zn, Ga, Ge, As, Se, Br)have electron configurations and energy level diagramsanalogous to those in the second and third periods, apartfrom an altered principal quantum number for the outerelectrons and an additional closed 3ps3d10 shell (cf. Table13). The 4s and 4p shells are completed at Kr, and thisends the first long period of the periodic system with its(10 + 8) elements.

150 Building-Up Principle: Periodic System [III, 3

Now the performance begins again, just as at the begin-ning of the first long period. When more electrons areadded with a corresponding increase of nuclear charge, it isnot (as might be expected) the 4d or 4f shells that are firstfilled, but rather the 5s shell, which lies lower for thesenuclear charges (cf. Fig. 57). This gives the elements Rband Sr. The 4d shell lying at about the same height asthe 5s shell is then filled for the succeeding elements, Yto Pd (similar to the 3d shell). When this is completed,

the next lowest shells, the 5s and 5p shells (see Fig; 57),are filled, and give the elements Ag to Xe, which correspondexactly to the elements Cu to Kr. This completes thefifth period, or second long period, with (10 + 8) elements.

The inner f shell is still unoccupied (see Table 13 andFig. 57). The 4f shell is first occupied after the 6s shell hasbeen broken into by Cs and Ba, and the 5d shell by La.The rare earths then follow, exactly 14 in number, thisbeing the maximum number of electrons in the 4f shell.When this is filled, the 5d shell is filled from Lu to Pt, andthen the 6s and 6p shells. The resulting elements, A u toRn, correspond once more to the elements Cu to Kr.With Rn, the sixth period of the periodic system with its32 ( = 8 + 10 + 14) elements is completed. The elementsfollowing from the unknown element 87 to uranium corre-spond to the first elements of the fourth, fifth, and sixth,periods. The elements for which a building-up of inner shellstakes place are grouped within dotted lines in Table 13.’

Inert gases (except for He) always occur with the closingof an sep6 group, as seen from Table 13. For n > 2, inert

4 It should be noted that the elements Y to Pd do not correspond exactly tothe elements Sc to Ni (although in both cases a d shell is built up). since,owing to the approximately equal heights of the and d shells under considera-tion (3d and 4s : 4d and 5s), a sort of competition occurs between the two whichleads to different results for different principal quantum numbers. Conse-quently the ground terms are not always completely analogous to one another.For example, the ground state of Pd is the 4&O %‘o state, whereas the groundstate for Ni occurring one period (18 elements) earlier is 3dW V’. This cor-responds to the chemical behavior. The elements of these columns of the peri-odic table have by no means such similar properties as have, for example, thehalogen6 or the alkalis.


gases do not follow the completion of a whole shell witha given value of n, because of the fact that the d orbits lieconsiderably higher than the s and p orbits for the samevalue of n-in fact, just about as high as the s and p orbitshaving n one higher (see Fig. 57). On account of this, theexcitation potential for closed s and p shells is large (inertgases), whereas it is small for elements for which a wholeshell with n > 2 is filled (for example, Pd, no inert-gascharacter).

The order of the different inner shells for an element ofhigh atomic number is normal (Fig. 57, right). For ex-ample, when electrons are added one after another to thelowest possible orbits for a uranium nucleus (Z = 92), thenineteenth electron comes into a 3d orbit and not into a 4sorbit, as with K (Z = 19). The normal order for the nine-teenth electron is already reached by SC (Z = 21). This isshown by the second spark spectrum of SC, which has a 2Dstate for its ground state and not a *S state; that is, theoutermost electron (the nineteenth) is here a d-electron andnot an s-electron (and similarly in other cases).

A more detailed discussion of the periodic system is givenin the following books: Grotrian (8) ; Pauling-Goudsmit (9) ;White (12). It has, however, been made quite clear thatthe whole periodic system of the elements can be unambiguouslyderived by using the building-up principle in conjunction withthe Pauli principle. The length of the long and shortperiods is given exactly, together with the existence of therare earths, which had previously appeared to contradictthe periodicity. The rare earths correspond to the build-ing-up of an inner shell, similar to the Fe, Pd, and Pt groups.

CHAPTER IV

Finer Details of Atomic Spectra1. Intensities of Spectral Lines

The intensity of an emission spectrum line correspondingto the transition from n to m is given by the product:

W”“hu’n,,,

where IVn~ is the number of transitions taking place persecond in the light source, and hi’,,,,, is the energy of theradiated quantum. Wn”( is the product of N,,, the numberof atoms in the initial state n, and A”I” the number of transi-tions per second of an atom. A”“I is the so-called Einsteintransition probability. Thus a knowledge of the two magni-tudes A”“I and N, is important in the calculation of in-tensities.

The intensity of absorption for the transition m to n(that is, the absorbed energy per unit time of the frequencydnn) is:

N,,,B”“hu’ nmP*where N,,, is the number of molecules in the initial state m,pr is the radiation density of the frequency u’,,~, and Brim isthe transition probability for absorption.

The transition probability for absorption is proportional tothe transition probability for emission. According to Ein-stein, we have :

w, 1)where g,, and g,,, are the statistical weights of the states nand m.

The transition probabilities can be calculated, accordingto wave mechanics, from the eigenfunctions belonging tothe atomic states taking part in the transition (see p. 52).

152

IV, 1] Intensities of Spectral Lines 153

From these eigenfunctions can also be obtained the selec-tion rules, which will be summarized in the following,

General selection rules (dipole radiation). The selectionrule for the total angular momentum I is ∆J = 0, f 1, withthe restriction that J = 0 -l+cT = 0 (++ means “cannotcombine with”). This holds for any type of coupling (seesection 3 of this chapter). For the component M of J in apreferred direction (for example, the direction of a magneticfield), the following rule holds: AM = 0, f 1, with therestriction that M = 0 -I+ M = 0 for AJ = 0 (see p. 104).For Russell-Saunders coupling, which holds at least ap-proximately in the great majority of cases and which wehave always used above, ∆S = 0 (prohibition of inter-combinations, p. 94). Under the same conditions, theselection rule for the orbital angular momentum Z isAL = 0, f 1. At the same time, Al must be f 1 for theelectron making the quantum jump (see p. 85).

Special selection rules (dipole radiation). Transitions in whichthe quantum numbers of only one electron change are always byfar the moat intense. Transitions in which two or more electronsjump at the same time are considerably weaker but are not forbiddenby any strict selection rule. In order to formulate the selectionrules for such transitions, it is useful to divide the terms of anatom into even and odd terms, according as Cl; is even or odd.The summation is to extend over all the electrons of the atom.The odd terms are distinguished from the even by a superior oadded to the term symbol, or sometimes the subscripts g(= gerade, meaning “even”) and u ( = ungerade, meaning “odd”)are used. For example, the ground state of the 0 atom,ls22sS2@ SP, is an even state, and may be written simply aP orsometimes “PO. The ground state of the N atom, 1922~~2~~ 4S~,l,is an odd state, and may be written ‘Soal, or 4S8/2u.

The division of the terms into odd and even has the followingwave mechanical meaning. As indicated before, the number ofnodal surfaces which go through the origin of co-ordinates (thenucleus) is equal to I for the eigenfunction of a single electron;that is, the number is even or odd according as 1 is even or odd.This means at the same time, however, that the eigenfunctioneither remains unaltered (I even) or changes sign (I odd) by

1 This rule does not hold for a strong magnetic field or for quadrupoleradiation (cf. below).

154 Finer Details of Atomic Spectra [IV, 1

reflection at the origin (inversion),Z that is, when + x, + y, + zare replaced by - 2, - y, - z. When several electrons arcpresent, the total eigenfunction is approximately equal to theproduct of the eigenfunctions of the individual electrons, and ittherefore follows that the total eigenfunction is even or odd accordingas Cli is even or odd; that is, remains unaltered by reflection at theorigin or changes sign. This property of the eigenfunction holdseven when the 2; are no longer approximately quantum numbers(angular momenta), as a more detailed wave mechanical investi-gation shows [cf. Condon and Shortley (13)].

The transition probability between two states n and m is nowgiven by ~&&,,*z dr, and correspondingly for the other co-ordinates (see p. 53). The integrand is obviously an odd func-tion when I+%, and $,,, are either both even or both odd; that is, theintegrand, and with it the value of the integral, change sign bythe transformation of co-ordinates + 2, + y, + z to - x, - y,- z. Since, however, the value of an integral cannot possiblychange by an alteration of the system of co-ordinates, the aboveintegral must be exactly equal to zero (see similar argument,p. 69). On the other hand, if +,, is even and &,, is odd, or viceversa, the integrand will be an even function and the integral willgenerally differ from zero.

Thus the strict selection rule for dipole radiation is: Even termscan combine only with o d d , and odd only with even (Laporte rule).For the particular case of two electrons i and k. the Laporterule may be formulated: When Al; = f 1, AZk must be 0 or+ 2 or - 2, and vice versa. A special case of the Laporterule is the prohibition of the combination of two terms of the sameelectron configuration. (For &ample, according to this, the threelowest terms of the N atom, ls92st2pP ‘S, *D, “P cannot combinewith one another.)

Additional special selection rules are: (1) In a strong magneticfield (Pazchen-Back effect), AML = 0, f 1, and AMs = 0, butAJ need no longer be 0, f 1.Aj, =

(2) For (i, 51 coupling (cf. below),0, f 1 for the electron performing the quantum jump.

Forbidden transitions. As we have already noted, transitionsviolating the above selection rules do sometimes occur with verysmall intensity.transgressions :

The following are possible grounds for these

Case 1. The selection rule under consideration may be trueonly to a first approximation.

Case 2. A transition may be forbidden as dipole radiation butmay be allowed as quadrupole radiation or-magnetic dipole radiation,and may therefore appear, even though very weakly;

f The nodal surfaces for I = 3 (drawn schematically in Fig. 19, p. 41) willhelp to make this clear. It should be remembered that $ has opposite signson different sides of a nodal surface.


Case 3. The selection rules under consideration (for dipoleradiation) may be strictly true in the absence of electric or mag-netic fields. They may, however, be transgressed when suchfields are applied externally or are produced by neighboring atomsor ions (enforced dipole radiation).2a

Case 1. An example is the selection rule AS = 0, which holdsunconditionally only under the assumption of vanishing couplingbetween Z and S, and therefore holds less and less rigorously asthe coupling between L and S increases; that is, the higher theatomic number and thus the larger the multiplet splitting, thestronger will be the intercombination lines which appear. For anatomic number as high as that of Hg, these forbidden transitionsare rather intense (for example, the Hg line 2537 A).

Case 2. The second case comes into operation with the selec-tion rule AJ = 0, f 1 (J = 0 t, J = 0) and with the Laporterule. According to quantum mechanics, these selection rulesshould actually hold quite accurately. The fact that transitionsviolating them do appear, though with very small intensity, isdue to the possibility of quadrupole radiation or magnetic dipoleradiation (cf. Chapter I, p. 54). As stated earlier, quadrupoleradiation depends on the integral Sx~,,&,,* dr, which (as P isan even function) will always vanish except when & and &, areboth even o r both odd.dipole radiation.

A similar relation holds for magneticHence we have, for these two types of radia-

tions, exactly the opposite selection rule to the Laporte rule,namely: Even terms combine only with even, and odd only with odd.From this it follows directly that ordinary dipole radiation, on theone hand, and quadrupole radiation or magnetic dipole radiation,on the other, cannot take part simultaneously in one and thesame transition.

Further calculations show that, for quadrupole radiation, theselection rule for J is: AJ = 0, f 1, f 2, with the addition thatJ’ + J” z 2, where J’ and J” are the J values of the upper andlower states (that is, J = 0 t, J = 0; J = 3 + J = 4; J = 1t, J = 0). For magnetic dipole radiation: AJ = 0, f 1; andJ = 0 t, J = 0 (as for ordinary dipole radiation). For L, theselection rules are (Russell-Saunders coupling): AL = 0, f 1,f2 ( L = 0 t, L = 0) and AL = 0, f 1, respectively. Thoselection rule for S is: AS = 0, and this holds to the same degreeas for ordinary dipole radiation.

To illustrate, terms of the same electron configuration can com-bine with one another according to the selection rules for quadru-pole radiation as well as for magnetic dipole radiation, whereasthey could not combine according to the selection rules forordinary dipole radiation (see above).

The ratios of the intensities of magnetic dipole radiation andquadrupole radiation compared to electric dipole radiation arc,

hTwo further causes of violations of selection rules have recently beendiscussed: Case 4: Coupling with the nuclear spin [Mrozowski (158), see alsofootnote 3. p. 156 and Chapter V]. and Case 5: Simultaneous emission of twolight quanta [Breit und Teller (159)].


respectively, of the order 10e5 : 1 and lO-+ : 1, provided that thereis no intercombination.

Case 3. The occurrence of lines in an electric field which con-tradict the selection rules ∆L = 0, f 1, or A1 = f 1 is anexample of the third case (mforccd dipole radiation). Underthese circumstances the intensity of the forbidden lines mayeven become comparable to the intensity of the allowed lines.

It is important to note that the selection rules for the Zecmaneffect for quadrupole, magnetic d ipole , and enforced dipoleradiation differ from those for ordinary dipole radiation‘, and alsofrom one another. Consequently an investigation of the Zeemaneffect gives an unambiguous criterion for the kind of transition

under consideration. Details will not be given here [see Rubino-wicz and Blaton (65)].

In absorption, forbidden transitions can be observed by using asufficiently thick layer of absorbing gas. For example, the inter-combination line IS - sP of the alkaline earths can be observedin this way. The intensity of the corresponding line for Hg isso great that a very thin layer suffices for the observation (cf.f&ove). Because of the J selection rule, only the component

- aPX usually a (Cf. the energy level diagram of Hg,p. 202.) The forbidden lines of the alkalis, 1% - m “D (forsmall values of m), have also been observed, with small intensity,in absorption. Segrb and Bakker (68) have shown, from a studyof the Zeeman effect of these lines, that they are undoubtedly dueto quadrupole radiation, and not to enforced dipole radiation.On the other hand, Kuhn (69) has observed the higher membersof the same series in absorption in the presence of an externalelectric field, but they have not been observed in the absence of afield. Thus we are dealing here with enforced dipole radiation.,

In emission, transitions due to enforced dipole radiation aresometimes observed in electric discharges where electric fields arealways present (external fields or ion fields). Here, also, it ischiefly the higher members of the series that appear since thehigher terms are influenced much more strongly by the Starkeffect than the lower (see p. 118). With the alkalis, for example,the series 2P - mP, 2S - mS, 2S - mD are observed.

On the other hand, forbidden ‘transitions which are not causedby electric fields are more difficult to observe in emission. When

a By using considerably thicker absorbing layers of Hg (IO’-fold), the for-bidden line X2269.80, corresponding to the transition l& - VP,, may also beobserved [Lord Rayleigh (66)]. The occurrence of this line contradicts theselection rules for ordinary dipole radiation, as well as those for quadrupoleand magnetic dipole radiation. (The upper state is odd, and the lower even.)

According to Bowen [cited in (67)], the transition is apparently due to theinfluence of nuclear spin.‘So

The line x2655.58, corresponding to the transition- *PPo, has been observed in emission [Fukuda (141)]. It also contradicts

the above-mentioned selection rules.


the probability for a given transition is extremely small, thecorresponding upper state has a very long life (provided no otherallowed transitions take place from that state). Therefore inan ordinary light source, before an atom in such a metastablestate radiates spontaneously, it has the opportunity to collidemany times and thus to lose its excitation energy without radiat-ing (collisions of the second kind, p. 228).’ This influence ofcollisions can be kept sufficiently small only under special condi-tions; for example, at extremely low pressures or by the additionof a gas whose atoms or molecules either are not able to remove theexcitation energy of the metastable state or can remove it onlywith difficulty. Since the life of a state which is actually meta-&able to dipole radiation is of the order of seconds (as comparedto lo-* seconds for an ordinary excited state), it is almost im-possible in terrestrial light sources to reach a pressure so low as toavoid the effect of collisions-especially since, at low pressures,collisions with the wall of the vessel lead to loss of excitationenergy. However , suitable conditions are present in cosmiclight sources.

Bowen (70) first showed that the nebulium lines, which had beenobserved in the spectra of many cosmic nebulae but were longa complete mystery, were to be explained as forbidden transitionsbetween the deep terms of 0+ (4S, *D, “P), 0++ (“P, ID, IS), andN+ (“P, ID, IS). The deep terms of these ions are shown in Fig.58 (see p. 158). Transitions between them involving dipoleradiation are strictly forbidden by the Laporte rule, since they areterms belonging to the same electron configuration: ls22s22p3 for0+, 18292~~ for 0++ and N+. The pbsitions of these energy levelshave been known with great accuracy for a long time from al-lowed combinations with higher terms. Bowen showed that thewave lengths of the forbidden lines, calculated from the combina-tion of these terms, agree exactly (within the limits of experimentalaccuracy) with the wave lengths of the unexplained nebuliumlines. Thus it was proved that the nebulium lines result fromforbidden transitions in the 0 II, 0 III, and N II spectra, andit was no longer necessary to assume the presence of a new ele-ment in these nebulae. Actually, in cosmic nebulae the condi-tions are extremely favorable for the occurrence of these forbiddentransitions. It is estimated that the densities in the nebulae areof the order of 10-l’ to lob20 gr. per cc. Assuming a plausible

value for the temperature (approximately 10,000” K), the timebetween two collisions suffered by an atom is then 10’ to lo4seconds. Thus, when 0+, O++, or N+ ions, which certainly arepresent, go into these low metastable states by allowed transitions

4 If other allowed transitions are possible from this state, a forbidden transi-tion is even less likely to occur, since long before that transition the ordinarydipole transition to some other level would have taken place.

Finer Details of Atomic Spectra [IV, 1 from higher states, they remain there uninfluenced until theyradiate spontaneously. A large fraction of the more highly ex-cited ions must come eventually into these states, and practically

424.325

444.252

“p, -444*54s

% 444.661

0 II 0 III NIIFig. 58. Origin of the Most Important Nebular Lines (Transitions Be-

tween the Low Terms of 0 II, 0 III, and N II). let and doubletsplitting is drawn to a much larger scale than thevalues are written to the right. The 0 III lines, 5006.8the most intense nebular lines and are sometimes designated N1 and N,.

every ion goes from them to the ground state by radiation. Thisexplains why the nebulium lines are very intense in nebulae,whereas they are not observed in terrestrial light sources, in whichthe other allowed 0 II, 0 III, and N II lines appear strongly.

In the last. few years, additional weaker nebular lines have beenidentified by various investigatqrs in a similar manner as for-bidden transitions of S II, S III, Ne III, Ne IV, Ne V, A IV, A V,and Cl III. The identity of a few others still remams doubtful[see Bowen (71)].

In an analogous way, McLennan (72) and Paschen (73) haveexplained the green and red auroral lines as corresponding to for-bidden transitions (‘Sneutral 0 atom.s

---) ‘D and ‘D + “P, respectively) of the(Cf. the energy level diagram in Fig. 59.)

According to Condon *74), the intense nebulium lines iV1 andN2, ascribed to O++, are due to magnetic dipole radiation. Cor-responding to this, the component ‘Ds + apt does not appear(cf. the above selection rules). Since we are dealing at the same

‘Thin naturally suggests that the lines observed in the spectrum of thesolar ‘corona, which have not been found in any terrestrial light sources, maybe explained in a similar ray as forbidden transitions. However for manyyears they defied all attempts of identification. Only very recently Edlen(146) [see Swings (147)] succeeded in identifying them with forbidden transi-tions between the low term of Fe X, Fe XI, Fe XIII, Fe XIV, Fe XV,Ni XII, Ni XIII, Ni XV, Ni XVI, Ca XII, Ca XIII, Ca XV, A X, and A XIV.


time with an intercombination (singlet-triplet), the mean lifeof the upper state is even greater than for quadrupole radiationwithout intercombination. On the other hand, the green aurora1line, as well as the corresponding nebular lines ID2 - %, is dueto quadrupole radiation (∆J = 2, no intercombination).

The aurora1 lines have also been obtained in the laboratory insuitable light sources [McLennan and Shrum (75) ; Paschen (73)];for example, in discharges through argon with a small additionof oxygen. The destruction of the metastable atoms is con-siderably hindered by the argon. This artificial production ofthe green aurora1 line made possible the study of its Zeemaneffect. From this it follows definitely that a quadrupole transi-tion is involved [Frerichs and Campbell (76)]. Since it is asinglet transition, the normal Zeeman effect with three corn-ponents would have been expected for dipole radiation. Actu-ally, two additional components were observed at twice the dis-tance from the middle line-an effect in agreement with thetheory for quadrupole transitions.6

General remarks on the intensity ratios of allowed lines.Apart from the selection rules by which certain transitionsare completely, or almost completely, forbidden, certaintheoretical predictions can be made concerning the intensityratios of allowed lines. In a series of lines which differ inthe value of the principal quantum number for the upperstate, the intensity generally decreases regularly toward theseries limit. Theoretically, the variation in intensity canbe calculated according to wave mechanics (see p. 5O), andso far as these calculations have been carried out, there isagreement with experiment.

According to the earlier part of this chapter, the intensitydepends on the number of atoms in the initial state as wellas on the transition probability. In order to ascertain theintensity, two limiting cases may be distinguished:

Case 1. In the case of thermal equilibrium or the tempera-ture excitation of the spectral lines, if E, is the excitationenergy of the state. n above the ground state, the numberof atoms or molecules in the state n is proportional toe-Ea’kT (Boltzmann). However, this rule holds only so

a-se middle component does not appear for exactly transversal or longi-tudinal observations, but does occur for observations inclined to the directionof the magnetic field.

160 Finer Details of Atomic Spectra [IV, 1 IV, 1] Intensities of Spectral Lines 161

long as the statistical weight or a priori probability is 1(cf. p. 119). If the weight is gn, the probability of findingthe state n is g,, times as great; that is, the number of atomsin the state n is proportional to g,e-BnlkT. If m is a secondstate with excitation energy E,,,, then

where N,, and N,,, are the number of atoms in states n andm, respectively. If m is the ground state of the atom(E, = 0), the number of atoms in the state n becomes:

N,, = N, &! e-EnIkT

!h (IV, 3)

The intensity of the line v I(m is proportional to this quantityin the case of thermal equilibrium.

Case 2. In the case of many electric discharges whereexcitation results from collisions with electrons of all pos-sible velocities, the Boltzmann factor plays no very signifi-cant part. Or, expressing this in another way, the tem-perature of the electron gas is so high that e-B’kT can betaken equal to 1 for most of the states in question. Then

(IV, 4)

Thus, while the states of lowest excitation energy are themost frequent for temperature excitation (owing to theBoltzmann factor), in electric discharges the higher excitedstates are, within certain limits, approximately as frequent.In both cases, for states with practically equal excitationenergies, N,/N, = g,,/g,,, since e--BnlLT is then approxi-mately equal to e-BmlkT; that is, the intensities are deter-mined mainly by the statistical weights.

The first doublet of the principal series of the alkalisillustrates the point (for example, the D lines of Na).The lower state is single. The two components of theupper state, 2P3,2 and “Pin, have statistical weights 4 and 2.Owing to the approximately equal excitation energies, fortemperature excitation as well as in a discharge, the number

of atoms in the 2Ptp2 state is twice the number in the “I-‘rrzstate. The intensity ratio of the two lines in emissionshould therefore be 2 : 1, and this actually is observed.The same holds for absorption, since then the number oftransition possibilities is twice as great for one componentas for the other.

Sum rule. The generalization of these considerations forcomplicated cases is the Burger-Dorgelo-Ornstein sum rule: Thesum of the intensities oj all the lines of a multiplet which belong tothe same initial or final state is proportional to th.e statistical weight2J + 1 of the initial or f inal state, respectively. By way of illus-tration the following scheme for a 2P - =D combination may bederived [see Table 14; cf. Figs. 29(b) and 30, p. 74 and p. 75].The sum of the intensities of the transitions with *PI,2 are tothose with ?PJ12 as 5 : (1 + 9) = 2 : 4; that is, in the ratio ofthe statistical weights. Similarly for 2D, (5 + 1) : 9 = 4 : 6.Conversely, from these two relations the relative intensities maybe calculated.

TABLE 14

INTENSITIES FOR A ‘P - ZD TRANSITION

w+1 4 6

2Plll 2 5 -“Palz 4 1 9

From the sum rule the following general rules can be derived:(a) The components oj a multiplet for which J and L alter in thesame manner are more intense than those for which they alter un-equally. (b) The components belonging to a large J value are moreintense than those with small J. These rules are especially impor-tant for the practical analysis of a multiplet (cf. Figs. 31, 33, and34; see also section 4 of this chapter).

The sum rule is not sufficient for an unambiguous determinationof the intensities of compound triplets or higher multiplets. Insuch cases we must use the general theoretical intensity formulaederived by Sommerfeld, Honl, and de Kronig [see (5a) and (13) ],which naturally contain the sum rule. These formulae, as wellas the sum rule, hold only for Russell-Saunders coupling (smallmultiplet splittmg). Intensities in more general cases and fornon-Russell-Saunders coupling have been treated in recent in-vestigations but this work will not be dealt with here. A discus-

IV, 2] Series Limits and Related Topics 163162 Finer Details of Atomic Spectra [IV, 2

sion of the intensity rules for Zeeman components must also beomitted [consult White (12); Condon and Shortley (13)].

2. Series Limits for Several Outer Electrons, Anomalous Terms,and Related Topics

Series by excitation of only one outer electron. When theoutermost or valence electron for an alkali atom is raised toorbits with higher values of n and then allowed to return to alower state, there result different emission series whose limits cor-respond to the complete removal of the valence electron. Inabsorption, only one series of lines (doublets) is obtained, theprincipal series; whose limit gives directly the ionization potentialof the atom. The state of the ion resulting from the removal ofthe outermost electron has only closed shells; it is the %O groundstate (inert gas configuration, see Table 13). This state is single,and therefore the series limit is single; all term series go actuallyto the same limit. Similar relations hold for the alkaline earthand the earths (boron group), where, likewise, the removal of theoutermost electron leads to the ground state of the ion. How-ever, different relations hold for the elements of the carbongroup and the following groups. For these elements, the ionwhich is obtained by removing the outermost electron has anelectron configuration which gives excited terms in addition tothe ground state. For example, for C the remaining ion can be ina *P1l, or *Pa,* state; for N, in a “9, ID, or IS state; and so on.

We shall consider in more detail the case of the oxygen atom.According to the building-up principle (Chapter III), we canpredict the qualitative energy level diagram that will be obtainedwhen we add an additional electron to the lowest electron con-figuration of the ion, ls52ss2pa. I n this case the lowest electronconfiguration of the ion corresponds to three different terms, ‘S,zD, and 2P (as for N). Different term series are thus obtainedfor the neutral 0 atom according as the emission electron is addedto the terms ‘S or zD or *P in the different free orbits with variousn and I values. The number of terms is thus considerably largerthan for Be or B, for example.

If an s-electron is added to the % ground state of the 0+ ion,%S’ and Yi terms are obtained by vector addition of the 2 of theadded electron to the Z of the ion, and of the s to the S (see p.129 f.). For each of these terms there is an entire series corre-sponding to the different possible values of the principal quantumnumber (n Z 3).

If a p-electron is added to ‘S, a series of *P as well as a series of“2’ terms is obtained. According to the Pauli principle, for 5Pthe n value of the added p-electron must be at least 3 (seeTable 13); but for aP, n can also be 2. The state 2p *P is the

ground state 7 of the 0 atom. Similarly, aD and KD, or *F and SFseries are obtained by the addition of a d-electron or an f-electron,respectively. Parts of these series are shown graphically at theleft of Fig. 59 (terms not observed are indicated by dotted lines).

If an s-electron is added to the excited *D state of the ion havingthe same electron configuration (1#2#2@) as ‘S, there resultseries of ID and ‘D terms whose limit, however, lies above the

- - - - - - -

,-~--_--~--~--- ; ..---*:-$:* 5-e- sm. -26,000

~~~-4--- I- 4 ..m 8- a-1

a -

s - -

2 -

2-

O-

2a.ooo -

ao*ooo -

76,000

100,000 1o+-~~-~-~~----~--------------- 4

Fig. 59.Limits. The a values given are the true principle

Energy Level Diagram of the 0 Atom, with Different Series quantum numbers of the

emission e l ec t ron . . The term to the extreme right (2~2~6 ‘P) does not belongrbzmy;f the indicated series limits. Dotted lines indicate terms not yet

limit of the previously considered terms by an amount equal tothe excitation energy of the *D state of the ion (see Fig. 59, center),From *D, by adding a p-electron further series of terms are ob-tained: ‘P, “P, ID, “D, ‘F, aF; correspondingly, by adding ad-electron: IS, “S, ‘P, “P, ‘D, aD, ‘F, “F, ‘G, V. In an exactlysimilar manner, the series ‘P, “P, W, %‘, lP,‘aP, lD, “D, . . . resultfrom the *P state of the ion (Fig. 59, right), the series limit beingstill higher.

In general, the term values are so chosen for atomic spectrathat ionization with the ion left in its lowest state corresponds to aterm value 0 and terms of smaller energy are counted positive

7 For simplicity, only the symbol for the emission electron is given.


(see Chapter I). Terms corresponding to excited ion states whichlie above the first ioniration limit will thus be negative.

When it is necessary to distinguish terms belonging to serieswith different limits, the term type of the corresponding ion canbe included in the designation; for example, 2~~(~D)4p IDS, andsimilarly in other cases. All terms of the same multiplicity andthe same electron configuration resulting from a given term ofthe ion are called a polyad. For example, all triplet terms, 3P,aD, aF, of the configuration 2~~(~D)np of oxygen would be called atriad. They generally lie fairly close together. All the transi-tions between the terms of two polyads are called a supermultiplet[cf. Condon and Shortley (13)].

Term series going to different limits (such as have been amplifiedhere for the 0 atom) appear for all those atoms (and ions) thatpossess several terms for the lowest electron configuration of theion. A great many such cases have already been investigated,and each has confirmed the theoretical conclusion that the separa-tions of the series limits must be equal to the observed term differ-ences of the corresponding ion. The existence of these additionalterms leads to a considerably larger number of line series in emis-sion and absorption than is observed for simpler atoms. Forexample, according to the selection rules (AS = 0; AL = 0, f 1;AZ = f 1), the ground state of the 0 atom can combine with theterms ls?2s22pa(*S) 12s Y?; 2pa(‘S) nd aDo; 2p*(*D) ns aL)0; 2pa(zD)nd *So, *P, *Do; 2@(tP) ns 8pO; 2~3(~P) nd aPO,.aD”; whereas, forinstance, the ground states of Na and Mg can combine with onlyone term series, n 2P and nlP, respectively.

Series by excitation of two electrons; anomalous terms. Apartfrom the terms for which only one electron is excited, other termsare possible for which two (or even more) electrons are in shellsother than those for the ground state. Such terms are actuallyobserved and are called primed or anomalous terms. They werefirst observed for the alkaline earths and the alkaline-earth-likeions. In their spectra, were found multiplets which could notbe arranged in the normal triplet series and which did notshow the normal structure of a compound triplet. Fig. 31(d),p. 76, shows a spectrogram of an anomalous triplet of Ca, whichshould be compared with the normal compound triplet in Fig.31(b). The lower part of Fig. 60 shows the same schematically.The relationships between the separations and between the in-tensities for a normal compound triplet (see p. 78) are not ful-filled here. However, these and similar multiplets may be ex-plained (as indicated in Fig. 60) as due to a combination of two*P terms with not very different splitting (taking into account theselection rules for J and the intensity rules). If the explanationis correct, the energy level diagram shows that the separations ofthe components a to c and d to f must be exactly equal. This is

IV, 2] Series Limits and Related Topics 165

actually observed to be the case and the separation gives thesplitting 8P2 - apt of the lower term. It now appears that thissplitting and also the splitting 8P~ - ‘PO (separation of the linesc and e) agree exactly with those of thelowest sP term of the alkaline-earthmetal under consideration (Ca, in Fig. ~

J

60) which have been known for a longtime. The foregoing means that the aPt

lower state of this multiplet is the lowestsp aP state. The upper state is an ano-malous term which does not belong tothe normal term series and is designatedas SP’

The fact that thii anomalous termCombines with the known 8P term,although it is itself a P term,s contra-diets the selection rule AL = f 1, whichmust hold for terms for which only oneelectron has I + 0. It follows that theanomalous term corresponds to an excita-tion of two electrons. When this is thecase, AL = 0 is also possible, provided Y-that Al = f 1 for the one -electronmaking the quantum jump (transition

Fig. 60. Origin of anAnomalous Triplet of the

between even and odd terms). This con- Alkaline Earths. clusion is supported by a large number offurther arguments which cannot be taken up here [consultWhite (12)]. Agreement with experiment is obtained when theassumption is made that, in the aP’ term, both outer electrons areexcited for Be to 2p orbits, for Mg to 3p orbits, for Ca to 4porbits, and so on. According to the foregoing (p. 131), twoequivalent p-electrons give the terms: IS, aP, ID. Here we aredealing with the SP state since it can combine with the sp SPterm in the way shown in Fig. 60. Writing the symbols in full,for Be we have the transition ls22pz aZ? --) ls22s2p JP; for Ca,4p2 *P * 4s4p 3P [see Fig. 31(d)]. Since only 072e electron jumps,these transitions are allowed and are very intense. Owing to theLaporte rule, the p2 aP state cannot be reached by absorptionfrom the s2 ‘S ground state. It is probably also impossible toexcite it directly in a discharge by electron collision from theground state. Possibly it is reached through the sp 8P state bytwo successive electron collisions.

The two other terms, ‘S and ‘D, with the same configuration,p”, have likewise been found for Be and for other cases,- althoughtheir identification is not so certain since they are singlets..

* The values of J necessary to explain the splittii pattern show that theterm cannot be any other than a P term.


The triplet splitting for the anomalous ps aP term must be ofapproximately the same magnitude as that for the normal sp *Pterm since the p-electrons have the same principal quantumnumber. This is in agreement with experiment [cf. Figs. 60and 31 (d)].

Volt513.22 _

l3-

12.

ll-

10.

3.231’9.

0,

7,

6

6

4

3

2

1

0

5

’ ‘D 8PL%P-zL-

e m - ’Be+ls’2p ‘PO-+ - - -

-.--6 . ..--6 I:::; -30,000----5 ----5 --m6----1~---~ ----(----3 -20,000

----3 - tpld ‘De

- 31 ,zpss ‘Pfo

-10 ,000’ 1s lp” ID ‘S ‘Pa ‘D --a‘c_rr_8__np_-nd-_b_np__nd_ 0= c= 2s=; =5 --s--s=, - 2

-4--I -5-s-*

-4 -a -4-4 10,000

- 3- 3 - 2

- 3- 3 20,000

- 3

30,000- 2

40,000

60,000- 2

60,000

70,000

.-2----------------- ---- -

Fig. 61. Energy Level Diagram of Be I with Anomalous Term Series[Paschen and Kruger (78)]. The normal singlet and triplet series are drawnto the left (cf. Fig. 32 for Ca I); the anomalous term series, to the right. T h eterms drawn with dotted lines have not been observed. Apart from theterms G, ID, *P, for R > 2, the terms %S, *D, 1P are also possible for the con-figuration l&p np, but thus far have not been observed. For n > 2, termsof the configurations 2p M and 2p nd are also possible; of these, however,only the first member of each has been observed (indicated at the extremeright of the figure). n is the true principal quantum number.


The observed energies of these anomalous terms correspondalso with the theoretical expectations. They should lie, roughly,twice as high above the ground state as the normal sp BP term,since two electrons have been brought into the 2p orbit (for Be)instead of one. An inspection of the Be energy level diagram inFig. 61 shows that this is actually the case.

The following is another somewhat more accurate estimation ofthis excitation energy. If the explanation of anomalous terms iscorrect, we should expect for Be, for example, that the wavenumber of the transition W2s2p JP - ls22p* 3P would approxi-mately agree with the wave number of the line obtained whenone 2p electron is left completely out (that is, with the 1s22s2s - ls22p *P transition of the Be+ ion), since it can hardly beassumed that the 2p electron can influence the energy of the twoterms very differently. Actually, this relation is well fulfilled(see Fig. 61).

Thus the term 1922~~ of Be and the analogous terms of the otheralkaline earths and of the alkaline-earth-like ions lie ratherclose to the first ionization limit. Apart from the term ls22p2,analogues are obviously to be expected for which one electrongoes to higher orbits, 3p, 4p, and so on (that is, a whole series ls22pnp, corresponding to the series 1822s np). The limit of the formerseries is the ion term 192~; that is, an excited state of the ionquite similar to the foregoing, but with the difference that thisterm no longer has the same electron configuration as the groundstate of the ion. Two members of this series have been found forBe (see Fig. 61). These terms have negative term values-; that is,they lie above the lowest ionization potential. Due to this, only afew of them have been observed in this and similar cases. Beforean atom in such a state can radiate, pre-ionization (auto-ioniza-tion) usually takes place. (This topic will be discussed furtherat the end of the present section.) Series of terms correspondingto the above also result when ns or nd replaces np.

Similar anomalous terms have been found for many atoms andions. Relatively few occur for the lighter elements since they lie,for the greater part, above the lowest ioriization limit. However,these terms are very numerous for the heavier elements since, forthem, some of the outer shells frequently have not much moreenergy than the ground state, and hence the energy for the simul-taneous transition of two electrons to a higher shell is often notparticularly large. That they are so numerous also depends onthe fact that the corresponding ion has a large number of low-lying terms. This is one of the reasons for the essentially greatercomplexity of the spectra of the heavier elements as compared tothose of the lightei.

Excitation of inner electrons. Very closely connected withthe foregoing are the spectra resulting from the excitation of inner

168 Finer Details of Atomic Spectra [IV, 2_

14

13

1 2

1 1

60,OOi

6P -.

-I-4PPJ-

I

45,ooa

I 40,000

I

I35,000

I

I25.000

iI

16,000

1

O-

3-

)-

I-

l-

I -I’

IV, 2] Series-Limits and Related Topics 169

electrons. Such spectra have recently been investigated in detailby Beutler (79). They provide a connecting link between opticaland X-ray spectra. As is well known, ‘the latter correspond totransitions involving the innermost electron shells of an atom.Beutler found, in absorption, transitions from the ground state ofthe atom to states in which one of the electrons of the outermostclosed shell (which must be designated as an inner electron) goesto a higher orbit. He has designated these spectra I* spectra, asan extension of the usual designation for the ordinary spectra ofneutral atoms as I spectra (for example, Hg I). The essentialpoint is that, contrary to the case just treated, only one electronneeds to alter its quantum numbers in order to reach the corre-sponding excited state (Ib term) from the ground state. However,it must be an inner electron, and this difference distinguishes suchterms from normal terms. Since only one electron has to jump,these terms may be reached by absorption.

An illustration from the zinc spectrum will help to make thispoint clearer. The electron configuration of Zn in the groundstate is ls22s22p63s23p63P4s2. The normal spectrum resultswhen one electron goes from the 4s shell to higher orbits; anoma-lous terms result when both electrons go from the 4s shell tohigher orbits. The I* term series results when one electron goesfrom the closed 3d shell into higher orbits. Such terms lie veryhigh-appreciably higher than the ionization limit of the normalatom. The lowest state to be excited in this way is = - * 3@4$4p.Beutler found a whole series with np (n = 4, 5, * * e), and a cor-responding series with nj. Naturally, many terms belong to eachconfiguration (cf. Tables 10 and 11 on p. 132). Of these terms,only three (‘PI, 3P1, 301) can be observed in absorption from theground state (Go), because of the selection rules ∆J = 0, f 1(J = 0 -I+ J = O).s Term series with ns or nd in the place ofnp or nj cannot be observed because of the selection rule Al = f 1.

Fig. 62 shows the observed Zn Ib terms. All of the predictedterms except the nj 301 terms have been observed. The terms lieabove the lowest ionization limit of the normal atom. The energylevel diagram is drawn from this point up. The energy leveldiagram of the Zn+ ion is indicated at the right of the figure.The series limit (n -+ 00 ) of the 1” terms under consideration mustcorrespond to the 3ds4s2 state of the Zn+ ion. This is a term ofthe ion c2U) for which an inner electron is excited (according toBeutlcr, a II* term). A continuous absorption spectrum joinsthe series limit just as for a normal series, and corresponds toionization leaving the ion in the IIb state mentioned. In X-ray

9 The deviation from Russell-Saunders coupling is already so great that theselection rules AS = 0 and ti = 0, f 1 no longer hold strictly.


nomenclature this spectrum would be called an absorption spectrumfrom the Ma shell.

Beutler and his co-workers have already found similar absorp-tion spectra for a large number of atoms. The series limits donot necessarily correspond to a II* term of the ion. For K, for

20,000

25,000

8 0F4-1IG

1

.---.

6

5

1 0fi

-L-.

example, the upper states of a Id

series are 3~~44s ns, and this gives theordinary excited state 3~~4s of K+when n +m.

Naturally there also exist termswhich correspond to the excitation ofshells lying still farther in. Theyare correspondingly designated I”, Id,. . . . For Tl, absorption lines havebeen found whose transitions corre-spond to such terms. These spectrabridge the gap to X-ray spectra andmight well be called X-ray spectra.

Summarizing the results of thepreceding discussion, we conclude :in theory, term series of a neutral atomresult from the addition of an electronnot only to the ground state of a singlycharged ion but also to each excitedstate of the singly charged ion, whetheror not it has the same electron con-figuration as the ground state,whether it is normal or anomalous,or whether or not it belongs to the bterms. In general, this leads to agreat number of terms. The foregoing considerations of course alsoapply to the spectra of ions.

Term perturbations. Sometimes

Fig. 63.deviations from the normal posi-

Perturbed *FTerms of Al II Compared with

tions (expected according to the ordi-the Hydrogen-like Terms

nary series formula) are observed in4Rlnf and with the 4 Terms. certain line series belonging to atomsThe perturbing term is in- and ions with several emission elec-dicated by a dotted line. trons. These deviations are known

‘Pa and *F, terms of the Alas perturbations, As an example the

left of Fig. 63.II spectrum are given to the right and

For comparison, the terms 4R/na are drawn in thecenter of the figure. They should follow very closely the variationof the F terms of Al II, since F terms are usually hydrogen-like.10

10 The factor 4 enters the formula since we are dealing with the first spark spectrum.


We can see that this is largely the case for the IF3 terms through-out the entire region. On tha other hand, for the JF, terms thisis true only for large and small values of 72, whereas pronounceddeviations from the normal position appear in the region n = 5 ton = 7. There is actually one more term present than would beexpected.”

The reason for this phenomenon is a resonance process quiteanalogous to the Heisenberg resonance for He (p. 66), whichled to the energy difference between singlet and triplet terms.When it happens that two terms of different electron configura-tions of the same atom or ion have approximately the same en-ergy, the states influence each other. In the case of He, theeigenfunctions of the resulting states are mixtures of the eigen-functions of the two originally degenerate states [~7~(1)9,(2):electron 2 excited, and (p,(l)cp1(2): electron 1 excited; cf. p. 67].Similarly, here, a mixing of the eigenfunctions results. If $1 and& are the zero approximation eigenfunctions of the two states ofnearly equal energy with different electron configuration, theeigenfunctions of the two resulting states will be, to a first approxi-mation (as shown by more detailed calculations not given here):

$1 = aA + Wt and $11 = c#l+drtr

Thus each of the resulting states has, so to speak, both electronconfigurations (though not in equal amounts as for He, wherea = b = c and d = - a).. This mixing may also be regarded asan oscillation of the atom between the two states (the two elec-tron configurations). There is at the same time a shifting of bothterms away from each other, as for He. Theory shows that theseperturbations can occur only between terms which have equal Jand, in the case of Russell-Saunders coupling, equal L and S.In addition they must either both be odd or both even.

In fact, in the example of Al II an anomalous term(ls22sP2p63p3d aFo) is to be expected, and it will be of the sametype as the term of the normal 2~~3s mf aFo series and may liesomewhere between n = 6 and n = 7 (dotted line in Fig. 63,center). Its eigenfunction mixes with that of the neighboringnormal terms, and, furthermore, the latter will be displaced awayfrom the position of the perturbing term. The perturbing termitself forms the extra term. On account of the mixing of theeigenfunctions, we cannot ascribe an unambiguous electron con-figuration to terms in the region of perturbation.

Pre-ionization (auto-ionization). The phenomenon of pre-ionization or auto-ionization [Shenstone (81)] is very closely re-lated to perturbations. As we have already pointed out, many

-11 In addition, there is at the same time an abnormally large triplet splitting

of the 8F terms (not shown in Fig. 63).


of the terms resulting from the excited states of an ion (for example,practically all Ib terms) have negative values; that is, they liehigher than the lowest ionization potential of the atom or ion inquestion. They thus overlap the continuous term spectrumwhich joins the normal sequence of terms. This is shownschematically in Fig. 64. As in the case of perturbations, wehave here two different states of an atom which have the sameenergy: the discrete anomalous state, and the continuous ionizedstate with a corresponding relative kinetic energy of ion plus elec-tron (indicated by the dotted arrows in Fig. 64, right). As before,a mixing of the eigenfunctions takes place-that is, an oscillation

between the two states of equal4

energy.

t/However, when the system has once

oscillated from the discrete state into1.”II’ the continuous state lying at the sameet* height, a return oscillation is not pos-

sible, since the electron has alreadyleft the atom. This can also be ex-pressed in the following way: A radi-

Fig. 64. Pre-ionization of ationless quantum jump takes placethe Terms Lying above the First Ionization Potential of from the discrete state to the contin-an Atom or an Ion. To thele&r;h;$y~ JJJt$=rern~

uous state lying at the same height( s h o w n by the horizontal arrows in

. height as the continuum Fig. 64), and results in an ionizationwhich joins the series of of the atom. Analogous to a similarterms drawn to the right. phenomenon for molecules (pre-disso-

ciation), this effect should be calledpre-ionization but in the literature is usually referred to as auto-ionization.

In the case of perturbations, a shifting of the levels takes place.Similarly here, theory shows that a broadening of the discrete levelsis to be expected. Actual observations show that lines in whichsuch negative terms participate are in many cases considerablybroadened, although in some cases they are sharp (narrow). Itmay be shown theoretically that the greater the probability of aradiationless transition, the greater the broadening. A noticeablebroadening (greater than the normal Doppler breadth) will takeplace only when the probability of a radiationless transition isvery great compared with the probability of a transition to anenergetically lower state with radiation. This means at the sametime that emission lines which originate from levels broadenedin this way should be either very weak or entirely missing, ,a con-clusion that agrees completely with experiment. It was statedabove that negative terms are very difficult to observe in emission.

We shall now consider why some of the absorption lines arefairly sharp and some of the emission lines are relatively intenseeven when the above conditions for pre-ionization are fulfilled.

IV, 3] Other Types of Coupling 173

This has in principle, the same explanation as the facts that somenormal lines are strong and others weak, and that the continuous spectrum which joins the absorption series diminishes fairlyrapidly in intensity with decreasing wave length. The radiation-less transition probability depends on the eigenfunctions of thetwo states involved, in a similar manner to the transition proba-bility with radiation. There are also selection rules for radiation-less transitions. It should be noted that in the continua whichextend beyond the different term series, the angular momenta S,L, and Jretain their meaning unaltered and the property even-oddis also defined. The selection rules are the same as for perturba-tions (see above): ∆J = 0, ∆S = 0, ∆L = 0, and even terms do notcombine with odd. The discrete terms lying above the lowestionization potential cannot, therefore, go over by a radiationlesstransition into the continuum joining any arbitrary term se-quence; instead, they can go only into specific continua. If thesedefinite continua do not exist, pre-ionization cannot occur. Inaddition, the radiationless transition probability becomes smallerwith increasing distance from the series limit, since the eigen-function is a periodic function with a nodal distance (wave length)which becomes smaller and smaller with increasing distance fromthe limit. Therefore the value of the transition integral ap-proaches nearer and nearer to zero. This conclusion correspondsto the fact that absorption lines, whose upper states lie at a fairlygreat distance from a series limit, are very sharp.

Similar radiationless quantum jumps occur also in the X-rayregion. When a K-electron is removed from an atom by Kabsorption, the ion is left in a highly excited state (upper state ofKi and Kp). This state lies considerably higher than the lowestionization potential of the ion-actually higher than the ioniza-tion energy for the removal of an L-electron. Therefore, insteadof the atom emitting a K, quantum as a result of t.he transition ofan electron from the L shell to the K shell, the energy set free bythis transition can be used to liberate one of the remaining L-elec-trons. Such a radiationless quantum jump was first discoveredby Auger, and is called after him the Auger effect or Auger process.This name is sometimes used as a general term for all suchprocesses-for atoms as well as molecules.

3. Other Types of CouplingThus far we have always used Russell-Saunders coupling

(p. 128), which assumes that the interaction of the individual Ziand the individual s; is so strong between themselves that theycombine to give a resultant L and S. L and S then combine witha smaller coupling to give a resultant J. This assumption holdsfor a large number of elements, particularly for all the lighterelements, as may be seen from the fact that, for them, the multi-


plet splitting is usually small compared to the energy differenceof the levels having the same electron configuration but differentL. The splitting is likewise small compared to the energy differ-ence of corresponding levels which differ only in theirmultiplicities.

Because of its validity in so many cases, Russell-Saunderscoupling forms the basis for the usual nomenclature.

(j, j) Coupling. When we assume the opposite case to Russell-Saunders coupling-namely, not that there is a strong interactionof the li with one another and the si with one another, but ratherthat there is considerable interaction between each ti and the sibelonging to it-we obtain so-called (j, j) coupling: Each ticombines with the corresponding si to give a jc the total angularmomentum of the individual electron.‘* The individual ji are lessstrongly coupled with one another and form the total angularmomentum J of the atom. Such coupling can be writtensymbolically :

(z~s~)(zes2)(z~s3)“’ = (jlj& - *) = J (IV, 5)There is no definite L and S for this coupling. However, Jremains well defined. The same holds for M.

Let us consider, as an example, the configuration ps, whichgives a 8P~ I. B and a IPI state on the basis of Russell-Saunderscoupling. Assuming (i, ~1 coupling, however -the resultant isformed first from 1~ = 1 and s1 = 3. This gives j1 = 8 or 3.From the supposition of strong coupling between t and s, thesetwo states have very different energies. j, can take only onevalue, namely, 4, since I z = 0. Because the coupling between jland j2 is assumed to be small, we have, to a first approximation,two terms which have equal j, and which differ in the two abovej, values. The two states may be characterized briefly as($1, j,) = (3, a> and (3,s). To the same approximation, wehkewise have two terms for Russell-Saunders coupling: one lPand one SP term. (See Fig. 65, in which the two limiting casesa r e drawn to the extreme left and right.) When the small (j, j)interaction taken into account, a slight splitting of each of thetwo levels, (j,, j,) = (3, %) and (4,9 , n 0‘) i t two components occurs(two possible orientations of j2 with respect to jl).2 or 1; for (3, a>, J is 1 or 0.

For ($, #), J isFor Russell-Saunders coupling, when

we allow for the small (L, S) interaction, ‘P splits into its threecomponents, J = 0, 1, 2 (Fig. 65, left).

Thus we see that the number of terms is eventually the samefor both types of coupling and that the J values are the same also.

I* The component, of j in a magnetic field is mj. For the application of thePauli principle, in this case, it is more convenient to employ n, I, j, and m jthan it is to use n, Z, ml, and m. (cf. footnote 1, Chapter III).

IV, 3] Other Types of Coupling 175

Hence an unambiguous correlation is possible (dotted lines inFig. 65). Therefore terms can be designated in the Russell-Saunders manner in spite of the fact that they may have prac-tically (j, j) coupling. However, this method of designation hasthen only a very limited value. First of all, it no longer corre-

q, J J. .

A. 32-I-.-.

~~-‘b----%.c-2’////

“PI2-’l-------l()------(J - ‘in. 42

Fig. 65. Relative Positions of the Terms of a ps Configuration. To the left,Russell-Saunders coupling; to the right, (j, j) coupling.

sponds to the relative position of the terms. Second, the prohibi-tion of intercombinations ∆S = 0 and the selection rule ∆L = 0,f 1 no longer hold, since L and S are no longer definite quantumnumbers. The terms combine according, to the Laporte rule andthe selection rules: ∆J = 0, f 1; Aji = 0, f 1 (see section 1).

For cases in which pp, pd, or other configurations are presentinstead of the case of one p-electron and one s-electron, the rela-tionships are naturally much more complicated. Neither thesenor the completely altered g-formula for Zeeman splitting for(j, j) coupling will be considered further here. [Consult White(12); Condon and Shortley (13).]

Transition cases. Pure (j, j) coupling occurs relatively seldom.Instead, we usually have to deal with transition cases which cor-respond to the region at the center of Fig. 65. The figure showsthat in this region the splitting of the terms does not follow exactlyeither Russell-Saunders or (j, ~3 coupling. In Fig. 66 the posi-tions of the first excited 8P terms and the corresponding lP termsof the elements of the carbon group are given. These two termsare due to an electron configuration ps. Carbon has practicallypure Russell-Saunders coupling, as has Si. However, Ge, Sn,and Pb approach closer and closer to (j, j) coupling; this effect i sindicated especially by the term with J = 2, which moves fromthe neighborhood of the lowest term with J = 0 into the neigh-borhood of the uppermost term with J = 1 (‘PI) (see p. 175).

It must be emphasized that, when (j, ~1 coupling occurs forone term, it need by no means hold for the whole term system ofthe atom in question. This coupling holds preferentially forexcited states. Practically pure (j, j) coupling is present in theabove case of an excited state of Pb, but does not hold for itsground state. The outer electrons in the ground state have the


configuration p*; therefore the lowest, terms are, in order, aP, ID,*R, just as for C (see p. 142). The triplet splitting, it is true, isconsiderable, though not so large that the terms cannot be dis-tinguished according to Russell-Saunders. The same holds forSn and Ge, whereas their excited states (Fig. 66) already approachfairly closely the case of (j, j) coupling (the higher excited statesapproaching it, even more closely).

Thus with increasing atomic number first the higher excitedstates show a transition to (j, j) coupling, because, for an electron

with large principal auantum‘P

J, -_ _-_ __- . ..-. ..- 1 n u m b e r , the coupling with theother electrons is rather weak.

. - 2-:’ Even with fairly small atomicnumber this coupling may be

: weaker than the coupling of 1 and:

8’- s for this electron.’ Therefore aresultant j is first formed for thiselectron, which then interactsweakly with the angular momentaof the other electrons. In thecase of elements of the carbon

:group (shown in Fig. 66), only one

:a’additional electron is present with

‘pa, : ,-/-- I + 0 (namely, a p-electron).ap--:: *--..

*,w . ..-...-._.-.. := l!This electron forms its own j,*

4 C Si Ge Sn Pband (j, j) coupling results for

Fig. 66. Observed Relativelarge principal quantum numbers

Positions of the First Excited *Pof the emission electron. The

and ‘P Terms of Elements of the two j values of the p-electron inCarbon Group. Transition from the core correspond to the two Russell-Saunders to (j, J> cou-pling. The scale is different for

components of the zP ground term

the various elements, but has beenof the ion to which the terms of

so chosen that the separation be- the neutral atom converge.tween each uppermost and lowest If several electrons with I + 0term in the diagram is the samefor each element.

are present, as well as an emis-sion electron with high n, theformer will have Russell-Saunders

coupling with one another for a not too high atomic number;that is, they give an LC and an SC of the atomic core with a re-sultant Jc, which will then be weakly coupled with the j of theemission electron. This coupling can be written symbolically:

(111a...)(~d2...)(1, s) = (Lc, SC)& sj= (Jc, j) = J (IV, 6)

Such a case occurs for the excited states of Ne, for example, inspite of a rather small atomic number.

Still other modes of coupling are possible but will not be dealtwith here.

IV, 4] Interval Rule; Analysis of Multiplets 177

When approximation to (j, j) coupling makes it impossible toascribe definite Russell-Saunders term symbols to the observedterms in a given case, the latter are distinguished by their Jvalue, if necessary with a superior 0 added as an upper index to indicate that the term is odd. When the symmetry of the groundstate is known, whether a term is odd or even can easily be estab-lished on the basis of the Laporte rule, which holds absolutely forany type of coupling.

4 . The Interval Rule; Analysis of Multiplets

General remarks concerning the analysis of atomic spectra.According to what has already been said, the analysis of atomicspectra such as the alkali or alkaline-earth spectra, consisting ofsimple series, presents no difficulties. One needs only to identifyamong the observed lines those lines that belong to certain series,and then to relate these series according to the theoretical prin-ciples. However, the analysis of a complicated spectrum whenseveral outer electrons participate is by no means so simple. It isparticularly difficult for the beginner to understand how to pickout the regularities from the perplexing abundance of lines in sucha spectrum (cf.. Fig. 6, p. 7), how to assign the lines to definiteseries and definite terms, and how this can ever lead to an un-ambiguous result. We shall touch on these topics briefly in thissection.

First of all, the regularities whieh have-been discussed in earlierchapters and which form the basis of the analysis will be sum-marized.

1. It must be possible to arrange the lines in R y d b e r g series ofthe form already given (see also p. 197). The different membersof such a serie may lie in entirely different spectral regions.

2.. Lines belonging to one and the same series show the sameZeeman effect; only singlet lines show the normal Zeeman effect.

3. Apart from singlet lines, it should be possible to group thelines together as multiplets. [We are disregarding here the caseof (j, j) coupling.] The discovery and analysis of such multipletsis the first main task in the analysis of a spectrum. In this stepthe following points are of importance:

(a) In a multiplet. constant differences must occur betweenpairs of lines. This follows from the explanation given previouslyin connection with compound triplets (p. 78). For example, inthe quartet transition for C + shown in Fig. 34, the followingseparations must be exactly equal to one another: b - a = h - d,d - c = g - e; and, conversely, d - a = h - b, e - c = g - d.These separations correspond to term differences of the upper andlower states, respectively. When, therefore, the lines of a multi-plet are put, in a square array (see Table 15) such that lines ineach vertical row have lower states with equal J, and those in


each horizontal row have upper states with equal J, the differ-ences between the lines in two horizontal or two vertical rows mustbe exactly constant. The table shows that this is actually thecase, within the limits of experimental error. In the scheme onlythe diagonal from upper left to lower right and the two parallelsto it are occupied by lines, due to the selection rule ∆J = 0, f 1.

(b) According to our earlier discussion (p. 161), in a multipletthose transitions for which J and L alter in the same sense are themost intense; of these, the most intense is that with greatest J.Table 15 shows that this rule holds also for the C+ quartet.

TABLE 15

4P - ‘D TRANSITION FOR C+ [FOWLER AND SELWYN (59)]

(Wave-number differences are given in italic type. Numbers in paren-theses are es t imatee intensities.p. 80.)

Superior letters a, b, c, etc., refer to Fig. 34,

‘J%a

‘I~,/,

'Dwa

‘ha

‘PpII2 ‘PW, ‘l’wz14,729.79(2P 23.73 14,70&06(0)e

14.82 14.6114,744.61(2)~ 23.94 14,720.67(3)d 44.95 14,675.72(0p

25.10 25.0314,745.77(4)h 45.02 14,700.75(3)*

36.3014,737X&(6)/

(c) In the Zeeman effect, each multiplet level splits into 2J + 1components. The number of components for each line is givenby the splitting of the upper and lower terms and by the Selectionrules (II, 12) and (II, 13). Conversely, it is always possible touse the Zeeman splittmg to obtain the J values for the upper andlower states of the respective lines. Investigation of the Zeemaneffect is, however, not always practicable.

(d) When an investigation of the Zeeman effect is not practi-cable, an interval rule (discussed in the following) is employedin the determination of J.

Lande interval rule. Under the assumption of Russell-Saunders coupling, the ratios of the intervals in a multiplet canbe easily calculated in the following way: The magnetic field pro-duced by L is evidently proportional to dm, and the com-ponent of S in the direction of this field is dm cos (L, S).Therefore from (II, 7), the interaction energy is

HPH = Am I’- cos .(L, s)


where n is a constant. From Fig. 47 (p. 109) it follows [similarto equation (II, 19)] that

cos (L, S) =J(J + 1) - L(L + 1) - S(S + 1)

2mms+l)

Consequently the interaction energy is:

A J(J + 1) - L(L + 1) - S(S + 1)2

As L and S are constant, for a given multiplet term, t h e intervalsbetween successive multiplct components arc in the ratio of thedifferences o f the corresponding J(J + 1) values. But thedifference between two successive J(J + 1) values is 2J + 2.Conscquontly, for a multiplet term the interval between two successive components (J and J + 1) is proportional to J + 1. Thisinterval rule was first formulated by Lande. Deviations fromthis rule occur with increasing deviation from Russell-Saunderscoupling. According to the interval rule for example, the sepa-rations of the components of a ‘D term with J = 4, $, 2, .$ arc int h e ratio 3 : 5 : 7. For the ‘I) term of C+ (Table 15), these sepa-rations arc 14.72, 25.07, 36.30; and are in the ratio 2.94 : 5: 7.24.The interval rule is thus verified to a fair approximation in thiscase, and similarly in other cases.ls The multiplet intervals inall the illustrative diagrams have been drawn in accordance withthe Lande interval rule.

Example of a multiplet analysis. In order to locate multiplctsin a complicated atomic spectrum! it is necessary first, by syste-matic trial, to discover pairs of lines with exactly equal wave-number differences. As can be seen in Table 15, these pairsusually occur in double sets. When a number of such doublesets have been found, they must be arranged in a scheme similarto the one used in that table. For a given multiplet; only suchdouble sets come under consideration as have one line in common.In arranging the various pairs in the scheme, one must considerthat in all the horizontal rows the wave numbers of the linesdecrease or increase continuously from left to right; the sameapplies, correspondingly, for the vertical columns. Practically,it is usually easy to arrange the lines in such a scheme when thelines in the spectrum form separated groups (multiplets); how-ever, this is always theoretically possible even when differentmultiplcts overlap one another.

Table 16 gives such a scheme for a multiplet of Fe, whichis shown in Fig. 6 (p. 7). As can be seen, the wave-numberdifferences (given in italic type in the table) of pairs of lines, such

*a An exception is provided, for example, by He (see footnote 2, Chapter II).


as i - g and f -h - g and m

c, agree exactly.- d occur only once.

To be sure, the separationsNevertheless, that the lines

h and m belong to the multiplet follows from the fact that thesame differences appear for other Fe multiplets having the sameupper or lower states.

TABLE 16

Fe I MULTIPLET [LAPORTE (82)](Wave-number differences are given in italic type. Numbers in paren-

theses above the wave numbers of the lines are estimated intensities. Su-perior letters a, b, c, etc., refer to Fig. 6, p. 7.)

J ‘i i+l i+2 i+3 i-i -4

(4OPk 25.966.89

104.61(4W em4

k + 1 25,362.38 1 6 8 . 9 9 26,031.30916.66 H6.6S(wc @co’ @oP

k +2 25,646&I 188.91 25.815.77 267.75 26,[email protected] s94.46

w5 w-o* (125)’k+3 25.521.32 267.73 25,779.05 561.50 26J30.35

411.21 4u.19(5Y (15F cm)”

k+4 25.367.34 SbZ.Jd 25.719.16 cds.60 26.167.66

The types of terms combining with one another must now bedetermined. We know that J increases or decreases by 1 forsuccessive horizontal and vertical rows. The direction of in-creasing J is deteimined by observing the direction of increasingseparation of the lines in the horizontal and vertical rows, since,according to the interval rule, the multiplet intervals increasewith increasing J.given in Table 16.

The relative values of J are, therefore, those

far undetermined.They include a constant i or k, which is thusThe absolute values of J are obtained when

the ratio of successive intervals for the upper and lower states iscalculated. In the present case, the numbers, for the upper state,104.51, 215.53, 294.45, 411.20, are approximately in the ratio1 : 2 : 3 : 4; whereas those for the lower state, 168.92, 257.73,351.31, 448.50, are in the ratio 2 : 3 : 4 : 5. From this it followsthat i = 1 and k = 0. Consequently, the J values of the upperstate are: 0, 1, 2, 3, 4; those of the lower state are: 1, 2, 3, 4, 5.When L > S, the number of term components is 2S + 1. Inthe present case thii number is 5, and therefore S = 2. Thesupposition that, here, L > S follows from the fact that the twostates have an equal number of components, although they


have different J valucs.1’ With S = 2 and with the above Jvalues, we find that L = 2 in the upper state and that L = 3 inthe lower state. The transition is thus a SF - 50 transition.The intensities provide a check on the correctness of the J andL values (see above).

The foregoing considerations do not alter when the upper andlower states are interchanged; that is, when Table 16 is reflectedat the diagonal through the upper left comer. A decision as towhich is the upper or the lower state can be obtained only bycomparison with other multiplets of the same spectrum or byabsorption experiments. The arrangement actually used in thetable was verified in both ways. Since the Y values in a verticalrow in Table 16 decrease with increasing J in the upper state, itfollows that the upper state is an inverted term. The same holds,in a similar manner, for the lower state. Thus, for both terms,the components with smallest J lie highest.

After a large number of multiplets of the same spectrum havebeen analyzed in this way, we can arrange similar terms in Ryd-berg series: R/(m + a)” (see p. 55). Terms for which this ar-rangement is possible differ from one another only in the principalquantum number of one electron. The energy level diagram ofthe atom is thus obtained, and, when sufficient terms of a Rydbergseries are known, the ionization potential can be obtained veryaccurately by extrapolation to n = a0 . (Cf. Chapter VI, section.1.) When the carrier (emitter) of the spectrum is known, aqualitative energy level diagram may be constructed on the basisof the building-up principle, and then the observed combinationsmay be arranged in this diagram.

14 The number of components for L < S is 2L + 1. Two terms with equalS can, therefore, have the same number of components less than 2S + 1 onlywhen they have the same L; that is, the same J values.

CHAPTER V

Hyperfine Structure of Spectral LinesWhen individual multiplet components are examined with

spectral apparatus of the highest possible resolution (inter-ference spectroscopes, large concave gratings in the higherorders), it is found that in many atomic spectra each ofthese components is still further split into a number ofcomponents lying extremely close together. This splittingis called hyperfine structure. The total splitting is only ofthe order of ‘2 cm-1 (that is, in the visible region of thespectrum approximately 0.4 A) and is in many cases con-siderably smaller. In Fig. 67(u), (b), and (c) we give as illus-trations the “lines” : 4122 A of Bi I (photogram), 5270 Aof Bi II, and 4382 A of Pr II.

As we. have seen in the preceding chapters, the assump-tion of orbital and spin angular momenta of the individualelectrons of an atom explains completely the multipletstructure thus far mentioned. It is, however, difficult toimagine an additional degree of freedom of the extra-nuclear electrons of an atom which would account for thestill further splitting (hyperfine structure) just mentioned.We are therefore led to assume (following Pauli) that thishyperfine structure is caused by properties of the atomicnucleus. This assumption is confirmed by a more thoroughinvestigation of the phenomenon.

The influence of the nucleus may be due either to its mass(isotope effect), or to a new property, an intrinsic angularmomentum or nuclear spin, which can be considered similarto the electron spin. Both influences have been found.

1. Isotope EffectAs is well known, most chemical elements consist of a

number of isotopic atoms, each of which has an approxi-182

V, 1] Isotope Effect 183

mately whole-number atomic weight. Different isotopesof an element have the same number and arrangement ofextra-nuclear electrons, and consequently have the samecoarse structure for their spectra. They are, however, dis-tinguished from one another by their mass.

Isotope effect for the H atom. We have seen in Chapter Ithat, because of the simultaneous motion of nucleus andelectron about the common center of gravity, the Rydbergconstant depends on the nuclear mass. The H spectrumthus depends upon the nuclear mass. Urey and his co-workers first found (1932) that each of the Balmer linesH,, Hg, H,, and Hs has a very weak companion on the shortwave-length side at distances of 1.79, 1.33, 1.19, and 1.12 A,respectively. The wave lengths of the additional linesagree completely (within the limits of experimental error)with the values obtained from the Balmer formula whenthe Rydberg constant for a mass 2 is used instead of for amass 1 (p. 21). The calculated separations are 1.787,1.323, 1.182, and 1.117 A. The existence of the hydrogenisotope of mass 2 (heavy hydrogen) was first shown in thisway. It should perhaps be added that the heavier isotope ispresent to the extent of only 1 in 5000 in ordinary hydrogen.

(a) (b) (c)Fig. 67.

the " line "Hyperfine Structure of Three Spectral Lines. (a) Photogram of

[Zeeman,4122 A of Bi I, with 4 components. Total splitting 0.44 A

Back, and Goudsmit (83)].5270 A of Bi II, with 6 components.

(b) Spectrogram of the " line "

Goudsmit (84)].Total splitting 1.37 A Fisher and

components.(c) Spectrogram of the " line " 4382 A of Pr II, with 6

Total splitting 0.30 A [White (85)].

184 Hyperfine Structure of Spectral Lines [V, 1

Isotope effect for more complicated atoms. As soon as severalelectrons are present, the isotope effect can no longer be calculatedin such a simple manner as for the H atom. We shall discuss hereonly the qualitative results. The fine structure of the Li res-onance line, which is not a simple doublet, w a s explained anumber of years ago as due to the isotopic shift of Lib and Li7[Schiiler and Wurm (86) 3. This interpretation has been verifiedby the intensity ratio of the corresponding lines’in the hyperfinestructure pattern, which agrees with the abundance ratio of theisotopes.

Another case that was among the first to be explained is the NCspectrum, part of the lines of which consist of two components.Apart from the somewhat rare isotope Ne*l, Ne has two principalisotopes, Ne20 and Ne22, whose abundance ratio (9 : 1) agreeswith the intensity ratio of the two line components and to whichthe two line components are thus to be ascribed. This interpre-tation was further confirmed by the separation of the two isotopesby diffusion [Hertz (87)]. The separated isotopes show only theone or the other component of the doublet.

It might be expected that with increasing atomic number theisotope effect would become smaller, since the motion of the

nucleus becomes more and more un-ZnM

Zn=

J&L

important. However, it has actuallybeen found [Schiiler and Keyston(88) ; and others] that even for ele-

Zn=ments of rather high atomic numbera noticeable isotope effect is present,

1P ml-’which is of the same order of magni-

-189 -95 0tude as the influence of nuclear spin.

V(Cf. section 2 of this chapter.) As

Fig. 68. Isotope Effect an example, Fig. 68 shows schematic-for the 6215 A "Line " of ally the isotope effect of the 6215 AZn II (Schematic). Fre- “line " of Zn. The intensity of thequency. differences in units of10-a cm-1 referred to the

components is indicated by the heightof the vertical lines in the diagram.

~‘$~in’~b~&(,(h~oa$ It corresponds to the abundance ofsplitting < 0.2 cm-” Schuler the three 1 principal isotopes: Zn”,and Westmeyer (89) 5. Zn6s, Zn”. Worth noticing is the

fact that the lines of the three iso-topes lie equidistant, in the order of their masses.

In general, it is not always easy to separate the two effects(isotopy and nuclear spin). For this purpose the intensity of thecomponents is important. An unambiguous decision is alwayspossible when the Zeeman effect can be studied. For a pureisotope effect, each of the individual components will show theZeeman effect for the extranuclear electrons quite independently

1 The much rarer isotope’Zn”7 has been observed for another line, λ7479

V, 2] Nuclear Spin 185

of one another, whereas hypermultiplets, resulting from nuclearspin, should show an essentially different Zeeman effect (seebelow). Apart from this, it is naturally possible to make an un-ambiguous differentiation when the spectra of separated or partlyseparated isotopes can be investigated. In this way the isotopeeffect in the hyperfine structure of Pb has been carefully investi-gated by using leads from different radioactive origins (withdifferent atomic weights and therefore different proportions ofthe individual isotopes). [See Kopfermann (90).]

A quantitative explanation of the isotope effect is not simple,since, with the exception of the H atom, it is not given merely bythe altered Rydberg constant. A detailed wave mechanicalcalculation shows that, for the lighter atoms (Li, Ne, and so forth),an explanation can be obtained on the basis of different massesalone and is at least of the right order of magnitude [Hughes andEckart (91); Bartlett and Gibbons (92)]. However, for theheavier elements, the effect is traced back to the change of nuclearradius with mass [Pauling and Goudsmit (9); Bartlett (93)].

In this connection it is interesting to note that Schiiler andSchmidt (135) found in the case of samarium that the three evenisotopes Sm”O, Srnl”, Sm*a4 do not give equidistant lines as do theisotopes of Zn (Fig. 68) and practically all other elements. Theseparation of Sm160-Sm16* is double that of Smm-Smm Sincethe usual isotope shift for heavy nuclei is due to a regular increasein nuclear radius (cf. above), the large change between Sm150 andSml* points to a larger than usual increase in radius, which mayindicate a fundamental change in the building-up of the nucleusat this atomic weight.

2. Nuclear SpinIn many cases the isotope effect is not sufficient to explain

the hyperfine structure. The number of hyperfine struc-ture components is often considerably greater than thenumber of isotopes. In particular, elements which haveonly one isotope in appreciable amount also show hyperfinestructure splitting. This is, for example, the case with Biand Pr (cf. Fig. 67). Likewise, the number of componentsof different lines is frequently quite different for one andthe same element. These hyperfine structures can bequantitatively explained, however, when it is assumed (asfor the electron.) that the atomic nucleus possesses an in-trinsic angular momentum with which is associated a magneticmoment. This angular momentum can have different


magnitudes for different nuclei and also, of course, fordifferent isotopes of the same clcmcnt.

Magnitude of the nuclear spin and its associated mag-netic moment. If i t is assumed that wave mechanicsholds for nuclei, thc nuclear spin can he only an integralor half-integral multiple of h/2r. We write for it I(h./2r),where Z is the quantum number of the nuclear angularmomentum,* which can be integral or half integral. Forthe simplest nucleus, the proton, investigations of the II2molecule (spectrum, specific heat) have shown fhat itsspin I equals 4. The proton has therefore the same angularmomentum as the electron. Naturally, different v a l u e smight be expected for heavier nuclei since they contain,among other component parts, several protons.

A magnetic moment- is associated with the nuclear spin(as w i t h electron spin), since the nucleus is also electricallycharged and t h e rotation of electrically charged particlesgives rise to a magnetic moment. Classically, the magneticmoment resulting f rom the rotation of charges is (e/2nlc)p(see Chapter II). For an angular momentum p = 1 h/%rand nr = the mass of the electron, one Bohr magnetonresults (BM) If WC substitute the mass of the proton for~1 and if 1) = 1 h/2*, we obtain a magnetic moment. of1/1840 BM, which is called one nuclear magneton (NM).Therefore, classically, the magnetic moment of the protonshould be 4 NM, or 1840 times smaller than that of theelectron, which should similarly be 3 BM. Actually, thisrelationship holds for neither the proton nor the electron.Analogous t o the procedure with the extranuclear electrons,the discrepancy is formally explained by introducing anuclear g-factor and putting the magnetic moment of thenucleus equal to:

hg2Ttfl’27; = g-I NM

*The more accurate formula for the magnitude of the nuclear angularmomentum is dl(1 + 1) h/2r, just as for J (see p. 88). For the sake of sim-plicity, we shall use the expression I(/$&) in what follows.


where mp is the proton mass. Note that g is counted posi-tive when the magnetic moment falls in the direction of thenuclear spin (as is generally to be expected for the rotationof positive charges), and is counted negative when it fallsin the opposite direction.

Since the g values for the nuclei are numbers of the orderof 1, the magnetic moment of the nucleus is always about2000 times smaller than that of the electron.

Vector diagram allowing for nuclear spin. PreviouslyZ and S were combined to give the total angular momentumJ of the extranuclear electrons. Now J and Z must simi-larly be combined to give a resultant, in order to obtain thetotal angular momentum F of the whole atom, including nuclearspin. As before, the corresponding quantum number Fcan take values

F=J+I,J+I-l,J+i-2,~~~,~J-I~ (V, 1)This gives, in all, 2J + 1 or 2I + 1 different values, ac-cording as J < I, or J > I.Fig. 69(a) shows the addi-tion for the case of J = 2,

I=S,i”

I = 4. I t cor respondsF=% I=?$ A’

completely to the addition (a) I-92

‘II=?‘*

of L = 2 and S = 4 inJ=2 J=2 F=s J=2 J= 2,

4Fig. 37 (p. 89). IVDF=Y, 4-, F=!i

Because of the magneticmoment of the nucleus, acoupling between J and Z (b)results (similar to thatnoted previously between

qq

L and S) and produces aFig. 69. Vector Diagram and Cor-

responding Energy Level Diagram Al-precession of the vector lowing for Nuclear Spin. (a) Vector

addition of J and I to give the total an-diagram (Fig. 70) about gular momentum F for the case .I = 2,

the total angular momen-I = 31. (b) Energy level diagram forJ = 2, I = 35. TO the left. without al-

tum F as axis. Due to lowing for hyperfine structure splitting;to the right, allowing for it. The split-

this, a small energy differ- ting of the states with different F is

ence between states withdrawn iu accordance with the intervalrule (see Chapter IV, section 4) .

V, 2] Nuclear Spin 189188 Hyperfine Structure of Spectral Lines [V, 2

different F exists. However, since the magnetic momentof the nucleus is approximately 2000 times smaller thanthat of the electron, the precession is 2000 times slowerthan that of L and S about J (also indicated in Fig. 70),

Fig. 70. Precession ofthe Angular MomentumVectors about the TotalAngular Momentum F forthe Component F = 5 ofa % Term with I = 2.The solid-line ellipseshows the precession ofZ and J about F. Thedotted-line ellipse showsthe much faster preces-sion of L and S about J,taking place at the samet i m e .

From equation (V, 1), it followsthat in general the number of hyper-fine structure components of which anatomic term consists is different fordifferent terms o f the same atom.Terms with J = 0 are always single.If I = 4, all other terms show a split-ting into two components. If I is’greater than 3, terms with J < I have2J + 1 components, whereas thosewith J > I have 2I + 1 components(cf. above).

The greater the nuclear magnetic moment, the greaterwill be the splitting. The latter is also dependent on thetype of atomic state under consideration. For example,if the emission electron is in an s orbit, the splitting is muchgreater than for a p orbit with the same principal quantumnumber, since the electron in an s orbit approaches closer tothe nucleus. This dependency can be calculated in detailtheoretically, but will not be taken up further here [consultCondon and Shortley (13)].

and correspondingly the energy differ-ences are very much smaller. Theseare the small differences observed inthe hyperfine structure of spectrallines. Fig. 69(b) shows the energylevel diagram of the term with J = 2and I = 3.

Selection rule for F; appearance of a hypermultiplet.The same selection rule holds for the total angular momen-tum F [see Pauling and Goudsmit (9)] as holds for thetotal angular momentum of the extranuclear electrons:

AF= fl,O and F=O+F=O w, 2)

From this it follows that a hypermultiplet, although itssplitting is much smaller, will have a similar appearance toan ordinary multiplet (cf. Figs. 29 and 31, p. 74 and p. 76),particularly since the same interval rule holds for both.

In Fig. 71 (a), (b), and (c), energy level diagrams for thoselines of Bi I, Bi II, and Pr II are shown whose spectro-grams have already been given on page 183. The spm ofthe Bi nucleus is I = i. In the upper and lower states of

4 J=%

7mF

5 I-96>

: :. . ::

(a)Bii +?jA.: .*

(b)

F

I II III III I II 11T Y

Fig. 71. Energy Level Diagram Showing the Hyperfine Structure for theThree Spectral Lines Reproduced in Fig. 67. (a) Bi I X4122 line. (b) Bi IIx5270 line (upper and lower states must be interchanged). (c) Pr II X4382line.


the Bi I line X4122, J = 3; for the Bi II line X5270, J = 1in both states. This gives in the first case 4 components,and in the second case 7 components. The J values are notknown exactly for the Pr II line X4382, but must, at anyrate, be very large. With Z = i and the assumption thatthe splitting is exceedingly small in the lower state, 6 linecomponents result; of these, 4 consist of 3 unresolved com-ponents each, and one consists of 2 unresolved compo-nents [see Fig. 71(c)].

Determination of Z and g from hyperfine structure. As-suming the above theoretical relations and selection rules,we can, conversely, derive from the observed hyperfinestructure the magnitude of the nuclear spin of an atom.

In principle, the procedure is the same as that given above forthe analysis of an ordinary multiplet. However, here we havean advantage: usually the J values of the terms involved s areknown, as is also the fact that all the terms must have the sameZ value. Once again we have to arrange the hypermultiplet in asquare array (cf. Table 16, Chapter IV). If the same number ofcomponents is obtained for two terms with different J values,this number gives directly 2 1. Such is the case in the aboveexample of Pr II, where a. great many different lines have 6components, as shown in the diagram [Fig. 71(c)].follows that Z

It therefore= 3 [cf. White (85)]. If, however, the number of

components varies for different terms, the number of componentsmust be-equal to 2J + 1, as in the example of Bi [Fig. 71(u)and (b)]. In such cases, when there are more than 2 components,we first obtain F from the interval rule, and from this Z (seeChapter IV, section 4). For the Bi II line in the figure, the inter-vals in the lower state are 1.756 and 2.152, and in the upper state0.459 and 0.562, as derived from the observed pattern. They areboth approximately in the ratio 9 : 11; that is, the F values mustbe 4, 8, +-, as indicated in the figure. From the fact that thereare 3 components each, it follows that J = 1 and thereforeI =sv. A similar procedure could always be rather easily car-ried out if it were not for the overlapping of the lines-a situationthat often is complicated by the smallness of the splitting, thelimited resolving power of the spectral apparatus, and the finitewidth of the lines.

* Conversely, with a known nuclear spin, we can determine the J values ofunanalyzed multiplets by investigating their hyperfine structure.


When the value of the nuclear spin I has been obtained,. theg-factor and the magnetic moment of the nucleus can be denvedfrom the magnitude of the splitting by using the theoreticalformulae.

Zeeman effect of hyperfine structure. In a magnetic field aspace quantization of F takes place precisely as given above for J.The quantum number Mp of the component of the angularmomentum in the field direction can take only the followingvalues :

Mp = F, F - 1, F - 2, . . ., - F (V, 3)

The 2F + 1 values of MF correspond to states of different energiesin a magnetic field. Because of the precession of J and I about F(see Fig. 70), the direction of the magnetic moment of the extra-nuclear electrons lies, on the average, in the direction of F. Theenergy differences of the 2F + 1 states with different Mp arethus of the same order of magnitude as for the ordinary Zeemaneffect [cf. formula (II, 14)]. As before, the states are equidistant.With increasing field strength, the precession of F about the fielddirection becomes faster and the energy difference between thevarious term components becomes greater. Fig. 72 shows, to theleft, the splitting of the two hyperfine structure components of aterm with J = 3 and I = ;9 in a weak field. (The order of thecomponents with F = 1 is the inverse of the order with F = 2,

J=% I=%

F=l

Fig. 72. Zeeman Splitting of the Hyperfine Structure Components F = 1and F = 2 of a Term with J M,Z= $36. To the left, the splitting is i n aweak field; to the right, in a strong field.


MJ value has 21 + 1 equidistant components, the separationsbeing different in the upper and lower states. Therefore, con-sidering the selection rule AM1 = 0, each anomalous Zeemancomponent is split into 21 + 1 lines. This splitting does notdepend upon the field strength so long as the latter is sufficientlygreat to produce an uncoupling of J and I. Thus, simply bycounting up the number oj line components, the nuclear spin I canbe determined quite unambiguously from photographs in a suffi-ciently strong magnetic field. This elegant method for the deter-mination of nuclear ‘spin from hyperfine structure was first em-ployed by Back and Goudsmit for Bi. In a strong magneticfield each of the Zeeman components of Bi consists of 10 com-ponents due to nuclear spin, and therefore I must be equal to Q(a result already obtained above, though with less certainty, fromthe interval rule).

Statistical weight. It follows from the foregoing discussionthat a hyperfine structure term with a given F has a statisticalweight 2F + 1. In hypermultiplets, this statistical weight i simportant for the determination of intensity ratios, which inturn serve as a check on the analysis of hyperfine structure.

The total statistical weight of a term with a given value of J(that is, the total number of single components in a magneticfield, if nuclear spin is included) is:

w + 1) x (21 + 1)since we have seen that in the Paschen-Back effect each Zeemanterm (single without nuclear spin) splits into 21 + 1 components.The statistical weight is thus increased, by a factor 21 + 1, overthat previously given (p. 119) where nuclear spin was not allowedfor. As this factor is the same for all states of an atom, ourearlier discussion of intensities in ordinary multiplets still applies.

Determination of nuclear spin by the Stem-Gerlach experi-ment. Rabi and his co-workers, employing the foregoing con-siderations on the Zeeman effect of hyperfine structure, have de-veloped a very beautiful method for the determination of nuclearspin with the aid of atomic rays., For exam.ple with the alkalis,disregarding nuclear spin and using any arbitrary field strength,t,,~;;ll be a sphttmg of an atomic ray mto two rays (MJ = + +

= - +), because of the S ground state. If a nuclearspin is present, for a weak field, F (not J) is space quantized. Inthis case the magnetic moment of the atom has, on the average,the direction of F, and therefore the atomic ray i s spht mto2F + 1 (not 2J + 1) components, where F is the largest of thepossible F values (for J = + and I = #, there are 5 componentsinstead of 2). .

On the other hand, in a strong field the components of themagnetic moment (which are the deciding factors in the sphtting


since in the first case J is antiparallel to F.) On the basis of theselection rule AM, = f 1, 0 (MF = 0 t, Mp = 0 for AF = 0)which is analogous to (II, 12) and (II, 13), each individual com-ponent of a hypermultiplet gives a Zeeman splitting correspondingcompletely to the spectrograms previously given for the anom-alous Zeeman effect (Fig. 39). In actual investigations this effectis scarcely ever observed, since the hyperfine structure splitting.itself is generally close to the limit of possible resolution (seehowever the more recent work of Rasmussen (148) and Jacksonand Kuhn (149).

When the magnetic field is so great that the velocity of pre-cession of F about the field direction becomes greater than thatof J and I about F, a Paschen-Back effect takes place, as for ordi-nary multiplet structure. In the case of hyperfine structure onaccount of the weakness of the coupling between J and I thePaschen-Back effect occurs at very much lower field strengthsthan for ordinary multiplet structure. J and I are then spacequantized in the field direction independently of one another andwith components MJ (corresponding to M, above) and MI.The space quantization of J gives the ordinary Zeeman effectstudied in Chapter II, with line separations which, with sufficientfield strength, are considerably greater than those of the field-freehyperfine structure components. Each term with a given MJ ishowever, once more split into a number of components corre-sponding to the different values of MI. Since MI can takev&eSI,I-f,I-2, ..., - I, there are 2I + 1 components.This number of components is the same for all terms of an atomsince I is constant for a given nucleus.field is shown to the right of Fig. 72

The splitting in a strongfor the simple case J = 1

I = Q. -The splitting of the levels with different MI is smallcompared to the separation of the levels .MJ = + 3Mr= -3.

a n dIt is not due to the interaction of the nuclear spin

I with an external magnetic field H, since this is 2000 timessmaller than that of J with H; but is due to the interaction be-tween I and J, which is also present in a strong magnetic field andcontributes a term AMJMI to the energy, similar to the ordinaryPaschen-Back effect (p. 113). The 2I+ 1 components of aterm with a given MJ are thus equidistant. A is the constantdetermining the magnitude of the field-free hyperfine structuresplitting. The difference from the ordinary Paschen-Back effectIS that the term corresponding to the term 2hoMs of equation(II, 21) can be disregarded here for all practical purposes becauseof the factor 1/2000. This also accounts for the differencebetween Figs. 72 and 49.

For a transition which, without field, gives rise to one hyper-multiplet, the selection rules in a strong field are: AM= = 0, f 1Fil with (II, 12)] and AMI = 0 (corresponding to

z = 0). The first of these rules gives the ordinary anomalousZeeman effect if at first we disregard nuclear spin. Becuase ofnuclear spin, however, each of the magnetic levels with a certain


of the ray) take only two values, given by MJ = + + andMJ = - a. The magnetic moment connected with I does nothave any appreciable influence on the splitting of the atomic ray,because of the uncoupling of J and I.two rays takes place.

Thus a splitting into onlyUsing a strong inhomogeneous field, Rabi

and his co-workers first produce such a splitting into two rays,One of the rays is shielded off, and the remaining ray (whichmay have, for example, MJ = + +) contains atoms with MI = I,I - 1, ***, - I. All such atoms, however, have practically thesame magnetic moment and, therefore, practically the samedeflection. This ray is then sent through a second field, which isweak and extremely inhomogeneous. When the second field isso weak that no Paschen-Back effect can take place, I is no longeruncoupled from J, and there occurs a comparatively large splittinginto as many rays as there are Mp values in the ray. There arejust 2I + 1 values of Mp, since the states which had MJ = - 4in a strong field are no longer present. In this way the magnitudeof the nuclear spin is found simply by counting up the number ofcomponent atomic rays, as in the Zeeman effect for hyper-fine structure. For Na, Rabi and Cohen (94) have foundI = Q (see Fig. 72).

If the variation in the splitting pattern of the atomic ray in thetransition from weak to strong fields is investigated in greaterdetail, the nuclear magnetic moments may also be determined,since the uncoupling of J and I is reached sooner for smallermagnetic moment (cf. above). A more direct method consists inthe application of the ordinary Stern-Gerlach experiment eitherto atoms whose outer electrons have zero magnetic moment, or todiatomic molecules with zero magnetic moment which contain theatoms in question. Much more accurate results have been ob-tained more recently by Rabi and his coworkers (159) (151) (152)by means of the molecular beam magnetic resonance method.

The results of these procedures for individual nuclei will begiven here only for the proton and the deuteron, the nucleus ofthe heavy hydrogen atom. The proton, whose spin I = 9, givesa value of 2.7896 NM [see (151) and (152)], which is remark-ably high; whereas the magnetic moment of the deuteron, whose

P

spin 1. = 1, is only 0.8565 NM [see (151) and (152)].mentioned methods are listed in Table 17. For the sake

. of completeness, values obtained by band spectroscopic

. methods have also been included in the table..Importance of nuclear spin in the theory of nuclear structure.. It is clear that significant conclusions as to the structure of the

. nucleus may be obtained from the determination of nuclear spin. and the magnetic moment belonging to it, just as a fundamental* knowledge of the arrangement of the extranuclear electrons was

I- &u\ts: The nbtC\ear SpihS ObfaiMdr by +he

above bdcCly . . , n


2 0212325272 9

3 334

TABLE 17

OBSERVED VALUES FOR NUCLEAR SPIN

I Z- -

!,i 3 61 370

j!$38

?5 C?)0 3 91 410 4 2!-i003.i

O(1) 475i?i 4805%,5.2I?i3.’

0;;) 49ti

34 5 03;

ti 513i3$ 5200Bi 530 543532'$400 554i 56%0$4o- 57

-

Z-596 263

6.5676 970

71

7 2

7 374

75

77

7 8

7 98 0

81

82

8391

196 Hyperfine Structure of Spectral Lines

obtained from the evaluation of their angular momenta. How-ever, the relationships for nuclei are more difficult to find since,for each nucleus, only one nuclear spin can be determined-thatbelonging to the lowest state of the nucleus. Excited nuclearstates occur only for natural or artificial disintegration processesand cannot be investigated optically-or, at least, only withgreat difficulty. It is due to this that we have not yet mademuch progress with the systematization of the spin values occur-ring for different nuclei (representation by the spin of the indi-vidual nuclear components). For speculative work in this field,Schiiler (100), Lande (101), Bartlett (102), and Bethe andBacher (138) should be consulted.

Apart from conclusions regarding the spin and the magneticmoment of the nucleus and also the nuclear radius, the investiga-tion of hyperfine structure may provide information about apossible asymmetry of the nucleus, as was recently pointed out bySchiiler and Schmidt (139). In some cases there occur in hyper-multiplets deviations from the interval rule which are ascribed toa quadrupole moment of the nucleus-that is, to a deviation fromspherical symmetry.

CHAPTER VI

Some Experimental Results and Applications1. Energy Level Diagrams and Ionization Potentials

In earlier chapters, examples have been given of a numberof energy level diagrams obtained from analyses of corre-sponding line spectra. They were the energy level diagramsof the atoms H (Fig. 13); He (Fig. 27); Li (Fig. 24) ; K(Fig. 28); Be (Fig. 61) ; Ca (Fig. 32) ; C (Fig. 55) ; N (Fig.56); 0 (Fig. 59). In order to show at least one examplefrom each of the columns of the periodic table, the energylevel diagrams of Al I and Cl I are reproduced in Figs.73 and 74 (pp. 198 and 199).

In addition, the energy level diagram of Hg, which isimportant for many practical applications, is reproducedin Fig. 75 (p. 202). It is qualitatively similar to Ca (Fig.32), except that the triplet splitting is very much larger (cf.also the Hg spectrogram in Fig. 5, p. 6).

Finally, Fig. 76 shows the energy level diagram of Ni Ias an illustration of the complicated term spectrum of oneof the elements for which a building-up of inner shells takesplace (see p. 203).

If for an atom several terms T of the same series havebeen found, they can be represented by a Rydberg formula:

T = A _ tz - PW(m + a)2 WI, 1)

where T is measured against the lowest term (m = runningnumber, Z - p =and 60 f.).

number of charges of core; see pp. 55In order to calculate the two unknown con-

stants A and a, at least two members of the term series mustbe known, although more known terms are preferable. Form+ao,T= A; that is, the constant A empirically foundis the ionization potential of the atom or ion in question

197

198 Experimental Results and Applications [VI, 1

measured in cm-*. If an absorption series is observed forthe atom, A is the wave number of the series limit (cf. p.59). The results obtained in this way for the various ele-ments are given in Table 18 (pp. 200-201), which containsnot only the ionization potential of the normal atoms(column I) but also that of the single- and multiple-chargedions (columns II to V). Higher ionization potentials thanthe fifth are not included, although they are known in a few

VOlb6.96,

6

4

I

:

tS% ‘P% =Ppk ‘Da =Ds ;F,,, 6~19% m n A ?z n a n --=- -.9-?- ‘I- ?- 7- 3- g- ,- i-

15- “-- t/5,000

16,006

26,000

Q’Pk--------

1

Fig. 73. Energy Level Diagram of Al I [Grotrian (8)]. n is the trueprincipal quantum number of the emission electron. Only the normal doubletterms are indicated, all of which go to the same limit. Series of anomalousterms (doublets and quartets) have been observed by Paschen (64).

VI, 1] Energy Level Diagrams 199

cases. The evaluation of ionization potentials is partic-ularly important for practical a p

Energy level diagrams of atoms and ions with one, two,and three valence electrons are given fairly completely inGrotrian (8). Complete tables of all terms of atoms andions observed up to 1932 hare been collected by Bacherand Goudsmit (22), whose data have been used for mostof the energy level diagrams reproduced in this book.

3-

6-

4-

2-

I

hk~krlli~d

Clll-1-

5 0 . 0 0 0 .

_20,000-

30,000 -

40.000-

so*ooo~

60,000 -

io,oUo-

90,000 -

90.000 -

100.000-- - - - 3esPe~----+--

Fig. 74. Energy Level Diagram of Cl I [Kiess and de Bruin (103)].Terms belonging to the same electron configuration are drawn under oneanother. Apart from the ground state, the different multiplet componentsare not drawn separately in the diagram.

1 For the sake of completeness, there are included in Table 18 some valuesof ionization potentials which have been obtained by other methods (electroncollision measurements, and s o on) for rant of spectroscopic data. Uncertainvalues a r e indicated by ~.

200 Experimental Results and Applications [VI, 1 VI, 1] Ionization Potentials 201

TABLE 18 (Continued)

IONIZATION POTENTIALS OF THE ELEMENTS (IN VOLTS)

All values are based on the new conversion factor 1 volt = 8067.5 N(see p. 10). Uncertain or estimated values are indicated by ~.

TABLE 18

IONIZATION POTENTIALS OF THE ELEMENTS (IN VOLTS)’

Ail values are based on the new conversion factor 1 volt = 8067.5 cm-1(see p. 10). Uncertain or estimated values are indicated by ~.

I II III T V V----

340.156392.0

97.87113.7114.22126.43

--

154.28

166.565.01 ,62.267.8

~78

--

. . . . . . . . . . . . .*91.8 ;99.84 i

-64 ;-73.0 ;- 7 6 . 0 ;-

:-i- :. . . . . . . . . . . .:

---

93.4362.6173.11--

E l e m e n t- -

1 H2 He

13.595 - -24.581 54.405 -

3 Li 5.390 75.622 122.427 -4 Be 9.321 18.207 153.85 217.6715 B 8.296 25.119 37.921 259.316 C 11.265 24.377 47.866 64.4787x 14.545 29.606 47.609 77.48 0 13.615 35.082 55.118 77.28QF 17.422 34.979 62.647 87.142

10 ?\‘e 21.559 40.958 63.427 96.897

11 Nu. 5.138 47.292 71.650 -12 Mg 7.645 15.032 80.119 109.53313 Al 5.985 18.824 28.442 119.96114 Si 8.149 16.339 33.489 45.13115 P 10.977 19.653 .30.157 51.35616 S 10.357 23.405 35.048 47.29417 Cl 12.959 23.799 39.905 54.45218 A 15.756 27.619 40.68 - 6 1

19 li 4.34020 Ca 6.112. ..-.......... . . . . . . . . . . . . . . . .21 SC -6.722 Ti 6.83523V 6.73824Cr 6.76125 Mn 7.42926 Fe 7.8627 Co 7.87628 ?ii 7.633. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 (:u 7.72330 Zn 9.39231 c&l 5.99732 Ge 8.12633 As 10.5348 9.750&5 BI 11.84436 Kr 13.996

31.81111.868. . . . . . . . . . . . . . .

-12.9-13.6

14.2-16.7

15.63616.24017.418.2. . . . . . . . . . . . . . . .20.233

17.96020.50915.9320.221.691

-19.2-26.5

45.7 -51.209 67.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24.753 73.913

-27.5 43.237-26.5 -48.5- --

30.6-

---

-. . . . . . . . . . . . . . . .

--39.7’

30.734.21627.29734.0783.5.88836.94

-. . . . . . . . . . . . . . . .

--

64.145.750.12342.960-

48

II III

- 4 7-

. . . . . . . . . . . . . . . . .20.524.1024.332---

. . . . . . . . . . . . . ...*36.1038.21728.03030.65424.82530.611-

32.115

- 3 5-

. . . . . . . . . . . ...*..19.1719.70--------

,................-

34.529.832.125.56-

-. . . . . . . . . . . . . . . . .

29.5

IV

40-

. . . . . . . . . . . . . . .-

33.972----

. . . . . . . . . . . . . . .

-58.03740.74044.14737.817-

-48

-51-

. . . . . . . . . . . . . . .-

36.715--------

. . . . . . . . . . . . . . .-

-7250.838.9745.2-

-. . . . . . . . ...*...

-

V

--

. . . . . . . . . . . . .*- 7 7 i- i

+jl) ;61.12;- :- :. . . . . . . . . . . . :---

81.1355.6960.27-

-76

Element I

37 Rb38 Sr. . . . . . . . . . . . .39 Y40 Zr41 Cb42 MO

45 Rh46 Pd. . . . . . . . . . . . .47 Ag48 Cd49 In50 Sn51 Sb52 Te53 I54 Xe

4.176 27.4995.693 11.026. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- 6 . 6 12.46.951 14.03-7.383

-7.78.334

---

19.921.96016.90418.86714.629

-18.621.54319.01021.204

55 ca56 Ba. . . . . . . . . . . . .57 La58 Ce62 Sm63 Eu64Gd70 Yb74 w76 0s77 Ir78 Pt. . . . . . . . . . . . .79 Au80 Hg81 Tl82 Pb83 Bi86 Rn

7.5748.9915.7857.3328.649.007

10.4412.127

-58-

. . . . . . . . . . . I-

-

-

-

-

-

-

-

-

-. . . . . . . . . . . .

-

432-

69.756.0-

3.8935.2097.* . . . . . . . . . . . . .5.614

-6.575 . 65.676.166.257.98

-8.79 . 29 . 0. . . . . . . . . . . . . . .9.223

10.4346.1067.415-

10.746

32.45810.001,...............11.43-

-11.411.24-

12.11---

- 1 9 . 3,...............20.118.75220.42315.0416.7-

88Ra 5.278 10.145. . . . . . . . . . . . . ..*............ . . . . . . . . . . . . . . . .90Th - -

-

‘.......“....*

- :

i

.’


2. Magnetic Moment and Magnetic Susceptibility

Magnetic moment of an atom. According to the dis-cussion of the Zeeman effect in Chapter II, the magneticmoment oJ of an atom in a given state has the averagemagnitude

VJ = oFi 9POwhere cc0 is the Bohr magneton (p. 103) and g is the Land@

I Singlets I Triplets I

Ob‘ILIL - - - - - - - - - + - - - - - - - - - - - --~~-~-------~---~

SO,OOO-

70*000-

60,000-

Fig. 75. Energy Level Diagram of Hg I [Grotrian (8)].of the more intense Hg lines are given (cf. Fig. 6, p. 6).

The wave lengthsThe symbols 6p,

etc., written near each level, indicate the true principal quantum number andthe 1 value of the emission electron.

VI, 2] Magnetic Moment and Susceptibility 203

V”lk3

7.606

7

6

6

4

3

1

(

;W’4x II 3dW‘hl ,,)I 3dJla nd ad’ (I* &I* “P ad@ Id

5%‘(?) 5% 6% ml-’_I. _

-------se=I-H(?; *.p,~r”P------ - - - - v-p----

--(i’F

=.-::$;-

. -*.-

-

-

60,000

60,000

-4 40000.

3osooo

- 4’G

20,000

ZZi - 4’I’

-4’D-3%

-4’0

10.000

-1- ’ 4 % =54?,,---I-___- -L------------------o

Pi. 76. Energy Level Diagram of Ni I [Russell (104)]. In general, theindividual multiplet components arc drawn separately, except for a few posi-tions where too many terms nearly coincide with one another. Differentterms of the same electron configuration (drawn above one another) do notusually go to the same limit. The lowest series limit (lowest ionitationpotential) is indicated by a dotted line.


g-factor. ?JJ has the opposite direction to J. For Russell-Saunders coupling g depends on J, L, and S of the atomicstate under consideration in the way given by (II, 15).Russell-Saunders coupling holds to a close approximation forthe ground states of practically all atoms.

On account of the double magnetism of the electron, theinstantaneous direction of the magnetic moment does notgenerally fall in the direction of J for states with S -+ 0,but carries out a more or less rapid precession about thisdirection (cf. Figs. 47 and 48). However, the above meanvalue for the magnetic moment in the J direction can usuallybe used.2 For J = 0, the magnetic moment becomes 0.

In a magnetic field the atom arid its magnetic moment cantake only 2J + 1 different directions. The precession of J,as well as that of &J, about the direction of the magneticfield is faster, the stronger the magnetic field. The com-ponent of the magnetic moment in the field direction isMgpo (M = J, J - 1, J - 2, .a ., - J)?

The most direct determination of the magnetic moment isbased upon t h e deflection of an atomic ray in an inhomo-geneous magnetic field (see Stern-Gerlach experiment,Chapter II, section 3). From the magnitude of the splittingof the rays (corresponding to the different orientations),the velocity of the rays, and the value of the inhomogeneityof the magnetic field, the magnetic moment of the atomconsidered can be evaluated.’

Paramagnetism. When a gas which consists of atomspossessing a magnetic moment different from zero is in amagnetic field, the states with smaller energy (with negativeM) are more strongly occupied than the states with largerenergy, as a result of the Boltsmann distribution law. This

*For accurate investigations, the component of the magnetic momentwhich is at right angles to J must sometimes be taken into account.Vleck (36)].

[Cf. Van

8 Often the maximum value of these components, Jgh is given as the mag-netic moment of the atom (and not the component IX in the direction of J).

* Primarily one measures the components of the magnetic moment in thefield direction and not MJ itself (see footnote 3, above).


means that the atoms align preferentially with their magneticmoment in the field direction, as would be expected. Thestronger the magnetic field, the greater will be the energydifference for the various orientations in the field and,therefore, the greater the difference in the number of atomsoccupying each state. For a given field, the difference in‘these numbers will be greater, the lower the temperature,since the arrangement of the atoms will be less hindered byunordered heat motion. The fact that in the presence of amagnetic field, on the average, more atoms will align withtheir magnetic moments parallel to the field direction thanantiparallel to it results in a magnetic moment per unitvolume, P, whose action is added to. that of an externalfield and which can be experimentally determined. Thegas is paramagnetic. P (the intensity of magnetization)is proportional, to a first approximation, to the field strengthH, and is inversely proportional to the absolute temperatureT. The proportionality factor depends mainly on themagnitude of the magnetic moment of the atom considered.The theoretical formula for a not too large H and a not toosmall T is : 5

p = J(J + l)s2pfNLH _ BJ~NL~3kT 3kT WI, 2)

where N L is the number of molecules per cc. The coefficientof H

K = J(J + Og’dNt _ cca2Nr,3kT 3kT WI, 3)

is called the paramagnetic susceptibility.6 It is inversely .

c An additional term independent of temperature occurs in the more accu-rate formula. This term is due in part to the influence of the component of uperpendicular to J (mentioned in footnote 2, above), and in part to diamagnet-ism (see p. 207). In most cases, though not in all, this term is negligiblecompared to the main term given in (VI, 2).

6 Very often, instead of I, the molar susceptibility is given:G

x-If-P

where G is the molecular weight. and p is the density. In order to obtain x,in formula (VI, 3) N, the number of molecules per mol, is substituted for NL.


proportional to the temperature (Curie’s law). The para-magnetic behavior of a substance in the gaseous state canbe predicted according to (VI, 2) and (VI, 3) when J and ghave been determined from the spectrum; or, conversely,from (VI, 3) an experimental value for the magnetic momentPJ of an atom may be derived (cf. Table 19, p. 209).

Paramagnetic saturation. The Zeeman splitting in themagnetic fields practically attainable is so small that forroom temperatures (and, increasingly, for higher tempera-tures) the energy difference between the levels M = + Jand M = - J is exceedingly small compared to kT. Con-sequently, under these conditions, the difference in thenumbers of atoms occupying these two states is very small.At room temperature and H = 20,000 oersted, for thealkalis, for which J = S = 3 and g = 2, the ratio of thenumber of atoms oriented parallel and antiparallel to thefield is 100 : 99.1. The orienting of the atoms in the direc-tion of the field. increases with increasing field strength anddecreasing temperature. When M = - J for all atoms, afurther increase in the magnetic moment per unit volume(P) in the direction of the field is no longer possible-that is,a state of paramagnetic saturation is reached. At room tem-

Fig. 77. Dependence of Magnetiza-tion P on Field Strength a n d AbsoluteTemperature, H/T (Langevin). Thedotted horizontal line corresponds toparamagnetic saturation.

alized (cf. the above ex-ample). Saturation canbe reached only at verylow temperatures. How-ever, all substances whose

atoms have a magnetic moment different from zero arethen. in the solid state (see below).

i


Fig. 77 gives the dependence of the magnetization P onH/T, according to a formula of Langevin.’ Formula(VI, 2) applies only to the linear part of the curve near theorigin. It is seen that, with increasing H/T, P does notincrease above a limiting value corresponding to saturation.

Diamagnetism. Apart from the orienting effect whichthe magnetic field has upon the individual atomic magneticdipoles, the field also exerts an induction effect on all atoms;that is, a current flowing in a closed circuit is induced ineach atom, in accordance with the Faraday law of induction.This current, of course, arises from an acceleration or re-tardation of the electrons in their orbits superimposed onthe ordinary motion of the electrons. The superimposedinduction currents are so directed that their magneticmoment is in the opposite direction to the external field.Thus, in contrast to the paramagnetic directional effect, thediamagnetic induction effect produces a magnetic momentper unit volume antiparallel to the field. This effect is,however, very small and can be conveniently observed onlywhen no paramagnetic directional effect is present-that is,when the atom considered has no magnetic moment(J = 0). This is the case for all inert gases as well as formost molecular gases.

Paramagnetism of ions in solutions and in solids. V e r yfew atoms having J + 0 (that is, atoms having a magneticmoment) occur free in the gaseous state; thus the abovetheoretical results can be tested only for a few gases. Up tothe present time, measurements of susceptibility have beencarried out only on the vapors of the alkali metals. Thesemeasurements agree, within the limits of experimental error,with formula (VI, 3), where J = S = 3 and g = 2 [Gerlach(105)].

Because of the difficulty of investigating other para-1 This formula is derived according to the classical theory. Allowing for

the quantum theory makes necessary only a slight correction. [Cf. VanVleck (36).]


magnetic atoms in the gaseous state, we have to resort tothe investigation of paramagnetic ions in solutions and incrystals to test the theory. However, most salts and solu-tions of salts are diamagnetic, since the ions present in themhave the inert gas configuration, J = 0 and thus pJ = 0.Examples are Na+, Ca++, O--, Cl-, and so on. (See alsosection- 3 of this chapter.) However, ions with a magneticmoment (J + 0) do occur for those elements in which abuilding-up of inner shells is taking place (see Chapter III).The corresponding salts and their solutions are conse-quently paramagnetic. The rare earths, for example,usually occur as trivalent ions, which generally do not havethe inert- gas configuration (inner 4f shell is not filled; cf.Table 13, p. 141). The resulting state of the ion, usuallyhas L =#= 0, S + 0, J + 0 and, therefore, pJ =/= 0. Theparamagnetism shown by the salts and salt solutions of therare earths follows, very nearly, the Curie law :

constantK= T

From this observation it must be concluded that theseions in solution or in the crystal have the possibility oforienting themselves more or less freely, as have atoms inthe gaseous state. Such a conclusion is plausible in viewof the fact that, in solutions and, similarly, in crystals, theindividual ions are rather widely separated from one anothereither by the diamagnetic solvent or by other larger ionswhich are themselves diamagnetic (for example, Sod-- andwater of crystallization). There is also the further fact thatthe magnetic moment is produced by inner electrons. Even

quantitatively, there is close agreement between the ob-served susceptibilities and the values calculated, accordingto (VI, 3), from the f and g values for the ground state of

the ion. To be sure, the ground states of the rare-earth ionshave, for the most part, not yet been determined spectro-scopically; they can, ‘however, be theoretically predicted on


the basis of the building-up principle and the Hund rule(p. 135).

In Table 19 the ground states and the number of electronspresent in the incomplete 4f shell are given. The g values,as obtained from the Lande g-formula, a e listed also, to-gether with gdJ(J + 1). According to the foregoinggw+T is the magnetic moment of the ion in units ofpo. In the column next to these calculated values appearthe means of the experimental values for the magnetic mo-

TABLE 19

CALCULATED AND OBSERVED VALUES FOR THE MAGNETICMOMENTS OF THE RARE EARTH IONS

Ion

I&+++ ISOa+++ 4j aFuzp,+++ 4f ‘HI

Nd+++ 4p ‘I91211+++ 4ja ezI(

Sm+++ 4f” “HsnEu+++ 4f” ‘FeGd+++ 4p oSrlaTb+++ 4f” ‘F9

DY +++ 4p bH~sizHo+++ 4j” “18Er+++ 4j” ‘zlslt

Tm+++ 4p’ ‘Hoyb+++ .I 4j” ‘R/tLu+++ 4j” ‘So

State

-I

Q

/&I W UNITS OF ~0

CalculatedValue

gav=Fij

0.002.543.583.622.680 . 8 40.007.949.72

10.6510.61

9.607.564.530.00

ments. The latter have been obtained by different

ObservedValue

Diamagnetic2.513.533.55

1.463.378.009.35

10.5510.4

9.57.3s4.5

Diamagnetic

authorsfrom observations of magnetic susceptibilities of the solidsulphates M2(S01)3,8H20, according to formula (VI, 3).With the exception of Sm+++ and Eu+++, the agreement isvery satisfactory. The discrepancies in these two caseshave been completely accounted for by a refinement o f the


theory, proposed by Van Vleck and Frank [cf. 3 6 ) ] , whohave calculated as values for these ions 1.55 and 3.40.*

When the susceptibilities of the ions in the iron series from Soto Ni are calculated in the same way, at first no agreement withexperimental values is obtained. This apparent discrepancy isdue to the fact that, for these ions, the multiplet splitting in theground state is s o small that, at room temperature, not only thelowest component but also the higher components are present inappreciable amount. Consequently, we must calculate the mag-netic moment for all the components. The mean values obtainedby using suitable weighting (corresponding to the Boltzmannfactor) agree more closely with experimental values, insofar asaccurate data are available. [Cf. Hund (7); Van Vleck (36).]

Magnetocaloric effect; production of extremely lowtemperatures. The paramagnetism of ions (in particular,the fact that this paramagnetism follows the Langevin-Curielaw to very low temperatures even in the solid state) hasrecently led to an important application-namely, theproduction of extremely low temperatures (following asuggestion of Giauque and Debye).

When a magnetic field is applied to a paramagnetic sub-stance, there is at the first instant a uniform distribution ofthe magnetic moments-over all possible directions. In orderthat an atom, whose magnetic moment was originally anti-parallel to the field direction, may be able to align itself inthe field direction, energy must be taken away-the amountbeing greater, the stronger the field. This removal ofenergy is accomplished only by thermal collisions or, in thecase of solids, by heat vibrations. The heat energy isthereby raised, and consequently a rise in temperature takesplace when a magnetic field is applied. This is called themagnetocaloric effect . Corresponding to the fact already

* Their calculations involve making a more exact allowance for the fact thatthe instantaneous direction of the magnetic moment does not coincide with J(see p. 204) and that, further, other multiplet components of the ground stateare excited at the room temperature at which observations are made. Corre-spondingly, the values of the magnetic moment for Sm+++ and Eu+++ givenabove are only effective values and do not represent the magnetic moment ofthe free atom in the ground state, as the values for the other ions do for whichthe correction mentioned makes no appreciable difference.


mentioned that in the stationary state at room temperatureonly a very small percentage of the atoms take a preferreddirection in a magnetic field, the magnetocaloric effect is sosmall at room temperature that it cannot be observed.9With decreasing temperature, the amount of heat producedbecomes more and more noticeable compared to the totalheat content of the body. The effect has actually beenobserved at very low temperatures.

The converse of this process-that is, a cooling by adiabaticremoval of the magnetic field-has also been observed and isdue to the consumption of energy in reproducing the com-pletely unordered direction distribution of the individualelementary magnets. This cooling effect has recently beenused for the -production of extremely low temperatures.

To simplify matters, let us suppose that the ions haveJ = 3 so that only two magnetic sub-levels M = + 3 andM = - * are present. Due to the interaction with theinhomogeneous electric and magnetic field in the crystal(the field between the ions), there exists a small energydifference AI& between the two sub-levels, even for H = 0.Only those substances are useful for the magnetic coolingmethod for which A& is so small that, even for the lowesttemperatures reached by the ordinary methods (of the orderof l”K), approximately the same number of atoms are inthe two pub-states.

If now a sufficiently strong magnetic field H is applied,the splitting between the levels M = + 3 and M = - #becomes much larger [namely, 2poH; cf. equation (II, 7)],and therefore, if thermal equilibrium has been reached-that is, if the heat produced by the ordinary magnetocaloriceffect has been taken away-most of the atoms will be inthe state M = - 1in the state M

z and only a very small fraction (e&w’**)= + 3. If at this stage the field is removed,

at the first instant, even for H = 0, most of the atoms arein the state M = - 3 and energy has to be supplied from

’ A magnetocaloric effect is observed for ferromagnetir substances at roomtemperature. The theory of this effect is rather more complicated.


the heat vibrations of the crystal lattice in order to reachthe equilibrium distribution of the atoms between theM=+#andM= - 4 states-that is, the temperature ofthe substance is lowered. The decrease in temperature isconsiderable since, as can be shown, the energy of the latticevibrations is small compared to AEc. However, thismethod only works if AEc is sufficiently small, because forlarge AEo, even at zero field, the equilibrium distributiongives most of the atoms in the state M = - 4. Thus, onlysubstances such as the rare-earth salts that obey Curie’slaw to very low temperatures are suitable for. the process.In this way de Haas and Wiersma (106) have reached atemperature as low as 0.0044”K.

At zero field the ratio N+l,~ : N-,/t of the number of atoms inthe states M = + 3 and M = - ) is given by eVABJkT, whichfor suitable substances is appreciably smaller than 1 only fortemperatures << 1°K. If N+l,s : N-4, is < 1, it means that thesubstance has a magnetic moment and therefore conversely, bymeasuring the magnetic moment, N+l,a : N-112 can be measuredand the temperature determined according to’ the relation

ly+m _N-112

e-AEolkT m4) .

For the field H, N+*12 : Nbl,, = e-2~HlkT~ where T is the initialtemperature (about 1°K). For large H values, e-*mHIkTi is muchsmaller than eWAE@OlkTi. Since, at the first instant after remov-ing the field, N+1,2 : N--l,* is unchanged (= e-*mHIkTi), the ap-parent temperature T, [which corresponds to the N+1,2 : N-i,*value according to (VI, 4)] is much lower than Ti becauseAE, < 2poH. T, can be immediately calculated from

e-A.%lkTo = e-2roHlkTi

which means that0% 5)

Due to the fact mentioned above that the heat energy of thelattice vibrations is very small compared to A&,, the true tem-peratures obtained after equilibrium has been reached are notvery different from the T, values.10 It is seen from equation .(VI, 5) that T. is lower, the smaller AEo and the larger H.

10 Recently Heitler and Teller (142) have shown that at temperaturesbelow 1°K the heat exchange is so slow that usually equilibrium betweenlattice vibrations and A& is not reached. This would mean that the observed

VI, 3] Chemical Applications 213

3. Chemical ApplicationsPeriodicity of chemical properties. In Chapter III it

was shown how the periodicity of the spectroscopic prop-erties of the elements of the periodic system results from thebuilding-up principle together with the Pauli principle. Inother words, it was explained how, at certain intervals, ele-ments recur with qualitatively the same energy level dia-grams and, therefore, qualitatively the same spectra. Theseperiods coincide with the periods of chemical properties, onthe basis of which the periodic system was originally formu-lated. This circumstance-that chemically similar ele-ments are also spectroscopically similar-strongly suggeststhat the foundation of spectroscopic periodicity on the‘building-up principle likewise provides the foundation forchemical periodicity. Some general grounds for the. factthat such is actually the case will first be given.

The chemical properties of an element depend, withoutdoubt, on the behavior of the outer electrons of the atom,since when atoms approach, these outer electrons stronglyinfluence one another. This leads to chemical reaction,molecule formation, and the formation of liquids or solids.The inner electrons are mainly inoperative in chemicalprocesses, since they are much more tightly bound than theouter electrons (because of higher effective nuclear charge).The energies necessary to influence appreciably the innerelectrons are thus very much greater, as is shown by thehigher spark spectra and the X-ray spectra. Apart fromthat, the distance of the inner electrons of an atom from theelectrons of another atom is greater and, therefore, the ex-tent of the interaction is necessarily smaller than for theouter electrons.

Naturally the nucleus and the inner electrons do indirectlyinfluence the chemical properties of atoms. The nucleus islow temperatures refer to the orientation of J only (ratio h’+vs : IV-&,whereas the lattice vibrations still correspond to a higher temperature. But,since the energy of the lattice vibrations doss not form an appreciable partof the total heat content, one is yet justified in claiming that these low tem-peratures have been reached.


responsible, by its charge, for the total number of electrons of anatom, and the inner electrons affect the energy relationships ofthe outer electrons by a partial shielding of the nuclear charge.Apart from that, the nuclear mass can sometimes influencereaction velocity.

The chemical properties of an element are thus essen-tially the properties of the outer electrons of an atom l1 andmust depend on the arrangement of these electrons, ontheir quantum numbers, on the way in which their angularmomentum vectors are added together-that is, on justthose quantities which we can predict theoretically on thebasis of the building-up principle and which we can evaluateempirically with the help of spectra. The foregoing is thereal reason why spectroscopically similar elements are alsochemically similar, and why chemical periodicity andspectroscopic periodicity coincide.

In principle it must, therefore, be possible to derivetheoretically all the chemical properties of any atom, withthe help of the complete energy level diagram obtained fromspectra (including electron configurations). Up. to thepresent time, on account of mathematical difficulties, nogreat progress has been made toward the completion of thisprogram.

Although the complete theory is not yet developed, it isalready possible to draw, from the observed energy leveldiagrams of some of the elements, a number of conclusionsof importance in chemistry and to obtain an understandingof some of the characteristic properties of these elements.It is not the object of this book to give a complete treatmentof these applications. Instead, we shall discuss a fewcharacteristic examples from which it will be realized thateven the more complicated considerations of the previouschapters are of importance for a fuller understanding ofcertain chemical facts.

Types of chemical binding (valence). The chemical be-havior of an atom is characterized mainly by its valence

11 This connection was first recognized by Kossel.


number, or valency; that is, the number of univalent atomswith which an atom can enter into chemical combinationat the same time (or double the number of divalent atoms,and so on). An atom has often several valencies. Forexample, Cl has valencies of 1, 3, 5, and 7.

Two main types of chemical valence must be distin-guished. (1) It has been found that the members of onelarge group of chemical compounds-in particular, the in-organic salts-are built up from positive and negativeions. The forces which hold them together are the ordinary.Coulomb forces of attraction between positive and negativecharges. This type of compound is called an ionic com-pound or a heteropolar compound. The term electrovalentcompound is also used. In order to understand the forma-tion of these compounds, it is first of all necessary to con-sider in greater detail the ionization potential (position ofthe ground term). (2) In contrast to these ionic com-pounds are the compounds belonging to the second largegroup-for example, the elementary molecules Hzr OS, NI;most organic molecules; and others which are built up notfrom ions but from atoms. Because of this structure, theyare called atomic or homopolar compounds. The termcovalent compound is also used. An actual understandingof the forces holding these atomic compounds together wasfirst made possible by quantum mechanics. For this pur-pose it is necessary to take account of the term type of theground state of the atom, together with the. type and positionof the other low-lying terms.

The ionization potential. In Table 18 (p. 200) are listedthe first and higher ionization potentials obtained spectro-scopically for the elements. The dependence of the ioniza-tion potential of the neutral atom on the atomic number isgiven graphically in Fig. 78. The most noticeable regu-larity is that the curve has a steep maximum for the inertgases and a minimum for the alkalis. The opposite chem-ical behavior of these two groups of elements is due largely


to this fact. The underlying reason for it is that the alkalishave a single electron outside closed shells, whereas the inertgases have no electrons outside closed shells. An electroncan be removed from a closed shell only with difficulty;a single electron in an outer shell is, on the other hand,easily removable.

For a single electron outside closed shells, the nuclearcharge is so completely shielded. that Zeff is approximately 1.Therefore the energy of this outer electron corresponds ap-proximately to that for hydrogen in an orbit with the cor-responding principal quantum number. Apart from n = 1,these energies (term values) are small (of the order of 3volts), and consequently the ionization potentials o f thealkalis are also small, since for them n L 2. The decreasein ionization potential in the alkali group is explained bythe increase in n. On the other hand, for the inert gasesand also, though to a somewhat less degree, for the halogens(which have completely or nearly completely closed outershells), the nuclear shielding for an electron in such a shellis very much smaller, since all the electrons in the shell areat approximately the same distance from the nucleusTherefore the ionization potential is very much greater thanit would be for a single electron with the same principalquantum number.

The ionization potentials of the other elements lie be-tween those of the alkalis and the halogens. For the alka-line earths the first ionization potential is somewhat greaterthan for the alkalis, but the second ionization potential isstill comparatively small for the same reason that the firstionization potential of the alkalis is small. Therefore thealkaline earths can occur relatively easily as doubly chargedpositive ions (in contrast to the alkalis). Correspondingly,the elements of the third column may occur as triplycharged ions.

Electron affinity. While the alkalis, alkaline earths, andearths easily give up electrons to form positive ions, the

2 1 7

218 Experimental Results and Applications [VI, 3 VI, 3] Chemical Applications 219

halogens and the elements of the oxygen group readily takeon electrons to form negative ions. They have a positiveelectron affinity; that is, although they are electrically neu-tral, energy is liberated when the outermost shell is filledup by adding one or two additional electrons. The reasonfor this positive electron affinity is the same as for therelatively high ionization potential of the halogens-namely, the incomplete shielding of the nuclear charge inthe outermost shell of electrons. Thus, although the atomas a whole is obviously neutral, an additional electron (oreven two) can be held in the outer shell. In contrast, forthe inert gases no further electron can come into the outer-most shell (Pauli principle). It can at best go into an orbitlying farther out, in which, however, the nuclear charge isalmost completely shielded. Consequently, it is not heldin this orbit and the electron affinity of the inert gazes iszero. The alkalis, alkaline earths, and the earths are quitesimilar in this respect, and their electron affinity is alsopractically zero.

Experimentally an exact determination of the electronaffinity is rather difficult and usually only possible indirectly.We shall not go into the various methods for its determina-tion, but give simply a summary in Table 20 of the resultsthus far obtained.12

It is to be expected that the excited states of a negative ion willnot be stable since, as soon as the electron is in an orbit of higherquantum number than the ground state, the nuclear shielding ispractically complete and therefore the additional electron is nolonger held. Consequently discrete electron affinity spectra havenever been abserved. Even the continuous emission spectrumcorresponding to the capture of an electron by a neutral atom,such as a halogen atom, has not yet been observed with certainty[cf. Oldenberg (112)], although Franck and Scheibe (113) havebeen successful in showing the reverse of this process: a continuousabsorption spectrum by negative ions (electron affinity spectrum).Negative ions are present in high concentrations in solutions ofthe alkali halides. Scheibe (114) found that, in all solutions con-

1) Compare the corresponding table by Mulliken (111), and a paper on theelectron affinity of iodine by Sutton and Mayer (110).

Element

HFClBrI0S

TABLE 20

ELECTRON AFFINITIES

ELECTRON AFFINITY

Volts kcal./mol.

es6%0.749 I+%59 17.274.13 95.33.72 85.83.49 80.53.14 72.43.07 70.82.8 65

-

_-

Reference

(57) (153)(107)(154)(155)(156)( 1 5 7 )(108)

taining the I- ion, there occur two relatively small continuousabsorption bands whose separation is 7600 cm-l. This separa-tion agrees exactly with the doublet separation of theiodine atomin the ground state as found spectroscopically. According toFranck and Scheibe, the explanation is, therefore, that by lightabsorption an electron is separated from the iodide ion in thesolution. The energy required for this varies according as the Iatom remains in a 2P2,2 state or a 2P1,2 state (two series limits;see Chapter IV, section 2).doublet separation.

The difference is just equal to theOne would therefore expect two positions

of absorption (continua) corresponding to the two differentprocesses. This is found experimentally. The absolute positionsof the continua can also be correctly calculated by a more detailedtreatment of the process (allowing for hydration and so on).Analogous effects are found for Br- and Cl-. This is one of thefew cases in which the elementary act in a light absorption processin solution has been unambiguously explained.

Ionic compounds. The small ionization potentials of thealkalis, alkaline earths, and earths are responsible for theirelectropositive chemical character; the considerable electronaffinity of the halogens and the elements of the oxygengroup; for their electronegative character. Owing to theCoulomb attraction between ions, the elements of thesetwo groups form typical ionic compounds with one another.In solution they occur as positive and negative ions, re-spectively. The number of electrons that an atom caneasily give up or take on decides the number of partners with.


which it can form such ionic compounds, and is known ast h e heteropolar valency (electrovalenc). T h e m a x i m u mnumerical value of this valency depends on the number ofelectrons present outside closed shells, or on the number ofelectrons lacking to make up a closed shell. The alkalis whichhave one electron in an unclosed shell are correspondinglyunivalent. On the other hand, the alkaline earths (forexample, Ca) have two easily removable electrons whichare outside closed shells (cf. Table 18, p. 200). These twoelectrons can therefore be taken up either by an electro-negative atom lacking two electrons for a closed shell (forexample, 0 or S) or by two atoms each lacking one electron(for example, Cl). The alkaline earths are therefore di-valent. Thus it can be understood, for example, that CaOand CaClz are formed, but not H or CaC13. The singlypositively charged Ca+ can, however, form a moderatelystable compound with Cl-, as observation of the compoundCaCl shows. Similarly, the earths B, Al, Ga, In, Tl have amaximum valency-of 3, but can have valencies of 1 and 2,as observed for Ga and In. Correspondingly, the halogensare univalent, since they lack one electron for a closed shell;and the elements of the oxygen group are divalent, sincethey lack two electrons for a closed shell.

On the basis of the Pauli principle, a given number ofelectrons lying outside a closed shell or required to make upa closed shell recur periodically in the system of elementsa s the atomic number increases. The same is therefore truefor heteropolar valence. This shows very clearly the valueof the Pauli principle for an understanding of the periodicityof the chemical properties of elements.

In the early development of the subject it was generallyassumed that all chemical compounds were more or lessionic (Kossel); that is, that their component parts werebound together as ions.ls For example, according to thisassumption, in Ccl, four singly charged negative Cl ions

u Cf. van Arkel and de Boer (37).

VI, 3] Chemical Applications

should be bound to a C ion with four positive charges, andcorrespondingly in other cases. We know now, however,that apart from this ionic binding there. is also true atomicbinding, in which the components are bound to one another asatoms (see below). In principle, a given number of atoms(or ions), such as C + 4Cl, can form one and the same mole-cule either in a state with atomic binding or in one withionic binding. The molecule will be an ionic compound ora n atomic compound in the ground state according to whichstate has the lower energy. We can therefore say qualita-tively that an ionic compound is more probable, the smallerthe difference between the ionization potentials involved(for C’CIl,, for example, the work to remove the four outerelectrons of C) and the electron affinity of the negative ionor ions. A consideration of the following cycle may help tomake this clear.

- 290 kcal.cc14 h c+4c1

+ 3358 kcal. 2 J- 3068 kcal.c++++ + 4 Cl-

The energy required to transform C + 4 Cl intoC++++ + 4 Cl- is: 148.0 - (4 X 3.72 = 133.1 volts or3068 kcal. (cf. Tables 18 and 20). On the other hand, it isknown that 12.5 volts or 290 kcal. are required to split CC&into C + 4Cl. If the ground state of Ccl, really originatedfrom ions, (3068 + 290) kcal. should be set free when it isformed from ions. We can, however, calculate the theoret-ical amount of energy which would be set free by the com-bination of a singly positively charged ion with a singlynegatively charged ion, on the basis of the Coulomb law(V = e%J, using a plausible value for the smallest separa-tion of the two ions h This energy is at most 8 volts or185 kcal. If we had a fourfold positively charged ion anda fourfold negatively charged ion, the energy would be4x4= 16 times as great. However, in dealing withC++++ and 4 Cl-, the mutual repulsion of the Cl- ions mustbe taken into account. A more detailed calculation shows


that a value 12 times that for singly charged ions should beused. This gives 96 volts or 2200 kcal., which is consider-ably smaller than the above 3340 kcal. Ccl, cannot there-fore be an ionic compound.

Corresponding considerations for NaCl lead to quitedifferent conclusions. The analogous cycle is given below.

- 97 kcal.NaCl ____+ Na + Cl

+ 129.7 kcal. \ J - 32.7 kcal.Na+ + Cl-

The work required to transform Na + Cl -+ Na+ + Cl-is now only 32.7 kcal., and, since the heat of reaction forNa + Cl * NaCl is 97 kcal., 129.7 kcal. would be obtainedby combining Na+ + Cl- ---) NaCl. This is, however,quite a plausible value for the amount of energy that wouldbe liberated by bringing together two such ions. It isthereby shown that NaCl, in contrast to Ccl,, may verywell be an ionic compound. That it really is an ionic com-pound receives confirmation from the fact that the observedheat of reaction agrees quite well with the results of quanti-tative calculations by Born and Heisenberg for such anionic compound. This is apart from other evidence suchas the dissociation into ions in solution.

The foregoing considerations cover only the investigationof the question whether a free molecule in the gas state is anion molecule or an atom molecule. However, the samedo not necessarily apply to the compound in the liquidstate or in aqueous solution or in the crystal state. The freemolecule of HCl, for example, is certainly an atomic com-pound. However, in aqueous solution it is dissociated intoH+ + Cl- AgCl is an atomic compound in the vapor state,but in the solid it forms an ionic lattice. The reason forthis difference is that in the lattice several ions exert an at-tractive force on a given ion. Consequently, the amount ofenergy liberated per mol by the coming together of the ionsto form a lattice (lattice energy) is relatively much greater

VI, 3] Chemical Applications

than the energy set free in the formation of an ionic mole-cule in the gas state. Therefore in the solid state the ioniclinkage may sometimes give a lower energy state than theatomic linkage, though the reverse is true for the gaseousstate. The difference between an aqueous solution and thegas state (for example, for HCl) is due mainly to the hydra-tion of the ions-that is, to the fact that a number of water

1 molecules are arranged about each ion, their dipoles beingradially directed. This means a considerable gain inenergy for the ionic state and is the main reason for thedissociation into ions. In spite of this, Ccl, does not occurin the ionic form in the liquid, solid, or dissolved states, be-cause of the highly endothermic nature of the ionic stateof the free molecule.14

Atomic compounds (homopolar valence). The fact thatneutral atoms can attract one another strongly, as shownby the formation of such molecules as Hz and Nz, could notbe understood on the basis of the Bohr model. An explana-tion for this fact was first provided by quantum mechanics.In particular, the saturation of homopolar valencies wasdifficult to explain on the old theory (for example, that ahydrogen molecule no longer attracts a third hydrogenatom) in contrast to ionic binding where saturation is easilyexplained purely classically as electrical neutralization.

The first successful theoretical attack on the treatmentof homopolar chemical binding was made by Heitler andLondon (115). For the simplest case dealt with, tbat ofH,, they found that two normal H atoms attract each otheronly when the spins of the two electrons are antiparallel to eachother; whereas they repel each other when the spins are parallel.The value of the heat of dissociation of the molecule ob-tained theoretically agrees approximately with experiment.

According to this theory, the large binding energy is caused,not by the interaction of the spins, but by a resonance processsimilar to that for the He atom (see p. 67). At large separations

1’ Further examples and details may be found in Rabinowitsch and Thilo(38).


of the two atoms a degeneracy is introduced by the equivalenceof the two electrons, since the state: electron 1 with nucleus A,and electron 2 with nucleus B (see Fig. 79), has the same energy

E,2 as the state: electron 2 with nucleus A and0 electron 1 with nucleus B (exchange degen-

eracy). The eigenfunctions belonging to

0 o these states are: ~~(l)(pz(2) and (~~(2)(~z(l),A

Bwhere VA and (OB are hydrogen eigenfunctions

Fig. 79,1eyvpgen referred to A and B, respectively (see p. 39).. As the two atoms approach each other, an

exchange of the two electrons takes placewith increasing frequency [transitions from (PA (1) (o& 2) to(p~(2)cpz(l), and conversely]; that is, a periodic transition fromthe one state to the other results. Just as for He, this process can

be represented as the superposition of two stationary vibrations:‘k* = (PA(1h’B@) + PA@bB(1)$0 = P‘4h3(2) - vDA@h3U)

Wave mechanically the system can be in only one of the twostates, either the state characterized by $, or the state character-ized by #a. The former remains unchanged by the exchange ofelectrons 1 and 2 (is symmetric in the electrons) ; the latter changessign (is antisymmetric). Just as in the case of He, the two stateshave different energies E. and E,, the energy difference beinggreater, the greater the coupling (that is, the smaller the separa-tion of the nuclei). In contrast to He, here the state #, hassmaller energy. Fig. 80 shows the variation of E, and E. withchanging nuclear separation. In the same way as for He, theinfluence of the spin consists, not in its effect on the energy, butin its effect through the Pauli principle. According to thatprinciple, the total function must always be antisymmetric in allthe electrons (see p. 123). Therefore only the state #,, with energyE, can be realized without spin. But, as Fig. 80 shows, E. in-creases continuously with decreasing nuclear separation, whichmeans repulsion of the two atoms. However, since by includingthe spin function the total eigenfunction can be made antisym-metric even for the symmetric co-ordinate function (in the sameway as for He), the symmetric state $a with energy E, can yetoccur. E, first decreases with decreasing separation, and anattraction (molecule formation) takes place (lower curve in Fig.80). The minimum of E, (the potential energy for the motion ofthe nuclei) corresponds to the equilibrium position of the nuclei.Quantitative calculations yield the right value for the equilibriumdistance, known accurately from the Hz spectrum, as well as forthe heat of dissociation (separation of the minimum from theasymptote). According to the Pauli principle, #, can occur onlyw i t h the antisymmetric spin, function-that is, with antiparallel


spin directions of the two electrons (t +); whereas #a can occuronly with parallel spin directions (t t ). The former is a singletstate; the latter, a triplet.

-61 I-0.6 1.0 1.6 2 . 0

T(i8cm.)

Fig. 80. Dependence of the Potential Energy of Two H Atoms on NuclearSeparation.

The extension of the Heitler-London calculation for H2to more general cases has shown that the deciding factorfor the homopolar valence of an atom is the multiplicity ofits ground state or its low-lying terms, or, expressed inanother way, the number of unpaired electron spins. Ac-cording to Heitler and London, this latter number is directlyequal to the valency of the atomic state considered. It isequal to 2S where S is the quantum number of the resultantspin. We can therefore say that the valency is one less thanthe multiplicity.

Correspondingly, He and the other inert gases have avalency 0 in accord with experiment, since their groundstate is a singlet state. The alkalis have a valency 1(doublets). The alkaline earths should again have valency0 in the ground state (YS). However, there is an excitedtriplet state (S = 1) lying not far above the ground state,and therefore the alkaline earths can sometimes have a


homopolar valency I6 of 2.?P and, thus, a valency 1.

The earths have as ground stateThere is, however, a quartet

state lying not very high above the ground state(lsz2s2p2 ‘P); that is, a state with three free valencies.Carbon is divalent in the ground state jP. Again the ob-served tetravelency of C is traced back by Heitler andLondon to an excited state in which one electron is broughtfrom a 2s orbit to a 2p orbit, the ls%2ps 5Sstate (see p. 143),for which all four outer electrons have parallel spins. Thetheory is further developed in a corresponding manner forother elements.

The multiplicities and valencies for the ground statesand some of the excited states of the elements of thedifferent columns of the periodic system according to thisrepresentation are tabulated in Table 21. The valency inthe ground state is printed in heavy, boldface type. It isespecially worthy of note that, whereas the elements of the0 and F groups have a number of different valencies, inagreement with experiment, the elements 0 and F them-selves show only the valencies 2 and 1, respectively, of theground state. This is naturally explained by the fact that,to raise their multiplicity, an electron must be brought intoa shell with higher principal quantum number, whereas withthe other elements in the same columns this is not necessary.

TABLE 21

VIIHalogens

24631357

It is furthermore important to note that for any onecolumn either only even or only odd valencies occur, since

I* Practically, this homopolar valence is of no importance, since most com-pounds of the alkaline earths are ionic.

Chemical Applications

only odd or only even multiplicities occur. An alternationlaw holds for the homopolar valence just as for multiplici-ties: For an even number of electrons the valency is even,whereas for an odd number of electrons it is odd.

The saturation of homopolar valencies follows naturallyfrom this representation as a saturation of the spins-a pair-ing off in antiparallel pairs. If an additional H atom ap-proaches an Hz molecule having antiparallel spins, no addi-tional pair is formed and consequently there is no furthergain in energy-that is, no bonding action. More compli-cated cases can be treated in a similar way.

The Heitler-London mode of representation is thus inprinciple simple, but its use involves some fundamentaldifficulties which must now be mentioned. The Heitler-London theory is rigorously derived only for atoms whichare in S states a n d is true for these only when there are noother atomic states in the neighborhood. The calculationsfor P states do not lead to any simple results. Actually, Pstates occur quite frequently as ground states, and thereoften are, also, other states in the neighborhood of theground state (for example, for C, N, and 0).

Because of these difficulties, two further methods for thetreatment of homopolar binding have been worked out:the method of Slater and Pauling, and the method of Hundand Mulliken.

Slater and Pauling calculate the interaction of the indi-vidual electrons of the different atoms instead of the inter-action of the atomic states. From this point of view thechemical behavior of an atom depends not so much on theterm type as on the electron configuration. Terms with thesame electron configuration are treated as one state. Usingthis method of treatment, one can deduce the fact that cer-tain valencies always occur at a definite angle to oneanother; ‘for example, in H20 the two OH directions areapproximately at right angles to each other, and similarlyfor the three NH directions in NHa. The tetrahedralsymmetry of the four valence directions for C is obtained by


taking into account the electron configuration sp3 as wellas s2p2.

The Hund-Mulliken method attempts to explain chemicalbinding from a consideration of the -behavior of individualelectrons in the field of the different nuclei (many-centerproblem).

A detailed discussion of these modern valence theorieswill not be attempted here. [See Hund in (Id); Pauling-Wilson (32); Sponer (39).] All that we wished to showwas that the chemical behavior of an element depends on theterm types and electron configurations of its lower energy states-that is, on the angular momenta of the atom. Thus aknowledge of the energy level diagram of an element is ofgreat importance for an understanding of its chemicalbehavior.

Activated states and collisions of the second kind; ele-mentary chemical processes. So-called activated states ofatoms and molecules often play a very important part inchemical reactions. These are simply excited atomic ormolecular states. The excitation energies are the energiesof activation. Owing to their larger energy content, excitedatoms (or molecules) have in general a much higher reac-tivity than normal atoms. Furthermore, there is the addi-tional effect that excited states often have more free valen-cies than the ground state (see above). In such cases theatom is more reactive, the smaller the excitation energy tothis state. The inert gases are distinguished by a particu-larly high first excitation energy. They are therefore en-tirely unreactive in the ground state.‘6

A knowledge of the spectroscopically obtained excitationenergies of atoms (and molecules) is thus of particular im-portance for the understanding of elementary chemicalprockssks. A few examples will be considered briefly. A

*I The first excited state of He is at 20 volts. When helium has. beenbrought to this state-for example, by an electric discharge-it has a very highreactivity and can form a molecule with a second normal He atom. This isshown by the He* bands emitted by the discharge.


systematic treatment is possible only with the help of amore complete knowledge of molecular spectra and molec-ular structure than can be assumed here.

When a collision between two atoms or molecules occurs,we distinguish between collisions of the first and the secondkind. In collisions of the first kind a change of kinetic energyof translation into excitation energy takes place by collision(corresponding to excitation by electron collision) ; that is, aprocess :

A + B + kinetic energy + A + B* 0% 6)where A and B are two different (or identical) atoms inthe ground state and B* is the atom B in an excited state.The necessary kinetic energy may be present if the tempera-ture is sufficiently high or if the atoms are artificially ac-celerated, possibly as ions 1’ (excitation by atom or ionoollision). Collisions of the second kind (Klein-Rosseland)are more important for our purpose. They include notonly the exact reverse of collisions of the first kind; that is,a process:

A + B* + A + B + kinetic energy 0% 7)but also all other processes an which an atom or molecule givesup excitation energy by colliding with another partner; forexample :

A+B*+A*+B WI, 8)The conversion of excitation energy into chemical energy -for example, into the dissociation energy of a molecule-is afurther possibility.

The most thorough investigations of such collisions of thesecond kind have been made for Hg. When a number ofHg atoms have been brought into the excited 3P1 state byirradiating the Hg vapor with the 2537 A line (see Fig. 75,p. 202), the Hg vapor reradiates the 2537 line as fluorescence.If now T1 vapor, for example, is added to the Hg vapor, itis observed that T1 lines occur in the fluorescence spectrum

17 We might also think of atoms or molecules with a particularly high velocityresulting from a chemical reaction.


as well as the Hg 2537 line [Franck and Cario, see (lc)].Only those T1 lines occur whose excitation energy is less thanthat of the Hg *PI state. Apparently the collision process

W’PI) + Tl(*Pmj + HgWoj + T1* 0% 9)has taken place. Such a process is called sensitized fluores-cence. Any excess excitation energy of the Hg over thatof the metal atom is changed into kinetic energy of the twopartners after collision.

When gases whose excitation energy for fluorescence isgreater than that of the line 2537 A (for example, He, Hz, 02,CO, Nz) are added to the Hg vapor, naturally no sensitizedfluorescence appears, although an increasing quenching ofthe fluorescence takes place with increasing pressure. Thisquenching can have different origins. Either the wholeexcitation energy can be converted to inner energy by colli-sion without leading to subsequent radiation; or the Hg cango from the aP1 state to the metastable *Pa state by collision

and only the small difference in energy be transferred to thecollision partner; or, finally, a chemical reaction can takeplace.1* All these cases have been observed. By O,, forexample, a transfer of Hg(aPJ atoms into the 9 groundstate is brought about 19; by Nz, a transfer to the metastable8P0 state.20 The case of Hg vapor plus hydrogen has par-ticularly interesting and important chemical applications.In this case the quenching of the fluorescence of 2537 A isparticularly strong and the Hg(aPl) atoms are transferreddirectly to the ground state. At the same time atomichydrogen is found to be present. Two elementary processes

18 The case of the exact reverse (VI, 7) of the collision of the first kind mighthave been expected for the inert gases which cannot take up inner energy of theorder of the excitation energy of Hg(Tl). Actually, this process of changingthe total excitation energy into kinetic energy of the collision partner takesplace very seldom, and hence has not yet been proved with certainty. SeeHamos (118).

1’ Part of the o? reacts chemically with excited Hg: Hg* + OS 4 HgO + 0.See Bonhoeffer and Harteck (119).

l@ The evidence that metastable Hg atoms are produced may be obtained,for example, by investigating the absorption of Hg lines having this state asthe lower state.


are assumed in order to explain this:

Hg(“P1) + Hr + Hg(QSo) + H + H + kinetic energy (VI, 10)Hg()PI) + HZ + HgH + H + kinetic energy (VI, 11)

That the second process occurs as well as the first is shown,by the observation of the HgH spectrum [see (120); (121)].The excitation of the resulting HgH follows by a secondcollision process, as in sensitized fluorescence:

HgW’j + HgH + Hg(‘S) + HgH*

The process (VI, 11) is the prototype of an elementarychemical process for which the excitation energy of thecolliding partner is the deciding factor. The reaction(formation of HgH from Hg + He) would not be possible atordinary temperatures without excitation, since it would bemuch too strongly endothermic.

Exactly the same elementary processes (VI, 10) and(VI, 11) are possible with the met&able ‘PO state, andhave in fact been observed when N, as well as Hz was addedto Hg vapor, the Ns causing preferentially a transfer from‘PI * JPO.

Two general laws are of importance for these elementaryprocesses. The first one. states that for a collision of thesecond kind, the yield is greater, the less the energy which needsto be thereby converted to translational energy [see (122);(123)]. Thus we find, for the Hg sensitized fluorescenceof metal vapors, that those lines are particularly intensewhose excitation energy is approximately the same as thatof the Hg(sPl) state, or possibly of the Hg(“Po) state.Similarly, the strongest quenching on the Hg fluorescenceis exerted by those added gases which have a correspondingexcitation energy. A theoretical basis for this law has beengiven by Kallmann and London (124).

The second general law is that, for a collision, the totalspin of the two collision partners must remain unaltered beforeand after the collision [Wigner (125)]. For example,Beutler and Eisenschimmel(l26) found that in the collision


of excited Kr in the 3P state with normal Hg(QSo), tripletterms are preferentially excited in the Hg when the Krreturns to the singlet ground state by collision:

Kd’P) + W’S) -+ Kr(%) + Hg (triplet state)

Thus before and after collision the total spin S = 1.Collisions of the second kind in which Kr alters its multiplic-ity but not the Hg occur much less often. The basis forthis prohibition of intercombinations is the same as fortransitions within a single atom involving radiation (seep. 125). This prohibition also holds to the same approxima-tion as the ordinary intercombination rule, becoming lessand less strict for higher atomic numbers.

In all collision processes in which excited atoms take part,the lifetimes of the excited states are important sincethe collision must occur before a transition to the groundstate takes place with radiation. Therefore metastablestates are often more effective than states which are notmetastable- and which have a life only of the order of 10m8sec., particularly when not every gas kinetic collision iseffective or when an interaction of two excited atoms isnecessary for the process.

We have considered above what is really an elementarychemical process-namely, the dissociation of Hz by ex-cited Hg. We shall now consider two further examples ofimportant elementary processes in which excited atoms playa role.

Investigations of molecular spectra have shown that, byirradiation of O2 with light of wave length below 1750 A,a normal 0 atom in the 3P state and an excited 0 atom in the‘D state are produced [see Herzberg (127); cf. Fig. 59,p. 163). The resulting 0 atoms react with the moleculespresent, and this leads, for example, to ozone formation.However, this reaction has not yet been explained withcertainty. When hydrogen is added to the oxygen, the Oatoms can also react with H, molecules [cf. Neujmin-Popov(128)]. The reactions which might be expected to take


place are:O(3P) + Hz + OH + H - 1.3 kcal. WI, 12)O(lD) + HZ -+ OH + H + 43.8 kcal. (VI, 13)

Reaction (VI, 12) is weakly endothermic; it therefore doesnot generally occur, and this has in fact been shown in ex-periments by Harteck and Kopsch (129), who used atomic 0produced in an electric discharge. The observed Hz0 orH202 formation by irradiation of an OZ-Hz mixture is thusprobably due to reaction (VI, 13), in which the excitedmetastable O(lD) atom brings the activation energy di-rectly with it. Similar experiments have been done withNOz + H2 [Schumacher (130)]. We know from thespectrum that NO, decomposes into NO + O(3P) by irra-diation with light in the region 3800 A but, in contrast tothis, gives NO + O(lD) by irradiation with light < 2450 A[Hersberg (131)]. Thus again the above two reactionscan take place if hydrogen is added. It was found that nowater formation takes place by irradiation at the longerwave ‘length, although such formation does take. place byirradiation with the light of shorter wave length.

One of the photochemical reactions most often investi-gated is the formation of HCl from Hz + CL It is knownfrom the molecular spectrum that the primary process is:

Cl2 + J&v --j CU2P3/*) + CU2P1/t)Thus there result one Cl atom in the ground state and onein an excited metastable state [cf. the energy level diagramin Fig. 74 (p. 199)]. When the Cl atom collides with an Hzmolecule, the reaction Cl + Hz ---) HCl + H is possible.With Cl in the ground state this reaction is about one kcal.endothermic, and will therefore, in general, not take placeat room temperature. On the other hand, the reaction isexothermic for the excited Cl atom in the *PI,* state (excita-tion energy 2.5 kcal.) formed by irradiation of Cl2 withlight, and can therefore very well take place in this case.

Thus we have considered two elementary chemical reac-tions which are encountered in experiment and which can, in

Experimental Results and Applications [VI, 3

general, take place only with excited atoms. It is clear that afull discussion of these reactions was possible only after theexcitation energy of the states had been evaluated by meansof the somewhat complicated analyses of the correspondingatomic spectra.

In conclusion it should perhaps be mentioned that anaccurate determination of the heats of dissociation of themolecules Oz, N2, the halogen molecules, and others wasfirst possible after the atomic excitation energies had beenevaluated. A knowledge of the values of these heats of dis-sociation is obviously of extreme importance in discussingelementary chemical processes and in calculating the heatevolution of individual reactions.

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INDEX

IndexItalic page numbers refer to a detailed discussion of the subject. Section

numbers are included when the reference covers the entire section.Boldface page numbers refer to figures.The individual elements are given only under their respective symbols (for

example, “Hydrogen fine structure” is lii under “I-I fine structure”).

A

A (Argon), 140, 147,260A IV, V, 158Absorption, intensity of, 152Absorption spectra, series in, 3, 5, 25,

54, 169 ff., 218 ff.AC (Actinium), 141Action integral, 17.Activated states, 2 2 8 ff.Addition:

of J and I to F, 187of L and S to J, 87of magnetic moments in an atom,

109, 111of the li to Z, 82of the SJ to S, 92

Adiabatic law, Ehrenfest, 86, 114,121, 133

Affinity, electron, 216 ff.Ag (Silver), 140, 150, 195, 201AgCl, 222Al (Aluminum), 140, 147, 195, 198

200, 220Al II, 170 ff.Alkali atoms (see also each alkali),

54 f f . (I, 5), 71 ff., 92, 94,162,215 ff., 225

Alkali ions (see also each alkali), 92Alkali-like ions (spark spectra), 69 ff.Alkaline-earth-like ions, 70Alkaline earths (see also each alkaline

earth), 69 ff., 75 ff., 9 2 94,156, 216, 226

Alternation :of multiplicities, 81 ff., 94, 227of valences, 227

245

I

Analysis of multiplets, 177 ff. (IV, 4)Angstrom unit, 8Angular momentum, 16 ff., 19, 47 ff.,

58, 82 ff., 92, 96, 102 ff.,109 ff., 120

magnitude, according to quantummechanics, 101 ff.

of the electron (spin), 93 ff., 108 ff.,120 ff., 128 ff., 224 ff.

of the nucleus, 182, 185 ff. (V, 2)orbital, 17 ff., 47 ff., 58, 89 ff. , 153

Anomalous terms, 141, 164 ff.Anomalous triplet, 76, 164 165 ff.Anomalous Zeeman effect, 93, 97,

106 f f .Antisymmetric eigenfunctions, or

states, 67 ff., 123 ff., 224 ff.Applications, chemical, 213 ff. (VI, 3)A priori probability, 119, 160, 193Arc spectra, 59, 81As (Arsenic), 140, 149, 195, 206Atom colliiion, excitation by, 229Atomic binding, 215, 221, 223 ff.Atomic compounds (atom molecule),

215, 221 ff., 223 5.Atomic core, 68 ff., 62, 70, 92 5.Atomic model:

interpretation of the alkali series,56

Rutherford-Bohr, 13 ff.wave mechanical, 44

Atomic nucleus:deviation from spherical symmetry,

196influence on the spectrum, 1 8 2 ff.

(V, 1 and 2)

Index

Atomic nucleus (continued) :magnetic moment, 186 ff., 194spin, 186 ff. (V, 2)

Atomic number Z, 13Atomic ray (beam), 100 ff., 193 ff.,

294Atomic theory, elements of, 11 ff.Au (Cold), 141, 159, 195,261Auger effect, 173Auroral lines, 168 ff.Auto-ioniaation (pm-ionization), 167,

171 ff.Azimuthal quantum number (k, l),

17 ff., 27 ff., 38 ff., 45 ff., 58,73, 82 ff., 129, 123

B

B (Boron), 140, 142 269,229B III, 61Ba (Barium), 141, 159, 195,291Baimer formula, 11, 20, 55, 133Baimer series of the H atom, 5, 11 ff.,

22, 23, 27Balmer terms, 19 ff., 24, 26, 38, 55Band spectra, 4, 7Be (Beryllium), 91, 139 ff., 149, 166,

166 ff., 195,299Be II, 61Be IV, 26 ff.Bergmann series, 55 ff ., 64, 75, 77 ff.Bi (Bismuth), 141, 183, 189 ff., 193,

195,291Bi II, 183, 189Bohr building-up principle, 120 ff .,

128, 151, 181, 209, 213Bohr correspondence principle, 28Bohr frequency condition, 14, 20, 50,

52Bohr magneton, 103, 108, 186, 202Bohr-Sommerfeld elliptical orbits for

the H atom, 17, 18Bohr theory, 13 ff. (I, 2 and 3), 49,

36Boltzmann factor (Boltzmann distri-

bution), 119,169, 204Born's explanation of the # function,

34 ff., 42

Boundary conditions for the waveequation, 33

Br (Bromine), 140, 149, 195, 200, 219Brackett series of the II atom, 12Broadening of spectral lines, 114,

172 ff.Building-up principle, Bohr, 1 2 0 ff .,

128, 151, 181, 209, 213Burger-Dorgelo-Omstein sum rule,

161

C

c, velocity of light, 10C (Carbon), 146, 142, 143, 162, 175 ff.,

195,200C II, 80, 177 ff.C IV, 61C group of the periodic system, 81,

142, 162, 175 ff., 226Ca (Calcium), 76, 77, 149, 148,

164 ff., 195, 200, 229Ca II, 74Cb (Columbium), 140, 195, 201CC&, example of an atomic com-

pound, 220 ff.Cd (Cadmium), 140, 195, 201Ce (Cerium), 141, 201, 209Centrally symmetric non-Coulomb

field, 53, 120Charge distribution, 34 ff., 42, 43, 44,

52, 135, 136 ff.Chemical behavior and energy level

diagram of an atom, 228 ff.Chemical elementary processes, 228 ff.Chemical properties, relation to outer

electrons, 213Chemiluminescence, 2 ff., 228 ff.Circular orbits in the H atom, 16Cl (Chlorine), 140, 146, 195, 199, 200,

219 ff., 233Cl III, 153Closed shells, 127, 131, 135, 133,

146 f., 150 ff., 1 6 9 , 216 ff.,220

Co (Cobalt), 146,195, 200Collision crow section (yield), 231 ff.Collision processes, 228 ff.

Colissions:of the first kind, 2 2 9of the second kind, 229 ff.

Combination principle, Rydherg andRitz, 13, 72

Compound doublets, 73, 74, 75Compound triplets, 76, 78Conservation of spin, in collision proc-

esses, 2 3 1 ff.Continuous (term) spectrum, 22 ff.,

59, 172Continuum at the series limit, 22, 23,

55, 169,172Conversion factors for energy units,

8 f.table of, 10

Cooling by adiabatic demagnetiza-tion, 810 ff.

Co-ordinate functions, 124 ff., 224Core, atomic, 58 ff., 62, 70, 92 ff.,Corona lines, 153Correspondence principle, Bohr, 88Cosmic nebulae, 157 ff.Coulomb field (potential), 15, 37, 46,

120Coupled vibrations, 67Coupling (see also Interaction) :

types of, 123, 173 ff. (IV, 3)Cr (Chromium), 140, 149, 200Cs (Cesium), 6 0 , 62, 91, 141, 150,

195,291cu (Copper), 140, 149, 195, 200Curie law, 205 ff., 208, 210, 212

D

d-electrons, 86, 129 ff., 149d, Rydberg correction, 56D lines of Na, 5, 71. 74, 92, 106, 107

111, 116, 169 5.D terms, 66 ff., 73 ff.De Broglie waves, 29 ff ., 36 ff.Degeneracy; 46 ff., 67, 119, 224Derivation of terms of p’, 134Deuteron, 194 ff.Diamagnetism, 207Dielectric constant, 114

Diffraction:atomic rays, 30corpuscular rays, 3 0De Broglie waves at a slit, 37electrons (cathode rays), 30, 37molecular rays, 39

Diffuse series, 55 ff., 64, 72, 77 ff.Dipole, electric, 61 ff., 115Dipole moment, 51 ff., 115Dipole radiation, 51 ff., 153

enforced, 155 ff.magnetic, 64, 111, 154 ff., 158

Dirac theory of light emission, 50 ff.Dirac wave mechanical theory of the

electron, 91, 93 112Discharge, electric, 3, 59Displaced (anomalous) terms, 141,

164 ff.Displacement law (Sommerfeld-Kos-

sel), 81 ff., 92Diiciation:

heat of, 224 ff., 234into ions in solution, 222 ff.

Doppler effect for spectral lines, 4,172“Double” magnetiim of the electron,

9 6 , 108 ff., 112Doublets, 71 ff., 74

compound, 73, 74, 75Doublet structure:

of the alkali spectra, 71 ff., 74, 94,139

of the H spectrum, 95,133Dy (Dysprosium), 141,299

E

e, elementary charge, 9 ff.Earths (B. Al, etc.), 81, 142, 216, 226Effective nuclear charge (Z,tl), 68 ff.,

216Effective principal quantum number,

55, 60 ff.Ehrenfest adiabatic law, 86, 114,121,

133Eigenfunctions, 88 ff . , 38 ff . , 67,

153 ff., 171 ff.of the H atom, 38 ff., 40

Eigenvalues, 88,37 ff.Einstein transition probability, 152

I n d e x

Electrical discharges 3.59Electric density of an atom, 42, 43

44,62Electric field:

influence on the atom (Starkeffect), 214 ff.

transgression of selection rules in,86,113# 166

Electric moment (dipole moment),51 ff., 115

Electroluminescence, 2 ff.Electromagnetic waves, emission of,

51Electron affinity, 916 ff.

spectrum, 918 ff.t i le, 219

Electron as point charge, & ff.possible states in an atom, 127

Electron capture, 98Electron clouds, 44 ff 52,135,136 11..,Electron collision experiments, 26,

25, lQQ, 22QElectron configuration:

of the ground state, 121 ff., 123,288 ff. (III, 3), M6

table, 140-141term type from, 36 ff., 198 ff. (III,

2)Electron diffraction, 30,37Electron distribution, 34ff., & 43,

44,52, 136, 136ff.Electro-negative elements, 219Electron 17mass ,Electron orbits, IS, 16 ff., 18 ff., 44,

46,49,53, 121 ff.Electrons in an atom, 85 ff., 190 ff.,

129 ff.Electron spin, 66, 93, 103, 120, 124,

129,225and valence, 998 ff.

Electron structure of molecules, 117Electron volt, 9Electro-positive elements, 219Electrostatic interaction of the elec-

trons, iv, 125Electrovalency, 826,2lSff.

.Elementary c h e m i c a l 998 f f .processes,

Elements:of atomic theory, f f ff.periodic system of the, 81 ff., 123,

188 ff. (III, 3), MSfi.Elliptical orbits, 17, 18,58, 121Emission electron, 59,89 ff., 122,127,

141, 162Emission spectra, 3,55,152,X6Energy level diagrams (see also each

element), W ff. (I, 3)derivation of, I90 ff., 162 ff., 181

Energy levels, 14,~~Enforced dipole radiation, 166 ff.Equivalent electrons, 197,180 ff.Er (Erbium), 14i, 20%Eu (Europium), 141,195,2Ql, 2QQEven terms, 165 ff., 177Exchange degeneracy, 224Exchange of electrons, 87,69, 288 ff.,

224Excitation energy, importance of, for

elementary chemical proc-esses, 228 ff.

Excitation of inner electrons, 267 ff.Excitation potential, 15, 26Excited states, 16,25, ti, 127,298 ff.Exclusion of equivalent orbits (see

also Pauli principle), 293

F

f-electrons 150r, Rydberg correction, 56F (Fluorine), 140, ltiff., 195, 200,

219F, quantum number of the total

angular momentum withnuclear spin, 187 ff.

F, selection rule for, 188 ff.F, term symbol 55Fe (Iron), 7, 140, 279 ff., 200Fe group of the periodic system, 149Fine structure constant, 19.27Fine structure of the H-lines and

terms, 26ff., 46, 51,.96, 118Fixed terms 55Fluorescence, 2 ff., 97,889 ff.Forbidden lines (transitions), 23, 75,

79, 118, 164 ff.

Index

Franck-Hertz experiment% 16,25Frequency condition, Bohr, 1.4, 20,

50,52Frequency of light, 8, 14Fundamental series, 66 ff., 64, 75,

nff.

G

g-factor, Lande, 106 ff., 202,209for nucleus, 186 ff ., 190 fT.

g-formula, Laade, 109,175,2QQg, statistical weight, 162,160Ga (Gallium), 140,149,1Q5, #)o, 220Cd (Gadolinium), 141,2Q1,209Ge (Germanium), 140, 149, 175ff~

200Goudsmi t -Uhlenbeck assumption of

electron spin (see also Elec-trim spin), 98,loS

Graphical representation by energylevel diagrams, 98ff. (I, 3)

Ground state (ground term), 16, 25,56, mff., 127K, 204

of the elements, 140 ff.table, MO-141

Gyroscopic forces, 97 ff., 115

H

h, Planck’s constant, 9, 14I-I (Hydrogen), $12 ff. (I, 11, 20,23,

24, 26, 37 ff., 95, 136, 138,200,219

H atom:according to wave mechanics, 87 ff .,

95doublet structure (hydrogen fine

structure), 98, 133electron clouds (probability density

distribution), 42, 43,44isotope effect (heavy hydrogen),

18sStark effect, 117ff.

H eigenfunctions, 88 ff., 40,41H tie structure, 26ff., 46,51,96, 118H-like ions, 90 ff., 28,87 ff.H’, H’ = D (proton and deuteron),

194Hs, dissociation by excited Hg, 880 L

Kr molecule, 95$3 ff.Halogens, 81,145, US,=HCl, 222 ff., 22aHe (Helium), 64,6Sff., 76 ff., sb, 114,

lWff., 138, 15B, lss, =O,.225,228

according to wave mechanics, 68 ff,122 ff.

He II, 90 ff., 133He-like ions, 66 IHeat vibrations of crystal lattice, 212Heavy hydrogen, 133Heisenberg resonance, 66ff., 198%

129, 171Heisenberg uncertainty principle,

86 ff., 47”

Heitler-London theory of homopolarbinding, 998 ff.

Hertz oscillator, 51Heteropolar valence (compounds),

216, !ZtOHf (Hafnium), 141, IQ5Hg (Mercury), 6,75,7Q, 91,141,155,

195, 197, ml, 202, 999ff.Ho (Holmium), 141,1Q5,2QQHomopolar valence (compounds),

216,22S iKHund-Mulliken method for dealing

with chemical binding, 223Hund rule, 186,209Hydration of ions in aqueous solu-

tion, 219, 223Hyperbolic orbits, 99Hyperfine structure, 182 ff.Hypermultipleta, lsJ, 188 ff., 189

I

I (Iodine), 140, 195, 201, 219I- ion, absorption of the, 818 ff.I, nuclear spin quantum number,

186 ff.In (Indium), 14Q,lQ5,2Ql, 220Indeterminacy principle (see Uncer-

tainty principle, Heisenberg)Inert gas, 81, 146,160 ff., 215 ff., 225Inner electrons, 127,187 ff., 213“Inner” quantum number (see also

J), 73

a50 Index Index

Intensity of spectral lines, 60 ff., 111119,16.6ff. (IV, l), 193

Interaction (coupling) :between J and I, 187 5.between k and Sip 174 5.between Z and S; 90 ff., 95 5., 112,

M8ff., 173electrostatic, of the electrons, 67,

125of ji’a, 174 5.of k’s, 8.6 5., 128 ft., 173of the spins, 96,169, 173

Interaction energy, 178 5.Interatomic fields, 118Intercombination lines, 65, 68 5., 79.

94, 1855., 153, 155, 159Interference of De Broglie waves,

30 5., 37Interval rule, Lande 178 5., 196Inversion, 154Inverted muitiplet, 92, 135, 181Ion collision, activation by, 229Ionic compounds (ion molecules), 815,

919 5.Ionic lattice, 222Ionization, &, 59, 162, 172Ionization potential, 25, 21, 59, 66,

162, 181, 197 5., 215 5., 217of atoms and ions, table, 266-291

Ions, paramagnetism of, 607 5.Ir (Iridium), 141,195Irregular doublets, law of, 6~Isotope effect, 1895. (V, 1)Isotopes, 182

Jj, quantum number, 123. 174 5.j, selection rule for, 154, 175(j, j) coupling, 154, 174 5.J, quantum number (total angular

momentum of the electrons),7 3 5., 775., 875., .985.,196,119

J, selection rule for, 73 5., 79, 99,1535., 175

J values:for doublet terms, table, 73for triplet terms, table, 78

,

I

/

i

AA

A1

I

L

LLL

K

k, azimuthal quantum number, 17,17 ff., 38

k, selection rule, .%7 5.K (Potassium), 69, 62, 72, 146, 147,

170, 195,266K+, radial charge distribution in the

ground state, 136K electrons (K shell), 221 ff., 137Kr (Krypton), 146, 149, 195,206

I, azimuthal quantum number (orbit-al angular momentum ofan electron), 38 5., .+5 5., 58,73,8.95., 129, 128

!, selection rules for, 61, 58, 65, 153L, quantum number (resultant orbit-

al angular momentum), 73,89 5., 87,96, 112 5.

C, selection rule for, 73, 82, 85, 118,153

L shell (L electrons), 137, 146L value of terms of different electron

configurations, table, 8751, Lt shells, 142h (Lanthanum), 141, 159,195,261,

209Lande g-factor, 106 5., 202, 209Lande g-formula, 109, 175,299Lande interval rule, 178 5., 196Langevin curve, iWLaporte rule, 154, 157, 165, 175, 177larmor frequency, 103, 112Larmor precession, 98, 193,199,112Lattice energy, 222Lattice vibrations, 212i (Lithium), 64, 56, 57 5., 61, 71,

91, 122, 1695., 184, 195,200

&i+, radial charge distribution in theground state, 136

d++, 20 ff., 138i-like ions, 66, 61, 63ifetime of excited states, iii, 167,

232

Light, emission by an atom, 16 5.,505., 152

Light quantum, 14,34 5.Liiht sources, 9 5.Limit:

of a series of lines, IQ, 22,23,65,159of a series of terms, W, 57, 1625.

(IV, 2), 197Linear Stark effect, 117 5.Line series (see a l s o Series in line

spectra, 4,115., n, 2651,54 5., 64,69, 159, ‘66, 169

Line width, 4, 114, 172 5.Low temperatures, production by

adiabatic demagnetization,210 5.

Lu (Lutecium), 141,159,195,299Luminescence, g 5.Lyman series:

for the H atom, ld, %,27 5., 95of hydrogen-like ions, 91

M

m, magnetic quantum number, 39,47, 50

m, running number, 12,55,72,197mj, magnetic quantum number, 123,

174ml, magnetic quantum number, 1%

126 ff., 136 5., 133 5.tit- magnetic quantum number, 166,

126 5., 136, 133 5.M, magnetic quantum number, 98 5.,

163 b., 115 ff.M, selection rule for, 1045., 153M shell (M electrons), 127, 137,

146 ff.MI, quantum number, 1915.Mr, selection rule for, 19i?MI, quantum number, 1915.MI, selection rule for, 198MJ = M, quantum number, 1915.ML, quantum number, 116,117ML, selection rule for, 113&, quantum number, 118, 117MS, selection rule for, 11sMO M,, M, shells, 147

p, magne~;m;;tient (see Magnetic

pi, magnetic moment in the directionof J,. 109 5., 202, 209

pi, magnetic moment of Z. 109 5.p8, magnetic moment of s, 109 5.Ma (Masurium), 146Magnetic dipole radiition, 54, 111,

1545., 158Magnetic field, splitting of spectral

lines (Zeeman effect), 96,97 5.

Magnetic moment:of a ‘P,n s t a t e 111of the electron, 95 5., 108of the extranuclear electrons of au

a t o m , 8 3 , 965., 1005..202 ff. (VI, 2)

of the nucleus, 185 5., 191, 194Magnetie susceptibility, 908 5. (VI, 2)Magnetization P, SO5 ff.Magneto-caloric effect, 210 5.Magneton:

Bohr, 106,168,186,292nuclear, 186

Magnitude of angular momentumvectors, according to quan-tum mechanics, 1015.

Mass:of the electron, 17of the proton (H nucleus), 17

Matrix, 53Matrix element, 53, 105,125 5.Matter waves, 69 5., 36 5.Maximum number of electrons in a

shell, 167Metastable state, 66,138,157 b., 232Mg (Magnesium), 6,64,69,149, 147,

165, 195,206Mixing of eigenfunctions, 67 5., 2715.Mn (Manganese), 146,195,266MO (Molybdenum), 146,195,291Molecular spectra (band spectra), 4,7Molecule formation, theory of, 114,

815 5.Moment:

electric, 515., 115magnetic (see Magnetic moment)

Index

Momentum, 47 ff.probability distribution of, 48

Monopole, 53Moseley lines (Moseley diagram), 69,

62Multiplet analysis, example of, 179 ff.Multiplets:

higher, 79 ff., 177 ff.intensities in, 181 ff.inverted, 92, 136normal (regular), 92, 136

Multiplet eplitting, magnitude of, 91,95 ff., 126,129

* Multiplet etructure of line ape&a,71 ff.

Multiplicity, 79 ff., 89 ff ., 94 ff., 225 ff.and valence, 996 ff.law of alternation of, 81 ff., 94ff.,

227

19

n, principal quantum number, 16, 18,41, 51, 53j l!ul

n, selection rule for, 61n*, effective principal quantum num-

ber, 66,tM ff.n, radial quantum number, 17a, (a k) ’N (Nitrogen), 140, luff!, 153, 162,

195,200N II, 167 ff.N V, 61N group in the periodic system, 81,

226N (Avogadro number), 9 ff.Na (Sodium), 5,23,64 ff., 60,82, 71,

74, l40,146ff., IQ+ 200Ns+, radial charge distribution in the

ground state, wbN&l, example of an ionic compound,

smNd (Neodymium), 141,208Ne (Neon), 140,Z&184,1Q5,200Ne III, IV, V, 153Nebulae, cosmic, 157 ff.Nebulium lines, 157 ff.Negative terms, 164,172Ni (Nickel), 14Q,149,2Q3,210

Nodal surfaces of $, 41, 45,s~Non-Coulomb field, centrally sym-

metric, 58,180Non-equivalent electrons, 1%~ ff., 189Norm&ration of $, 86, 137Normal multiplet, 99, 135Normal state (ground state), 16, 25,

5 6 , Moff., 127ff., 14off.,204

table, MO-141Normal Zeeman effect, h ff.Nuclear angular momentum (sa &o

Nuclear spin), 155, 132,186ff. (V, 2)

Nuclear charge, e!Tective (Z,r!); 68 ff .,216

shielding of, 62,216 ff.Nuclear g-factor, 186ff., 1QQff.Nuclerv magneton, 188Nuclear m&e, e&t on hyperfine

structure (isotope effect),189 ff. (V, 1)

Nuclear radius, influence on hyper-fine structure, 135

Nuclear spin, 156, 182, 186 ff. (V, 2)determination of, 190 ff., 196 ff.influence on statistical weight, 119,

193table of observed v a l u e s 196

Nuclear structure, theory of, 194.ff.

0

0 wwm), 149 144 15% 164162 ff., lQ5, Zoo, 219,232 ff.

0 II, III, 157, 1J80 VI, tW,610 group in the periodic system, 81,

219,225Odd terms, 16s ff., 177One-electron problem, 16ff., 32ff.,

117Orbital angular momentum, 17 ft.,

47 ff., 58,8# ff., 153Orbits of the electrons, 16, 16ff.,

18 ff., 44,4-l, 49, 58, 191 ff.Order number, 12,55,72Order of a spark spectrum, 69

Ordinal number Z (atomic number), 13Orthohelium, 64 ff., 75,125, 133Ortho system of the alkaline earth&

70,7506 (Osmium), 141,201Overtones, 32 ff.

P

p-electrons, 86, 120,129 ff.p, Rydberg correction, 66P (Phosphorus), 140,147,1Q5,~P, magnetization, 906 ff.P terms, 56 ff., 64, 72 ff.*-componente, 104 ff.$ function, 89 ff.

nodal surfaces, 41,45,154normfdiaation of, 86,137physical interpretation, 84 ff ., ~$8 ff.

Pa (Protoactinium), 141,1Q5Pammagnetic eaturation, 806 ff.Paramsgnetiim, 304 ff.

of ions, 607 ff.P&a eyetern of the slme e&he,

69,75Parhelium, or pamhelium, 64 ff., 75,

125,133Peechen-Back effe&, 119 ff ., 122, iX3,

154analogue in electric field, 117of hyper6ne ,&ructure, 199 ff.

Paeden series of the II atom, IS, 24Pauli principle (Peuli exclusion prin-

ciple), 19Qfftl; (III, l), HOff.,133, 142, 145, 151,218,220,224

Pb (Lead), 141, 176% 135, 195,201Pd (Palladium), 140,150,201Pd group in the periodic eystem of tb

elements, 150Penetrating orbita, 53Periodlcity of chemical and twectra

ecopic properties of the elcmea& 122,123,Z/7,913 ff.MO

Periodic eyt3tem of the elements, 81 ff,123, 138 ff. (III, 3), 913 ff.

Perturb&on of term eeri* 170 ff.

fund series of the H atom, l&24hosphorescence., 3hotochemical reeztione, examples of,

EM ff.hoto effect., 98 *hotoluminesceace, 3by&A interpretation:of the J, function, 34 ff., 43 ff.of the quantum numbem, 89 ff.

(II,21lanck’s constant (h), 9, 14‘0 (Polo~um), 141‘olarization:of an atom in an electric field, 116 ~of spectral link, 51, 104ff.

‘olyads of terms, 264.+n&ble states of an electron in sn

atom, I98 ff.% (Praseodymium), 141, 195,209‘r II, hyperfme structure, lg3,13Q%xeesion :

in a magnetic field, 0%. 118 ff., 204of J and Z about F, 187,188,lQl ff.of S and L about J, 34, QQff., 96,

lQQ.112,133of the 1~ about L, 84 ff.of the magnetic moment, 109, 111,

191 ff., 304velocity of, 84, 98, 112, 192

?r+ioni&ion (auto-ionisation) , 167,171 ff.

?rincipal quantum number n, 16, 18,41,51,53, 120

edhtive n*, 66,60 ff.true, 69, 61 ff.

FrincipaI series, 64 ff .,’ 04,72,77,162Probability density Uribution, S4 ff .,

@,43,44,52,136,136 ff.Probabiity dletxibution of the mc+

mentum (velocity) in an Hstem, 47,48 ff.

Production of extremely low tempera-tureaq 910 B.

Prohibition of combinations:of a symmetric and an antisym-

metric state, 68 ff.of terms of the same electron con-

figuration, 164

954 Index

Prohibition of intercombinations (intercombination lines), 6!68ff., 79, 94, l%iiff., 1%155,159

for collision processes, 231 ff.Proton, spin and rnngneti;~~;Proton mass, 17Pt (Platinum), 141, 150, lQ5, AlPt group in the periodic system, 13

QQuadrupole, 53 ff.Quadrupole moment of nucleus, IQ6Quadrupole radiation, ,54, 254 ff., 15!Quantum conditions, 14, 16 ff., 29Quantum jump, 14,35, 153

radiationless, 17.9 ff.Quantum mechanics (wave mechan-

ics), 98 ff. (I, 4), lop ff., 110123 ff., 153 ff., 215,223 ff.

Quantum numbers (see also the indi-vidual quantum numbers),26ff., s?ff., e#l ff.

of a single electron in an atom, It?%,1.96 ff.

physical interpretation of, 89 ff.(II, 2)

Quantum states, 14 ff., 32,33, 52Quartets, 79, 60i 90 ff., 95, 178Quenching of fluorescence, 9.30 ff.Quintets, 79,91, 95, Hoff.

,R

Ra (Radium), 141,201Radial quantum number (n,), 17Radiation from an atom, according

to wave mechanics, 59 ff.Radiationless quantum jump, 179 ff.Rare earths, 82, 150,908 ff.Rb (Rubidium), 6Q,62,14Q, 150,195,

201Ra (Rhenium), 141, 195Recombination of ion and electron, 23Reduced mass, lbReflection at the origin, 154Regular multiplets, 99, 135Relativity theory, influence on the H

spectrum, 19,27,33, ‘118

l-44

k

D

3

1 1

. 1I

1, 1

1

aesSs

S

s

51SC

Resonance degeneracy, 67Resonance fluorescence (radiation), HResonance lines of an atom, 27,139Resonance process, 66 ff., 172 ff.Resultant orbital angular momentum

L, 73,89 ff., 87, 96, 112 ff.Resultant spin S of the extranuclear

electrons, 36, 99ff., 103ff.,l25,198ff., 225

Rh (Rhodium), 140,201Rn (Radon), 141,15Q,2QlRosette motion, 19Ru (Ruthenium), 140Runge rule, 108Running number (w), 12, 55, 72, 197Running term, 65Russel-Saunders coupling, ig8 ff ., 153,

155; 161, 169, 173ff., 204Rutherford-Bohr model of the atom,

1sRydberg constant, 11, 9Off., 55 ff.,

133Rydberg correction, &?,53,64Rydberg-Ritz combination principle,

lS, 27Rydberg Series (see also Series in line

spectra), Cc5 ff., 177,18&W

S-electrons, 86’ff., 127, 130, 133, Rydberg correction, 56I (Sulphur),. 140, 147, 195, 219I II, III, 153

200,

1, quantum number (resultant spinof the extranuclear elec-trons), 33,92 ff., 108 ff., 125,

198ff., 2 2 5‘ , selection rules for, 94, 195 ff., 153,

155terms, 56ff., 62ff., 64 ff., 79ff.,

Tiff., 89component , 104 ff.aturation:of valence, 223,227paramagnetic, -W R.

3 (Antimony), 140, 195, 201: (Scandium), 140, 149, ‘15Q, 195,

S&r&linger equation, 88 ff., 37,66Screening doubleta, law of, 68Se (Selenium), 140, 149, 195,200Selection rulea:

for (J j) coupling, 154,176for perturbations, 171for radiationless transitions, 173general, 97ff., 5Off., 258ff.special (see each quantum number)

Sensitized fluorescence, 980Separation energy of the electron, 15Series formulae, 55 ff.Series in line spectra, 4, 11 ff., 22,25,

51, 64ff., 64, 6Q, 159, 162,169

by excitation of an inner electron,167 ff.

by excitation of one outer electron,58 ff ., 64 ff ., 69 ff ., 162 5.,167 ff.

by excitation of t w o electrons,164 ff.

Seriesries limits, 12, $9, 23, 55 ff., 159,162ff. (IV, 2), 198

Sextets, 8&Q!, 95, 145sharp series, 55 ff., 64,72,77 ff.Shells:

closed, 137, 131, 135, 139, 146,150 ff., 169,916 ff., 220

of the extranuclear electrona of anatom, 1+7,166 ff., 151

Shielding of nuclear charge, 62,216 ff.Si (Silicon), 140, 147, 175 5., 200Singlets, 765., 805., 905., 95, 104,

125Slater-Pauling quantum mechanical

treatment of valence, 997 ff.Sm (Samarium), 141,195,2Ql, 2OQ"Smearing out” of the electrons (see

also Electron clouds), 44Sn (Tin), 140,175 5., 195,201solar corona, lines in, 153Solutions:

dissociation into ions, 222 5.hydration of ions in, 219,223light absorption by, 918 5.

Sommerfeld fine structure constant,19, 27

kmmerfeld fine structure formula,19,27, 91

kmmerfeld “inner ” quantum num-ber (see af80 J), 73

Sommerfeld-Kossel displacement law,81 ff., 92

Space degeneracy, 47,5Q, 100Space quantization, 5Q, 96 ff., 112 ff.,

191 ff.Spark spectra, 69, 81Spectra, examples of (see also indi-

vidual elements), 5, 6, 7, 23,74, 76, 97, 163

Spectral analysis, 4 ff.Spectrograms (see Spectra)Spin:

conservation of, in collision proc-esses, 931 ff. .

of the electron, 66, 99, 166, lz0,X24,129,225

resultant, of the extranuclear elec-trone(-?rg)

Spin eigenfunction, 194 5., 224Spin interaction, 98,199Splitting of spectral lines and terms:

in a magnetic field (Zeeman effect),96 5. (III, 3), 1915.

in an electric field (Stark effect),114,116 5.

in hypermultipleta (hyperfine struc-ture), 189 5.

in multiplets, 715.Br (Strontium), 140,15Q,lQ5,2QlStark effect, 114,115ff.

linear (H ‘atom and E-like ions),117 5.

stationary states (quantum states),17, 315., 69, 68

Stationary vibrations, 67,224Statistical weight, 1185., 125, 152,

159ff., 193Stern-Gerlach experiment, 100, 101,

10s d., 204String, vibrating, 39 5.Subordinate series (see Diffuse series;

sharp series)sum rule, 161Supermultiplet, 1 6 4

256 Index

Susceptibility, magnetic, ,862 ff. (VI,2)

Symbols :for electron configurations, 35,

1.86 ixfor terms, 56, 79 ff., 126 ff., 139

Symmetric eigenfunctions, or states,Mff., Lwff., 2.94 ff.

T

T, term, IS ff., 55 ff.Ta (Tantalum), 141,195Tables (see list, p. xiv)Tb (Terbium), 141, 195, 299Te (Tellurium), 140, 195,201Temperature excitation of spectral

lines (thermal radiation),8 ff.; 169 ff.

Term perturbations, 170 ff.Terms, l&66, 79 ff., 120

anomalous, 141, 164 ff.even and odd, selection rule, 164negative, 164, 172number of, 86, 161, 170odd and even, selection rule, 164of equivalent electrons, 130 ff.

tables, 132, 134of non-equivalent electrons, lm,

ll90 ff.table, 132

of the same electron configuration(relative energies), 186

representation of spectral lines by,* ldff.,55ff.

Term series, 13, 66 ff., 94, 71, 169 ff.for several outer electrons, 169 ff.

Term splitting in a magnetic field(Zeeman effect), 86ff. (II,31, 97, 156, 159, l&16.,191 ff.

Term symbols, 53, 79 ff., 136 ff., 139Term systems (see Singlets; Doublets;

etc.)Term types:

from electron configuration 36 ff.,128 ff. (III, 2)

Term types (CO&~H&) :of the ground states of the elements

in the periodic system,13838. (III, 3), 140, 141

Th (Thorium), 141, 201Thermal equilibrium, 159 ff.Ti (Titanium), 140,290Tl (Thallium), 141, 195, 201, 220,

229ff.Tm (Thulium), 141,195,209Total angular momentum F, in-

cluding nuclear spin, 187ff.Total angular momentum J of the

extranuclear electrons, 87 ff .,98ff., 106ff., 153

Total eigenfunction, 123 ff., 154Total spin S of the extranuclear

electrons, 33, 96ff., 109ff.,123 ff., 225

Transition moment, 66 ff ., 105Transition probability, 60 ff., 63 ff.,

105,26,??for radiationless transitions, 179 ff.

Transitions, forbidden, 28, 75, 79,118,164 ff.

Triad of terms, 164Triplets, 76, 76 ff ., 99 ff ., 95, 125

anomalous, 76,164, 165 ff.compound, 78 ff.

True principal quantum number, 69,61 ff.

Two-electron problem, 66 ff.

lJ

U (Uranium), 141Uncertainty principle, Heisenberg,

36 ff., 47Uncoupling:

of J and I in a magnetic field, 192of Id and s<, 16.8of L and S by a magnetic field

(Paschen-Backeffect),11Cff.of L and S by an electric field,

117Units, 8 ff.

Index

V

V (Vanadium), 140,195,2tNlValence, 914 ff., %?0,223 ff.

types of, 216Vector diim allowing for nuclear

spin, 187 ff.Velocity distribution in an atom

(wave mechanical), 47,48Vibrating string, 32 ff.Vibrations, coupled, 67

W

W (Tungsten), 141,195Wave equation, Schrodinger, W ff.,

37, 66Wave function $, 32 ff.Wave length of De Broglie waves, 69Wave mechanics (quantum mechan-

ics), 3886. (I, 4), 104 ff.,110, 123ff., 153 ff., 215,223 ff.

Wave number, 8 if., 14

.

X

X (Xenon), 140, 150, 195,291X-ray spectra, 1, 169 ff., 173X-ray terms (see X-ray spectra)

Y

Yb (Ytterbium), 141,209Yield, in collision processes, %l tl.Yt (Yttrium), 140,150,195,201

ZZ, atomic number (nuclear charge),

13&ff, effective nuclear charge, 88 if.,

216zeeman effect, wff. (II, 3), 97, 156,

159, 134 ff., 191 ff.anomalous, 93,97,106 fl.normal, 97,103, 194of hyperiine structure, 19 1 ff.

Zn (Zinc), 140, l-49, 134, 195,209Zn Ib, 163ff.Zr (Zirconium), 140,201

Atomic Spectra Atomic Structure - Socrates

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