Recent Developments Photoelectric Effect - NISERbedanga/RevModPhys.5.34.pdf · 2015. 8. 21. · EXTERNAL PHOTOELECTRIC EFFECT and not to visible light or to electrical effects. Hallwachs'

JANUARY, 1933 REVIEWS OF MODERN PHYSICS VOLUME 5

Recent Developments in the Study of the External Photoelectric Effect

LEQN B. LlNFORD, » Palmer Physical Laborator'y, Prirscetots Urtioersity

TABLE OF CONTENTS

L Introduction

A. HistoricalB. General Experimental Facts

II. General Theories of the Photoelectric Effect

A. Classical Theories.B. Quantum Mechanical Theories. . . . . . . . .

1. The new electron theory of metals. .2. Wentzel's formulation .3. Frohlich's theory for thin films. . . . .4. Theory of Tamm and Schubin. . . . .

3435

363838404142

S. Penney's theory .6. General remarks.

III. Special Photoelectric Phenomena

A. The Effect of Temperature on the PhotoelectricEmiseion .

B. Saturation Phenomena.C. Preparation of Photoelectric Surfaces. . . . . . . , . . . .D. Space Charge Effects.E. The Spectral Selective Effect of Composite

SurfacesF. The Vectorial Effect.

44475455

5658

I. INTRoDUcTIQN

A NY process whereby an electron absorbs anincident quantum of radiation, thereby re-

ceiving sufficient energy to free itself from itsconstraints may be said to be a photoelectric ef-fect. The electrons thus set free may move in asuitable electric circuit and produce a currentwhich would not flow in the absence of radiation.Depending on the nature of the incident radiationand on the absorber, the processes and the result-ing effects differ greatly. For convenience theymay be classified as follows:

(I}The external photoelectric effect. The ejec-tion of electrons from surfaces (mostly metallic)by visible, ultraviolet or infrared radiation.

(2) The x-ray photoelectric effect. The ejectionof electrons by Roentgen rays.

(3) Photoionization. Ejection of electrons fromgases or vapors.

(4) Photoconductivity. Freeing of electronswithin certain insulators by light, so that theybecome conducting.

(5) The "Sperrschicht" photoelectric effect.The ejection of electrons from one side to theother of a rectifying boundary such as cuprousoxide formed on copper.

The external photoelectric effect was the firstto be discovered and studied, and the presentdiscussion will be limited to this field. No attemptwill be made to make a complete survey of thegreat amount of work done on this subject, as thishas been done in a complete and excellent mannerby Hughes and Duaridge. ' The preliminary draftof this paper was made before the book was pub-lished, and has since been revised so as to elimin-ate considerable duplication of material. It ismore the purpose of this article to considercritically some of the better known effects in thelight of quantum mechanical theories which werepublished after much of the previously mentionedbook was written.

A. HistoricalWhile experimentally investigating Maxwell's

electromagnetic theory of light, Hertz' noticedan effect which later proved to be an externalphotoelectric effect. He found that when the lightfrom the spark of his primary oscillatory circuitfell on the electrodes (particularly on the negativeelectrode) of the spark gap of the secondary or re-ceiving circuit the distance which the spark wouldjump at the second gap was increased. He showedthat the effect was due to the ultraviolet light,

*National Research Fellow in Physics. Now in Depart- ' A. L. Hughes and L. A. DuBridge, Photoelectricments of Mathematics and Physics, Utah State Agri- Phenomena, McGraw Hill, New York (1932).cultural College. ' H. Hertz, Ann. d. Physik 31, 983 (1887).

34

EXTERNAL PHOTOELECTRIC EFFECT

and not to visible light or to electrical effects.Hallwachs' found that metals insulated fromground became negatively charged when exposedto ultraviolet light, and thus aasociated this effectwith the emission of negative electricity. Becauseof his work, the external photoelectric effect issometimes called the Hallwachs effect. Lenard'and J. J. Thomson' showed that the emitted par-ticles were the same as the cathode rays of aGeissler tube and thus that the effect was theemission of electrons.

The first theoretical advance was made byEinstein' in 1905 when he postulated that lightmust be absorbed as well as emitted in quanta ofenergy kv, and thus if one quantum be absorbedby one electron the following relation must hold:

ep= j~&max= e Vmaz {&)

Here v is the frequency of the incident light; h isPlanck's constant; ep* is the minimum energyrequired to remove an electron from its place in

the metal to field free space outside, and is calledthe work function of the surface; m is the massof the electron; e is the velocity of the fastestejected electrons; and V, is the retarding po-tential in e.s.-units required to stop the fastestelectrons. This equation was satisfactorily veri-fied several years later by the work of HughesfCompton, ' Richardson and Compton, s andMillikan. "

The frequency vp of light which will just ejectelectrons, but with zero velocity is called thethreshold frequency and is defined by

hvp= ep.

The wave-length Xp corresponding to the thresh-

s W. Hallwachs, Ann. d. Physik 33, 301 (1888).' P. Lenard, Wien. Ber. 108, 649 (1899);Ann. d. Physik

2, 359 (1900),s J.J. Thomson, Phil. Mag. 48, 547 (1899).' A, Einstein, Ann. d. Physik 17, 132 (1905).~ sp is expressed in ergs per electron. For convenience in

photoelectric formulas, energies will be in ergs per electron

og per quantum. To change to electron volts multiply by(c/10se)Pq~ (6.288&0.006) P 10"where Pq is the factor forchanging from absolute to international volts (R. T. Birge,Rev, Mod. Phys. 1, 1 (1929)).

' A. L, Hughes, Phil. Trans. Roy, Soc, 212, 205 (1912).s K. T. Compton, Phil. Mag, 23, 579 (1912).' O. W. Richardson and K. T. Compton, Phil. Mag. 24,

575 (1912)."R. A. Millikan, Phys. Rev. P2 j 7, 355 (1916).

old frequency is called the long wave-lengthlimit.

B. General experimental facts

To describe completely the photoelectric prop-erties of a surface, it is necessary to know itsthreshold frequency; its spectral distributionfunction, or the emission per unit absorbedenergy as a function of the frequency of the inci-dent light; the velocity distribution of the ejectedelectrons; and how these functions vary with thetemperature, the applied field and, if polarizedlight is used, with the orientation of the electricvector of the light.

On modern theories the electrons lose most oftheir energy on leaving the surface of the metal,and it would be expected that the nature of thesurface would greatly affect its photoelectricproperties. Experiments show that the presenceof foreign substances to the extent of only a smallpart of a monomolecular layer may change theentire photoelectric character. One of the mostdiScult phases of the experimental work is ob-taining and maintaining surfaces free from un-wanted contamination.

The spectral distribution function {one isshown in Fig. 3) as experimentally determined iszero for frequencies less than about vp, rises withincreasing frequency, and may have one or moremaxima in the spectral region which can bestudied. Characteristic of all curves is that theyapproach zero tangentially. This necessarilymakes any direct determination of the thresholdfrequency from experimental data uncertain. Re-cent theory has shown a method of plotting dataso as to remove this uncertainty.

An alternate method of determining the workfunction of a surface is to measure the photo-electric current reaching a collector against aretarding field. Then from the observed V, andEinstein's equation one calculates the work func-tion. In this case the current approaches zeroasymptotically and thus leaves uncertainties asto the value of V, and therefore the workfunction.

The velocity distribution of the ejected elec-trons is difficult to measure and especially if thedirection is to be taken into account. Except forthin films of metal, the energy distribution func-tion has not been treated theoretically with much

LEON B. LI NFORD

success. Such factors as collisions of the excitedelectrons before leaving the metal and the effectof the finite structure of the metal surface make atheoretical treatment with present methods al-most impossible.

Much interesting information about the photo-electric effect can be obtained from photoelectricsurfaces consisting of films on some suitable base.In addition, many new problems arise in con-nection with such surfaces. The materials usedfor these films are usually alkali or alkaline earthmetals alone or in combination with other sub-stances, usually dielectrics. The most frequentlyused dielectrics are hydrogen and oxygen. Theoutstanding characteristics of these compositesurfaces are low work function and high photo-electric efficiency, or large emission per unitabsorbed energy. These characteristics are theones desired in photoelectric cells for technicaland commercial purposes.

Elster and Geitel" showed that alkali films

produced very sensitive photoelectric surfaces.These surfaces have been studied in great detailby Elster and Geitel, Ives and coworkers, and bySuhrmann and Theissing. They found"" thatwith increasing thickness of film the work func-tion changed continuously from the value for thebase metal to a minimum, less than either metalalone, and finally increased to the value for thesolid alkali. Brady" found for potassium, rubid-ium and caesium on silver, that the minimumwork function occurred when the film was 1 to 2

atoms deep, that maximum emission occurredwhen films were about 5 atoms thick and thatfilms more than 12 atoms thick had the character-istics of the solid metal. These thicknesses weredetermined by distilling the alkali metal onto thesilver surface at a known rate.

Ives and Olpinis found that the minimum

work function, reached by an alkali metal film

as it forms on a clean surface, is equal to theresonance potential of that metal. This meansthat the minimum threshold frequency is equal

"J.Elster and H. Geitel, Ann. d. Physik 42, 564 (1S91).~ H. K. Ives and A. R. Olpin, Phys. Rev. |2j 34, 117

(1929).~ R. Suhrmann and H. Theissing, Zeits. f, Physik 55, 701

(1929).t4 J.J.Brady, Phys. Rev* L23 37, 230 (1931);L2j39, 546

(1932); P2g 41, 613 (1932).

to the frequency of the first line of the principalseries of the metal in vapor form. A satisfactoryexplanation of this effect has not been given.

Some of the surface films and complex surfacesshow a marked spectral selectivity, which meansthat they show a very high photoelectric effi-

ciency in a limited spectral range. Another char-acteristic which may be shown by such surfacesis vectorial selectivity, or a dependence of thephotoelectric efficiency on the orientation of theelectric vector of polarized light with respect tothe surface. The efficiency is usually greaterwhen a component of this vector is normal to thesurface than when it is in the plane of the surface.

Finally, some of these surfaces show abnormalcurrent voltage characteristics. If the photo-electric surface or cathode is slightly negativewith respect to the collector or anode, all theemitted electrons would be expected to flowacross from cathode to anode. In practice withpure metal cathodes, an accelerating potential ofless than 5 volts is sufficient to produce a currentwhich remains almost constant as the voltage isincreased. This is called the saturation current.The current from some of the complex surfacesincreases with the applied accelerating potentialuntil it reaches the order of 100 volts.

II. GENERAL THEORIES OF THE PHOTOELECTRIC

EFFECT

With this short sketch of some of the outstand-ing experimental features of the photoelectric ef-fect, a summary of some of the more successfulgeneral theoretical treatments will be given. Mostof these general theories must of necessity bebased on simple models and therefore apply onlyto clean surfaces for emission at absolute zero.Special refinements of the general theory as ap-plied to particular phenomena and to complexsurfaces will be discussed in the third part ofthis paper.

A. Classical theoriesPrevious to the Sommerfeld electron theory of

metals, one of the best pictures of the interior of ametal was that the free electrons, about I peratom, formed a "gas" with a Maxwell-Boltzmanndistribution of velocities. This theory accountedfor the high electrical and heat conductivities ofthe metals„but gave too high a value for the heat


I=AT'e "~sr, (3)

where I is the current density from the surface,A is a constant, T is the absolute temperature, kis Boltzmann's constant, and r is a constant and,depending on assumptions made in derivations,either q or 2. The value 2 is the one obtained on

~ O. %. Richardson, Phil. Mag. 24, 570 (1912)."J.J.Thomson, Phil. Mag. 2, 674 (1926).'r A. W. Uspensky, Zeits. f. Physik 40, 456 (1926).'"See page 200 of reference 1."O. W. Richardson, Phil. Mag. 23, 594 (1912).

capacities. Several rather unsuccessful attemptswere made to overcome this difficulty. Photoelec-tric theories based on the old picture were able toaccount for the sm@1 change in threshold fre-quency and emission with temperature. Theseresults follow from the fact that the change invalue of the most probable electron energy, overthe temperature range used in photoelectricexperiments, is small compared with the workfunction.

Theoretical spectral distribution functionshave been calculated by Richardson, " J. J.Thomson" and Uspensky. " Richardson's curvemeets the axis with finite slope at the thresholdfrequency. The other two are tangent at v= vp,

but all three rise more rapidly with the frequencynear the threshold than the experimental curves.

Richardson�'s

equation has a maximum atv= (3/2)vp and Thomson's at v=2vp. Selectivemaxima have been found quite generally in thisregion. The maximum of Uspensky's curve is atv=9vp. Hughes and DuBridge'" point out thatby varying Thomson's assumption as to the wayin which the absorption coefficient for light varieswith the frequency, spectral distribution func-tions can be obtained which fit the data well forfrequencies near the threshold. These equationshowever have no maximum. Thus no singletheory follows the experimental data over a widefrequency range.

Without any special assumptions as to thetheory of metals, or the actual process of photo-electric emission Richardson" has given an equa-tion for the photoelectric emission from a surfaceexposed to radiation from a black body, this hasbeen called the complete emission. Richardson'sequation for thermionic emission is

the newer theory of metals, and is now acceptedas correct.

Eq. (3) can be derived by determining the rateat which electrons leave a surface enclosed in ablack body cavity without any assumptions as tothe mechanism of the emission. "When the elec-tron emitter is within the cavity the electronsmay be ejected either by thermionic emission orby photoelectric emission due to the black bodyradiation. This is an equilibrium process and ifequilibrium conditions could be maintained itwould be expected that the same equation. wouldhold if the electron emitter were elsewhere andexposed to the radiation from a black body.Deviations from this equation should be then ameasure of the deviations from equilibrium con-ditions. Evidently the constant A would bemuch smaller and Twould represent the tempera-ture of the black body.

This equation has been verified experimentallyby Roysp using black body radiation and bySuhrmann" who calculated the complete emissionfrom experimentally determined spectral distri-bution functions. If F(v) represents the spectraldistribution function and E (v, T) the energydistribution function of black body radiation,Planck's law, then the complete emission I, isgiven by

I,= F(v)E(v, T)dv.

By means of graphical integration the completeemission can be determined for various tem-peratures.

When I,/T' was plotted against 1/T, bothRoy and Suhrmann found the lines were straightand that the slope gave a good value for the workfunction. Later Suhrmann" reported that he ob-tained better straight lines if r in Eq. (3) wastaken as about 4 rather than 2. There seems tobe no theoretical justification for this highervalue.

"For a good derivation of Richardson's equation onthese assumptions, see S. Dushman, Phys. Rev. P2g 21, 623(1923).Richardson's equation in this form is not rigorouslytrue if pp is a function of temperature. For clean metals thedeviations are small. This form of the equation is exactenough for the present discussion.

» S. C. Roy, Proc. Roy. Soc. A112, 599 (1926).» R. Suhrmann, Zeits. f. Physik 33, 63 (1925).» R. Suhrmann, Zeits. f. Physik 54, 99 (1928).

38 LEON B. LI NFORD

B. Quantum mechanical theories

(1) The 33eto electron theory of mefab. With theapplication of the Pauli exclusion principle andthe resulting Fermi statistics to the free electronsin metals, Sommerfeld and others were able toformulate a new theory which would account forthe electrical and heat conductivities of metals,as well as other principal characteristics, withoutgiving too high a heat capacity. This with quan-tum mechanical methods gave a new and power-ful method of treating the problems of emission ofelectrons from metals. "

A brief summary of some of the results of thenew theory of metals will be helpful in under-standing the photoelectric theories.

Consider the surface of the metal which isemitting electrons to be in the xy-plane and theoutward normal the positive s-direction. Letg, q, f be the z, y, s-components of velocity of anyelectron. To calculate the emission from unit areaof the surface it is necessary to find the numberof electrons N(W)dW whose normal componentof energy W=fmps is between W and W+dW,which strike unit area of the surface in unit time;multiply this by the probability D(W) that anelectron with normal component of energy W willbe transmitted through the surface and escape,and integrate over all values of W. Thus thecurrent density is

I= N(W)D(W)d W. (5)

On the basis of Fermi statistics the velocitydistribution function of electrons in a metal is

f(h, ~, t)dHnd(2m3 d$dqdi

(6)k3 exp ( I ~~333(p+ils+gs) —eI/kT)+I

where f(g, il, f)dgdi7di is the number of electronsper unit volume with z, y, z-components of

"L. Nordheim, Phys. Zeits. 30, 177 (1929) gives anexcellent discussion of the theory of metals, thermionic andautoelectronic emissions, and a review of the early theoriesof the photoelectric effect.

~ Although energy cannot have a component in thestrict sense of the word, the value of W as defined and notthe total kinetic energy s = $m(P+s'+P) determineswhether or not the electron can leave the surface. Forconvenience in writing, the above terminology will be used,as is usual in treating this subject.

8vr333 (2me) &deF(e)de=

$3 e(c=e)jsT+1(8)

The striking difference between the two theoriesis that on the classical theory the electrons areat rest at absolute zero whereas on the newtheory they have energies up to the relativelylarge value e. With increasing temperature, onthe old theory all electrons take on increasingenergies, whereas on the new theory only a few ofthe fastest increase their energy above e.

Fowlers3 used the distribution given by Eq.(8) for a preliminary theory of the photoelectriceffect. It was soon pointed out that only thenormal component of energy is effective in

escape and Nordheim~ calculated the requiredfunction N(W)dW, which is

N(W)d W= {4irm/k3)kT. ln(1+e &~ '»'~)dW (9)

This can be reduced to the three approximateforms for different values of (W—e)/kT:

N(W)d W= (4s rent/133) (e —W)d W'

for (W—e)/kT(&0, {10a)= {4 re/k')kTdW

for (W—s)/kT=0, (10b)= (4sm. /k3)kTe &w—'»s~dW

for (W—e)/kT»0. (10c)'~ R. H. Fowler, Proc. Roy. Soc. Alls, 229 (1928).

velocity between g, il, g and $+dg, q+dy, p+df,e is the energy of the fastest electrons of theFermi distribution at O'K and is given by

e = hi = (k'/Sm) (3n js)I, (7)

where re is the mass of the electron and n thenumber of free electrons per unit volume of themetal. The value of e ranges from the equivalentof about 2 electron volts for the alkalies to morethan 10 volts for some of the heavy metals.

The value of f(p, il, i) at O'K as given by Eq.(6) is constant and equal to 2'/k3 for energies ofthe eleCtrOn e= sm(@+31'+g') (e and fOr energieSgreater than this critical value the function iszero. For higher temperatures, f($, q, g) dropsfrom the value 2m3/k3 to zero exponentially in theregion e-=s having half value at e= e.

Expressed in terms of the energies of the elec-trons the distribution function is

EXTERNAL PHOTOELECTRIC EFFECT 39

At absolute zero these reduce to

N(W)dW= (4s.i'/h')(» —W)dW for W&», (11a)

=0 for W&». {11b)

The difference in form of the two functions isshown in Fig. 1 where they are sketched for bothabsolute zero (solid line) and for a temperatureabout 1000'K (broken line). W and» are plottedon the same scale, but the ordinates F(») andN(W) are on entirely different scales.

In order to determine the transmission coeffi-cient" D(W) for a surface, a particular form ofpotential barrier retaining the electrons in themetal must be chosen, and then the wave func-tion set up for the electrons both inside and outof the metal. A common method is to write thewave function normalized for unit current den-sity moving normally toward the surface fromwithin the metal; then set up functions with un-

determined coefficients to represent the trans-mitted beam leaving the metal, and the reflectedbeam returning to the interior. The coefFicientsmust be so adjusted that the total wave functionand its first derivative be continuous everywhere.These conditions are sufficient to determine thecoefficients of the terms representing the trans-mitted and reflected beams. The transmissioncoefficient is the square of the absolute magnitudeof the coefficient of the term representing thetransmitted beam, since the corresponding mag-nitude for the incident beam has been normalizedto unity. The resulting transmission coefficient asa function of the normal energy W of the electronis dependent on the form of the potential barrierchosen.

The simplest form of potential barrier at ametallic surface is an abrupt rise of potential toits value at infinity, OAB (Fig. 2). A potentialwhich more nearly corresponds to the actual caseis

V= hi, —e/4z for s &s', (12)

which is obtained if it be assumed that for dis-tances greater than s' (of the order of atomic di-

'4 Besides the review of the early work on this subject inreference 23, a more recent and detailed discussion is givenby E.U. Condon, Rev. Mod. Phys. 3, 43 (1931).Among themore recent articles may be mentioned N. H. Frank and L.A. Young, Phys. Rev. P2g 38, SO (1931);W. Wetzel, Phys.Rev. f2138, 1205 (1931);and V. Rojansky and W. Wetzel,Phys. Rev. t 2j 38, 1979 (1931).

I

I

IIIIIIII

Car W

Fir. 1. Fermi distribution functions for electrons atabsolute zero (solid lines) and 1000'K (broken lines).F(e)d» is the number of electrons with energy between»and»+d~. N(W)dW is the number of electrons withnormal component of energy between W and W+dWwhich strike unit area of the surface in unit time. ~

mensions) the emerging electron is subject to theforce due to its own image in the metal. This po-tential curve is shown by ODB {Fig. 2). In bothcases the difference in potential between the

hV, = E',

h+a" ~a

Fin. 2. Two types of surface potential barrier, showingthe effects of an accelerating Seld. OAB is a step-likebarrier in zero field and OAC in accelerating Seld. ODB andOEC are the corresponding barriers for an image Seld.

' It is impossible for N(W), see Eq. (9), at T&0'K to beless than it is at T=0 for any particular value of the normalcomponent of energy W. The area under this curve is thenumber of electrons striking the surface per second and thismust increase with the temperature. Nordheim~ sketchedthe curve for T&0 so that it dropped below the curve forT 0 at a value of W somewhat less than i, and thenapproached the T=O curve from below, as W decreases.A similar error has been made by other writers. ' "

40 LEON B. LI NFORD

interior of the metal and field free space outsideis kv, = e,. From previous discussion it is evidentthat e, is the sum of e, the energy of the fastestelectrons of the Fermi distribution, and ep thework function which is the energy needed to takean electron with energy e to field free space out-side the metal. Or

the electrostatic field in the neighborhood of theelectron. ) (3) Calculate N'(W)d W', which is thenumber of excited electrons with energies normalto the surface between W' and W'+dW' whichstrike unit area of the surface in unit time. Theemitted current density will be

eg= e+ep,'hv„= hv+hvp. (13) I= N'( W')D(W')d W'.p

(14)

Calculations by Nordheim and others show thatfor both barriers D(W) = 0 for W &hv. and D(W)rapidly approaches unity for W) hv, . ForW —kv corresponding to 0.1 electron voltD{W)&0.99.

If an accelerating field is applied to the surface,the potential barrier OAB becomes OAC, andODB becomes OEC. In the former case the ap-plied field does not affect the maximum height ofthe barrier whereas in the second case the heightof the barrier and thus the effective work functionis decreased. As in the case of no external fieldD(W)-p1 for W'greater than the maximum heightof the barrier, but in this case for W less than themaximum height of the barrier, D(W) remainsfinite though small. This probability that anelectron will be transmitted through a regionwhere its classical energy would be negative,from one classically allowed region to another,accounts for the emission of electrons from coldmetals under intense fields, as well as some otherphenomena which could not be accounted for onclassical theory.

By substituting the values of N(W)dW and ofD(W) into Eq. (5) and integrating over properlimits, both Richardson's thermionic equationand the experimentally determined equation forthe autoelectronic effect can be derived.

The problem of photoelectric emission is morecomplicated, as one must calculate the effect ofthe incident light on the velocity distribution ofthe electrons, and use this new function to deter-mine the emission. The following are the essentialsteps: (1) Set up the equation for the light in themetal. (2) Determine the probability that anelectron with velocity components $, q, 1 will

absorb a quantum of light of frequency v so thatthe electron will have a final component ofenergy, normal to the surface, W. (This will in-volve among other factors g, q, l, v, the intensityand polarization of the light, and the nature of

So far it has been impossible to obtain an exactsolution on this scheme, so the present theoriesare the results of various methods of approxima-tion. Some of the better ones will be outlinedbriefly.

(2) Wentzel's forrrtttAztion. WentzePp made thefirst solution in which the interaction of theradiation and electrons was treated quantummechanically. He assumed the free electrons in arectangular block of metal with sides li ls ls andset up the wave equation in the box as

—tt s(ski/p) (me +a) tv'p p

where

(15a)

up= (8/l&lslp) sin k&x sin k&y sin kpz (15b)

k;=~n;(1;; j=1, 2, 3; nt=1, 2, ~ ~ (15c)

e= h'k' j8s'm k'= kP+ks'+kas. (15d)

n G. Wentzel, in Sommerfeld's 60. Geburtstag Fest-schrift, Probleme der Modernen Physik, edited by P.Debye, p. 79, Leipzig (1928),

e is here the kinetic energy of the electron beforeexcitation by the light. He then assumed that thelight is classically damped inside the metal withthe extinction coefficient a and thus the electricvector of the light wave in the metal where thecoordinate z is negative, is given by the formula

E=Epe 'icos 2vrt'vt —{Kr)j. (16)

By this assumption he tried to overcome thedifficulty arising from conservation of momentumconsiderations. The point will be discussed ingreater detail in connection with Tamm andShubin's theory.

%entzel applied the electric vector as a per-turbation on the wave equation of the electrons,Eq. (15a), and obtained the new wave function

4 =4p+4i.


The first order perturbation term Pi is given

by the equation

k' k Bfi—hei+———mc fi8~ 2~ 8t

ekeas-i(Kt) —print, (Ep grad fp) (18)

8%m v

From the theory {fi {'gives the probability that

an electron with initial energy e will absorb aquantum of energy kv. This must be integratedover the volume and summed for all initialenergies e.

He then assumed that all electrons with energysufficient to escape would do so regardless of theirdirection of motion, thus that the transmissioncoefficient D is unity for e+kv) kv, and arrivedat the result

I~[v P —(vo —v) 's jv "sEgs+(1/14)[v "—(vp —v)" jv "[3Es+2E,'+2E 'j+ ~ ~ ~ (19)

E„E„,E. are the components of the electricvector, E, being the component normal to thesurface.

Houston" correctly pointed out that only thoseelectrons which, after the last collision in themetal, have an energy component normal to thesurface greater than p. can escape. He arrived atthe formula

1

I~ — —1+—

E& +Ev v —vp v —vp

+ " . (20)3v v v

Kentzel~ recalculated his formula on the basisof Houston's correction and obtained

I v i s{(i/3)v[v3~2 —(v, —v)

—(I/&) [~"—("—)'"jIE'+ (21)

Although Eqs. (21) and (22) are quite differentin form, the distribution curves they representare similar.

The essential characteristics of the spectralsensitivity curves as given by Eqs. (20) and (21)are: (1) The curves leave the axis at v=vp ata finite angle; (2) rise to a maximum for the fre-

quency v in the range v &v, )vp, (3) theequations show that E, is more effective in eject-ing electrons than E, or E„.Although these sensi-

tivity curves meet the axis at a finite angle atv= vp, their shape for greater frequencies is verynear that of the experimental curves.

~%. V. Houston. As quoted on page 495 by E. Q.Lawrence and L. B.Linford, Phys. Rev. l 2/30, 482 (1930)."See page 496 of reference 26.

(3) Froklick's theory for tkie fibns. Frohlich~pointed out that our know'ledge of the behaviorof light within a metal is hmited. He chose towork with a film, thin enough that the absorp-tion of light in the metal could be neglected, andthus eliminated the term involving the absorp-tion coefficient o. from the equation for the electricvector of the light in the metal (see Eq. (16)).

He divides the space in three regions, (1) inthe film, (2) outside the film, on the side of theincident light (positive z-direction), and (3) theregion back of the film (negative z-direction); andderives the following wave equations for thethree regions, where the film is a rectangle ofsides li, ls, and thickness 2lp.

(1) il = 2a cos ksz sin k&z sin k&y, (22a)

(2) f= be'&' sin kix sin ksy, (22b)

(3) /=be "*sin kixsinksy, (22c)

where kk;/2s. m is the classical velocity in the x',

y, or z-direction as j takes the value 1, 2 or 3, and

P'= kps —Ss'mv, /k (23a)

b=2g cos kplse '"'3 (23b)

a = (2lils4) j. (23c)

PP H. Frohlich, Ann. d. Physik "I, 103 (1930).

He then allowed the light to act as a perturba-tion, as did Wentzel, and calculated the numberof electrons which are excited to energies anddirections such that they can escape from thesurface and obtained the following equations.

LEON B. LI NFOR D

For vp&v&v„

whereI= (e v„vE, /16sh2)v 4Ic4&((v —vp)/v, )+As((v —vp)/v„) + ' ' ' }

~i= ——+ —————— —+-— — +(24a)

For v&v,

where

——+—+ 3+-——+ —+—— — + .

I= (e' v,~vZ, 2/16m hs) v 4I8t+Bs+ ~ I,

—+ -+-—+ ——————— — +'(24b)

Bs= —2+- + —————+ ———— — +.

The general features of the spectral sensitivitycurves are the same as those of Kentzel andHouston. The distribution of velocities is differ-ent. This theory predicts that the most probableenergy of ejection is close to the maximum energy,with relatively few of the slow electrons. Thiseffect was noted by Lukirsky and Prilezaev, s" whofound that the thinner the film, the more nearlydid the observed energy distribution approachthe form predicted by this theory.

The positions of the spectral maxima for someof the alkali metals as calculated by Frohlich fromhis theory are in good agreement with the experi-mental values. The values of v were calculated onthe assumption of 1 free electron per atom usingEq. (7). The values for the thresholds were notthe best now available„but were of proper orderof magnitude. The frequency for maximum emis-sion v, as calculated was within 10 percent ofthe observed values. The most convincing agree-ment was for potassium where the data for thethreshold and spectral maximum w ere taken from

one spectral distribution curve for a distilledfilm of potassium on platinum reported by Suhr-mann and Theissing. " The calculated v,„. was

about 5 percent lower than the experimentalvalue.

(4) Theory of latm and Schsibin. Tamm andSchubin" independently made essentially thesame calculations as did Frohlich and carriedthem further. In addition their interpretation of

~~ P. Lukirsky and S. Prilezaev, Zeits. f. Physik 49, 236(1928).

~o L Tamm and S.Schubin, Zeits. f. Physik 08, 97 (1931).

many of the results is better. They point out thatan electron in field free space cannot absorb allthe energy of a light quantum and at the sametime conserve both energy and momentum. If theelectron is in an electric field, momentum can betransferred by means of the field to its source, orin this case to the metal lattice. Because of themass of the metal ions, the excess momentum canbe transferred with little loss of energy to the ions.

There are two types of fields in which conduc-tion electrons may be found. One is the field ofthe potential barrier at the surface of the metaland the other is the field of the crystal lattice.Since the latter fields are small compared withthe former, the lattice or volume effect can beneglected in calculating the effect due to the sur-face potential barrier or the surface effect.

The method Nentzel used to satisfy momen-tum conditions is not free from criticism. It istrue that a uniformly damped wave can be ex-panded in a Fourier series such that formally theconservation of energy and momentum in aphotoelectric process can be satisfied. Tamm andSchubin~ and later Bloch "call attention to thefact that the absorption of light in a metal isprincipally photoelectric, and therefore essen-tially a discontinuous process. The effect of thediscontinuity in the absorption of light on theresults of the theory is difficult to determine, soit is preferable to have a theory free from thisdefect.

Since Frohlich assumed no damping of theelectric vector of the light and that the electrons

"' F. Bioch, Phys, Zeits, 32, 881 (1931).


Px„x., x,= exp ji(2+ix„x., x,t—Kix—Xmy —Ksz)g. ux„x„x.{x,y, z), (26)

where K'= Ki2+Ks'+Ksm= yi, y = 87r'm/h and uis a periodic function of the coordinates, with aperiod equal to the lattice constant. X' is an in-verse length and proportional to the energy of thestate in question.

If a quantum of energy hv is absorbed by anelectron in the state K and raised to the state h

the energy equation

Pk„k, , k~= VX„X,, Ki+V (2'I)

~ R. Suhrmann and H. Theissing, Zeits. f. Physik 52, 453(1928).

were free in the metal, his result gives the emis-sion due to the surface effect. Tamm and Schubincall attention to the error in Frohlich's proposedextension for solid metals by summing up theeffect for each layer with an absorption coefficientfor both the light and electrons. Such a processwould assume surface type excitation in the in-

terior of the metal. They recalculate the surfaceemission for a thick block of metal and obtain

I= {e'v,[ 8, [12/2ch2i' cos8) ', {vd,) (3A —v)

-(~--)(3~+-) ln 0"+~')/(l~--l') jl (»)where 6= v —vp and 8 is the angle of incidence ofthe light. The shape of this spectral distributionfunction, Eq. {2S), is similar to that experimen-tally determined by Suhrmann and Theissing"for thick layers of potassium.

The above correlation indicates that while onlya very small part of the light is absorbed at thesurface of a metal, it contributes most of thephotoelectrons for frequencies near the threshold.Whereas most of the light is absorbed in the in-terior of the metal from which place few electronsescape. This may be due either to inelastic colli-sions of electrons in the metal before escaping, orto the excitation of lower energy electrons in theinterior than on the surface. The latter effectwould lead to a large number of electrons excited,but to an energy less than that required toescape.

To determine the conditions for absorption oflight within a metal, Tamm and Schubin used theBloch eigenfunction for electrons in the periodicfield of a metal lattice.

wherevp = 2(viv~) —vi,

i i= 4(cr/3)'l

(29)

(30)

This gives a second threshold frequency vp'

which can be called the threshold for the volumeeffect. It happens that with many metals it is alittle less than twice the threshold for the surface

24

PS

O

E4J

5300 4700 4I00 3500 2900 2300Wave -length (Anilstroms)

Fio. 3. How the observed spectral distribution functionmight be built up by the surface effect (broken line) andthe volume effect (dotted line). The solid line shows theemission observed by Suhrmann and Theissing from athick potassium film. (Tamm and Schubin. )

effect and for the alkalis at a frequency higherthan the first maximum of the spectral distribu-tion curve. The volume threshold is not sharp norcan its position be predicted accurately, therebeing a small volume effect for frequencies be-tween the surface and volume thresholds. Atfrequencies greater than the volume threshold,the emission from the interior of the metal, iflarge enough, will cause a secondary rise in thespectral distribution function. This effect hasbeen found on alkali metal surfaces. Fig. 3 from

must be fulfilled. Besides this the following dif-fraction condition is imposed as a result of thediscrete energy levels in the metal

n=1 2k;=K;&2m.n;/a (28)j=1, 2, 3,

where a is the lattice constant of the metal.The smallest frequency which can be absorbed

will be given by Eq. {28)when one n; = 1 and theother two are zero. The lowest frequency ip'

which can give electrons of energy hv, consistentwith the above conditions is

LEON 8. LI NFORD

Tamm and Schubin shows how the observedspectral distribution functions would be built upon their theory from a surface effect (broken line)and a volume effect (dotted line). The solid line isfrom the data of Suhrmann and Theissing" for athick potassium film.

(5) Penney's theory. Penney attempted a moreexact formulation for the emission from a thinfilm by using a model with lattice structure andintroducing an absorption coefficient for the lightin the film. The most striking result was the pre-diction of sets of discrete energy levels and bandsof allowed electron energies separated by energyvalues not allowed. These energy bands whichcannot be occupied ranged from e to above e,.Theexistence of such bands cannot be verified bypresent experimental methods. Other results aresimilar to those of the previous theories.

(6) General remarks. It appears that the workof Tamm and Schubin gives the best generaltheory of the photoelectric effect. While admit-tedly incomplete, it offers an explanation formany observed phenomena. It will be used forcomparison and discussion of particular phenom-ena in the next section.

As a general summary of the quantum me-chanical theories, the following features are evi-dent. All theories predict a spectral distributionfunction which leaves the axis at v= vs with finiteslope. All are based on the Fermi distributionat O'K. They predict a maximum at a frequencysuch that v) v „)vs. Tamm and Schubin's com-plete theory predicts a maximum, a minimumand then a second increase. For the heavy metalsthis second threshold is farther in the ultravioletthan investigations have been carried. All of thetheories predict that the component of the elec-tric vector normal to the surface 8, is much moreeffective in ejecting electrons than are the com-ponents in the plane of the surface. To the ap-proximation given, the formulas of Frohlich andof Tamm and Schubin depend on E. only.

Finally, it might be mentioned that the criti-cism of the theory of Tamm and Schubin raisedby Frenkel, ~ and his alternate explanation for theposition of the spectral selective maximum pre-sented in the same paper have been withdrawn. "

"W. G. Penney, Proc. Roy. Soc. A133, 407 (1931).~ J. Frenkef, Phys. Rev. L27 38, 309 (1931).'~ I. Tamm, Phys. Rev. $2739, 170 (1932).

III. SPEcIAL PHQTQELEcTRIc PHENoMENA

Although able to account for the generalfeatures of the photoelectric effect, there aremany special phenomena, especially those asso-cited with coated surfaces, which are not ex-plained by the general theories. Frequently someapplication of a quantum mechanical theory orsome classical consideration is helpful in under-standing specific characteristics of emission.Some of these considerations should later be in-corporated in a better general theory.

A. The efFect of temperature on photoelectricemission

In the quantum mechanical theories consid-ered, the electrons were assumed to have thevelocity distribution characteristic of 0 K, andthe work function es= kvs was defined as es= e, —e.

For temperatures above absolute zero there areelectrons with energy greater than e. Since thereis no sharp upper limit to these energies to use as acharacteristic energy, the definition of the workfunction as given above will be retained for alltemperatures. As a result, electrons can beemitted by light of frequency less than the thresh-old frequency if the metal is above absolute zero.The work function will be said to depend ontemperature only if the energy, required to re-lease an electron whose normal component ofenergy is e, changes with the temperature; or inother words, if e, depends on T.

While working on potassium films Lawrenceand Linford~ noted the difference between thetheoretical curves of Wentzel and Houston, Eqs.(l9) and (20) and the observed tangential ap-proach to zero of the spectral distribution func-tion. They reported that the magnitude of thedifference could be explained by the temperatureeffect.

Recently, Fowler'7 has derived a theory which

gives a different method of plotting experimentaldata from which the threshold for O'K can bedetermined. He worked through the theory withthree initial assumptions, and among other thingscalculated I„ the emission produced by light ofthe threshold frequency as a function of tempera-ture. The assumptions were:

~ E. O. Lawrence and L. B. Linford, Phys. Rev. {2736,482 (1930).

~r R. H. Fowler, Phys. Rev. t 27 38, 45 (1931),


(1}That all electrons with energy greater thane, would escape. The result for T=O is similar tothe first term of Wentzel's first equation, Eq. (19).He found that I -T and that for T)0 thecurve approached zero with finite slope. The dis-placement was smaller than that observed, so theresult was discarded.

(2) That all electrons with normal componentof energy greater than e would escape. For T= 0the result was similar to the first term of Hous-ton's equation and Wentzel's corrected formula,Eqs. (20) and (21). In this case I T'.

{3)Fowler then introduced a term to accountfor the probability of excitation, and obtained aresult for T= 0 similar to the first term of Froh-lich's formula, Eq. (24a). On this assumptionI„TI~'.

In the last two cases the approach of thespectral distribution curve to zero is tangential.The temperature corrections are so nearly correctthat present experimental data are not sufFi-

ciently accurate to indicate either one as superior.Young and FranlP' suggested a different

probability of escape from the surface thanFowler's third assumption, which formulationleads to I„T".So far it has not been testedwith experimental data, but it is doubtful whetherthe difference is great enough to detect.

In calculating the temperature dependentspectral distribution function on the second as-sumption, Fowler was interested in the functionfor frequencies near the threshold. This allowedhim to make the simplifying assumptions thatthe probability of excitation of any electron isindependent of the frequency of the light andthe energy of the electron. He then calculated thenumber of electrons which with the addition ofthe energy hv would have sufficient energy normalto the surface to escape.

By integrating Eq. (6) for the distribution ofvelocities of the electrons, over all values of $ andi1 one obtains the number N(g)di of electrons perunit volume whose z-component of velocity liesbetween f and f'+dg. The number N of electronsper unit volume which can be given sufficients-component of energy to escape, by light offrequency v will be N(I)df integrated over all

'~ L. A. Young and N. H. Frank, Phys. Rev. [2g 38, 838(1931).

velocities such that +mP~h(v —v), or

2m3 +mN=-k $ — y — fmp h(»N-»)

d$dgdf(31)

exp I gm(P+g'+P) —ej/kT]+IThis integral must be expanded differently as

p=(kv —hvs)/kT is greater or less than zero.For p~0

~(2m) '~' O'T'N=

h3 (hv, —hv) &

for p —0

g2» g 8»

X e»—+——22 32

(32)

m (2m)'~2 k'T'N=

h3 (kv, —hv)&

3-2

X —+-p' —e-» ——+——1

6 2s 32

When T~O

(33)

N {kv—kv0)'/(kv —kv)& for v & vs, (34a}

when v= vo

N T'.

for v( vs (34b)

(35)

The assumption that the emission is propor-tional to the number of electrons with normalcomponent of energy sufficient to escape if theentire energy hv is added to the normal com-ponent of energy may be expressed by I ¹

Then from Eqs. (32) and (33)

I(h —h )I/T'=Af(p)=Af((h —h o)/hT), (36)

where A is a constant independent of v and Tand f(p) represents the terms in the squarebrackets of Eqs. (32) and (33) as y is greater orless than zero.

Since v, —v is large for frequencies near thethreshold, small changes in v will leave (v, -v)Inearly constant, so that it can be absorbed in A.Define q(p) = log f(p) and take the logarithm ofEq. (36).

log (I/Ts) =B+y(p) =B+q ((hv —hvo)/kT). {3&)

LEON B. LI NFORD

If q(p) is now plotted as a function of p, thecurve might be considered a modified spectraldistribution function whose shape should bethe same for all surfaces. If on the same scale,log 1/T' from experimental data be plottedagainst hv/kT, one should obtain a curve of thesame shape but shifted parallel to itself from thetheoretical one. The vertical shift necessary tobring the curves into coincidence is a measure of9, which depends on the units of current andlight intensity, and the probability that a quan-tum of light will eject an electron. As theseshould be practically independent of the tem-perature for any one surface, the curves for thesame surface at different temperatures shouldrequire the same vertical shift to bring them tothe theoretical curve.

Since the theoretical curve is plotted as a func-tion of p= (hv —hvp)/kT and the experimentaldata against hv/kT, the required horizontal shiftin the curve is kvp/kT from which vp and thework function can be calculated.

Fig. 4 shows the results of the analysis of datafor outgassed palladium by the above method asreported by DuBridge and Roehr. " The solidline is the theoretical curve and the points repre-sent the experimental data for the various tem-peratures after they have been shifted to coincidewith the theoretical curve.

DuBridge~ has pointed out that the abovemethod requires the determination of the relativeintensities of the light source for the variousfrequencies. He suggests that one plot p(p)against log

~p~. Then experimentally determine

the emission I at various temperatures with lightof a single frequency, and plot I/T' againstlog 1/T (= —log T). To make the curves coin-cide there will again be a vertical shift of littleconsequence, but the horizontal shift will belog (hv —hvp)/k since log p= log (hv —hvp)/k—log T. Thus from the frequency v of the inci-dent light, vp can be calculated.

To show the internal consistency of themethods and the degree to which the two meth-ods of analyzing the data agree, the results for theemission of clean palladium are given in Table I.

~~ L. A. DuBridge and W. W. Roehr, Phys. Rev. L2j 39,99 (1932).

~ L. A. DuBridge, Phys. Rev. |2)39, 108 (1932).

I Al3LF. I.

Fowler's method~Temperature Work

of surface function('K) (volts)

305 4.96400 4.97550 4.97730 4.97830 4.98925 4.98

1005 4.961078 4.97

DuBridge'sWave-lengthof incidentlight (A)

248223992378234523022225

method"Work

function(volts)

4.964.954.944.944.984.98

Average 4.97 volts Average 4.96 volts

The two methods give essentially the same re-sult. The internal consistency of Fowler's methodseems somewhat better. In either case the errorin a threshold determination from the data issmall compared with the error which would beintroduced into the data by a small amount ofcontamination on the emitting surface. For aclean surface the choice of method may be gov-erned by the convenience in obtaining the data.

Table I as well as other data" for clean surfacesshows no trend in threshold with temperature be-tween room temperature and 1100'K. A trendof 1 percent or 0.05 volts in the threshold forpalladium would be detected easily. The evidenceis that for the metals studied, the height of thepotential barrier kv is constant to within 1

percent.The vertical shift required to bring the experi-

mental curve over Fowler's theoretical curve is8, Eq. (37), and the anti-logarithm of 8 is pro-portional to A, Eq. (36). In an experiment on onesurface, A is proportional to the probability thatan incident quantum will eject an electron. Thereflection coefficients of most metals are sensiblyconstant so that considerable changes in A meanchanges in the emission efficiency. Thus analysis

by Fowler's method separates the changes inspectral sensitivity curves into changes in thresh-old and changes in emission efficiency. If 8, andtherefore A, varies with the temperature, Fow-ler's method of analysis shows this as a change invertical shift of curves taken at different tem-peratures. DuBridge's method of analysis fails asa dependence of 8 on T will change the shape ofcurve as plotted, making a superposition impos-sible. Changes in 8 with impurities on the surface,


+'ACOV

+C4

Ol

FII

Te,my. oK

+ 305400

x 550730

o 860+ 925

10051075

5 10hV- hV,

KT

15 20

Fir. 4. Analysis of photoelectric data, for clean palladium, by Fowler's method. The points representing the data forvarious temperatures have been shifted to coincide with the theoretical curve. (DuBridge and Roehr. )

and with temperature are found in the data ofWelch and Warner.

Welch" determined the spectral distributionfunctions for various metals at different timesafter renewing the surface by filing. Evidently thechanges were due to gas or vapor collecting on thesurface. The work functions of the metals in-creased with time in amounts varying up to 0.16volts, except that for germanium which de-creased slightly. Without exception the verticalshifts corresponded to decreases in the emissioneSciency to about one-half.

Warner" used Fowler's method to analyze datafor a tungsten filament. From his diagram (his

Fig. 2) the change in vertical shift correspondedto an emission efficiency at 790'K about one-tenth that at 1100'K. This was ascribed to im-purities on the tungsten.

In the above cases the changes in emission effi-

ciency were large and therefore evident from thespectral distribution curves. Where the changesare small, and accompanied by changing thresh-olds, this method of analysis should prove ofvalue.

For convenience in analyzing data by eithermethod, DuBridge~ has given a table of valuesof logip

~p, j

aild of p(p) for values of p from —8.0to +50.0.

4' G. B.Welch, Phys. Rev. $2J 32, 6S7 (1928). Determi-nation of work functions by Fowler's method, Phys. Rev.P2] 40, 470 (1932).The author wishes to thank Dr. Welchfor subsequently furnishing the data on the vertical shifts.

o A. H. Warner, Phys. Rev. P2g 38, 1871 (1931).

B. Saturation phenomena

As mentioned in the introduction, certaincoated surfaces require abnormally high acceler-ating potentials in order to produce a saturation

LEON B. LI NFORD

(38)eo —e,= e(eE) .

Lawrence and Linfords' found this approximatelytrue for shifts in threshold of a thick potassiumfilm on tungsten. For small fields some surfacesshowed greater shifts in the threshold than thetheory predicts,

Deviations from Schottky's equation forthermionic emission have been investigated byBecker and Mueller~ and by Reynolds. "Largedeviations from Eq. (38) were found by Linford"on thoriated tungsten, and Huxford" on oxidecoated surfaces.

Seeker and Mueller~ showed that if the ap-parent work function as a function of the field is

"H. E. Ives, Astrophys. J. 60, 209 (1924)."R. Suhrmann, Naturwiss, 16, 336 (1928).~ %.S. Huxford, Phys. Rev. L2j 38, 379 (1931).4~ %.Schottky, Phys. Zeits. 15, 872 (1914)."S.Dushman, Rev. Mod. Phys. 2, 381 (1930) gives an

excellent review of the subject of thermionic emission.4~ K. T. Compton and I. Langmuir, Rev. Mod. Phys. 2,

123 (1930).The sections on contact potentials and electronemission in accelerating electric fields pp. 144-160, containa discussion of some of these phenomena.

"P.Debye, Ann. d. Physik 33, 41 (1910)."J.A. Becket and D. W. Mueller, Phys. Rev. L2j 31,

341 (1928)."N. B. Reynolds, Phys. Rev. 12) 35, 1S8 (1930).~ L. B. Linford, Phys. Rev. f2 j 36, 1100 (1930).

current, whereas clean surfaces saturate with lowfields. Ives~ reported that it required higher ac-celerating fields to saturate the current from thin,than thick alkali metal films. Suhrmann44 notonly found the same effect, but also that when

using light near the threshold, a higher accelerat-ing field was required to produce saturation thanwhen light of considerably higher frequency wasused. Huxford" found both of these effects truefor oxide coated cathodes of the type used forthermionic emitters.

This phenomenon is closely allied with devia-tions from Schottky's equation" for thermionicemission in accelerating fields, shown by activatedsurfaces. Schottky's equation is derived on theassumption that electrons leaving the metal sur-face are subject to an image 6eld, Eq. (12), andhas been verified" ~ for clean surfaces.

Assuming the image force law, Debye" showedthat in an accelerating field of E e.s.-units cm —',the effective or apparent work function e, isgiven by:

known, the surface field against which the elec-trons must escape may be calculated from theformula:

de./dE= —s,e. (»)Where E is the applied field in e.s.-units and si isthe distance from the surface to the point wherethe applied field equals the surface field, and thusto the point where the field acting on the electronis zero. The field they calculated was about theimage 6eld at distances less than 2X10 ' cm,but much larger at greater distances. At 10~ cmthe calculated field was about 1000 volts cm ' ascompared to the image field of 3.6 volts cm '.

For convenience in work with photoelectricdata, Eq. (39) may be written

dv, /dE, = —ate/300k (40)

where v, is the effective threshold and E, is theapplied field in volts. cm '.

To account for lack of saturation Langmuir"proposed the theory that the substance on thesurface formed a non-uniform 61m, that is, cer-tain areas or patches were more densely coveredthan others. Since the work function of a 61mcovered surface depends on the thickness of the61m, some patches would have a lower work func-tion than others. This theory has been used byvarious workers to explain emission data.

In order to make quantitative comparisonwith experimental data, it is necessary to calcu-late the local fields due to the patches, and thendetermine the emission to be expected with thesefields superimposed on the image field.

If two metals A and B with work functions eh

and e&, be brought in contact they will show acontact potential difference Vs —Vh. Energy con-siderations show that:

eh eB= VB Vh+PhB (41)

where PhIi is the Peltier coefficient at the junc-tion of the two metals, and is so small comparedwith the other terms that it can be neglected.Thus the contact potential difference betweentwo metals can be said to be equal to the differ-ence in their work functions. The metal of lowerwork function is the more electropositive. Thepresence of other metals in the circuit between Aand B does not affect the above relations.

~' I. Langmuir, Gen. Elec. Rev. 23, S04 (1920).

EXTERNAL PHOTOELECTRI C EFFECT

If a surface has areas with a work functiondifferent than the rest of the surface, there will bea contact potential difference between the twoclasses of areas, and there will result local electro-static fields. The contact potential of the surfaceas a whole may be considered only with referenceto another surface at a sufficient distance that thelocal fields are negligible. A contact potential sodetermined will determine the work function ofthe surface in zero field. This contact potentialwill be the surface average of the contact poten-tials of the various patches.

With no accelerating field, electrons whichescape through the areas of lower work function,which are the more electropositive areas, must doso against not only the image 6eld, but also thepatch 6eld. Those escaping from the more electro-negative areas travel against the image field de-creased by the patch field. The potential barriersover the two types of areas are sketched in Fig.5 for the case where the patch fields extend outfrom the surface to a much greater distance thanthe image 6eld, and assuming equal areas of thetwo types of patches. The line OB represents thepotential of field free space away from the sur-face, and is chosen as zero potential. The curvesabove OB are for the patches of higher, and thosebelow for the patches of lower work function. MNrepresents the contact potential difference be-tween the two types of patches. MB and NBrepresent the potentials above the centers of therespective patches, and these added to the poten-tial of the image 6eld ADB give the resultingpotential barriers AEB and A FB. Above otherpoints on the surface, the potential curves liebetween these two. The height of OB determinesthe threshold in zero field, but it is evident thatno electrons with just sufficient energy to escapecan do so through the more electronegativepatches. The effective work function of the centerof the more electropositive patches in zero fieldis greater by an amount OM than their workfunction would be, were there no patch fields.

If an accelerating field be applied, its potentialis represented by the line OC, Fig. 5, with a slopeequal to the magnitude of the field. The resultingpotential barriers (broken lines) are obtained byadding the potential of the field to those of thebarriers in zero field. The change in work func-tion is the difference in maximum heights of the

barriers with and without field. The diagramshows that the effective work function of theelectropositive patches (curves AEB and AEC) isdecreased by an amount much greater than theeffective work function of the electronegativepatches (curves A FB and AFC). The work func-tion of a clean surface with an image field barrier(curves ADB and ADC) is decreased slightlymore than that of the electronegative patches.Thus in accelerating fie1ds the effective work

-0.2.

N

~-01o

~ 0.0C0

~ 01

C.

'02

~ 03A 2 4 6 5 10

Detun~g fromm Surt'uca (c~«10')

FIG. 5. Image field potential barrier as modified bypatches and an accelerating field. ADB is the imagepotential. N and M are the potentials of the patches ofhigher and lower work functions, respectively, and thepotential of the patch fields are NB and 3fB.The resultingpotential barriers are A FB and AEB. The broken linesshow the barriers in an accelerating field of 1000 volts.cm '. (Calculated for MN or V0=0.36 volts and patchdiameter b =LS X 10 ' cm. See p. 50 and Fig. 6.)

function of the electropositive patches deter-mines that of the surface, and most of thethermionic electrons are emitted through thesepatches.

For small changes in applied field, the changein maximum height of the potential barrier hs„is to the first order, equal to the change in thepotential of the applied field at the potentialmaximum. This in turn is equal to the product 1of

the change in applied 6eld, d,E, the electroniccharge e, and the distance zi from the surface tothe potential maximum, or where the surfaceand applied fields are equal and opposite, orde, =zied, K This taken to the limit of smallchanges in field is Eq. (39).

Compton and Langmuir assuming a checker-board arrangement of patches b cm square, andsuch that there is a contact potential differenceVs between adjacent patches, calculated the

LEON B. L1NFORD

patch fields and expressed the result in a Fourierseries. Assuming the patches to consist of clustersof closely packed thorium atoms, certain con-siderations led them to choose b=10 ' cm andVs=1.9 volts. The thermionic emission as afunction of the accelerating field as calculated forsuch a surface, followed the Schottky law for lowfields, and showed increasing deviations for highfields, which is exactly opposite to the observedeffect. They concluded that the patch theorycould not account for the observed emission.

Independently Decker and Rojansky~ workingwith thermionic data and Linford~ working withphotoelectric data showed that the experimentalresults could be accounted for, if proper valuesof Vp and b were chosen. Values of b were about10~ cm while Vp depends on the amount ofabsorbed material. The assumptions as to thearrangement of patches were the same as thoseof Cornpton and Langmuir. A slight error in

their expression for the potential, due to patches,Vs, in the space above the surface was corrected.The corrected expression is:

sum of the image field E; and the patch field,and is given by

E,=E,+E,=e/4z'+(8 Vp21/trb)e —'1~*" (42a)

%'ith the values of the constants Up=0. 36volts and b= 1.8X10~ cm the observed and cal-culated fields were almost identical. Becker andRojansky found that similar values of the con-stants gave fields whose emission characteristicswere in agreement with thermionic experimentaldata.

Since the work function of thoriated tungstenis about 1.S volts less than that of tungsten,the small contact potential difference betweenpatches shows that the thorium atoms are dis-tributed over the entire surface, and the patchesare due to differences in the density of covering.The size of the patches is the same order of mag-nitude as the size of the tungsten crystals in thefilaments used. Dr. Seeker has suggested thatthere is a definite correlation.

The general features of the patch fields areshown in Fig. 6 where the surface fields as func-

8VpV,=lV.+—,r.(-1)' -p L- I(2j+»

f ij,

+(24+1)'I '"m.s/b)

cos (2j+1)m.x/b cos (24+1)sy/b

2j+1 2k+1

j, k=0, 1, 2,

where the origin of coordinates is taken at thecenter of an electropositive patch.

The photoelectric method of determining thesurface fields depends on measurements of thethreshold. The place where electrons can escapewith minimum energy is above the center of apatch of low work function, so the fields deter-

mined by this inethod will be those above thecenter of such patches. To calculate the field E„above the center of an electropositive patch, setx=y=0 in Eq, (42), and then E„=—dU„/dz.In the correlation with the photoelectric dataonly the first term of the expansion was used

and thus j=k=0. The surface field, E, is the

~ J. A. Becker and V. Rojansky in unpublished work

kindly communicated to the author.ss L. B. Linford, Phys. Rev. [2j 37, 1018 (1931).

1O'

elg-6

o1

sl

10' 10' lQ'Distance fram Surface (crn)

10' B

Fic. 6. Patch and surface fields plotted on logarithmicscales. AB is the image field. The broken line is the patchfield for Va -—0.36 volts and b = L8 X 10~cm. The solid line,above, which merges with AB at high and low fields is theresulting field above a patch of low work function. Thecircles are the photoelectrically determined fields (Linford)from which the patch constants were determined. Thetriangles are thermionically determined fields (Becker andMue)ler), both for thoriated tungsten. The crosses arefields above an oxide cathode calculated from photoelectricdata of Huxford. The dotted line and the solid line aboveit show the patch field and resulting surface field with thepatch constants as assumed by Compton and Langmuir,Va=1.9 volts and b=10~ cm. The arrows indicate thepoints on the curves where the distance from the surfaceis one patch diameter.

EXTERNAL PHOTOELECTRIC' EFFECT

tions of the distance from the surface are plottedon logarithmic scales. The straight line AB is theimage field. The dotted curve is the patch fieldcalculated on the assumptions of Compton andLangmuir of Va= 1.9 volts and b= 10~ cm. Thesolid curve above this, and merging with theimage field at about 10 ' cm from surface, is theresulting surface field. By using the valuesV0=0.36 volts and b= 1.8X10~ cm, the patchfield is shown by the broken line, and the re-sulting surface field is the solid line merging withthe image field for high fields and the patch field

for lower values. In each case the arrow indicatesthe distance from the surface equal to one patchdiameter.

The circles show the observed values of thesurface field from photoelectric data" for whichthe constants were determined. The trianglesshow the fields for 70 percent thoriated tungstencalculated by Becker and Mueller~ from thermi-onic data.

The following facts are evident from Eq. (42a)and Fig. 6: (1) That the patch field at the surfaceis proportional to Vo and inversely proportionalto b. (2) That out from the surface the field de-creases to about one-tenth value in a distance~ib, to one-hundredth in distance b and at 2b toabout one-ten thousandth. The diagram showswhy the values of the constants chosen by Comp-ton and Langmuir predicted deviations from theSchottky equation at high fields.

It must be remembered that the above calcula-tions are based on an ideal model, and the con-stants evaluated are therefore average values. Incase the patches were not all of about the samesize, but consisted of small scale fluctuations

superimposed on larger patches, a two-humpedcurve would result which would be of the form ofthe uppermost solid line of Fig. 6. Actually such acase is indicated by the photoelectric data ofHuxford" for oxide coated cathodes when calcu-lated in terms of fields by Eq. (40). The points areindicated by the crosses, Fig, 6. Though the datadoes not go to high enough fields to be certain, asuperposition of patches of about 10 ' cm diam-eter on patches with a diameter of 2 or 3X10~cm would account for the observed fields.

To explain the lack of saturation of the photo-electric current shown by certain surfaces at lowfields, one needs only to postulate patches with b

about 10~ or 10 ' cm and Us of the order of afew tenths of a volt. Light near the thresholdfrequency can eject electrons only from the areasof lower work function. Small accelerating fieldswill make relatively large changes in the workfunction of these areas, and since the light isnear the threshold frequency the relative changein the emission will be large.

If light of much higher frequency is used, emis-sion will occur from all areas. The threshold ofareas of high work function will be affected some-what less by the applied field, than would a cleansurface. The relative change in emission fromareas of lower work function will be less than withlow frequency light, due to the fact that a givenabsolute change is a smaller proportion of thetotal emission. If the higher frequency light isin the neighborhood of the spectral maximum theabsolute change in emission with apparent workfunction will be smaller. The net result is a muchsmaller relative change in total emission withapplied field in light of high frequency than inlight near the threshold frequency.

Extremely large changes in eRective workfunction with applied field have been reported byNottingham. ~ One alkali metal film on a heavymetal showed a decrease in eRective work func-tion of 1.9 volts with an accelerating potential of4 volts. No details are given, but if plane parallelelectrodes d cm apart were used, the field whichcaused the shift was 4/d volts cm '. Referenceto Eq. (39) shows that if a change of 1.9 volts inwork function is produced by a 4/d ~olts cm 'field, the mean distance zt from the surface towhere the surface and applied fields are equalmust be of the order of -',d or about midway be-tween the electrodes. No uniform surface chargedistribution can produce surface fields at such adistance, This would indicate that there wereinhomogeneities in the film with linear dimen-sions of the order of the distance between elec-trodes. These inhomogeneities might be due toa non-uniform deposition of the alkali film. Voilethe author was working with Lawrence on thinpotassium films on tungsten filaments so thatpatches of a size comparable with the distancebetween electrodes were impossible, a like eRectdid not exist. The sharp discontinuity in the rate

~ V . B. Nottingham, Phys. Rev. $2) 3S, 669 (1930}.

S2 LEON B. LINFORD

of change of the threshold with field, betweenthat characteristic of a retarding field, and a rateabout twice that predicted on image field theory,was used to determine zero field between anodeand cathode.

Suhrmann and Theissing" reported that theyfound evidence for the existence of areas ofwidely different work function on a potassiumfilm on platinum. They used Suhrmann's"method of calculating the complete emission I, byEq. (4) from the observed spectral distributioncurves. Using Eq. (3) they plotted log I/P'against 1/T. The theory predicts a straight line,of slope —esjk. They found that the slope of thecurve changed with T. In the temperature range1200'K & T&2000'K, where T is the temperaturecharacteristic of the black body radiation, theslope of the curve corresponded to a work func-tion of 2.02 volts. For the range 2400'K&T&4000'K the slope indicated 2.98 volts. Theirexplanation was that radiation characteristic oflow temperature could eject electrons only fromareas of lower work function, whereas the lightfrom a higher temperature source could ejectelectrons from all areas, and therefore the workfunction would be more characteristic of the en-tire surface.

Replotting the data with r=2, the slopes cor-respond more nearly to 2.46 and 2.93 volts, re-spectively, and with r=5 the plot is a goodstraight line. Suhrmannss reported that values ofr higher than 2 gave better results.

An analysis of their data by Fowler's methodshowed that for the two thicker films, their Figs.7 and 8, the thresholds for room and liquid airtemperatures are practically the same. The verythin film, much less than 1 atomic layer, theirFig. 6, showed a work function 0.2 volts lower atliquid air temperature. Since for such films smallvariations in the amount of alkali produce largechanges in work function, the observed differencecould be the result of the condensation of a littlemore potassium on the surface when cooled. Theshapes of all of the curves were very nearly thatof Fowler's theoretical curve, which would beimprobable were there large changes in effectivework function with the frequency.

4' R. Suhrmann and H. Theissing, Zeits. f. Physik 73, 709(1932).

The curvature of the log I/T' us. 1/T plotsmight be due to the inaccuracy of Richardson'sequation as applied to photoelectric emission.The spectral distribution function that Richard-sonis calculated by equating the right-handmembers of Eqs. (3) and (4) showed a steeperslope near v= vs than that observed. If the com-plete emission calculated from Richardson'sspectral distribution function would give straightlines for the log I/T' ns. 1/T curves, the observedspectral distribution functions would be expectedto give the shape reported by Suhrmann andTheissing even though the surface has only onework function.

Since this article was prepared, Nottingham"has presented an alternate theory to accountfor the observed changes in the apparent workfunction, of certain complex surfaces, with theapplied field. As previously stated these cannotbe explained by the image field theory.

He presents observations of his own, and citesothers, on the thermionic and photoelectricemissions of different complex surfaces, usingsmall accelerating and retarding fields. When hestudied the thermionic emission of a thoriatedtungsten filament as a function of the field, hefound that the observed velocity distributionwas Maxwellian, but it was a distribution char-acteristic of a temperature higher than that ofthe filament. In addition, he pointed out thatthe A coefficient of Richardson's equation (Eq.(3)) which measures the emissivity, decreasedwith increasing fields. Thus with acceleratingpotentials of more than 6 volts, its value was ofthe order of one-tenth of its value for a cleansurface.

To account for these phenomena, he postulateda potential barrier consisting of an image field

potential similar to 3PDO, Fig. 2, for distancesfarther from the surface than the layer ofthorium atoms; a potential minimum at thislayer; and finally a potential maximum ofapproximately parabolic shape between this layerand the surface of the underlying metal. Thispotential maximum is postulated to be higherthan AB. If the collector is at the potential of

4" %. B. Nottingham, Phys. Rev. I 2] 41, 793 (1932).The discussion of this paper has been added in proof, thusit is impossible to present diagrams which would make theexplanations clearer.


AB, electrons with just sufficient energy to reachthe collector would have to pass through thebarrier.

If a retarding field be applied, the potentialbarrier out past the thorium layer would ap-proach a line with positive slope, instead of aline like AC, Fig. 2, with negative slope. If thepotential of the collector is higher than the topof the barrier between the film and the basemetal, which will be called the "film barrier, "any electron with energy enough to reach thecollector will pass over the film barrier. Underthese conditions the observed emission was, asit should be, the same as though no film werethere. If the potential of the collector is droppedto that of the emitter, the slower electrons will

have to pass through the film barrier which hasa transmission coefficient less than unity. Thefaster electrons will still be able to travel overthis barrier. This will result in the escape of alarger proportion of the high-velocity electrons,and thus the velocity distribution would becharacteristic of a temperature higher than thatof the filament. Nottingham was able to con-struct a film barrier of atomic dimensions whichwould produce the required filtering out of theslow electrons.

He then used this model to explain his photo-electric results previously mentioned. "He foundthat in retarding fields, the change in thresholdwas as predicted by the Einstein equation,(Eq. (1)); and that in moderately large fields,the shifts approached those predicted by theimage field theory. With the collector near thepotential of the emitter, the changes were toorapid to be explained by the image field theory,and not enough to follow Einstein's equation.

His explanation assumed that Einstein's lawwas followed as long as the potential of thecollector remained above that of the film barrier.As soon as the collector became more negativethan this, the slower electrons had to penetratethrough the film barrier. Due to the fact that afinite current must flow to be measured, hepointed out that a measurable current wouldflow only when there were electrons excited toenergies greater than the minimum required toreach the collector, after passing the film barrier.This effect would become more pronounced, themore the potential of the collector was dropped,

and thus the change in the observed thresholdwould be less than the change in the potentialof the collector.

The explanation works well so far, but failswhen accelerating fields are applied. ContinuingNottingham's explanation to the region of smallaccelerating fields, the highest point, of the imagefield barrier outside the film, would be reducedaccording to the image law, and as before theobserved change in the threshold would be lessthan the decrease of the image barrier. This isnot what is observed, as his data, as well as allother data known to the author, show shiftsequal to or greater than those predicted on theimage field theory.

Nottingham's recent article shows the diagramof the photoelectric apparatus used. The emitterwas a nickel cylinder inside a cylindrical col-lector. The former was withdrawn from thecollector in order to clean and evaporate sodiumonto it from the side. In preparing very thinfilms, it would be difFicult, if not impossible, tocoat the cylinder uniformily. When the films arethin, small changes in thickness mean largechanges in work function. Thus patches of theorder of the diameter of the emitter could bepresent, which as stated before would accountfor the effect.

The application of this theory to the thermi-onic effect is out of the scope of this paper, butthe following observations might be interesting.It is evident that the film barrier theory willwork in the region of low fields, but its applica-tion to emission in fields of the order of magni-tude used by Becker and Mueller is doubtful,as it would probably predict an effect similar tothat predicted in the photoelectric case. Thispoint should be investigated in greater detail.

It should also be noted that the patch theorywill account for the filtering out of a part of theslower electrons emitted thermionically, and forthe small value of the A in Richardson's equationin large fields. To do this one must rememberthat the patches must be subdivided into areaswhich have the various potential barriers be-tween the extremes shown by AFB and AEBFig. 5. To obtain the emission, it is necessaryto calculate it for each type of barrier and sumover the surface as did Becker and Rojansky. ~If the potential of the collector is above F, all

LEON B. LI NFORD

areas of the surface will emit normally. If thepotential of the collector is between I' and B,the electropositive areas will be able to emit asbefore; but from the electronegative patches,only those electrons with energy sufficient toget over the barrier outside the film can escape.Thus there would be an abnormally large pro-portion of higher speed electrons. As the po-tential of the collector is dropped still more, thefield becomes accelerating, then an increasingproportion of the emission comes from thecenters of the more electropositive patches. Thiseffectively reduces the emitting area of thefilament and thus reduces A.

In comparing the respective merits of the twotheories, one finds that the film barrier theorycan explain, in a qualitative way, the photo-electric emission in very small fields, and breaksdown in larger accelerating fields. The patchtheory explains the phenomena in large fields,

and with reasonable assumptions, explains theexisting data at low fields. As will be discussedin section E, the assumption of a potentialbarrier, with low transmission coefficient, throughwhich emitted electrons must pass, would leadto a small photoelectric emissivity, whereas alarge one is observed for film covered surfaces.In the field of thermionics, the film barrier theorywill account for observed phenomena in low

fields, and the patch theory explains the emissionin large accelerating fields. The possible applica-tion of the film barrier theory to emission in

high fields is doubtful, whereas the patch theorycan account in a qualitative way for many ofthe characteristics of emission in low fields. Morecareful experimental and theoretical investiga-tion of this subject is needed.

C. Preparation of photoelectric surfaces

Various methods have been used to preparephotoelectric surfaces for investigation. Twomethods commonly used are, evaporation in high

vacuum; and for metals of high melting point,heating by radiation, conduction or electronbombardment. Two more methods have beenreported recently, and the results are sufficientlygood to warrant consideration.

The first method is outgassing by exposure to

ultraviolet light. Millikan~ noted an increase inphotoelectric emission when metals were exposedto strong ultraviolet light. This effect has beenidentified as a removal of gas and has beenstudied by Winch'" particularly on thin unbackedfilms of gold. He found that continued illumina-tion of the gold film shifted the threshold to thered to a final value of 1.164X10's sec ' as com-pared with the value reported by Morris~ forgold after prolonged heat treatment. Winchshowed that: (1) Light of less than the thresholdfrequency was ineffective in outgassing. (2)The back side of the film outgassed slowly andat a relative rate comparable with the fraction ofeffective light transmitted through the film. (3)When the external field was reversed so that theelectrons were returned to the surface the out-gassing was accelerated. It appears that thephotoelectrically ejected electrons knock the gasmolecules from the surface.

Similar effects were noted with solid gold and aribbon filament of silver. This method has beenused by Dillon to outgas single crystal zinc. "

The effectiveness of electrons in removingsurface gas was reported by Suhrmann, " whoshowed that electron bombardment would re-move hydrogen from silver and gold which couldnot be removed by heating.

Rentschler, Henry and Smith~ have reportedthat good photoelectric surfaces can be producedby sputtering. The metal to be investigated wasprepared in wire form, it and all parts were thor-oughly outgassed and then the metal sputteredfrom the wire onto a metal sheet to act as cathode.The work functions of tungsten and tantalumsurfaces so prepared agreed with thermionicvalues. This method can be used for preparingsurfaces of several metals which are difficult toclean by heating.

They found that thick films of certain metals

~' R. A. Millikan, Phys. Rev. [1329, 85 (1909); [1j 30,287 (1910); [1$34, 68 (1912)."R. P. 4%inch, Phys. Rev. [2]38, 321 (1931).

ss L %. Morris, Phys. Rev. [2j 37, 1263 (1931},ob-tained a value of L172X10" cm ' for the threshold ofoutgassed gold.

"J.H. Dillon, Phys. Rev. [2j 38, 408 (1931).'-' R. Suhrmann, Zeits. f. Physik 33, 63 (1925); Zeits. f.

Elektrochemie 37, 681 (1929).~ H. C. Rentschler, D. E. Henry and K. O. Smith,

Phys. Rev. [2j40, 1045 (1932).


such as thorium showed higher work functionsthan the thermionic work function for thin films

of the same metals on tungsten. This is evidentlythe same effect as found photoelectrically forthin alkali films. That thorium films of a certainthickness on tungsten have a thermionic workfunction less than thicker films has been shown

by Brattain. ~

D. Space charge 6EectsThe possible contribution of space charge to

the effective surface fields, and the resulting ef-fect on photoelectric emission is of interest.Under conditions of high thermionic emission,space charge becomes an important factor.

Bartlett and Waterman" have calculated thesurface fields on the assumption that spacecharge is the largest factor even at low emissions.In so doing they neglect the image forces whichare undoubtedly present, and use Poisson'sequation, which assumes continuous distributionof charge, in regions of low electron density. Atlow temperatures, space charge cannot be a factorin the production of fields at distances of the orderof 10 ' cm from the surface, as will be shownin the following paragraphs, and therefore theiranalysis does not apply at such temperatures.

Zwickker" made a similar calculation, tryingto work with both space charge and image eHectssimultaneously. It was impossible to make arigorous solution, but from approximations, heconcluded that space charge was of minor im-portance in the surface fields,

The effect on photoelectric emission at abso-lute zero can be investigated quite easily. Re-gardless of the source of the fields, there is a po-tential barrier with a height equal to the workfunction es above the energy e of the fastest elec-trons in the metal. Without additional energynone of the electrons can exceed the critical dis-tance from the surface to the place where theheight of the potential barrier is e.* With no

~ W. H. Brattain, Phys. Rev. f2j 35, 1431 (1930).'s R. S. Bartlett and A. T. Waterman, Phys. Rev. f2j

37, 279 (1931). R. S. Bartlett, Phys. Rev. f2J 37, 959(1931);f2J 3S, 1566 (1931).A. T. Waterman, Phys. Rev.f2j 38 1497 (1931).

66 C. Zwickker, Physica 11, 161 (1931).~ Quantum mechanically there is a small probability

that they will exceed this distance by a small amount, butthis effect is negligible.

electrons, there will be no space charge fartherfrom the surface than the critical distance, andthus space charge will not contribute to the po-tential barrier in this region. Closer to the sur-face the space charge may be an importantfactor.

Since the image law is known to hold for cleansurfaces at large distances, and space charge can-not contribute in this region, the fields will be as-sumed to be image fields at distances greater thanthe critical distance. To account for observedwork functions, this distance must be of the orderof 10 ' cm, at which place the field would beabout 4X10' volts cm '. The effective change inthreshold in an accelerating field depends on thenature of the surface fields equal to and less thanthe applied field. Since experimentally appliedfields are not as large as the surface fields at theplace where space charge can contribute, thechanges in threshold are independent of spacecharge.

In the limiting case pf low photoelectric emis-sion the space charge of the excited photoelec-trons is small, and thus the emission character-istics observed at absolute zero would not dependon space charge.

As the temperature rises, an increasingly largenumber of electrons have sufFicient energy to ex-ceed the critical distance. Assuming the fields

beyond the space charge region at absolute zeroto be largely image fields, they should be inde-pendent of the temperature. Any contribution byspace charge to the fields in this region at highertemperatures would cause an increase in thework function. Analysis of photoelectric data byFowler's method has shown that the work func-tion changes less than 1 percent between roomtemperature and 1100'K.Thus space charge is atmost a minor factor in determining photoelectricemission properties in accelerating fields.

An effect reported to be due to space charge ina retarding field was found and investigated byMarx and Meyer" in a cell using a thick potas-sium film as cathode, They measured the maxi-mum potential which the anode would attain by astring electrometer, and found that when mono-

"E.Marx, Naturwiss. 17, 806 (1929); Phys. Rev. f2)35, 1059 (1930); A. E. H. Meyer, Ann. d. Physik 9, 787(1931);and a theoretical discussion, E. Marx and A. E. H.Meyer, Phys. Zeits, 32, 153 (1931).

LEON B. LINFORD

chi'omatic light was used, Einstein's relation,Eq. (1), was satisfied. If light of high frequencybe used the anode will reach a definite potential,and if light of lower frequency but still above thethreshold frequency be added, the anode poten-tial will drop an amount proportional to (vi —vs)

nsv&/n&vs where v& and v& are the two frequenciessuch that vi) v2&vs and n, and n2 are the numb-ers of electrons ejected by the light of frequencyvi and v& respectively.

The theoretical explanation given is based onthe change in the distribution of space chargebetween the electrodes due to the addition of thelight of lower frequency, resulting in an increasein the energy required to eject an electron fromthe cathode with energy sufficient to reach theanode. This decrease in maximum potential wasfound to depend on the ratio es/ni and not on theabsolute magnitude of either. When consideredfor low light intensities, the electron density be-tween the electrodes would be so small thatPoisson s equation used in the treatment wouldnot apply because of the corpuscular naturerather than continuous distribution of electriccharge. Since the experimental facts can be ex-plained in a simpler manner, further theoreticaldiscussion seems unnecessary.

Following the first published note which gaveno experimental details, Olpin~ proposed the fol-

lowing explanation. The equilibrium potential in

monochromatic light would be reached when thenumber of electrons with energy sufficient toreach the anode equaled the number releasedfrom the anode by scattered light. %'ith potas-sium in the cell there would be a thin film on theanode which would make it photoelectricallysensitive, and probably more sensitive than thecathode. If light of lower frequency be added noadditional electrons can reach the anode, but thescattered light of lower frequency will releaseelectrons from the anode, and its potential mustdrop until the necessary additional number ofelectrons ejected from the cathode by the higherfrequency light can reach the anode. Olpin couldnot obtain the effect using a photoelectricallyinsensitive anode.

The complete report of the experimental workof Marx and Meyer showed that they had com-pletely shielded the anode from scattered light

'~ A. R. Olpin, Phys. Rev. f2( 35, 112 (1930).

and thus thought they had eliminated the possi-bility of an effect due to scattered light. Sincethey had potassium in the cell, a photoelectricallysensitive film would be deposited over all theglass surfaces. These would act exactly as theanode in Olpin's explanation. Photoelectronsejected from the cathode could not reach theanode unless the surrounding glass were at ashigh a potential as the anode, and thus theentire inside of the tube would act effectively aswould the anode were it exposed. Thus Olpin'sexplanation of the effect with the above extensionappears to be sufficient.

E. The spectral selective effect of compositesurfaces

Fowler" proposed an explanation for the posi-tions of the spectral selective maxima shown bycomposite surfaces. These surfaces are character-ized by both an electropositive substance and adielectric on the surface. He assumed that thepotential barrier of such a surface might be ideal-ized as in Fig. 7, having two maxima with aminimum or valley between. If an electron with

Fio, 7. Potential barrier assumed for selective transmission.(Fowler. )

energy e strikes the surface from within, itsprobability of transmission through the barrier isgreater if its classical kinetic energy e —e& in thepotential valley is such that the correspondingde Broglie wave-length will resonate in the widthof the valley ls. Fowler calculated the conditionfor resonance to be:

s —e2 ——n'k'/8ml2s, n = 1, 2, ~ ~ . (43)

He showed that for n= 1 and a reasonable width

"R. H. Fowler, Proc. Roy. Soc. A12S, 123 (1930).


of the potential valley a selective transmissionwould be predicted in the visible.

Olpin7 has tried to test quantitatively thetheory and has used many complex surfaces,some of which definitely showed evidence of smallcrystals formed on the surface, and others werenoncrystalline. For those surfaces which showedcrystal formation he calculated the distance be-tween alkali atoms along lines which passed asfar as possible from the electronegative atoms ofthe dielectric substance. Electrons in escapingwould select such paths since the potential bar-riers would be lowest.

From the experimentally observed frequenciesproducing the selective maxima, and tacitly as-suming that the kinetic energy of the electron inthe potential valley be equal to the energy of theincident light quantum, he calculated the widthof the valley which would cause the electrons tobe transmitted selectively.

For the various alkali hydrides the ratios of thewidths of the valley so calculated to the distancebetween successive alkali atoms along the chosenpath varied from 0.98 to 1.01. The alkali oxideand sulphide surfaces and others were more com-plicated since different compounds with differentcrystal structures can be formed. The variouscrystal structures shown by any two substances,e.g. , caesium and oxygen, gave spacings whichwere correlated with the different selectivemaxima shown under different conditions bycells containing these elements, The numericalagreements were close in all cases.

In this application of Fowler's theory to com-plex surfaces the exact numerical agreementswould not be expected for the following reasons.Olpin mentioned the first one in his report.

(1)The width of the valley was correlated withthe distance between the successive potentialmaxima. To be correct this latter distance mustbe reduced by the width of the potential barrierat the potential corresponding to the energy ofthe electron. In the case of the hydrides therewas no secondary rise of the spectral distributioncurve for electron energies a volt above the selec-tive energy. Since the transmission coefficient ispractically unity for electrons with energy suffi-cient to escape over all barriers, the barriers mustbe a volt higher than energy of electrons which

A. R. Olpin, Phys. Rev. f23 38, 1745 (1931).

can be transmitted selectively, to prevent asecondary rise of the spectral distribution curvewithin a volt of the selective maximum. From thepotential curve along the path of emission shownby Olpin, an electron with energy a volt less thanthe maximum of the potential barrier would bein the barrier for a considerable portion of thedistance, and the width of the valley would beless than assumed. In general this would apply toall surfaces.

(2) The kinetic energy of the electron was as-sumed to be that of the incident light quantum.This could be true only if the energy of the elec-tron before excitation was equal to the potentialat the bottom of the valley.

(3) Fowler obtained Eq. (43) from the follow-

ing relation by neglecting 8

(e eg) ply= &&i'

+8�&7$= 1& 2,

p'= 8s-2m/h'. (44)

In the case of the hydrides, the heights of variousparts of the potential curve can be estimated, andthe value of 8 may be as large as ~/3. Since thevalue n= 1 was used, neglecting h could cause anumerical error as large as a factor of 5/3 in

Eq. (43).It is difficult to understand how the above

mentioned corrections could compensate to givethe reported numerical agreements for such awide variety of surfaces.

Zachariasen" quoted recently determinedcrystal constants for the hydrides from x-raydata, and stated that the new values destroyedthe correlation reported by Olpin. With the ex-ception of Li H, the newer values are 1.08 to 1.10times the older ones, thus reducing the correctionnecessary for the part of the electron path whichis in the barrier, making the correlation moreprobable.

There is a general difficulty in applyingFowler's theory to the complex surfaces. A prom-inent characteristic is their high emission per unitlight intensity. Films of pure alkali metals whichhave thresholds and spectral maxima in aboutthe same regions as do certain complex surfaceshave much lower emission efficiencies. For suchsurfaces the transmission coefficiertt is practicallyunity. Fowler's theory as applied to complex

&' S. H. Zachariasen, Phys. Rev. L23 38, 2290 (1931).

L EON B. LI NFORD

surfaces requires the electrons to travel throughseveral potential barriers of considerable magni-tude, and even with selective transmission itwould not be expected to be unity.

A theory which would account for the selectivemaximum by means of a selective absorption oflight by electrons which may be ejected wouldovercome this difficulty. Two possibilities aresuggested by the work of Tamm and Schubin. "

They emphasized the fact that electrons couldbe excited only when in an electric field, and thatthe greater part of the electrons ejected from aclean surface by light near the threshold fre-quency were in the region of the surface fieldswhen excited. At the same time only a small partof the light is absorbed at the surface where theefliciency of emission is high. Most of the light isabsorbed by low energy electrons, in the interiorof the metal and these cannot escape.

The theory predicts a definite maximum forthe surface effect, and its position in relation tothe threshold is not far different than that ob-served for the hydrides using the value of v for thealkali itself as an approximation. Where nodefinite crystalline structure is formed on thesurface, the fields between the atoms or groups ofatoms of the two different kinds, would be moreof the order of the surface fields. The thickness ofsuch composite films is great enough to absorb aconsiderable proportion of the incident light andthis in a region where surface type excitation withits high efficiency might occur.

With a crystalline structure in the complexsurface there would be periodic fields in thecrystal, which like the lattice fields in the interiorof the metal would give rise to definite energylevels of the electrons. These energy levels woulddepend on the structure and constants of thecrystals. The volume effect for such crystalsmight account for some of the maxima. On such atheory electrons would not need to pass throughpotential barriers, with the resulting lower trans-mission coefficient.

More detailed study as to the structure of thecomplex surfaces and the theory of photoelectricexcitation of electrons in such surfaces is neededin order to explain satisfactorily their extremesensitivities as well as the positions of theirspectral selective maxima.

F. The vectoxial effect

Recent work of Ives" and Ives and Briggs" hasexplained the dependence of the emission fromthin alkali films deposited on heavy metals onthe angle of incidence and the polarization of thelight and on the optical constants of the twometals. They have been able to correlate theemission with the energy density of radiation inthe alkali film.

Wiener's work has shown that the interferenceof incident and reflected light waves producesstanding waves above a surface of discontinuity.A node in the standing wave system would befound at the surface of a perfect electrical con-ductor.

Since metals have finite conductivities, thenodes are displaced, giving finite values of theelectric vectors of the light at their surfaces.From the work of Fry, "the magnitude and direc-tion of the electric vector at the surface can becalculated from the optical constants of the metal,and the angle of incidence |I and the polarizationof the incident light.

The energy density at the surface is propor-tional to the square of the magnitude of theelectric vector, or its intensity. Since the area ofthe cathode covered by a defined beam of lightis proportional to sec 8, the electric intensitymust be multiplied by this factor before com-parison with experimental data. In the subse-quent discussion of comparisons with experi-ments, the electric intensities will be assumed tohave been multiplied by sec 8 whether specificallyso stated or not.

The experimental work was done on alkalimetal films reported to be about one atom thick.From their emission characteristics and com-

parison with Brady's results, "it is probable thatthey were thicker, but less than 10 atomic layers.This change would not affect the validity of Ives'conclusions. The films investigated were not vis-ible on the surface, and thus one can assume thatthey do not affect appreciably the optical con-stants of the underlying metal. The thickness ofthe film at most is a small part of a wave-length

r~ H. E, Ives, Phys. Rev. t 2g 38, 1209 (1931).~'H. E. Ives and H. B. Briggs, Phys, Rev. $2) 38, 1477

(1931)."T. C. Fry, J.Opt. Soc. Am. 15, 137 (1927);16, 1 (1928).


of light, and therefore the electric vector at thesurface of the alkali film can be assumed to bethat at the surface of the underlying metal.

%hen light enters the alkali film the compon-ents of the electric vector parallel to the surfaceare continuous, but the component normal, to thesurface must be multiplied by the factorQ= {N+iEs) ', which depends on N the index ofrefraction and Es the extinction coefFicient of thealkali metal.

For notation the electric vector of the standingwave at the surface of the metal will be denotedby Ell if the incident light is polarized parallelto the plane of incidence, E, if polarized perpen-dicular to this plane, and E, if the light is incidentnormal to the surface. The two latter quantitieswill be the same in the alkali film, but the formerwill be different. To calculate Ell' the electricvector in the alkali film for parallel polarizedlight, the component of the electric vector normalto the surface outside must be multiplied by Qbefore obtaining the resultant.

Ives" worked with a potassium film on plati-num and measured the photoelectric emissionfor both polarizations as a function of the angleof incidence. Fig. 8 shows the data for light of5461A. The solid lines show the calculated valuesof the electric intensities at the surface of theplatinum for both polarizations, [ E~ l (

' and)E.

)'.

The broken line shows the intensity calculatedfor the interior of the potassium film, ~EuThe experimental data are shown by the circlesand crosses. At 4359A, the data for the observedemission fell on a curve above that for jEll ~'.

The optical constants for potassium are notknown for this spectral region, but extrapolationsshow that

~ Q ~)1, thus qualitatively satisfying

the data.A similar comparison of experimental and cal-

culated data for rubidium on glass showed goodagreement.

To demonstrate the importance of the opticalconstants of the underlying metal, Ives andBriggs" investigated thin films of sodium onsilver. Sodium was chosen because the films weremore stable due to its lower vapor pressure. In-vestigation of sodium films on platinum showedno sudden changes in emission with the fre-quency, and since the optical constants ofplatinum show no large variations over the

frequency range studied, it was evident thatthose of sodium have no abrupt changes.

Silver was chosen as the underlying metal be-cause it has a transmission band at 3160A, andits optical constants show widely different valuesin the three spectral regions, defined roughly asnear 3160A, at longer, and at shorter wave-lengths. In this way the emission from one alkalifilm can be studied as the optical constants of the

0' X' 60'Angle of' jpcld&znce.

Fio. 8. Comparison of calculated intensity of electricvector with photoelectric emission, for various angles ofincidence. The solid lines show the electric intensity abovesurface of platinum for both polarizations, and the brokenline shows the intensity in the potassium film. The photo-electric data are the circles and crosses. Wave-length oflight 5461A. (Ives. )

underlying metal are changed by changing thewave-length of the incident light.

The electric intensities at the surface of thesilver as calculated for the frequency range stud-ied, for light of normal incidence, and for both

J.EON B. LI N FORD

polarizations at 60' incidence showed severalinteresting characteristics.

At 3260A or 100A longer wave-length, thanthe transmission band

~ Ei~ t' shows a sharp min-imum and ~Kjs a maximum. tE, ~' has a defin-ite maximum at about 3300A. The emissioncurves show these same characteristics. That theemission cannot be correlated with the reflectingpower of silver is shown by the fact that similarcurves for the reflecting power all show minimabetween 3100A and 3200A.

Between about 3100A and 3300A the curvefor )K)s is higher than that for )E, js whichpredicts that for this range the emission due toperpendicularly polarized light at 60' incidenceshould be greater than that at normal incidence.At other wave-lengths it should be less. If aneffect of this kind exists, curves of emission as afunction of angle of incidence for perpendicularpolarization would show a maximum, instead of amonotonic decrease from 0' to 90' incidence asshown in Fig. S. Such angle curves were taken atintervals from 3022A to 3650A. Curves for wave-

lengths greater than 3300A showed no maximum,the others did. At the shortest wave-lengths in-

vestigated the maximum became less pro-nounced, and at 3022A the emission at 60' wasthe same as at normal incidence. Experimentaldifficulties made investigation at shorter wave-lengths impractical.

Thus the emission from thin alkali films followsthe changes in intensity of the electric vector in

the film as calculated from the optical constantsof the underlying metal and of the film. In thisway the vectorial selective effect of such films isexplained completely without the assumption ofa greater efficiency of emission for the componentof the electric vector of the light normal to thesurface, than for the components parallel to thesurface.

Fleischmann" has found that thin potassiumfilms on glass show an absorption band in thevisible if the incident light not at normal inci-dence is polarized parallel to the plane of inci-dence and no absorption if polarized prependicu-lar to this plane. He correlates this with thevectorial effect. From the work of Ives and Briggsit would appear that the absorption of light as

n R. Fleischmann, Naturvriss. 19, 826 (1931}.

well as the photoelectric emission depends onthe intensity of the electric vector in the alkalifilm.

Ives and Briggs" investigated the emissionproperties of sodium films, during their formationon silver, with light of frequency near thetransmission band of silver at 3160A. In theirprevious work" they had studied the emissionfrom comparatively thick films, and found thatthe emission characteristics could be correlatedwith the intensity of the electric vector abovethe surface of the silver. They concluded thatthe photoelectrons originated in the alkali films.

In the more recent work, they found that filmfree silver was photoelectrically insensitive tolight of this frequency; and that when a littlealkali (much less than enough to form a layerone atom deep) was allowed to deposit on thesilver, it emitted photoelectrons. As before,these films were studied with light incidentnormal to the surface, and with both polariza-tions with the light incident at 60'. The emissioncharacteristics found were unlike those of thethick film, but they could be correlated verywell with the absorbing power of the layer ofsilver just inside the surface. Thus it wouldappear that most of the photoelectrons originatedin the surface layer of the silver.

By studying the films as their thicknessesincreased, it was found that the emission char-acteristics changed gradually to those found fora thick film, indicating that as the film becamethicker, an increasing proportion of the photo-electrons originated in the alkali film. This workgives very good evidence that a large proportionof the photoelectrons have their origin very closeto the surface of the emitter.

The experimental extension of this work to thebulk alkali metals will depend on the determina-tion of their optical constants. It seems reason-able that the relations found to hold so well forthin films should be true in general. If so, thiswork gives the foundation for better generaltheories of the photoelectric effect. It points outthat the magnitude of the electric vector at thesurface as calculated from the standing wavepattern is an important factor in the emission,

"H. E. Ives and H. B. Briggs, Phys. Rev. 12j 40, 802(1932).


and the theories should be modified to includethis.

The fact that the emission depends on the in-tensity and not on the orientation of the electricvector at the surface is in disagreement with allthe quantum mechanical theories, which predictmuch greater efficiency for the component normalto the surface. This cannot be explained by colli-sions of the electrons with the lattice before leav-ing the surface, as on this assumption Fowlerobtained a wrong result for the temperaturecorrection.

Momentum considerations may play an im-portant part in determining the direction of theincrease in velocity of the electron when it ab-sorbs a quantum. The change in momentum ofthe electron is of the order of 10' times themomentum of the incident quantum. This extramomentum must be transferred by means of theelectrostatic field to the lattice ions. If, as Tammand Schubin' point out, the emission for fre-quencies near the threshold is principally fromthe region where the surface fields are large, andsince these fields are unidirectional the change in

momentum o& the electron must likewise be uni-directional. How much the probability of excita-tion of the electron depends on momentum con-siderations and how much on the direction of theelectric vector should be investigated in greaterdetail.

Although the recent theories and experimentshave done much to unify and explain many of thephotoelectric phenomena, there is still a greatamount of careful and well directed experimentalwork to be done, which must be coupled withmore detailed theoretical investigations, in orderto produce a satisfactory unified picture of thephotoelectric effect.

In conclusion the author wishes to thankProfessor E. U. Condon for suggesting the prepa-ration of this article and for his continued help;Doctors J. A. Becker, V. Rojansky, A. R. Olpinand W. H. Brattain for their helpful suggestionsand criticisms during the preparation of thispaper. The author also wishes to express hisappreciation to the National Research Counciland the Palmer Physical Laboratory for makingthis study possible.

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