Appendix - Springer978-1-4615-1793-1/1.pdf · Appendix A 1 Stereoviews and Crystal Models A 1.1 Stereoviews The representation of crystal and molecular structures by stereoscopic

Appendix

A 1 Stereoviews and Crystal Models

A 1.1 Stereoviews

The representation of crystal and molecular structures by stereoscopic pairs of drawings has become commonplace in recent years. Indeed, some very sophisticated computer programs have been written which draw stereoviews from crystallographic data. Two diagrams of a given object are necessary, and they must correspond to the views seen by the eyes in normal vision. Correct viewing requires that each eye sees only the appropriate drawing, and there are several ways in which it can be accomplished.

1. A stereoviewer can be purchased for a modest sum. Two suppliers are:

(a) C. F. Casella and Company Limited, Regent House, Britannia Walk, London Nl 7ND, England. This maker supplies two grades of stereoscope.

(b) Taylor-Merchant Corporation, 212 West 35th Street, New York, NY 10001, U.S.A.

Stereoscopic pairs of drawings may then be viewed directly. 2. The unaided eyes can be trained to defocus, so that each eye sees

only the appropriate diagram. The eyes must be relaxed and look straight ahead. This process may be aided by placing a white card edgeways between the drawings so as to act as an optical barrier. When viewed correctly, a third (stereoscopic) image is seen in the center of the given two views. It may be found helpful to close the eyes for a moment and then open them wide, without attempting to focus on the diagram, and let them relax.

3. An inexpensive stereoviewer can be constructed with comparative 487

488

Cut 3

Q

I t- - - - - - 6.4cm - - --1 I I I

~1. B. I 3cm I

: 1 __ £.old __

E u

"<t

I "01 01 IL,

A

---------- 11cm ----------

APPENDIX

FIGURE ALl. Simple stereoviewer. Cut out two pieces of card as shown and discard the shaded portions. Make cuts along the double lines. Glue the two cards togethet with the lenses EL and ER in position, fold the portions A and B backward, and fix P into the cut at Q. View from the side marked B. (A similar stereoviewer is marketed by the TaylorMerchant Corporation, New York.)

ease. A pair of planoconvex or biconvex lenses each of focal length about 10 cm and diameter 2-3 cm are mounted in a framework of opaque material so that the centers of the lenses are about 60-65 mm apart. The frame must be so shaped that the lenses can be held close to the eyes. Two pieces of cardboard shaped as shown in Figure ALI and glued together with the lenses in position represents the simplest construction.

A 1.2 Model of a Tetragonal Crystal

The crystal model illustrated in Figure 1.30 can be constructed easily. This particular model has been chosen because it exhibits a 4 axis,

Al STEREOVIEWS AND CRYSTAL MODELS

Q

\ \

\ \ \ \ \ / \ /

/ / I

/ /

/

B I

/ /

/ /

/ /

I

/

/ \

\ \ \

\ \ \ \ \ \

/ \ E _____ - - - - _F K

FIGURE A1.2. Construction of a tetragonal crystal of point group '12m:

NQ = AD = BD = BC = DE = CE = CF = KM

= lOcm;

AB = CD = EF = GJ = 5 cm;

AP=PQ =FL= KL=2cm;

AQ = DN = CM = FK = FG = FH = EJ = 1 cm.

489

490 APPENDIX

which is one of the more difficult symmetry elements to appreciate from plane drawings.

A good quality paper or thin card should be used for the model. The card should be marked out in accordance with Figure A1.2 and then cut out along the solid lines, discarding the shaded portions. Folds are made in the same sense along all dotted lines, the flaps ADNP and CFLM are glued internally, and the flap EFHJ is glued externally. The resultant model belongs to crystal class 42m, and should be compared with Figure 1.30.

A2 Crystallographic Point-Group Study and Recognition Scheme

The first step in this scheme is a search for the center of symmetry and mirror plane; they are probably the easiest to recognize. If a model with a center of symmetry is placed on a flat surface, it will have a similar face uppermost and parallel to the supporting surface. For the m plane, a search is made for the left-hand-right-hand relationship in the crystal.

The point groups may be classified into four sections:

(I) No m and no I: 1,2,222,3,32,4,4,422,6,622,23,432

(II) m present but no I: m, mm2, 3m, 4mm, 42m, 6, 6mm, 6m2, 43m

(III) I present but no m: I, 3

(IV) m and I both present: - 4 6

2/m, mmm, 3m, 4/m, -mm, 6/m, -mm, m3, m3m m m

The further systematic identification is illustrated by means of the block diagram in Figure A2.1. Here R refers to the maximum degree of rotational symmetry in a crystal, or crystal model, and N is the number of such rotation axes. Questions are given in ovals, point groups in squares, and error paths in diamonds. It may be noted that in sections I, II, and IV, the first three questions (with a small difference in II) are similar. The cubic point groups evolve from question 2 in I, II, and IV.

Readers familiar with computer programming may liken Figure A2.1 to a flow diagram. Indeed, this scheme is ideally suited to a computer-

~ ("')

(II

~ ~ r 5 0 ~

;I>

.."

==

("') .." 0 ~ 0 ~

0 c:: .."

r/l .., c:: 0 -< ;I>

No

l I.";';~' I

z 0 ~

tTl

("') 0 0 z a 0 z r/l

("') :t

tTl a:::

FIG

UR

E A

2.1.

Flo

w d

iagr

am f

or p

oint

-gro

up r

ecog

niti

on.

tTl ... :::

492 APPENDIX

aided self-study enhancement of a lecture course on crystal symmetry, and success with the method has been obtained. *

A3 Schoenflies' Symmetry Notation

Theoretical chemists and spectroscopists use the Schoenflies notation for describing point-group symmetry but, although both the crystallographic (Hermann-Mauguin) and Schoenflies notations are adequate for point groups, only the Hermann-Mauguin system is satisfactory for space groups.

The Schoenflies notation uses the rotation axis and mirror plane symmetry elements with which we are now familiar, but introduces the alternating axis of symmetry in place of the inversion axis.

A3.1 Alternating Axis of Symmetry

A crystal is said to have an alternating axis of symmetry Sn of degree n, if it can be brought from one state to another indistinguishable state by the combined operation of rotation through (360/ n) degrees and reflection across a plane normal to the axis. It must be stressed that this plane is not necessarily a mirror plane. t Operations Sn are nonperformable (see pages 28 and 32). Figure A3.1 shows stereograms of ~ and S4; we

+0

0+

(a) (b)

FIGURE A3.1. Stereograms of point groups: (a) S2' (b) S4.

• M. F. C. Ladd, International Journal of Mathematical Education in Science and Technology, 7, 395-400 (1976). .

t The usual Schoenfiies symbol for 6 is C3h (31m). The reason that 31m is not used in the Hermann-Mauguin system is that point groups containing the element 6 describe crystals that belong to the hexagonal system rather than to the trigonal system; 6 cannot operate on a rhombohedral lattice.

A3 SCHOENFLIES' SYMMETRY NOTATION 493

recognize them as I and 4,* respectively. The reader should consider what point groups are obtained if, additionally, the plane of the diagram is a mirror plane.

A3.2 Notation

Rotation axes are symbolized by en, where n takes the meaning of R in the Hermann-Mauguin system. Mirror planes are indicated by subscripts v, d, and h; v and d refer to mirror planes containing the principal axis, and h indicates a mirror plane normal to that axis. In addition, d refers to those vertical planes that are set diagonally, that is, between the crystallographic axes. The symbol Dn is introduced for point groups in which there are n twofold axes in a plane normal to the principal axis of degree n. The cubic point groups are represented through the special symbols T and O. Table A3.1 compares the Schoenflies and Hermann-Mauguin symmetry notations.

TABLE A3.I. Schoenflies and Hermann-Mauguin Point-Group Symbols

Schoenflies Hermann-Mauguina Schoenflies Hermann-Mauguina

Cl D4 422 C2 2 D6 622 C3 3 DZh mmm C4 4 D3h 6m2 Cn 6 4 Ci , S2 1 D4h -mm

Cs ' SI m(2) m

S6 3 D6h

6 S4 4

-mm m

C3h , S3 6 Du '12m

C2h 2/m D3d 3m

C4h 4/m T 23

C6h 6/m Th m3

C2v mm2 0 432 C3v 3m

Td '13m C4v 4mm

Oh m3m C6v 6mm Coov 00

D2 222 Dooh oo/m(oo)

D3 32

a 2/m is an acceptable way of writing 1. , but 4/mmm is not as satisfactory as"± mm. m m

* Note that, among N, 4 (S4) is unique, in that it is not equivalent to any other symmetry element or combination of symmetry elements.

494 APPENDIX

A4 Generation and Properties of X-rays

A4.1 X-rays and White Radiation

X-rays are electromagnetic radiations of short wavelength, and are produced by the sudden deceleration of rapidly moving electrons at a target material. If an electron falls through a potential difference of V volts, it acquires an energy of eV electron volts. If this energy were converted entirely into a quantum hv of x-rays, the wavelength A would be given by

A = hc/eV (A4.1)

where h is Planck's constant, c is the speed of light, and e is the charge on the electron. Substitution of numerical values in (A4.1) leads to the equation

A = 12.4/V (A4.2)

where V is measured in kilovolts (kV). Generally, an electron does not lose all its energy in this way. It

enters into multiple collisions with the atoms of the target material, increasing their vibrations and so generating heat in the target. Thus, (A4.2) gives the minimum value of wavelength for a given accelerating voltage. Longer wavelengths are more probable, but very long wavelengths have a small probability and the upper limit is indeterminate. Figure A4.1 is a schematic diagram of an x-ray tube, and Figure A4.2 shows typical intensity versus wavelength curves for x-rays. Because of the continuous nature of the spectrum from an x-ray tube, it is often referred to as "white" radiation. The generation of x-rays is a very uneconomical process. Most of the incident electron energy appears as heat in the target, which must be thoroughly water-cooled; about 0.1 % of the energy is usefully converted for crystallographic purposes.

A4.2 Characteristic X-rays

If the accelerating voltage applied to an x-ray tube is sufficiently large, the impinging electrons excite inner electrons in the target atoms, which may be expelled from the atoms. Then, other electrons, from

A4 GENERATION AND PROPERTIES OF X-RAYS 495

o t

c ~~~---=---:----,

- - - ..... ~ - - - - ---~I~------/

E :x

FIGURE A4.1. Schematic diagram of an x-ray tube: W, heated tungsten filament; E, evacuated glass envelope; C, accelerating cathode; e, electron beam; A, target anode; X, x-rays (about 6° angle to target surface); B, anode supporting block of material of high thermal conductivity; I, cooling water in; and 0, cooling water out.

higher energy levels, fall back to the inner levels and their transition is accompanied by the emission of x-rays. In this case, the x-rays have a wavelength dependent upon the energies of the two levels involved. If this energy difference is !::..E, we may write

4

.~ 3 x co

~ iii c: Cl)

.~ 2 Cl)

> '';::;

.!!1 Cl)

a:

A = hc/!::..E

20kV

0.4 1.2 1.6

A axis, A FIGURE A4.2. Variation of x-ray intensity with wavelength A.

(A4.3)

496

4

'" X 3 <0

~ 'iii c Q)

C Q)

> .~

a; a:

o 0.4

APPENDIX

Ka.

K{3

A axis, A FIGURE A4.3. Characteristic K spectrum superposed on the "white" radiation

continuum.

This wavelength is characteristic of the target material. The white radiation distribution now has sharp lines of very high intensity superimposed on it (Figure A4.3). In the case of a copper target, very commonly used in x-ray crystallography, the characteristic spectrum consists of Ka (A. = 1.542 A) and Kf3 (A. = 1.392 A); Ka and Kf3 are always produced together.

A4.3 Absorption of X-rays

All materials absorb x-rays according to an exponential law:

I = 10 exp( -Ilt) (A4.4)

where I and 10 are, respectively, the transmitted and incident intensities, Il is the linear absorption coefficient, and t is the path length through the material. The absorption of x-rays increases with increase in the atomic number of the elements in the material.

A4 GENERATION AND PROPERTIES OF X-RAYS

I

E " If)

400

300

.~ 200 -::L

100

o 0.4 0.8 1.2 o

A axis, A

I I I I I I I

V 1.6

FIGURE A4.4. Variation of /l (Ni) with wavelength A of x-radiation.

497

The variation of f1. with A is represented by the curve of Figure A4.4; f1. decreases approximately as A3. At a value which is specific to a given atom in the material, the absorption rises sharply, This wavelength corresponds to a resonance level in the atom: a process similar to that involved in the production of the characteristic x-rays occurs, with the exciting species being the incident x-rays themselves. The particular wavelength is called the absorption edge; for metallic nickel it is 1.487 A.

A4.4 Filtered Radiation

If we superimpose Figures A4.3 and A4.4, we see that the absorption edge of nickel lies between the Ka and Kf3 characteristic lines of copper (Figure A4.5). Thus, the effect of passing x-rays from a copper target through a thin (0.018 mm) nickel foil is that the Kf3 radiation is selectively almost completely absorbed. The intensities of both Ka and the white radiation are also reduced, but the overall effect is a spectrum in which the most intense part is the Ka line; we speak of filtered radiation, to indicate the production of effectively monochromatic radiation by this process. The copper Ka line (X = 1.542 A) actually consists of a doublet, a1 (A = 1.5405 A) and a2 (A = 1.5443 A); the doublet is resolved on photographs at high () values, but we shall not be

498

'" ·x '" :: ·iii c: ~ c:

'" > . .., '" Qj

a::

4

3

2

o 0.4 o

A axis, A

APPENDIX

400

300

., E <.)

200 ui ·x '" :i.

100

FIGURE A4.S. Superposition of Figures A4.3 and A4.4 to show diagrammatically the production of "filtered" radiation.

concerned here with that feature. The value of 1.542 A is a weighted mean (2Aal + Aa,)/3, the weights being derived from the relative intensities (2: 1) of the al and az lines.

The absorption effect is important also in considering the radiation to be used for different materials. We have mentioned that Cu Ka is very commonly used, but it would be unsatisfactory for materials containing a high percentage of iron (absorption edge 1.742 A) since radiation of this wavelength is highly absorbed by iron atoms and reemitted as characteristic Fe K spectrum. In this case, Mo Ka (A = 0.7107 A) is a satisfactory alternative.

AS Crystal Perfection and Intensity Measurement

AS.1 Crystal Perfection

In the development of the Bragg equation (3.16), we assumed geometric perfection of the crystal, with all unit cells in the crystal

AS CRYSTAL PERFECTION AND INTENSITY MEASUREMENT 499

FIGURE AS.1. Primary extinction: The phase changes on reflection at Band C are each :fr/2, so that between the directions BE and CD there is a total phase difference of :fr.

Hence, some attenuation of the intensity occurs for the beam incident upon planes deeper in the crystal.

stacked side by side in a completely regular manner. Few, if any, crystals exhibit this high degree of perfection. Figure AS.1 shows a family of planes, all in exactly the same orientation with respect to the x-ray beam, at the correct angle for a Bragg reflection. It is clear that the first reflected ray BC is in the correct position for a second reflection CD, and so on. Since there is a phase change of Jr /2 on reflection, * the doubly reflected ray has Jr phase difference with respect to the incident ray (BE). In general, rays reflected nand n - 2 times differ in phase by Jr, and the net result is a reduction in the intensity of the x-ray beam passing through the crystal. This effect is termed primary extinction, and is a feature of geometric perfection of a crystal. In the ideally perfect crystal, I IX IFI.

Most crystals, however, are composed of an array of slightly misoriented crystal blocks (mosaic character) (Figure AS.2). The ranges

FIGURE AS.2. "Mosiac" character in a crystal; the angular misalignment between blocks may vary from 2' to about 30' of arc.

* This :fr 12 phase change is usually neglected since it arises for all reflections.

500 APPENDIX

of geometric perfection are quite small. Even crystals that show some primary extinction exhibit mosaic character to some degree, and we may write

(A5.I)

Generally, the mosaic blocks are very small « 10-4 cm), and m is effectively 2.

Another process which leads to attenuation of the x-ray beam by a crystal set at the Bragg angle is known as secondary extinction. It may be encountered in single-crystal x-ray studies, and the magnitude of the effect can be appreciable. Consider a situation in which the first planes encountered by the x-ray beam reflect a high proportion of the incident beam. Parallel planes further in the crystal receive less incident intensity, and, hence, reflect less than might be expected. The effect is most noticeable with large crystals and intense (usually low-order) reflections. Crystals in which the mosaic blocks are highly misaligned have negligible secondary extinction, because only a small number of planes are in the reflecting position at a given time. Such crystals are termed ideally imperfect; this condition can be developed, or enhanced, by SUbjecting the crystals to the thermal shock of dipping them in liquid air. The effect of secondary extinction on the intensity of a reflection can be brought into the least-squares refinement (page 4I6ff) as an additional variable, the extinction parameter ~. The quantity minimized in the refinement of the atomic parameters is then

(A5.2)

A5.2 Intensity of Reflected Beam

The real or imperfect crystal will reflect x-rays over a small angular range centered on a Bragg angle fJ. We need to determine the total energy of a diffracted beam ,€;(hkl) as the crystal, which is completely bathed in an x-ray beam of incident intensity 1o, passes through the reflecting range.

At a given angle fJ, let the power of the reflected beam be d,€;(hkl)/dt. The greater the value of 1o, the greater the power. Hence,

d,€;(hkl)/dt = R(fJ)/o (A5.3)

A5 CRYSTAL PERFECTION AND INTENSITY MEASUREMENT 501

-860 6 axis

FIGURE AS.3. Variation of reflecting power R( 8) with 8 arising from "mosaic" character: 80 is the ideal Bragg angle, and ± 880 represent the limits of reflection.

where R( 0) is the reflecting power. Figure A5.3 shows a typical curve of R( 0) against O. The area under the curve is called the integrated reflection J(hk/):

i680 J(hkl) = -680 R(O)dO (A5.4)

Using (A5.3), we obtain

J(hkl) = (1/10)i680 [d'iS(hkl)] dO -680 dt

(A5.5)

If the crystal is rotating with angular velocity w(= dO/dt),

J(hkl) = w'iS(hkl)/lo (A5.6)

where 'is(hkl) is the total energy of the diffracted beam for one pass of the crystal through the reflecting range, ±(jOo. Since intensity is a

502 APPENDIX

measure of energy per unit time, we have

'l;(hkl) = lo(hkl)t (A5.7)

and, from (4.57), we obtain

(A5.8)

where C(hkl) includes correcting factors for absorption and extinction, and for the Lorentz and polarization effects (page 503ff). Because of the proportionality between energy and intensity (A5. 7), although we are actually measuring the energy of the diffracted beam, we usually speak of the corresponding intensity.

A5.3 Intensity Measurements on Photographs

X-ray intensities are measured on photographs from the blackening of the photographic film emulsion.

The optical density D of a uniformly blackened area of an x-ray diffraction spot on a photographic film is given by

D = 10glO(Io/ I) (A5.9)

where 10 is the intensity of light hitting the spot and 1 is the intensity of light transmitted by it: D is proportional to the intensity of the x-ray beam 10 for values of D less than about 1. In practice, this means spots which are just visible to those of a medium-dark gray on the film.

An intensity scale can be prepared by allowing a reflected beam from a crystal to strike a film for different numbers of times and according each spot a value in proportion to this number; Figure A5.4 shows one such scale. Intensities may be measured by visual comparison with the scale, and, with care, the average deviation of intensity from the true value would be about 15%.

In place of the scale and the human eye, a photometric device may be used to estimate the blackening. In this method, the background

. . . . . . . . FIGURE A5.4. Sketch of a crystal-intensity scale.

A5 CRYSTAL PERFECTION AND INTENSITY MEASUREMENT

. . ~. .'

(a)

•

•

•

•

•

• •

• •

• •

• •

• • (b)

503

• •

• •

• •

• •

• •

FIGURE A5.5. Spot integration: (a) typical diffraction spot, (b) 5 x 5 grid of points.

intensity is measured and subtracted from the peak intensity. This process is carried out automatically in the visual method. Carefully photometered intensities would have an average deviation of less than 10%.

The accuracy of film measurements can be enhanced if an integrating mechanism is used in conjunction with either a Weissenberg or a precession camera in recording intensities. In this method, a diffraction spot (Figure AS.Sa) is allowed to strike the film successively over a grid of points (Figure AS.Sb). Each point acts as a center for building up the spot. The results of this process are a central plateau of uniform intensity in each spot and a series of spots of similar, regular shape: Figure AS.6 illustrates, diagrammatically, the building up of the plateau, and Figure AS.7 shows a Weissenberg photograph comparing the normal and integrating methods with the same crystal.

The average deviation in intensity measurements from carefully photometered, integrated Weissenberg photographs is about S%. The general subject of accuracy in photographic measurements has been discussed exhaustively by Jeffery. *

A5.4 Data Processing

A5.4.1 Introduction

From (AS.8) and (AS.9), we see that certain corrections are necessary in order to convert measured intensities into values of lFol2. We

• See Bibliography, Chapter 3.

504 APPENDIX

(a)

(b)

(c)

6 axis

FIGURE A5.6. Spot integration: (a) ideal peak profile, (b) superposition, by translation, of five profiles, (c) integrated profile showing a central plateau.

shall write

Io(hkl) ex: ALp 1F;.(hkl)l~el (A5.1O)

and 1F;.(hkl)I = K lFo(hkl)lrel (A5.H)


-(a)

- - --

(b)

FIGURE A5.7. Weissenberg photographs: (a) normal, (b) integrated.

506 APPENDIX

where A is an absorption factor (including extinction for the purpose of this discussion), L is the Lorentz factor, p is the polarization factor, and K is the scale factor which places the !Fol values on to an absolute scale; it includes, implicitly, the proportionality constant of (A5.8). The Lorentz factor expresses the fact that, for a constant angular velocity of rotation of the crystal, different reciprocal lattice points pass through the sphere of reflection at different rates and thus have different times-of-reflection opportunity. The form of the L factor depends upon the experimental arrangement. For both zero-level photographs taken with the x-ray beam normal to the rotation axis and four-circle diffractometer measurements, L has the simple form of 1/sin 28.

The radiation from a normal x-ray tube is unpolarized, but after reflection from a crystal the beam is polarized. The fraction of energy lost in this process is dependent only on the Bragg angle:

p = (1 + cos2 28)/2 (A5.12)

Application of the Land p factors, where absorption and secondary extinction are negligible, is essential in order to bring the !Fo1 2 data onto a correct relative scale. The scale factor K can be determined approximately by Wilson's method (page 305) and refined as a parameter in a least-squares analysis.

A5.4.2 Standard Deviation of Intensity

The net integrated intensity I and background B are measured, most conveniently in diffractometry, with a step-scan moving-window method. * The standard deviation in I arising only from statistical counting fluctuations is given by

(A5.13)

where r is the ratio of the time spent in measuring I to that spent in measuring B, typically 1.5.

• I. J. Tickle, Acta Crystallographica B31, 329 (1975).


A5.4.3 Absorption Corrections

The absorption of x-rays by matter is governed by the equation

1= loexp(-Il,t) (AS.14)

where I is the diffracted beam intensity, 10 is the incident beam intensity, J.l is the linear absorption coefficient, and t is the thickness of specimen. Hence the transmission of the x-ray beam through a crystal is given by

(AS. IS)

where t; and td are the incident and diffracted beam path lengths. If the shape of the crystal is known exactly, then it is possible to correct for absorption by calculating

(AS.16)

where dV is an infinitesimal volume of crystal (Busing and Levy). * Frequently, however, the crystal faces are not well defined and it is

necessary to resort to empirical methods for estimating the transmission factor.

Empirical Absorption Correction. The incident and diffracted x-rays for a general reflection with <P = <Po will intersect the transmission profile at <Po - 0 and <Po + 0, where

o = tan- 1(tan f} cos A) (AS.17)

Hence, 0 = 0 and X = ±90°. The transmission profile used is that with f}

nearest to the equi-inclination angle v, where

v = sin-1(sin f} sin X) (AS.18)

The transmission T is given either as the arithmetic mean or as the geometric mean of the estimated incident and reflected ray transmissions:

or T = [Ty( <p - 0) + Ty( <p + 0)]/2

T = [Ty(<p - 0) X Ty(<P + 0)]1/2

* W. R. Busing and H. A. Levy, Acta Crystallographica 10, 180 (1957).

(AS.19)

(AS.20)

508 APPENDIX

X-ray beam

Ewald sphere

o FIGURE AS.S. Geometry of absorption correction.

Transmission Profiles. The transmission is measured for axial reflections (X = ±900) as a function of cP (Figure A5.8). The transmission is given by

(A5.21)

The variation of T with () is neglected as it has the same effect as a small isotropic temperature factor.

A set of profiles of T as a function of cP are obtained for different values of () and applied in data processing as detailed above.

A5.4.4 Scaling

Fluctuations in the incident x-ray beam intensity and possible radiation damage to the crystal may be monitored by measuring four standard reflections of moderate intensity at regular intervals, say, hourly. Two of these reflections should have X at about 0° and two at about 90°, with each pair about 90° apart in cp. The average of these intensities relative to the average of their starting values is smoothed and used to rescale the raw intensity data. If S is this scale factor, then the total correction applied is now

(A5.22)

(A5.23)

A6 TRANSFORMATIONS 509

A5.4.5 Merging Equivalent Reflections

Where more (n) than one symmetry equivalent of a given reflection is measured, the weighted mean is calculated:

(A5.24)

where (A5.25)

A chi-square test may be used to detect equivalents which may have a systematic error:

(A5.26)

where there are n - 1 degrees of freedom. If X2 exceeds X~-l (a =

0.001), then the equivalent with the greatest weighted deviation from the mean, Wj I~ - ~I, is rejected and the test repeated on the remaining equivalents. If n = 2, then the smaller intensity value is rejected.

The merging R value is defined by*

R = ~hkl [~j I~ - ~I] m ~hkl [~j~]

(A5.27)

A6 Transformations

The main purpose of this appendix is to obtain a relationship between the indices of a given plane referred to two different unit cells in one and the same lattice. However, several other useful equations will emerge in the discussion.

In Figure A6.1, a centered unit cell (A, B) and a primitive unit cell (a, b) are shown; for simplicity, only two dimensions are considered. From the geometry of the diagram,

• Also known as R;n"

A=a-b B=a+b a = A/2 + B/2

b = -A/2 + B/2

(A6.1)

(A6.2)

(A6.3)

(A6.4)

510 APPENDIX

b

O~~----------------------~B~

R

a

A __ --------------------------__

FIGURE A6.1. Unit-cell transformations within one and the same lattice.

We have encountered this type of transformation before, in our study of lattices (page 66).

The point P may be represented by fractional coordinates X, Y in the centered unit cell and by x, y in the primitive cell. Since OP is invariant under unit cell transformation.

R = XA + YB = xa + yb

Substituting for A and B from (A6.1) and (A6.2), we obtain

(X + Y)a + (-X + Y)b = xa + yb

whence

x = X + Y

y = -X + Y

Similarly, it may be shown that

X = x/2 - y/2

Y= x/2 + y/2

(A6.5)

(A6.6)

(A6.7)

(A6.8)

(A6.9)

(A6.1O)

A7 COMMENTS ON SOME ORTHORHOMBIC AND MONOCLINIC SPACE GROUPS 511

The vector to the reciprocal lattice point hk is given, from (2.15), by

d*(hk) = ha* + kb* (A6.11)

and that to the same point, but represented by HK, is

d*(HK) = Ha* + Kb* (A6.12)

The scalar d* . R is invariant with respect to unit cell transformation, since it represents the path difference between that point and the origin* (see page 183ff). Hence, evaluating d* . R with respect to both unit cells and using the properties of the reciprocal lattice discussed on pages 73ft, we obtain

hx + ky = HX + KY

Substituting for x and y from (A6.7) and (A6.8), we find

Hence, (h - k)X + (h + k)Y = HX + KY

H=h-k

K=h+k

(A6.13)

(A6.14)

(A6.15)

(A6.16)

which is the same form of transformation as that for the unit cell, given by (A6.1) and (A6.2): it shows that unit-cell vectors and Miller indices transform alike. Generalization of this treatment to three dimensions and oblique unit cells is straightforward, if a little time consuming.

A7 Comments on Some Orthorhombic and Monoclinic Space Groups

A7.1 Orthorhombic Space Groups

In Chapter 2, we looked briefly at the problem of choosing the positions of the symmetry planes in the space goups of class mmm

( 222) ·h f h·· f . - - - WIt respect to a center 0 symmetry at t e ongm 0 the umt mmm

• The full significance of this statement can be appreciated in the light of Chapter 4.

512 APPENDIX

cell. We give now some simple rules whereby this task can be accomplished readily, while still making use implicitly of the ideas already discussed, including the relative orientations of the symmetry elements given by the space-group symbol itself (see Tables 1.5 and 2.5).

Half-Translation Rule

Location of Symmetry Planes. Consider space group Pnna; the translations associated with the three symmetry planes are (b + c)/2, (c + a)/2, and a12, respectively. If they are summed, the result (T) is (a + bl2 + c). We disregard the whole translations a and c, because they refer us to neighboring unit cells; thus, T becomes b 12, and the center of symmetry is displaced by T 12, or b 14, from the point of intersection of the three symmetry planes n, nand a. As a second example, consider Pmma. The only translation is a12; thus, T = a/2, and the center of symmetry is displaced by al4 from mma.

Space group Imma may be formed from Pmma by introducing the body-centering translation!, !, ! (Figure 6.23b). Preferably the halftranslation rule may be applied to the complete space-group symbol. In all, Imma contains the translations (a + b + c)/2 and a12, and T = a + (b + c)/2, or (b + c)/2; hence, the center of symmetry is displaced by (b + c)/4 from mma. This center of symmetry is one of a second set of eight introduced, by the body-centering translation, at t t ! (half the I translation) from a Pmma center of symmetry. This alternative setting is given in the International Tables for X -ray Crystallography; * it corresponds to that in Figure 6.23b with the origin shifted to the center of symmetry at t t !. Space groups based on A, B, C, and F unit cells similarly introduce additional sets of centers of symmetry. The reader may care to apply these rules to space group Pnma and then check the result with Figure 2.37.

Type and Location of Symmetry Axes. The quantity T, reduced as above to contain half-translations only, readily gives the types of twofold axes parallel to a, b, and c. Thus, if T contains an al2 component, then 2x (parallel to a) == 21> otherwise 2x == 2. Similarly for 2y and 2z> with reference to the bl2 and c/2 components. Thus, in Pnna, T = b12, and so 2x == 2, 2y == 21, and 2z == 2. In Pmma, T = a12; hence, 2x == 21,

2y == 2, and 2z == 2. The location of each twofold axis may be obtained from the symbol

• See Bibliography, Chapter 1.

A7 COMMENTS ON SOME ORTHORHOMBIC AND MONOCLINIC SPACE GROUPS 513

of the symmetry plane perpendicular to it, being displaced by half the corresponding glide translation (if any). Thus, in Pnna, we find 2 along [x, t i], 21 along [t y, i], and another 2 along [1, 0, z]. In Pmma, 21 is along [x, 0, 0], 2 is along [0, y, 0], and another 2 is along [1, 0, z]. The reader may care to continue the study of Pnma, and then check the result, again against Figure 2.37.

General Equivalent Positions

Once we know the positions of the symmetry elements in a space-group pattern, the coordinates of the general equivalent positions in the unit cell follow readily.

Consider again Pmma. From the above analysis, we may write

I at 0,0, ° (choice of origin)

mx the plane (i,y, z), my the plane (x, 0, z), a the plane (x, y, 0)

Taking a point x, y, z across the three symmetry planes in turn, we have (from Figure 2.34)

mx 1 X,y,Z ~ 2 - x,y, z

my ~ x,y,z

~ ! + x,y, i

These four points are now operated on by I to give the total of eight equivalent positions for Pmma:

±{x, y, Z; ! - x, y, Z; x, y, Z; ! + x, y, i}

The reader may now like to complete the example of Pnma. A similar analysis may be carried out for the space groups in the

mm2 class, with respect to origins on 2 or 21 (consider, for example, Figure 4.16), although we have not discussed many of these space groups in this book.

514 APPENDIX

A7.2 Monoclinic Space Groups

In the monoclinic space groups of class 2/ m, a 21 axis, with a translational component of b/2, shifts the center of symmetry by b/4 with respect to the point of intersection of 21 with m (Figure S6.4b). In P2/e, the center of symmetry is shifted by e/4 with respect to 2/e, and in P2de the corresponding shift is (b + e)/4 (Figure 2.33).

A8 Vector Algebraic Relationships in Reciprocal Space

A8.1 Introduction

The reciprocal lattice was introduced earlier in a geometrical manner, as we find that treatment suitable for the beginner. With practice and familiarity in reciprocal space concepts, a vector algebraic approach has the appeal of conciseness and elegance, and we introduce this method here.

A8.2 Reciprocal Lattice

In considering the stereographic projection, we showed that the morphology of a crystal can be represented by a bundle of lines, drawn from a point, normal to the faces of the crystal. This description, though angle-true, lacks linear dimensions. The representation may be extended by giving each normal a length that is inversely proportional to the corresponding interplanar spacing in the real lattice, so forming a reciprocal lattice.

The noncoplanar vectors a, b, and c have been used to delineate a unit cell in the real (Bravais) lattice (see page 59ft). The corresponding vectors for the unit cell of a reciprocal lattice, a*, b*, and c*, will be defined by

* b X c a = ----v-' * c X a b =--

V '

where the unit-cell volume V is given by

V=a·bXc

c* = a X b V

(A8.I)

(A8.2)

A8 VECTOR ALGEBRAIC RELATIONSHIPS IN RECIPROCAL SPACE

x" axis

xaxis

zaxis

zOn;, ! C ~r.-r--------------------~G

R c" I I I I I

D

515

FIGURE AS.I. TricJinic unit cell, showing corresponding reciprocal lattice unit-cell vectors.

In Section 2.4, particularly equation (2.11), we included a constant K. In this appendix, we take the value of K as unity (as in Section 2.4.1), so that the reciprocal lattice has the dimensions of length- 1 and is independent of the wavelength of x-radiation.

The magnitude of a* is given by (see Figure A8.1)

* area OBGC be sin a 1 a = =

V V OP (A8.3)

where 0 P is the perpendicular from the origin 0 of the Bravais unit cell to the plane ADFE which contains the point P. Similar relationships may be written for b* and e*, in terms of OQ and OR respectively. Hence, the reciprocal lattice vectors a*, b*, and c* are normal to the planes be, ea, and ab respectively in the Bravais unit cell. From Figure A8.1, it is now easy to see that [cf. (2.13) et seq.]

a . a* = b . b* = c· c* = 1 (A8.4)

and

a . b* = a . c* = a* . b = etc. = 0 (A8.5)

516 APPENDIX

AS.2.1 Interplanar Spacings

Any vector d*(hkl) from the origin to the point hkl in a reciprocal lattice can be written as

d*(hkl) = ha* + kb* - lc* (A8.6)

The vector r from the origin to a point x, y, z in the unit cell of a Bravais lattice is given by

r = xa + yb + zc (A8.7)

The scalar product d*(hkl)· r leads to the equation of a plane in the Bravais lattice, normal to the direction of d*(hkl) (see Figure A8.2). From (A8.6) and (A8.7), the scalar product gives

hx + ky + lz = a constant, K (A8.8)

For K = 0, the plane passes through the origin; for K = 1, it is the first plane from the origin of the family of (hkl) planes. These two cases are expressed also in equations (1.11) and (1.12), since x = X la, and so on (page 12).

o Ifr----!

d(hkl)

o

I I ,

FIGURE AB.2. Planes for K = 0 and K = 1 in a Bravais lattice, showing the corresponding d and d* vectors and the vector r to P(x, y, z), any point in the plane K = 1.

AS VECTOR ALGEBRAIC RELATIONSHIPS IN RECIPROCAL SPACE 517

Since D, the foot of the perpendicular from 0, lies in the same plane as P, the termination of r from the same point 0, it follows that

d*(hkl) . d(hkl) = 1 (A8.9) or

d*(hkl) = d(~kl) (A8.1O)

which may be compared with equation (2.11), the starting point of the geometrical treatment of the reciprocal lattice. Hence, any point hkl in the reciprocal lattice may be said to correspond to a family of planes (hkl) in the Bravais lattice, with interplanar spacing d(hkl).

It may be noted that if P is a Bravais lattice point, then

r = Va + Vb + We (A8.11)

where V, V, Ware the coordinates of the Bravais lattice point [d. equation (2.1)]. The scalar product d*(hkl) • r may now be written as

hV + kV + IW = K (A8.12)

from which it follows that K is an integer, as defined above. The coordinates V, V, W describe the direction, or directed line, [UVW], as discussed on page 59.

We can now appreciate a fundamental difference between the stereographic projection and the reciprocal lattice constructions. Equation (A8.6) places no restrictions on the values of h, k, and I. Implicitly, they are prime to one another, but this limitation is not essential. Suppose that h, k, and I may be written as mh', mk', and ml', where h', k', and I' are prime to one another and m is an integer. If we carry out the same analysis as before, equation (A8.1O) becomes

1 1 m d*(h'k'l') = d(h'k'I') (A8.13)

or

d*(h'k'I') = 1 d(h'k'I')/m

(A8.14)

But d(h'k'[')/m is d(hkl) and, since the definition of Miller indices

518 APPENDIX

identifies each family of lattice planes uniquely (see page 72), equations (A8.6)-(A8.1O) apply to all (hkl). In the stereo graphic projection, h, k, and I are prime to one another. It is clear that the normals to mh, mk, and ml (m = 1, 2, 3, ... ) would all intercept the sphere (Figure 1.20) in the same point.

AS.2.2 Volume of a Parallelepipedon

The volume of a parallelepipedon of edge vectors a, b, and c is given by

v = a' (b X c) (A8.1S)

Let a, b, and c be expressed in terms of an orthogonal set of unit vectors i, j, and k, such that

Then

v = a' (b X c)

a = ali + a:z.i + a3k

b = bli + b:z.i + b3k (A8.16)

= (ali + a2i + a3k) . (blC2k + blcJ.i + b2clk + b2C3i + b3cJ + b3c2i)

(A8.17)

The right-hand side is the expansion of the determinant

al a2 a3 V = bl b2 b3

CI C2 C3

(A8.18)

Since we can interchange rows and columns of a determinant without changing its value, we can write

al a2 a3 V 2 = bl b2 b3

al bl CI

a2 b2 C2

a3 b3 C3

(A8.19)

A8 VECTOR ALGEBRAIC RELATIONSHIPS IN RECIPROCAL SPACE 519

Multiplying the determinants according to the rule for matrices, we obtain

alaI + aZaZ + a3a3

blal + bzaz + b3a3

alb l + azbz + a3b3

bIb I + bzbz + b3b3

aici + aZcZ + a3c3

blCI + bzcz + b3C3

cla l + cZaZ + C3a3 clb l + czbz + C3b3 CICI + CZCZ + C3C3

or

Evaluating, we obtain

a·a a·b a·c

V Z b· a b· b b· c

c·a c·b c·c

(A8.20)

(A8.21)

V Z = aZbzcz + ab cos y bc cos a ca cos f3 + ac cos f3 ba cos y bc cos a

- ca cos f3 bZac cos f3 - cb cos a bc cos a aZ - CZ bc cos yab cos y

(A8.22)

Simplifying gives

v = abc(1 - cosZa - cosZf3 - cosZy + 2cos a cos f3 cos y)lIZ (A8.23)

AB.2.3 Reciprocity of Unit-Cell Volumes

Following (A8.2), we may write

V* = a* X b* . c*

and from (A8.1),

I V* = V 3 {(b X c) X (c X a)·(a X b)}

I = V 3 {(b Xc· a)c - (b Xc· c)a} . (a X b)

I = V 3 {Vc - O}·(a X b)

I

V

520 APPENDIX

Hence,

V*V = 1 (A8.24)

AS.2.4 Angle between Bravais Lattice Planes

The angle <P in Figure A8.3 between planes (hI kIll) and (h2k212) may be most readily obtained from the supplement of the angle between the corresponding reciprocal lattice vectors d*(hlklll) and d*(h2k212). We have

Hence, for the orthorhombic system,

(A8.26)

From (A8.26), we see that the angle between (111) and (111) in the cubic system is cos- 1( -l), or 109.47°. The equation (A8.26) can be generalized by using the full form for dt . di, which incorporates the cross-terms such as (kll2 + llk2)b*c* cos a*.

o FIGURE AB.3. The interplanar angle cp, calculated in terms of the reciprocal lattice

vectors to the two planes concerned, for any crystal system.


AS.2.5 Reciprocity of F and I Unit Cells

In Figure A8.4, we select a primitive unit cell by means of the transformation

ap = bF/2 + cF/2

b p = cF/2 + aF/2

Cp = aF/2 + bF/2

From (A8.I) and (A8.27), we have

* _ bp X Cp _ 1 [(CF aF) (aF bF)] ap - - - - + - X - + -Vp Vp 2 2 2 2

or

since VF = 4Vp • Hence,

ap = -a; + b; + c;

(A8.27)

(A8.28)

(A8.29)

The negative sign before a; is needed to preserve right-handed axes from the product (CF X bF). Similar equations can be deduced for bp and cp.

aF

FIGURE AB.4. An F unit cell, with the related P unit cell outlines within it.

522 APPENDIX

Turning next to a body-centered unit cell, the equations similar to (A8.29) are

(A8.30)

Writing (A8.29) as

ap = -2a;"/2 + 2bj../2 + 2c;"/2 (A8.31)

we see that an F unit cell in a Bravais lattice reciprocates into an I unit cell in the corresponding reciprocal lattice, where the I unit cell is defined by the vectors 2a;", 2b;", and 2c;". If, as is customary in practice, we define the reciprocal of F by vectors a;", b;", and c;", then only those reciprocal lattice points for which h + k, k + I, and 1 + h are each integral belong to the reciprocal of F. In other words, Bragg reflections from an F unit cell have indices of the same parity.

A8.3 X-ray Diffraction and the Reciprocal Lattice

We have shown (Section 5.3) that the path difference p for scattering from two centers (0 and A of Figure 5.3) is, from (5.4),

p = A.r· S (A8.32)

If 0 and A are also lattice points, with 0 as the origin, then if the waves scattered from 0 and A are to be in phase, r· S must be integral; that is, p = nA.. Thus,

(Va + Vb + Wc)·S = n (A8.33)

where V, V, and Ware the coordinates of the lattice point A at the end of rand n is an integer. This equation holds for any integral change in V and/or V and/or W. Hence,

a· S = h

b· S = k

c· S = 1

(A8.34)

where h, k, and 1 are integers: (A8.34) is a vectorial expression of the Laue equations (page 133).


h=1

h=O

xaxis

o

FIGURE A8.5. Planes for h values of 0, 1, and 2 in a Bravais lattice, with the corresponding scattering vectors.

In Figure A8.5, three planes normal to the x axis are shown. For the plane h = 0, a' S = 0, which means that the projection of S on a is zero: it may be compared with the zero layer of an oscillation photograph taken with the crystal rotating about a (see page 150ff). For a . S = 1, we have a similar plane making an intercept of 1/a along the x axis. Hence, a . S = h represents a family of parallel equidistant planes normal to a. When the equations (A8.34) are satisfied simultaneously, scattering in the hkl spectrum, or from the (hkl) family of planes, occurs.

AS.3.1 Bragg's Equation

The equations (A8.33) may be written as

Hence,

(a/h)·S = 1

(b/k)·S = 1

(ell) . S = 1

(a/h - b/k) . S = 0

(A8.35)

(A8.36)

which means that S is normal to the vector (a/h - b/k). From Figure A8.6, it follows that this vector lies in the (hkl) plane. Also S is normal to (a/h - e/I) and to (b/k - e/I); hence, S is normal to the plane (hkl). This result can be seen in another way.

524 APPENDIX

zaxis

yaxis

o

xaxis

FIGURE A8.6. An (hkl) plane in a Bravais lattice; N is the foot of the perpendicular from the origin 0 to the plane.

In Figure AS.7, S is shown as the bisector of the angle between sand So, normal to (hkl). The magnitudes of OA and OB are each 1/)", and the angles AOC and BOC are each equal to n/2 - (J(hkl). Hence, S/2 = (1/),,) sin (J(hkl), or

2 . S = ;: sm (J(hkl) (AS.37)

The interplanar spacing d(hkl), shown in Figure AS.6, represents the projection of a/h (or b/k or ell) on to S. Hence,

d(hkl) = (a/h)· SIS (AS.3S)

Using (AS.34) with (AS.37) and (AS.3S), we have

(a/h)· S = 1

d(hkl) = l/S = M[2 sin (J(hkl)]

AS VECTOR ALGEBRAIC RELATIONSHIPS IN RECIPROCAL SPACE 525

c

(hkl)

o FIGURE AS.7. Relationship of the scattering vector S to the corresponding (hkl) plane.

or 2d(hkl) sin 8(hkl) = A (A8.39)

which is Bragg's equation. Figure A8.7 illustrates the idea of "reflection" from a plane, but subject to (A8.40). The reader is invited to redraw the Ewald sphere (Figure 3.25) with a radius of I/A (A = 1.54 A, say, with an appropriate scale) inside the limiting sphere, center 0, radius 2/A, and with CO = -so and CP = s to show that OP (OC in Figure A8.7) is identified with S.

A8.4 Crystal Setting

We shall consider the problem of bringing a given reciprocal lattice plane (equatorial plane) to a position normal to the crystal rotation axis, prior to taking an oscillation photograph. We shall assume that our crystal has a well-developed morphology, such as a prismatic (needleshaped) habit (Figure 3.4).

The crystal is set up on a goniometer head, with its prism axis along the axis of rotation. Two arc adjustments, A and B, and two sledges, C and D (Figure 8.2) enable the crystal to be set, initially to better than 5°, and arranged so as to rotate within its own volume.

A8A.1 Setting Technique

A method of Weisz and Cole, as modified by Davis, will be considered: it has the advantage that each arc can be treated independently of the other.

A 15° oscillation photograph is taken with the arcs A and B at 45° to the x-ray beam (Figure A8.8a) at the midpoint of the oscillation range,

Incident X-ray beam

(a)

Long eXPOSure

True zero

Layer-line

rt e){ft05Ure 5,,0 ..

(b)

Incident

X-ray beam ------ - ----

(e)

FIGURE AS.S. Crystal setting, (a) Goniometer arcs set at 45° to the x-ray beam. (b) Appearance of a double oscillation photograph-the traces are outlined by the spots from the Ka and KfJ radiation and the Laue "streaks"; the top right-hand corner (x-ray beam coming toward the observer) is clipped for identification. (c) Ewald sphere-the dashed line is the true equatorial circle, and the full line represents the longer exposure, related to the equatorial circle by rotation about the line OPE'

A9 INTENSITY STATISTICS 527

using unfiltered radiation and an exposure time sufficient to produce intense reflections. The goniometer head is then turned through exactly 180° and a second oscillation photograph taken on the same film, but with an exposure time of about one-third that of the first. The form of the double oscillation zero layer-line curve is shown in Figure A8.8b.

From Figure A8.8b, the distance of the curve above (Pd or below the true zero layer-line position at () = 45° is tan- 1 (ddD), where Dis the diameter of the film. This value is also that of the ~ reciprocal lattice coordinate of a possible reflection at PL' Thus, the angle of elevation of the reciprocal lattice vector OPL is ch = tan-l(~/OPL) =

tan- 1(ddDV2), taking K = A, as in Section 3.5.3. Similarly, OR =

tan- 1(dR /DV2). For values of 0 < 4°, tan 0 = 0 to 0.1%. Hence, for D = 57.3mm, 0 = 0.707do, with d expressed in mm.

In order to apply the corrections, we return the goniometer head to a reading at or near the center of the range for the longer exposure. With the photograph marked as shown in Figure A8.8b, consider the more intense curve. The correction OR is applied to arc A, lying in the NE-SW direction (N toward the x-ray source). The direction of movement of the arc is such that a reciprocal lattice vector at () = 45°, imagined to be protruding from the crystal in the NE direction (the reciprocal lattice origin is transferred to the crystal at this point) will be brought to the equatorial plane. The correction OL is applied to arc B in a similar manner.

A9 Intensity Statistics

A9.1 Weighted Reciprocal Lattice

The weights (intensities or amplitudes) associated with the reciprocal lattice points show five different characteristics that we shall consider in turn.

A9.1.1 Laue Symmetry

We discussed on page 45 the fact that x-ray diffraction patterns are centrosymmetric. Hence, the symmetry of this pattern, in terms of both position and intensity, corresponds to one of the 11 centrosymmetric (Laue) point groups. Where the crystal is centro symmetric , the Laue symmetry of the diffraction pattern is also the point group of the crystal.

528 APPENDIX

Where the crystal is noncentrosymmetric, the diffraction pattern is centrosymmetric only insofar as the Friedel law applies, that is, the effects of anomalous scattering are negligible. This condition is easily satisfied with, say, a compound of C, H, N, and ° irradiated with Mo Ka, but not necessarily with, say, a compound containing C, H, N, 0, and Br irradiated with Cu Ka (see Table 6.8).

A9.1.2 Systematic Absences

Certain groups of reciprocal lattice points have zero weight because of the space-group symmetry, irrespective of the contents of the unit cell. This topic was discussed sufficiently for our purposes in Chapter 2.

A9.1.3 Accidental Absences

A small proportion of possible reflections, although not of zero intensity, are often sufficiently weak not to be recorded with significance in the x-ray diffraction pattern. This effect depends upon both the nature of the atoms present and their relative positions in space. These reflections are often called "unobserved," because they do not produce visible blackening of an x-ray photograph. With diffractometer data, another criterion has to be adopted. Reflectiens may be so classified if, typically, / < 3a(I), where a(/) is the standard deviation, from counting statistics, of the intensity l. The "unobserved" data are often omitted, without powerful reason, from the structure analysis. In a good structure determination, the unobserved data should, at least, be checked against the corresponding IPcI values to ensure that no significant reflections have been classified erroneously or measured badly.

A9.1.4 Enhanced Averages

Some groups of reflections have enhanced average intensities. We touched upon this topic on pages 306 and 368, where we saw that such reflections were dependent upon the crystal class. We shall now look at this effect in more detail, and, in particular, show that the enhancement factor E is the same for the space groups P2/m, P2/c, P21/m, and P2dc, all of crystal class 2/ m.

The structure factor equation for these space groups may be written


as

N/4

F(hkl) = 2: 4gj cos 2n(hxj + lzj + n/4) cos 2n(kYj - n/4) (A9.1) j~l

where N is the number of atoms in the unit cell and n is an integer (0, I, k, and k + I, respectively, for the four space groups being considered). The average value of F(hkl) depends upon the averages cos 2n(hxj + IZj + n/4) and cos 2n(kYj - n/4). If we assume a random distribution of atomic positions and provided that N is not small, then since -1:;:;: cos (angle) :;:;: 1, these averages are zero and, thus, F(hkl) is zero. From (A9.1),

N/4

IF(hklW = 2: 16gJ cos2 2n(hxj + IZj + n/4) cos2 2n(kYj - n/4) j~l

N/4 N/4

+ 2: 2: 16gigj cos 2n(hxi + IZi + n/4) cos 2n(kYi - n/4) i~l j~l

x cos 2n(hxj + IZj + n/4) cos 2n(kYj - n/4) (A9.2)

The terms under the double summation will take both positive and negative values, and for a sufficiently large number N of similar, randomly distributed atoms this sum will tend to zero. Now we have

N/4

IF(hklW = 2: 16gJcos2 2n(hxj + IZj + n/4)cos2 2n(kYj - n/4) j~l

(A9.3)

From reasoning similar to that used before, the average value of cos2(angle) = !. Hence,

NI4 N

IF(hklW = 2: 4gJ = 2: gJ = S (A9.4) j~l j~l

which is the Wilson average (page 305). If we consider the hOI reflections,

N/4 N

IF(hOIW = 2: 8gJ = 2 2: gJ = 2S (A9.5) j~l j~l

530 APPENDIX

Similarly, we can shown that the average value IF(OkOW is also 2S, but that for any other class of reflection, equation (A9.3) gives the value S. Thus, we have shown that the enhancement (epsilon) factors for crystal class 21m are ehOI = ehOO = eOOI = eOkO = 2, otherwise e = 1.

A graphical derivation of these results is afforded by the stereogram of crystal class 21m, Figure 1.39. If we consider that the four points shown are gj vectors in the structure factor equation, it is clear that projection on to the m plane (hOI data) or on to the 2 axis (OkO data) leads to superposition of the gj vectors: each behaves in projection as though it has doubled weight (e = 2). Results for different crystal classes may be obtained in either of the ways shown (Table 7.1).

A9.1.5 Special Distributions

The weighted reciprocal lattice, in entirety or in special planes or rows, may approximate to one or more of certain distinctive distributions. Two of them depend upon the presence or absence of a center of symmetry in the crystal. This information can be very important in the determination of the space group, since information from systematic absences may be inconclusive.

The structure factor equation can be written conveniently as

NI2

l\. = L 2gj cos 2n(h . r j ) j=l

(A9.6)

for a centrosymmetric crystal. The terms hand rj are the vectors ha* + kb* + lc* and xja + yjb + zjc, so that h . rj = hXj + kYj + lZj. We have seen that the mean value 2gj cos 2n(h . rj) is zero. The variance of this term may be given as

l-j = 4gj[cos2 2n(h . rj) - cos 2n(h . rYl = 2gJ (A9.7)

Cramer's central limit theorem in statistics states that the sum of a large number of independent random variables has a normal probability distribution, a mean equal to the sum of the means of the independent variables, and a variance equal to the sum of their variances. We have shown in Section A9.1.4 that F(hkl) = 0; this result follows directly from Cramer's theorem.


A normal probability distribution of F is given by

(A9.8)

Since P = 0, and a2 is given by

N/2 N

~ = L ~ = L 2gJ = L gJ = S (A9.9) j=l j=l

the centric probability distribution function is given by

(A9.1O)

For noncentrosymmetric crystals, the corresponding acentric distribution function is

(A9.11)

From the relationship

(A9.12)

where S, as well as containing a temperature factor will here be assumed to include e, and with 1F12 on an absolute scale, it follows that

(A9.13) and

(A9.14)

Equations (A9.13) and (A9.14) are independent of the complexity of the structure; they may be used for deducing a number of useful statistical parameters, such as those in Table 7.2.

As an example, we will determine the value of lEI for a noncentrosymmetric crystal. The mean, or expectation, value of a variable X, distributed according to a probability function P(X), is given generally by

x = J XP(X) dX / J P(X) dX (A9.15)

532 APPENDIX

where the integration is carried out over the limits of the variable. Hence,

f IEI2 exp( -IEI2) dlEI lEI = -0-00------L lEI exp( -IEI2) dlEI

(A9.16)

To solve these integrals easily, let IEI2 = t, so that d lEI = dt/2t1l2. Thus, the numerator becomes

LOO 2/2 - 1 exp( -t) dt (A9.17)

This integral is the gamma function rG) = !r(!) = !-vn, or 0.89. It is easy to show that the value of the denominaor is unity; hence 0.89 is the value of lEI, as given in Table 7.2.

The cumulative values in the same table can be obtained from the same distribution equations. Let the fraction of lEI values less than or equal to some value p be N(p), given by

LIEI

N(p) = 0 P(IEI) dlEI (A9.18)

For the noncentrosymmetric crystal, we have

LIEI

N(p) = 2 0 lEI exp( -IEI2) dlEI (A9.19)

or (A9.20)

For p = l.S, N(l.S) = 0.89S. Hence, the number of lEI values greater than 1.S in the acentric distribution is O.1OS, or 1O.S%, as listed in Table 7.2. Many other useful results can be obtained quite simply by means of the two distribution equations.

Integrals of the type in (A9.16) are easily evaluated through the properties of the gamma function f(n). We define

f(n) = r tn - 1 exp( -t) dt

AID ENANTIOMORPH SELECfION 533

The following results may be used directly:

r(n) = (n - 1)! for n integral and greater than zero

r(n) = (n - 1)r(n - 1)

We may question the upper limit of 00 in equation (A9.16). It is easy to show that the maximum lEI, E(OOO), is given by VNTi, for a unit cell of N similar atoms. If we have a molecule of 25 similar atoms in general positions of space group P21 , E(OOO) is 5.00; g is the symmetry number (number of general equivalent positions) of 2 for the space group. From (A9.20), we see that the fraction of lEI values greater than 5.00 is 1.4 X 10- 11 . Hence, the error in taking the upper limit of 00 is totally negligible, but the convenience is considerable.

A10 Enantiomorph Selection

In those noncentrosymmetric space groups, such as P21 and P21212b

that contain no inversion symmetry (enantiomorphous space groups), it is always possible to specify two enantiomorphic arrangements of the atoms in the structure that will lead to the same values of IFI. For example, in the structure in Figure 1.3, which has two molecules per unit cell in space group P21, the two arrangements would be related by inversion through the origin, and will be referred to as the structure (S) and its inverse (I). From the structure factor theory discussed earlier, we can write

F(hh = A(hh + iB(hh (AlO.I)

for the structure, and

(AIO.2)

for its inverse. From the inversion relationship, we know that F(hh and F(h)j are complex conjugates; hence,

A(hh = A(h)j (AlO.3)

and

B(hh = - B(h)[ (AlO.4)

534 APPENDIX

For either the structure or its inverse, we can choose B(b) to be positive, so that the corresponding phase angle q,(b) lies in the range 0 ~ q,(h) ~ lr. This procedure was followed in the structure analysis, of tubercidin (Section 7.2.9), where the phase of symbolic reflection a (138) was restricted to a value between 0 and lr, specifically 3lr/4.

In P2 l 2 l 2l , another noncentrosymmetric space group of frequent occurence in practice, the zonal reflections Okl, hOI, and hkO are centric, and may be given phases equal to mlr /2. The value of m takes the same parity as the index following zero, working in a cyclic manner. Thus, an origin and an enantiomorph could be specified in this space group by the selection

520

011 11 3 0

11 0 0

+lr/2} +lr/2

+lr/2

+lr/2

Origin

Enantiomorph

A detailed practical treatment on the origin and enantiomorph for all space groups has been given by Rogers. *

It is important not to confuse the specifying of the enantiomorph with the selection of the absolute configuration of a structure: in both cases, the same type of space group is involved. Selection of the enantiomorph is essential to a correct application of direct methods to a structure with an enantiomorphous space group. However, the solution of the structure may correspond to either the absolute configuration or its inverse. This dilemma has to be resolved by further tests, usually involving anomalous scattering (see page 335ft).

A 11 Analytical Geometry of Direction Cosines

A 11.1 Direction Cosines of a Line

In Figure All.1, let PI be any point Xl, Yl, Zl referred to rectangular X, y, and Z axes. Draw lines from P1 perpendicular to the X, y, and Z axes to cut them at A, B, and C, respectively. Thus, OA = Xl> OB = Yl> and OC = Z1' The direction cosines of 0 P1 are given by cos X I = X d 0 PI,

* D. Rogers, in Theory and Practice of Direct Methods in Crystallography, New York, Plenum Press (1980).

All ANALYTICAL GEOMETRY OF DIRECTION COSINES 535

z axis

c

/

x axis

FIGURE A11.l. Direction cosines of a line, referred to rectangular axes x, y, and z.

COS 1/Jl = yt/OPv and cos Wl = zt/OP1• Hence,

cos2 Xl + cos2 1/Jl + cos2 Wl = (xi + yi + zi)/opi (All.l)

But since Xv Yv and Zl are the projections of OPl on to the x, y, and z axes, it follows that

xi + yi + zi = opi Hence,

A 11.2 Angle between Two Lines

(All.2)

(Alt.3)

On another line OP2, let a point X2, Y2, Z2 be marked off such that the lengths of OPl and OP2 are equal, say r. We have for OP1 , from above,

Xl = r cos Xv Yl = r cos 1/Jl, and Zl = r cos Wl

and for OP2 ,

X2 = r cos X2, Y2 = r cos 1/J2, and Z2 = r cos W2

If the origin is shifted from 0 to Pv then the coordinates of P2 become

x~ = X2 - Xl> y~ = Y2 - Yv z~ = Z2 - Zl (Alt.4)

536

so that the length PI P2 is given by

(P1P2)2 = (X2 - X1)2 + (Y2 - Y1)2 + (Z2 + Zl)2

= r2(cOS2X1 + COS2 X2 - 2COSX1COSX2

+ COS2 ""1 + COS2 ""2 - 2 cos ""1 cos ""2

+ COS2 WI + COS2 W2 - 2 cos WI cos W2)

Using (A11.3), we have

APPENDIX

(A11.S)

(P1P2)2 = 2r2[1 - (cos Xl cos X2 + cos ""1 cos ""2 + cos WI cos W2)]

(AI1.6)

In the isosceles triangles 0 PI P2

(AI1.7)

Therefore,

(AI1.8)

Comparing (AI1.6) and (AI1.8), we obtain

cos (J = cos Xl cos X2 + cos ""1 cos ""2 + cos WI cos W2 (AI1.9)

The same result can be achieved by solving

(A11.1O)

following Section 7.5.1.

A12 The Stereographic Projection of a Circle Is a Circle

Consider Figure AI2.1. Let AB be the trace of a small circle, center 0, on a vertical section of a sphere of arbitrary radius. Lines such as PA or AX generate a cone about the axis PQ. The trace AX of its right section is one diameter of an ellipse; AB is the trace of one circular section of this ellipse. Symmetrically inclined to the axis PQ is the

Al2 THE STEREOGRAPHIC PROJECTION OF A CIRCLE IS A CIRCLE 537

z

Sr-----------------r-~~~_j~----~T

P

FIGURE A12.1. Construction to show that the stereographic projection of a circle is a circle.

conjugate circular section, trace XY. Draw BZ parallel to the primitive ST. Then

UP = @ (equal arcs PZ, PB) (A12.1)

But triangles PAB and PXY are congruent, so that

(A12.2)

Hence

XY II BZ(IIST) (A12.3)

and, therefore, UV is a diameter of a circle on the primitive. It follows that all angular relationships on a crystal are reproduced on the stereogram, so that it becomes a symmetry-true (angle-true) representation.

538 APPENDIX

A13 Setting a Crystal for Precession Photography

As with other methods of single-crystal x-ray photography, production of a good precession photograph requires the crystal to be aligned accurately with respect to the mechanical parts of the camera and with the x-ray beam direction. There are two principal aims of this setting procedure:

(a) to set the crystal so that an axis of the direct lattice (usually a, b, C, or a reasonably simple [UVW] direction) is parallel to the x-ray beam when the precession angle p, is set at zero;

(b) to align a reciprocal lattice axis parallel to the horizontal, or dial, axis of the camera.

Of these two conditions, (a) is absolutely essential to the setting, and (b), although not essential, is highly desirable, as it allows for subsequent rotation of the crystal in order to pick up a second axis to satisfy condition (a) while maintaining the constant horizontal reciprocal lattice row common to both films. This condition will usually facilitate a complete survey of the reciprocal lattice of the crystal in the minimum of settings.

A13.1 Setting a Crystal Axis Parallel to the X-ray Beam

Optical examination of the crystal will usually suggest a possible crystallographic axial direction (page 119ff). A useful tip to remember is that a principal axis will often be found perpendicular to a broad face of the crystal. Stable crystals are usually mounted, using adhesive, onto a glass fiber while unstable crystals, such as proteins, are mounted in the presence of mother liquor inside sealed capillary tubes. For precession photography a crystallographic axis should be perpendicular or closely perpendicular to the fiber or glass capillary. The final alignment of the axis can be undertaken with the crystal mounted on the camera using the goniometer arcs (Figure 8.2). The camera is fitted with a telescope by means of which the crystal can be centered, that is, adjusted on the axes such that it rotates within its own volume, and the desired axis aligned approximately along the x-ray beam. On setting the precession angle to a given p, value, the axis will describe a cone defined by this angle (page 166ff). A zero-level precession photograph taken with unfiltered x-radiation (usually eu or Mo) will be characterized by Laue

A13 SETIING A CRYSTAL FOR PRECESSION PHOTOGRAPHY 539

• '1 1 .. I ;- . . I ' ~ : , i;', '1' , 17 . ····Y .' 1 "1' ., ' ~F

70~ --'- ,," . K I·· ' j.:o · . ... I" ::- --:---- -+--+-t1 --i '[j~ -:f' ! ., ;;(" W I T .'. ::;:t . . . . : '~ 1"-i W ·l ;" . * .!,. ' + IW " ·~ · · · · " ± . 60::--~' --;- ... _._- ,- " -"1 .' I· . I ~. . · Y I . • v J;; 'I r -f + ~ ' ~Ll- " ·1

Bmso :-;····-I~·······- '-",-- - . . ........ '. - --'-..-1' C L.; .. _ -1;-1 IS-b.,~. I-ii ~ I-:Y. I, . + .. '1., "1· . ..1·.·.···, _. - -'- -~ ....., .....,... "'" Vn /f • . II i .'i"J ~O j,;.1 . i

FIGURE A13.1. Chart and setting instructions for precession photography. The graph shows the variation of {jjmm with E/deg for various values of il (indicated for each curve), and with a crystal to film distance (M) of 60 mm. The horizontal arc (A) and dial (D) corrections are both clockwise for the situation given: d, direct beam spot (may be found by drawing Laue streaks toward the middle of the photograph); c, center of precession circle (radius of precession circle is 2M tan il). Assume that the face of the arc is horizontal, with the inscribed angular scale pointing upward toward the observer. (a) Dial correction (d above c): for the situation in the diagram, the dial correction Ev/deg derived from Dv/mm using the chart is clockwise. (b) Arc correction (d left of c): similarly, EA/deg the horizontal arc correction derived from c5dmm is clockwise for the situation shown. (c) Corrections corresponding to other relative positions of d and c are made by analogy with (a) and (b). (d) The diagram and explanation assume that the observer is looking at the x-ray film, with beam coming toward him and with the dial on the right-hand side (as with most cameras). (e) The desired setting can usually be achieved by means of a fairly small il setting. A il of 10° is recommended for most cases, with 2r = 10.6 mm (hole) for s = 30 mm.

streaks radiating from the point d (Figure A13.I), where the direct x-ray beam would strike the film when il = O. The ends of the Laue streaks define a circle, center c (Figure A13.I). When the crystal axis is perfectly aligned, points d and c coincide. Otherwise the situation is as shown in Figure AI3.l. Two adjustments are required, defined in this diagram by a horizontal component OA and a vertical component OD' The values of 0 are measured as shown from d to the circumference of the circle, taking the larger of the two distances in each case.

540 APPENDIX

The Vertical Correction E~, Defined by Do on the Film

This correction is applied to the dial axis (D) of the precession camera. The conventional direction for the correction to be made (clockwise or anticlockwise) is explained in Figure A13.l. The reading of DD mm is converted into E~ by use of the chart* shown in Figure A13.l. Values of ED are given for selected fi values, other fi values being accessible by interpolation.

The Horizontal Correction E~, Defined by DA on the Film

This correction applies strictly to a goniometer arc (A) which is horizontal (or parallel to the x-ray beam direction) when fi = O. The conventional direction for the correction to be made is explained in Figure A13.l. The reading DA is converted to E~ by use of the same chart. If the goniometer arcs are significantly off-parallel and offperpendicular to the x-ray beam, EA should be resolved into appropriate components depending on the cosine and sine of the offset angular value.

Application of the ED and EA corrections as explained should result in coincidence of points d and c. Further correction may be required and implemented by taking a second photograph.

A13.2 Setting a Reciprocal Lattice Row Horizontally

A reciprocal lattice row which is offset by a few degrees (usually no more than ±5°) from the horizontal may be leveled quite satisfactorily. It is achieved simply by adjusting the vertical goniometer arc by the required angle. Again, if the arcs are not horizontal and vertical, the procedure becomes more difficult. Corrections larger than about 5° may be worth attempting so as to ensure a good sequence of precession photographs, but can usually be carried out only by physically pushing over the crystal on its mount (glass fiber or capillary) or by completely remounting the specimen.

A13.3 Screen Setting

When undertaking the above settings from zero-level precession photographs, a screen should be inserted (cf. Section 3.5.5). If a small fi value (5° or 10°) is used, the screen may simply be a hole of radius r cut

* D. J. Fischer, American Mineralogist, 37 (1952).

A14 SYNCHROTRON RADIATION 541

into a metal plate. The film-screen distance s is calculated as usual from the relationship s = r cot p. A value of 2r = 10.6 mm is recommended for p = 10° at s = 30mm.

A14 Synchrotron Radiation

A synchrotron is a large-scale particle accelerator used in modern research, designed primarily as a tool for fundamental studies in physics, but with many spin-off applications, including x-ray crystallography. The term synchrotron (from the Greek word meaning"simultaneous") is used for cyclic resonance proton and electron accelerators, the latter capable of providing a high-intensity x-ray source. The orbit of the particles in such a machine is produced by means of magnetic fields that increase, with time, proportional to the increased momentum of the particles, while the radius during acceleration remains constant. This design requires the magnetic force to operate only over narrow ranges around the orbit, alternating between radially increasing and decreasing flux fields. Particles are accelerated in a linear accelerator prior to injection into the synchrotron (Figure A14.1). Acceleration of the particles is accomplished by resonators placed in the gaps between the magnets. In an electron synchrotron, the injection of the electrons takes place at relativistic energies of the order of 10 Me V, at which energy their speed is close to that of light. The higher the given energy of the accelerated particles, the greater the radius needed for the ring electromagnet of the synchrotron. For an energy of 100 GeV the radius is 200 m (or 660 ft). Consequent to the high-energy acceleration of the negatively charged electrons, theory predicts the emission of a strong electromagnetic radiation, known as synchrotron radiation (SR), with a wide spectrum extending from radio waves to x-rays. At energies of 10 GeV, each electron gives off radiation equivalent to about 30 Me V per revolution. The synchrotron installation at Daresbury, Cheshire, generates a maximum energy of 2 GeV. The high-intensity, finely collimated polychromatic x-radiation from a synchrotron source can be used in a great variety of applications in modern crystallographic research.

A typical SR spectrum is shown in Figure A14.2, which should be compared to that of the conventional sealed x-ray tube in Figure A4.2. The photon intensity is given in units of photon S-1 for (1) a horizontal angular aperature of 1 mrad (1 mrad = 3.4 min of arc), (2) a 1 A beam

542 APPENDIX

FIGURE A14.1. Diagrammatic representation of a synchrotron device; the diameter of the storage ring would be typically ca 20 m.

current, and (3) a 0.1 % spectral bandwidth after performing a vertical integration over the full angular divergence of the radiation above and below the orbital plane. The peak intensity in these units is approximately 5 X 1013. The flux attainable in practice depends on the multiplying factors set by the values of the dependent parameters listed above. Thus, the horizontal aperture of an experimental workstation may be less than 1 mrad for topography, typically 5 to 10 mrad for the majority of spectroscopic experiments, but up to 40 mrad for the "high-aperture" port used for time-resolved measurements. A bandwidth of 0.1 % represents a good resolution (0.1 A at 100 A) for many experiments. The available flux will change proportionately if this resolution is varied. Initially, sufficient radio-frequency power has been installed to achieve a circulating electron beam of up to 0.37 A at 2 GeV. At a later time, this may be upgraded to 1 A. The stored current and, hence, photon flux gradually decline as electrons are lost by scattering from closed orbits.

A14 SYNCHROTRON RADIATION

X Ray

1013 r-------''-----........ ----........ -----,

X-Ray microscopy

X-Ray absorption & diffraction • • structure studies

10keV

1 10-1

X-Ray .lithooraPh~

1keV

10 1

Energy 100eV

100 10

Wavelength ;..

543

FIGURE AI4.2. Spectral curves in the x-ray region from a normal bending magnet and a wiggler magnet for a 2-GeV I-A beam in the synchrotron radiation source, and the types of experiment used with the wavelength regions indicated.

The beam lifetime (time to fall to ca 1/e of the initial intensity) is expected to be ;3;8 hour.

The "wiggler" comprises three high-field bending magnets in series which, installed in a straight section of the ring, introduce an "out and back" swing of the electron beam; in the tighter curvature path the electrons radiate a spectrum extending to shorter wavelengths. The wiggler is designed specifically to produce the highest possible intensities for experiments at 1 A or below.

Tapping this x-ray source involves highly sophisticated technology tailored for particular applications, which include rapid data collection from unstable samples (e.g., proteins), poorly diffracting samples such as fibrous polymers or extremely small crystals, time-resolved studies of solid-state reactions or transformations including enzyme-catalyzed processes, and a study of crystal defects using x-ray topography.

Solutions

Chapter 1

1.1. (1,3.366).

1.2. (a) (120). (b) (164). (c) (001). (d) (334). (e) (043). (f) (423).

1.3. (a) [511]. (b) (352). (c) [111]. (d) [110].

1.4. (523); (523) and (523) are parallel, and [UVW] and [UVW] are coincident.

1.5. (a) See Figure S1.1. (b) cia = cot loo101 = cot 29.40 = 1.775. (c) In this example, the zone circles may be sketched in carefully, and the stereogram indexed without using a Wulff's net. Draw on the procedures used in Problems 1.3 and 1.4. (The center of the stereogram corresponds to (001), even though this face is not present on the crystal.) By making use of the axial ratio, the points of intersection of the zone circles with the Y axis may be indexed, even though they do not all represent faces present. Reading from center to right, they are 001, 013, 035, 011, 021, and 010 (letter symbols indicate faces actually present). Hence, the zone symbols

.100

.01 0 F-+---*-+--1-~~ ---*,d-~ .010

.100

FIGURE Sl.l

545

546 SOLUTIONS

-0 0-

(01 (bl

FIGURE S1.2

and poles may be deduced. Confirm the assignments of indices by means of the Weiss zone law.

1.6. (a) mm. (b) 2/m. (c) 1.

1.7. See Figure S1.2. (a) mmm

1.8. m.m.m == 1. {OW}

(b) 2/m

2.m == 1. {flO} {1I3}

2/m 2 4 4 42m 4 4 8 m3 6 12 24

1.9. (a) 1. (b) m. (c) 2. (d) m. (e) 1. (f) (2). (g) 6. (h) 6mm. (i) 3. (j) 2mm. (Did you remember to use the Laue group in each case?)

1.10. (a) 2. (b) m.

1.11. (a) 6m2 D3h {lOfO} or {OlIO}

4 (b) -mm

m D4h {lOO} or {lIO}

(c) m3m Oh {lOO} (d) 43m Td {111} or {Ill} (e) 3m C3v

(f) 1 C1

6 (g) -mm

m D6h

(h) mm2 C2v

(i) mmm D2h

(j) mm2 c2v

Chapter 2

2.1. (a) (i) 4mm, (ii) 6mm. (b) (i) Square, (ii) hexagonal. (c) (i) Another square can be drawn as the conventional (p) unit cell. (ii) The symmetry at each point is degraded to 2mm. A rectangular net is produced, and may be described by a p unit cell. The transformation equations for both examples are

a' = a/2 + b/2, b' = -a/2 + b/2

SOLUTIONS 547

Note. A regular hexagon of points with another point at its center is not a centered hexagonal unit cell; it represents three adjacent p hexagonal unit cells in different orientations.

2.2. The C unit cell may be obtained by the transformation a' = a, b' = b, c' = -a/2 + c/2. The new dimensions are c' = 5.763 A and (J' = 139.28°; a' and b' remain as a and b, respectively. v,,(C cell) = VAF cell)/2. (Count lattice points per unit cell.)

2.3. (a) The symmetry is no longer tetragonal, although it represents a lattice (orthorhombic) . (b) The tetragonal symmetry is apparently restored, but the unit cell no longer represents a lattice because the points do not have the same environment. (c) A tetragonal F unit cell is obtained, which is equivalent to I under the transformation a' = a/2 + b/2, b' = -a/2 + b/2, c' = c.

2.4. 28.74 A (F cell); 28.64 A. Students familiar with matrices may note that the second result can be obtained by transforming [312] in the second cell to the corresponding direction in the first cell, [411].

2.5. It is not an eighth system because the symmetry of the unit cell is not higher than L It represents a special case of the triclinic system with y = 90°.

2.6. (a) Plane group c2mm (see Figure S2.1).

(O,O,!, !)+ Limiting conditions

8 (f) x, y; x,)'; i,y; x,y hk: h + k = 2n 4 (e) m O,y; O,Y 4 (d) m x,O; x,O 4 (c) 2 1 1- I 3 As above + 4,4, "4, "4

hk:h = 2n, (k = 2n) 2 (b) 2mm 0, ! 2 (a) 2mm 0,0

(b) Plane group p2mg. See Figure S2.2. If the symmetry elements are arranged with 2 at the intersection of m and g, they do not form a group. Attempts to draw such an arrangement lead to continued halving of the "repeat" parallel to g.

2.7. See Figures S2.3 and S2.4.

4 (e)

2 (d) 2 (c) 2 (b) 2 (a)

(100) p2gg (010) p2 (001) p2gm

1

x, y, z; x, y, z; x,! + y,! - z

~,o, 4; !, to 0,0, t O,!,O t 0,0; I I I

"2,2, '2 0,0,0; 0, t!

b' = b, c' = c a' = a, c' = c/2 a' = a, b' = b

x,! - y, ~ + z; hkl: None hOI: 1= 2n OkO: k = 2n

} As above + hkl: k + I = 2n

The two molecules lie with the center of their C(1}-C(1)' bonds on any pair of special positions (a)-(d). The molecule is therefore centrosymmetric and planar.

548

o 0 0 0

o 9 ____ • __________ ~---~ 0

I 0 0 I

o 0- - - - • - - - -~ ~ - - - - .- - - - 0 0

o 0 0 0

d, , P

I

Ib

q'

dt P

FIGURE S2.1

,

t I

< I

I I

L I

:> I

FIGURE S2.2

, , + o 0-----------0-----------0

0+ 0+

FIGURE S2.3

~C(1)O ~

FIGURE S2.4

I

d+ p

,b q,

d '

'P

SOLUTIONS

SOLUTIONS 549

The planarity implies conjugation involving the C(1)-C(1)' bond. (This result is supported by the bond lengths C(1)-C(1)' = 1.49 A and C-C (in ring) = 1.40 A. in the free-molecule state, the two rings rotate about C(l)-C(l)' to give an angle of 45° between their planes.)

2.8. Each pair of positions forms two vectors, between the origin and the points ±{xt - x2), (Yt - Y2)' (Zt - x2)}: one vector at each of the locations

2.9.

2x, 2y, 2z; lX, 2y, 2z; 2x, 2y, 2z; lX, 2y, 2z

and two vectors at each of the locations

2x, !, ! + 2z; O,! + 2y,!; lX, t ! - 2z; O,! - 2y, !

Note: -(2x, t ! + 2z) == lX, !, ! - 2z.

x,y, z"':::"!!""'2p - x, -! + y, z~ !j - a

{ i,y,z

-1 (or 0) + 2p - x, 2q - y, 2r - z <--::-;;- - ! + 2p - x, 2q + ! - y, z

The points i, y, z and 2p - x, 2q - y, 2r - z are one and the same; hence, by comparing coordinates, p = q = r = O. Check this result with the half-translation rule.

2.10. See Figure S2.5. General equivalent positions:

x,y, z;

!+x,y,z;

Centers of symmetry:

x,Y, z; ! + x, y, z;

~ - x, ~ - y, z; i, ! + y, z;

t t -:1 - X,:1 - y, z

i,! + y, z

~,~, 0; ~, ~,O; ~, 1;0, i,~, 0 ! 1 1. I J. 1- J. 1 1- J. J. 1 4,4,2, 4,4,2, 4,4,2, 4,4,2

~ ~ ~--, I I I I I :-CD+ I I I 1

~---r---------r---------t---~

'11_(1)+ 0 : 0 :-(1)+ - (1)+1 1

I 1 I L---~---------r---------T---~

CD 1 I CD I - . +1 0 I 0 - . +1

I '-CD+ 1 I I I ~---r------ ___ L ____ -----~---~

1m I 1m n:v+ I n:v+ 1 I I

~ ~ ~ FIGURE S2.5

550 SOLUTIONS

0+ 0+ 0+

FIGURE S2.6

Change of ongm: (i) subtract t to from the above set of general equivalent positions, (ii) let Xo = x - t Yo = Y - t Zo = z, (iii) continue in this way, and finally drop the subscript:

±(x, y, z; i, y, z; ! + x, ! - y, z; ! - x, ! + y, z)

This result may be confirmed by redrawing the space-group diagram with the origin on L

2.11. Two unit cells of space group Pn are shown on the (010) plane (see Figures S2.6). In the transformation to Pe, only c is changed:

c'(Pe) = -a(Pn) + c(Pn)

Hence, Pn == Pc. By interchanging the labels of the x and z axes (which are not constrained by the twofold symmetry axis), we see that Pe == Pa. However, because of the translations of ! along a and b in Cm, from the centering of the unit cell, Ca *" Ce, although Ce == Cn. We have Ca == Cm, and the usual symbol for this space group is Cm. If the x and z axes are interchanged in Ce, the equivalent symbol isAa.

2.12. P2/e (a) 21m; monoclinic. (b) Primitive unit cell, e-glide plane .l b, twofold axis lib. (c) hOI: I = 2n.

Pea2) (a) mm2; orthorhombic. (b) Primitive unit cell, e-glide plane .l a, a-glide plane .l b, 2) axis lie.

Cmem (a) mmm; orthorhombic. (b) C face-centered unit cell, m plane .l a, e-glide plane .l b, m plane .l c. (c) hkl: h + k = 2n; hOI: 1= 2n.

P42)e (a) 42m; tetragonal. (b) Primitive unit cell, 4 axis lie, 2) axes lIa and b, e-glide planes .l [110] and [110]. (c) hhl: 1= 2n; hOO: h = 2n; OkO: (k = 2n).

SOLUTIONS 551

P6322 (a) 622; hexagonal. (b) Primitive unit cell, 63 axis lie, twofold axes lIa, b, and u, twofold axes 30° to a, b, and u, in the (0001) plane. (c) 00/: I = 2n.

Pa3 (a) m3; cubic. (b) Primitive unit cell, a-glide plane .l e, b-glide plane .l a, e-glide plane .l b (the glide planes are equivalent under the cubic symmetry), three fold axes II [111], [111], [111], and [111]. (c) Okl: k = 2n; hOI: (l = 2n); hkO; (h = 2n).

2.13. Plane group p2; the unit-cell repeat along b is halved, and y has the particular value of 90°.

2.14. (i) See Figure 2.4 (10) (ii) Tetragonal, P

2.15.

(iii) Monoclinic, C

(iv) Triclinic, P

aii = b;/2 + c;/2

bii = -b;/2 + c;/2

b;V = b;/2 + c;/2

c;v = -b;/2 + c;/2

The transformation equations in (iv) are not unique. For example, those in (ii) could be used.

m .l Y 4 along z

[~ ! n [! ~ n ~ [! ~ ~]

R3 represents a twofold rotation axis along [110].

552 SOLUTIONS

2.16. Multiplying the matrices and adding the translation vectors, we obtain

[1 0 0] [1/2] o 1 0 + 1/2 o 0 1 1/2

which corresponds to a 21 axis along [!, t z]. The space-group symbol is Pna21 ;

class mm2.

2.17. Since 2·3 == 6 (see, for example, Figure 1.39), we obtain

for the 63 axis. Using the half-translation rule, we see that the center of symmetry must be displaced by c/4 from the point of intersection of 63 with m. Hence, for 1 at the origin, we obtain

for the m plane. The coordinates of the general equivalent positions in this space group are

±[x, y, z; y, x - y, z; y - x, x, z; x, y, 1/2 - z;

ji,x - y, 1/2 - z; y - x, x, 1/2 - z].

Chapter 3

3.1. (a) Tetragonal crystal, Laue group i mm ; optic axis parallel to the needle axis (c) of m

the crystal. (b) Section extinguished for any rotation in the ab plane; section normal to c is optically isotropic. (c) Horizontal m line. Symmetric oscillation photograph with a, b, or (110) parallel to the beam at the center of the oscillation would have 2mm symmetry (m lines horizontal and vertical).

3.2. (a) Orthorhombic. (b) Axes parallel to the edges of the brick. (c) Horizontal m line. (d) 2mm (m lines horizontal and vertical).

3.3. (a) Monoclinic, or possibly orthorhombic. (b) If monoclinic, yllp. If orthorhombic, pllx, y, or z. (c) (i) Mount the crystal perpendicular to p, either about q or r, and take a Laue photograph with the x-ray beam parallel to p. If monoclinic, twofold symmetry would be observed. If orthorhombic, 2mm, but with the m lines in general directions on the film which define the directions of the crystallographic axes normal to p. If the crystal is rotated so that x-rays are perpendicular to p, a vertical m line would appear on the Laue photograph of either a monoclinic or an orthorhombic crystal.

SOLUTIONS 553

(ii) Use the same crystal mounting as in (i) and take symmetric oscillation photographs with the x-ray beam parallel or perpendicular to p at the center of the oscillation. The rest of the answer is as in (i).

3.4. a = 9.00, b = 6.00, c = 5.00 A. a* = 0.167, b* = 0.250, c* = 0.300RU. d(146) = 0.726 A; hence 2 sin 8(146) > 2.0. Each photograph would have a horizontal m line, conclusive of orthorhombic symmetry if the crystal is known to be biaxial; otherwise, tests for higher symmetry would have to be carried out.

3.5. (a) a = 8.64, C = 7.51 A. (b) nmax = 3. (c) No symmetry'in (i). Horizontal m line in (ii). (d) The photographs would be identical because of the fourfold axis of oscillation.

3.6. Remembering that the fJ angle is, conventionally, oblique, and that in the monoclinic system fJ = 180° - fJ*, fJ* = 85°, and fJ = 95°.

3.7. Refer to Figure S3.1. Let hmax = maximum value of h required. Since we are concerned with large d* values, we are looking for it = 0.2 A or 1/ it = 5 RU.

a* = l/a = 0.20 RU = 1/5 RU

Let d* required = a*hmax• Since a(a*) is inclined at rp = 70° to the x-ray beam, 8

X-Ray Beam

R-60mm a

Film _~ _______ +-_________ .3-__

X = R tan 40°-

FIGURE S3.1

554

must be 20° (see Figure S3.1). Thus,

Since 0 = 20°

d* = a*hmax = hmax/5 = 2 sin OJ),.

:.A = 2 sin 0 x 5/hmax ;;;. 0.2

10 x sin 20/hmax ;;;. 0.2

:.hmax ~ 10 x 0.3420/0.2

~ 17.1

Since h is an integer hmax = 17. Correspondingly,

A = 2 sin 20 x 5/17

A = 0.2012 A

Position on film x = 60 tan 40° = 50.35 mm. The half-width of the plate = 125/2 = 62.5 mm. Hence the 17,00 reflection will be recorded.

Chapter 4

SOLUTIONS

4.1. The coordinates show that the structure is centrosymmetric. Hence, A'(hk) is given by (4.62) with 1= 0, B'(hkl) = 0, and the structure factors are real [F(hk) = A'(hk)]:

F(5, 0) = 2( -gp + gQ)' F(O, 5) = 2(gp - gQ)

F(5,5) = 2( -gp - gQ)' F(5, 10) = 2( -gp + gQ)

For gp = 2gQ , 1jJ(0, 5) = 0 and 1jJ(5, 0) = 1jJ(5, 5) = 1jJ(5, 10) = :rr.

4.2. The structure is centrosymmetric. Since I = 0 in the data given

A(hkO) = 4cos2:rr[ky + (h + k)/4]cos2:rr(h + k)/4

11';,(020)1 11';,(110)1

y = 0.10

86.5 258.9

y = 0.15

86.5 188.1

Hence, 0.10 is the better value for y, as far as one can judge from these two reflections.

4.3. The shortest U-U distance is between 0, y, ~ and 0, y, ~ and has the value 2.76 A.

4.4. (a) P21 , P21/m. (b) Pa, P2/a. (c) Ce, C2/e. (d) P2, Pm, P2/m.

4.5. (a) P21212. (b) Pbm2, Pbmm.

SOLUTIONS

(c) Ibm2 (/cm2); Ib2m (lc2m); lbmm (/cmm) hkl: h + k + I = 2n Okl: k = 2n, (l = 2n), or 1= 2n, (k = 2n) hOI: (h + I = 2n) hkO:(h + k = 2n) hOO: (h = 2n) OkO; (k = 2n) 00/: (I = 2n).

4.6. (a) (i) hOI: h = 2n; OkO: k = 2n. No other independent conditions. (ii) hOI: I = 2n. No other independent conditions.

(iii) hkl: h + k = 2n. No other independent conditions. (iv) hOO: h = 2n. No other conditions. (v) Okl: 1= 2n; hOI: I = 2n. No other independent conditions.

(vi) hkl: h + k + I = 2n; hOI: h = 2n. No other independent conditions.

555

Space groups with the same conditions: (i) None. (ii) P2/c. (iii) Cm, C2/m. (iv) None. (v) Pccm. (vi) Ima2, I2am. (b) hkl: None; hOI: h + I = 2n; OkO: k = 2n. (c) C2/c, C222.

4.7. Pmna. Consider the nature and orientation of the symmetry planes in the aeb setting, and how they change by transformation to the abc setting.

Chapter 5

5.1. Following (5.23), we have the molecule at displacements p and -p from the origin. Hence the total transform is

GT(S) = Go(S) exp(i2Jrp . S) + G~(S) exp( -i2Jrp . S)

From Section 4.8, Go(S) = IGol exp(i.p), and G~(S) = IGol exp( -i.p), where .p is a phase angle. Hence,

GT(S) = IGol {exp(i2Jrp . S + .p) + exp( -i2JrP . S - .p)}

= 21Gol cos(2Jrp . S + .p)

As in Section 5.4.3, the maximum value of the transform is 21Gol, at points where cos(2Jrp· S + .p) is equal to unity. In this case, however, such points do not lie in planes, and, consequently the fringe systems are not planar but curved.

5.2. The atoms related by the 2\ screw axis will have coordinates x, y, z and i, ! + y, ! - z. From (5.21), we have

nl2

G2,(S) = L .fj{exp[i2Jr(hxj + kYj + Iz)] j~l

+ exp[-i2Jr(-hxj + kYj -Izj + k/2 + 1/2)]}

556

nl2

L 2/j cos 21rkYj j~l

SOLUTIONS

x {exp[i21r(hxj + IZj)] + exp[i21r( -hxj - IZj + kl2 + 112)]}

To obtain a special condition, we must consider that part of reciprocal space for which h = I = 0, whence

nl2

G21(S)h~I~O = 2 L /j cos 21rkYj{1 + exp(j1rk)} j~l

Evidently, G21(Sh~I~O = 0 for k odd (compare pages 208, 209).

5.3. Figure S5.1 shows the nodal lines of the s-s fringe systems. Since the transform is positive at the origin, we can allocate ± regions to the transform as shown. Hence, the following signs for the most intense reftexions may be assigned:

240 - 370 + 250 470 + 710 + 530 720 + 590 140 + 670 240 +

5.4. In Figure S5.2, the given three points are plotted in (a). A transparency is made of (a) inverted in the origin. The structure (a) is drawn three times on the transparency, with each of the atoms of the inversion, in turn, over the origin of (a), in the same orientation. The completed diagram (b) is the required convolution, and the six triangles outlined in (b) all produce the same set of nine vectors (three at the origin).

FIGURE S5.1

SOLUTIONS 557

-0'2

-0,1 • -0'2 -0'1 0'1 02 0'3 0'4

yaxis (a)

.0'1

0'2 • )( axis

(b) yaxis

! )( axis

FIGURE SS.2

5.5. The field figures of Figure PS.2 have been contoured in Figure SS.3. The same triangles are revealed, giving the six sets of atom coordinates, as follows:

0.15, 0.10; -0.15, -0.10; -0.05, 0.30

0.05, 0.20; -0.05, -0.20; -0.20, -0.20

0.10, -0.10; -0.10, 0.10; -0.20, -0.30

0.05, -0.30; 0.15, 0.10; -0.15, -0.10

0.25, 0.00; 0.05, 0.20; -0.05, -0.20

0.10, -0.10; -0.10, 0.10; 0.20, 0.30

The sixth set was that given in Problem 5.4, but each of the six sets gives one and the same vector set.

558 SOLUTIONS

10 14 21 I> 14 7 10

12 20 7 22 In 10 6

7 14 22 8 10

20 '8 \ 15 7 , H2

"::-" '.)/ , 16

II 2) 9 ~:--"ii ~ ~-..~3 9 23 II_yaxis

/ -..-.. II> 23 II 32 ',4 17 9 9

/ , 7 IS 8\ 20

10 22 14 7

II) In 7 22 20 12

10 14 21 14 7 10

! x axis

FIGURE S5.3

Chapter 6

6.1. A(hkl) = 4 cos 2.n-[0.2h + 0.11 + (k + 1)/4] cos 2.n-(1/4). Reflections hkl are systematically absent for 1 odd. The c dimension appertaining to P21/c should be halved; the true cell contains two atoms in space group P21• This problem illustrates the consequences of siting atoms on glide planes. Although this answer applies to a hypothetical structure containing a single atomic species, in a mixed-atom structure an atom may, by chance, be situated on a translational symmetry element. See Figure S6.1.

oT-7- xaxis

3 or-~t\--"O .. ----:-___ -...tJO-___ -'--. True repeat

o ~ 0 ~ Space group P21

/. z aXIs

FIGURE S6.1

SOLUTIONS 559

FIGURE S6.2

*6.2. There are eight Rh atoms in the unit cell. The separation of atoms related across any m plane is ~ - 2y, which is less than bl2 and thus, prohibited. The Rh atoms must therefore lie in two sets of special positions, with either 1 or m symmetry. The positions on 1 may be eliminated, again by spatial considerations. Hence, we have (see Figures S6.2 and S6.3*).

4 Rh(I): ±{Xl> t z,; ! + x" t ! - z,}

4 Rh(2): ± {X2' t Z2; ! + X2' t ! - Z2}

6.3. Space group P2dm. Molecular symmetry carinot be 1, but it can be m; (a) Cllie on m planes, (b) N lie on m planes, (c) two Con m planes, and four other C probably in general positions, (d) 16 H in general positions, two H (in NH groups) on m planes, and two H (from the CH, that have their C on m planes) on m planes. This arrangement is shown schematically in Figure S6.4a. The groups CH" H" and H2 lie above and below the m plane. The alternative space group P2" was considered,

FIGURE 56.3

o Rh aty =.i

o Rhaty=~ Ie Bat y = k J

10 B at y=lJ

* R. Mooney and A. J. E. Welch, Acta Crystallographica 7, 49 (1954).

560 SOLUTIONS

but the structure analysis· confirmed the assumption of P21/m. The diagram of space group P2tfm shown in Figure 6.4b is reproduced from the International Tables for X-ray Crystallography, Vol. I, edited by N. F. M. Henry and K. Lonsdale, with the permission of the International Union of Crystallography.

/1 /1

,/

I CH,

H,

FIGURE S6.4a

-0 0- -0 0+ +0 0+

o

-0 o· -0 0+

Origin at 1; unique axis b

Limiting conditions 4 f 1 x, y, z; i,Y,z; i,! + y, z; x,! - y, z. hkl: None

hOl: None OkO: k = 2n

2 e m x, 1, z; i, ~J Z

} 2 d i !, 0, !, 111 2,2' 2 As above +

2 c i 0,0, !, O,!, !. hkl: k = 2n 2 b i !, 0, 0; !, !, 0.

2 a i 0,0,0; O,!,O. Symmetry of special projections

(OOI)pgm; a' =a, b' =b (100)pmg; b' = b, c' = c (010)p2; c' =c, a' =a

FIGURE S6.4b

• J. Lindgren and I. Olovsson, Acta Crystallographica 824, 554 (1968).

SOLUTIONS 561

6.4.

XCI = 0.23 XCI = 0.24

hhh IFol IF" I K[ IF" I IF" I K2 IFol

111 491 341 315 317 329 222 223 152 143 160 150 333 281 145 180 191 189

K[ R[ K2 Rz 0.641 0.11 0.671 0.036

Clearly, XCI = 0.24 is the preferred value. Pt-CI = 2.34 A. For sketch and point group, see Problem (and Solution) 1.11(a).

6.5. Au(hkl) = 4 co~ 2.7r[hxu - (h + k)/4] cos 2.7r[kyu + (h + k + 1)/4]. Xu = t Yu = 0.20 (mean of t t ft.).

6.6. Since Z = 2, the molecules lie either on I or m. Chemical knowledge eliminates 1. The m planes are at ±(x, t z), and the C, N, and B atoms must lie on these planes. Since the shortest distance between m planes is 3.64A, F[> B, N, C, and H[ (see Figure S6.5a) lie on one m plane. Hence, the remaining F atoms and the four H atoms must be placed symmetrically across the same m plane. The conclusions were borne out by the structure analysis. * Figure S6.5b shows a stereoscopic pair of packing diagrams for CH3NHz·BF3 . F[, B, N, C, and HI lie on a mirror plane; the Fz, F3 , H4 , H5 , and H2 , H3 , atom pairs are related across the same m plane.

6.7. (a) IF(hkl)1 = IF(iiiJ) I IF(Okl) I = IF(OiJ) I IF(hO/) I = IF(iiol) I

(b) IF(hkl) I = IF(iiiJ) I = IF(hkl) I = IF(iikl) I IF(Okl) I = IF(OiJ) I = IF(Okl) I = IF(Okl) I IF(hOI)1 = IF(iiol) I

(c) IF(hkl)I = IF(iiiJ) I = IF(iikl) I = IF(hkl)1 = IF(hkl) I = = IF(hiJ) I = IF(iikl) I = IF(iikl) I

IF(Okl) I = IF(oiJ) I = IF(Okl) I = IF(Okl) I IF(hO/)1 = IF(iiol) I = IF(iiO/) I = IF(h 01) I

6.8. (a) Pa; [t v, 0]. P2/a: [t v, 0] and (u, 0, w). P2221 ; (0, v, w), (u, 0, w), and (u, v, ~).

(b) (u, 0, w) is the Harker section for a structure with a twofold axis along b, whereas [0, v, 0] is the Harker line corresponding to an m plane normal to b. Since the crystal is noncentrosymmetric, the space group is either P2 or Pm. If it is P2, then there must be chance coincidences between the y coordinates of atoms not related by symmetry. If it is Pm, then the chance coincidences must be between both the X and the z coordinates of atoms not related by symmetry.

* S. Geller and J. L. Hoard, Acta Crystallographica 3, 121 (1950).

562 SOLUTIONS

F,

B

H,

FIGURE S6.5a

FIGURE S6.5b

6.9. (a) P2dn (a nonstandard setting of P2dc; see Problem 2.11 for a similar relationship between Pc and Pn). (b) Vectors: 1: ±{t ~ + 2y,~} double weight

2: ±{! + 2x, t ~ + 2z} double weight 3: ±{2x, 2y, 2z} single weight 4: ±{2x, 2.9, 2z} single weight

SOLUTIONS

zaxis

xaxis .~ __________________________ ,1

0.20

• .0.70

0.30 •

FIGURE S6.6

0.80 •

Section v = !: type 2 vector-x = 0.182; z = 0.235. Section v = 0.092: type 1 vector-y = 0.204. Section v = 0.408: type 3 or 4 vector-x = 0.183; Y = 0.204; z = 0.234.

4S: ±{0.183, 0.204, 0.235} and ±{0.683, 0.296, 0.735}

563

You may select anyone of the other seven centers of symmetry as origin, in which case the coordinates determined will be transformed accordingly. The positions are plotted in Figure S6.6. Differences in the third decimal places of the coordinates determined from the maps in Problems 6.9 and 6.10 are not significant.

6.10. (a) The sulfur atom x and z coordinates are S(0.266, 0.141), S'( -0.266, -0.141). (b) Plot the position -S on tracing paper and copy the Patterson map (excluding the origin peak) with its origin over -S (Figure S6.7a). On another tracing, carry out the same procedure with respect to -S' (Figure S6.7b). Superimpose the two tracings (Figure S6.7c). Atoms are located where both maps have positive areas.

6.11. (a) P(v) shows three nonorigin peaks. If the highest is assumed to arise from Hf atoms at ±{O,YHf,!}, then YHf=O.ll. The other two peaks represent Hf-Si vectors; the difference in their height is due partly to the proximity of the peak of apparent lesser height to the origin peak-an example of poor resolution-and partly due to the particular Y value of one Si atom. (b) The signs are, in order and omitting 012,0 and 016,0, + - - + + -. p(y) shows a large peak at 0.107, which is a better value for Ym, and smaller peaks at 0.05, 0.17, and 0.25. The values 0.05 and 0.25 give vectors for Hf-Si which coincide with peaks on P(v). We conclude that these values are the approximate Y coordinates for Si, and that the peak at 0.17 is spurious, arising from both the small number of data and experimental errors therein.

564

en I B "0

~~ ~ ~ 0 0 c

:§> 0 0 o ~ 0 0

~ o

0 0

o o 0

0 0

0 0

0 0 0

0

0 0 0

0 0

0 0

0

0

0

0

© 0

~ \ ~ 0 .:< '" I

0 0 &. E

<0"0 c '" :§l g

o~ o~--------~o-~O-----------~-- ' o 0

0 0

0 0 c5a

'6

SOLUTIONS

SOLUTIONS 565

*6.12. Since the sites of the replaceable atoms are the same in each derivative, and the space group is centrosymmetric, we may write F(M,) = F(M2) + 4(IM] - 1M2)

(a) NH4 K Rb TI

+ + * Indeterminate because IFI is unobserved. + + +

+ + + + t + + t Indeterminate because IFI is small.

+ + + + Omit from the electron density synthesis. + + + +

+ + + +

(b) Peaks at 0 and ~ represent K and AI, respectively. The peak at 0.35 is due, presumably, to the S atom. (c) The effect of the isomorphous replacement of S by Se can be noted first from the increases in IF(555)1 and IF(666) I and the decrease in 1F(333)1. These changes are not in accord with the findings in (b). Comparison of the electron density plots shows that X S/Se must be 0.19. The peak at 0.35 arises, in fact, from a superposition of oxygen atoms in projection, and it is not altered appreciably by the isomorphous replacement. Aluminum, at 0.5, is not represented strongly in these projections.

6.13. 1F(01O)1 = 149, IF(OfO) = 145, IP(OlO) = 55°, IP(OfO) = -490

6.14. In Figure S6.8, the six intersections 1,1', 2,2', and 3,3' are strongest in the region *- - -*. The phase angle IPM calculated from (6.95) would lie in this region; the centroid phase angle IPB would be biased slightly toward 1 (see also Figure 6.41).

l' FpH, I circle

I F pH, I circle

--- ,. -- '~'--~--\-t--I

-FH2 rH~FH3

FIGURE S6.8

566 SOLUTIONS

6.15. Since cos(hx - 1/» expands to cos hx cos I/> + sin hx sin 1/>, this reduces to sin hx for I/> = n/2. Hence, we can write

in the form

~ 1 1JI(x) = ~ + 2 L hcos(hx - 1/»,

h~l

I/>=!!. 2

This equation resembles the crystallographic Fourier series closely; see (6.44).

6.16. i. The total mass of protein per unit cell is Z x 18,000 x 1.66 x 10-24 g, where Z = number of protein molecules per unit cell. Since there is an equal mass of solvent in the unit cell, the total mass per unit cell is 2 x Z x 18,000 x 1.66 x 10-24 g. The volume of the unit cell is Vc = a x b x c = 40 x 50 x 60 X

10-24 cm3 (1 A = 10-8 cm). Since density = mass/volume, we can estimate the crystal density as

(2 x Z x 18,000 x 1.66 x 10-24)/(40 x 50 x 60 x 10-24) g cm-3

which for a biological molecule we can assume to be approximately 1.0 gcm-3 .

Thus,

(2 x Z x 18,000 x 1.66 x 10-24)/(40 x 50 x 60 x 10-24) = 1.0

Solving for Z we obtain Z = (40 x 50 x 60)/(2 x 18,000 x 1.66) or Z = 2.008 = 2 (to nearest integer). ii. In space group C2 (see page 92ff) there are four general equivalent positions. If Z = 2, the molecules must occupy special positions, which in C2 are confined to twofold axes. Hence the protein molecule must exhibit point-group symmetry 2.

6.17. From (6.108) and Figure 6.45,

FH(+) = F~(+) + iFi'!(+)

FH(-) = F~(-) + iFi'!(-)

where F~( +) and FH( -) are the normal structure factor components. Fi'!( +) and Fi'!( -) are the anomalous structure factor components. From Figure S6.9 it is clear that the moduli IFH( +)1 and IFH( -)1 are equal, but that I/>H( +) * I/>H( - ).

6.1S. The answer to this problem can be modeled on equation (6.108), but since the structure is centrosymmetric it cannot be a protein (why?). For the centrosymmetric structure (6.105) becomes

FPH(+) = Ap(+) + A~(+) + iAH(+)

SOLUTIONS

where

o

FIGURE S6.9

Np

Ap( +) = 22: fi cos 27r(hxj + kYj + Iz) j~l

NH

A~( +) = 22: fi' cos 27r(hxj + kYj + IZj) }=1

NH

A~( +) = 22: tif/' cos 27r(hxj + kYj + Iz) j=1

Clearly there is no change for iikl and so

567

where T = A~(+) and B = Ap(+) + A~(+); and cJ>(hkl) = cJ>(iikl) = tan-leT/B).

6.19. When Friedel's law breaks down the diffraction symmetry reverts to that of the point group. The JF(hkl) I equivalents then become

(a) C2 :

(b) Pm: (c) P2 l 2l 2l :

(d) P4:

(point group 2) hkl, iikl (point group m) hkt, hkt (point group 222) hkt, hkl, iikl, iikt, (point group 4) hkt, khl, iikl, kiil

568 SOLUTIONS

Bijvoet pairs can thus be formed between any of those listed above with any of the remainder from the Laue group:

(a) C2: (b) Pm: (c) P2\2\2\: (d) P4:

(hkl, hkl) with (hkl, hkl) (hkl, hkl) with (hkl, hkl) (hkl, hkl, hkl, hkl) with (hkl, hkl, hkl, hil) (hkl, khl, hkl, khl) with (kill, hkl, khl, hil)

Strictly speaking, paris related by hkl, hkl should be discounted, since they are, of course, Friedel pairs

6.20. From the Bragg equation, 1.25 = 2d(I11) sin 0(111). Since d(I11) = a/V3, 0(111) = 12.62°. Differentiating the Bragg equation with respect to 0: bA = 2d(111) cos 0(111)<50. Remembering that 0 must be in radians, bA = 0.024 A.

6.21. Substituting the coordinates in the structure factor equation: F(hkl) = 4(fNa + fH-fD- cos 2.n:1/2j. There are two cases.

I even: I odd:

F(hkl) = 4[fNa+ + fH-fD-] F(hkl) = 4[fNa+ - fH-fD-]

(111)

NaH NaD

(220)

NaH NaD

x-rays neutrons

(using f3 in place off)

33.9 2.9

33.9 27.6 27.6 -1.3 -0.03 4.1

Chapter 7

7.1. Use 705,61'7,814; 426 is a structure invariant, 203 is linearly related to 814 and 61'7, and 432 has a low lEI value. Alternative sets are 705, 203, 814 and 705, 203, 61'7. A vector triplet exists between 814, 426, and 432.

7.2. IF(hkl) = W(hil) I = IF(hkl)1 = W(hkl) I k = 2n: ljJ(hkl) = -1jJ(hkl) = -1jJ(hkl) = ljJ(hkl) k = 2n + 1: ljJ(hkl) = -1jJ(hil) = .n: - ljJ(hkl) = .n: + ljJ(hkl)

7.3. Set (b) would be chosen. There is a redundancy in set (a) among 041, 162, and 123, because F(041) = F(04f) in this space group. In space group C2/c, h + k must be even. Hence, reflections 012 and 162 would not found. The origin could be fixed by 223 and 13'7 because there are only four parity groups for a C-centered unit cell.

7.4. From (7.32) and (7.33), K = 4.0 ± 0.4 and B = 6.6 ± 0.2A2. (You were not expected to derive the standard errors in these quantities; they are listed in order to give some idea of the precision of the results obtained by the Wilson plot.) The rms displacement (U2)1I2 = 0.29 A.

7.5. The shortest distance is between points like 1. y, z and t y, z. Hence, from (7.41), d2(CI' .. Cl) = a2/4 + 4y2b2, or d(Cl" . Cl) = 4.64 A. Using (7.49), [2da(d)]2 = [2aa(a)/4]2 + [8y 2ba(b)f + [8yb2a(Y)f, whence a(d) = 0.026A.

SOLUTIONS 569

7.6. By (7.25) 4Jh = -2.2°, and by (7.28) 4Jh = -5.9°.

7.7. The number of symmetry-independent reciprocal lattice points contained within 0< 8 < 8max is given by equation (6.46):

where v" = volume of unit cell in A3; A = wavelength in A; G = unit-cell translation constant (Table 4.1), 1 for a primitive cell, and m is the number of symmetryequivalent reflections in the appropriate Laue group, 8 for an orthorhombic crystal. For the given problem,

N = (33.51 x 30 x 50 x 40 sin3 8)/(1.53 x 8)

N = 74466.7 sin3 8

i. For 8 = 10°, N = 389 or 2N = 778 to nearest integer. ii. For 8 = 20°, N = 2979. Number of measurements in 10-20° shell = 2979 - 389 = 2950. iii. For 8 = 25°, N = 5620. Number of measurements in 20-25° shell = 5620 - 2979 = 2641. Resolution is defined as dmin = A/2 sin 8max •

i. For 8max = 10°, d min = 4.3 A. ii. For 8max = 20°, d min = 2.2 A.

iii. For 8max = 25°, dmin = 1.8 A.

Chapter 8

8.1. (a) The I-I vector lies at 2x, t 2z. Hence, by measurement, x = 0.422 and z = 0.144, with respect to the origin O. (b) cos 2n[ (0.422h)

hkl (sin 8)/ A 2fl + (0.144/») F] !Fa I

001 0.026 105 0.618 65 40 0014 0.364 66 0.995 66 37 300 0.207 82 -0.100 -8 35 106 0.175 86 -0.224 -19 33

The signs of 001, 0014, and 106 are probably +, +, and -, respectively. The magnitude of fi(3OO) is a small fraction of !Fo(300)1, and the negative sign is unreliable. Note that small variations in your values for fi are acceptable; they would probably indicate differences in the graphical interpolation of k (c) 9.83 A.

570

8.2.

SOLUTIONS

A simplified ~2 listing follows:

h k h-k IE(h)IIE(k)IIE(h - k)1

0018 081 0817 9.5 011 024 035 5.0

026 035 0.5 021 038 059 0.4

0310 059 0.4 024 035 059 9.6 038 059 0817 7.2

081 011,7 6.0 081 011,9 10.2

0310 059 081 7.9 081 011,9 9.2

In space group P2da, s(hkl) = s(iiki) = (-It+ks(hkl), which means that s(hkl) =

(-I)h+ks (iikl). The origin may be specified by s(081) = + and s(011,9) = +.

Sign determination

h k h-k Conclusion

011,9 081 038 s(038) = + 011,9 081 0310 s(031O) = + 038 081 011,7 s(011,7) = + 0310 081 059 s(059) = -059 038 0817 s(0817) = -038 059 021 s(021) = -0310 059 021 s(021) = -0817 081 0018 s(OOI8) = -

Let s(035) = A 059 035 024 s(024) = -A 035 024 011 s(011) = -035 011 026 s(026) = A

The two indications for s(021) and the single indication for s(026) will have low probabilities and must be regarded as unreliable. Within this limited data set, no conclusion can be reached about the sign of A, and both + and - signs are equally likely. Reflection 0312 does not interact within the data set.

8.3. The space group is P2dc (from Table 8.4). Hence, we must recall that F(hkl) =

F(iiki) = (-I)k+IF(hkl); Figure S8.1 shows the completed chart. The ~2 listing should look something like the following. An • indicates a sign change with respect to the hkO quadrant, and N means that no further relationships were derived by considering the reflexion so marked as h.

SOLUTIONS 571

1.79 2.16 1.79

1.45 1.96 2.26 1.96 1.45

2.54 2.54

3.12 1.92 1.92 3.12

1.51 1.51

1.85 1.75 1.85

--f---1.84 1.84

1.08 i<

1.08 +k

1 2 3 4 5 6 7 8 9 10 11

1

1.84 1.84 2

1.85 1.75 1.85

3

1.51 1.51 4

3.12 1.92 1.92 312

5

2.54 2.54 6

1.45 2.26 196 145 7

1.79 2.16 1.79

8 • h

FIGURE S8.1. The completed chart. Negative signs are shown as bars over the lEI values.

Reference number h IEhl k IEkl h-k IEh-kl IEhiIEkIIEh-kl

1 300 1.75 040 1.08 340 1.85 3.5 2 840 1.79 540 1.92 6.0 3 570 3.12 270 1.84 10.1 4 700 2.26 570 3.12 270' 1.84 13.0 5 800 2.16 670 2.54 270' 1.84 10.1 6 340 1.85 540 1.92 7.7 7 411,0 1.51 311,0' 1.51 4.9 8 040 1.08 840 1.79 4.2 9 730 1.96 040 1.08 770 1.45 3.1

10 540 1.92 270 1.84 6.9 11 040 N 12 340 1.85 770 1.45 411,0' 1.51 4.1 13 540 N 14 840 N 15 270 N 16 570 N 17 670 N 18 770 N 19 411,0 N

572 SOLUTIONS

An origin can be chosen as 0, 0 by making, for example, 270 (eoe, and occurring 4 times) and 540 (oee, and occurring 3 times) both +. From the l:2 listing we have:

Reference number

10 7 5 6

Conclusion

s(730) = + s(SOO) = -s(670) = + s(340) = -

s(040) = A s(300) = A

2 s(S40) = A 3 s(570) = A 4 s(700) = -A S s(S40) = -A 9 s(770) = A

11 s(411,0) = -A

Comments

s(411,0) = -s(411,0)

Sign propagation has now stopped. Set s(040) equal to symbol A.

One symbol, A, has been used. In this case, it would be necessary to test, by electron density summation, both possible signs for A. Of course, in a more extended set of lEI values multiple indications could make this test unnecessary. No l:2 relationship is noticeably weak, and the above solution could be regarded as acceptable. Alternative results, based on other choices of origin, may be equally correct.

Index

Absences in x-ray spectra accidental, 203, 528 see also Systematic absences

Absorption correction, 198, 507 Absorption edge, 497 Absorption of x-rays, 496ff Accidental absences, 203, 528 Alternating axis of symmetry, 492 Alums, 360 Amorphous substance, 3 Amplitude of wave, 189ff Amplitude symmetry and phase symmetry, 377 Analyzer (of polarized light), 118 Anatase, 52 Angles

between lines, 535 between planes, 520 see Bonds; Interaxial angles; Interfacial an-

gles; Phase angle Angular frequency, 188 Anisotropic thermal vibration, 207,418 Anisotropy, optical, 119, 122; see also Biaxial

crystals; Uniaxial crystals Anomalous dispersion, 201 Anomalous scattering, 335ff

and atomic scattering factor, 335 and diffraction symmetry, 388 and heavy atoms, 341 and phasing reflections, 344 and protein phasing, 345ff and structure factor, 341

Argand diagram, 19lff, 383, 385 Arndt-Wonnacott camera, 158 Asymmetric unit of pattern, 28, 81ff Asymmetry parameters, 429 Atom, finite size of, 195 Atomic scattering factor, 194ff

and anomalous scattering, 335 and electron density, 267

573

Atomic scattering factor (Cont.) factors affecting, 195ff and spherical symmetry of atoms, 200 temperature-corrected, 195 variation with (sinO)/i.., 195

Averaging (Patterson) function, 278ff Axes: see Coordinate, Crystallographic, Inver

sion, Oblique, Optic, Rectangular, Rotation, Screw axes

Axial angles: see Interaxial angles Axial ratios, 14 Azidopurine monohydrate, electron density of,

275

Beevers-Lipson strips, 359 Bertrand lens, 128 Bessel function, 227, 387 Biaxial crystals, 119, 125

idealized interference figure for, 128, 129

optical behavior, 125ff refractive index, 125 see also Anisotropy, optical; Crystal; Op-

tically anisotropic crystals Birefringence, 122ff Birefringent crystals, 119, 122 Bisdiphenylmethyldiselenide, 288ff

Patterson studies of, 290ff Body-centered (I) unit cell, 63ff, 203 Bonds, lengths and angles, 425ff

tables of, 434, 435 Bragg angle, 135 Bragg diffraction, 88, 134ff

and Laue diffraction, 137ff see also Bragg reflection

Bragg equation, 137, 523 Bragg reflections, 135, 137

order of, 137 see also Bragg diffraction

574

Bravais lattices, 62ff and crystal system, 62, 63, 68 interaxial angles of unit cells in, 68 and space groups, 80ff table of, 68 unit cells of, 62, 63ff see also Lattice

2-Bromobenzo[b]indeno[1 ,2-e]pyran (BBIP), detailed structure analysis of, 448ff

Carbon monoxide, molecular symmetry, 48 Center of symmetry, 31, 36, 221, 490, 530

alternative origins on, 370ff and x-ray diffraction pattern, 45, 140, 199

Centered unit cell in three dimensions, 62ff in two dimensions, 61ff

Centric reflections: see Signs of reflections Centrosymmetric crystals

structure factor for, 20 I ff see also Crystal; Signs of reflections in cen

trosymmetric crystals Centrosymmetric point group, 36, 45

and x-ray patterns, 45, 140, 199 Centrosymmetric zones, 21 Iff Change of hand, 32 Change of origin, 104ff, 235 Characteristic symmetry, 37 Characteristic x-rays, 496ff Classification of crystals

optically, 119ff by symmetry; see Crystal class; Crystal

systems Closure error, 330, 331 Complex conjugate, 193 Complex plane, 191ff Conformational analysis, 428ff Contact goniometer, 16, 17 Contour map of electron density, 1, 2, 274,

275, 46lff, 477ff Convolution, 243ff Coordinate axes, 5ff; see also Crystallographic

axes Copper pyrites crystal (42m), 42

Copper x-radiation, 496 Corrections of measured intensities, 503ff

absorption, 198, 507ff Lorentz, 198,506 polarization, 198, 506 temperature factor, 195ff, 274, 305ff

Cramer's central limit theorem, 530

Crystal, 3 density measurement, 253, 448 external symmetry of, 26ff geometry, Iff habit, 124, 126

INDEX

ideally perfect, ideally imperfect, 499, 500

internal symmetry of, 59ff law of constant interfacial angles of, 17 mosaic character of imperfect, 499 optical classification of, 119ff perfection of, 498ff as a stack of unit cells, 189 symmetry and physical properties of, 28 unit cell of, 11, 61ff see also Biaxial crystals; Centro symmetric

crystals; Crystalline substance; Uniaxial crystals

Crystal class, 34 and E-factor, 368 restrictions on symmetry of, 34 see also Classification of crystals; Point

group Crystal morphology, 18 Crystal setting, 525ff, 538ff Crystal systems, 36ff

and Bravais lattices, 62ff and characteristic symmetry, 36 crystallographic axes of, 36 and idealized cross sections, 125 and Laue groups, 46, 49 and optical behavior, 119ff and point group scheme, 36 recognition of, 168ff and symmetry in Laue photograph, 45ff,

140ff table of, 37 see also Cubic, Hexagonal, Monoclinic, Or

thorhombic, Tetragonal, Triclinic, Trigonal crystal systems; Systems

Crystalline substance, 2, 3, 59 Crystallographic axes, 8, 9, 37

conventional, 36, 37, 41 ff for cubic system, 18 for hexagonal system, 12, 32, 37 interaxial angles of, 9, 37, 68 and optic axis, 123, 125 see also Coordinate axes; Reference axes

Crystallographic point groups, 31ff and crystal systems, 36 and Laue groups, 45

INDEX

Crystallographic point groups (Cont.)

and noncrystallographic point groups, 48 notation for, 36ff, 43ff, 490ff restrictions on, 34, 35, 79 and space groups, 80 and special position sites of space groups,

84, 95, 256 stereograms for, 33, 34, 41, 43ff study and recognition scheme for, 490ff tables of, 38, 39, 493 see also Point groups

Cubic crystal system Bravais lattices for, 63, 70 crystallographic axes for, 18, 37 see also Crystal systems

Data bases, 435ff Data processing, 503ff De Moivre's theorem, 191 Density of crystal

and contents of unit cell, 253 measurement of, 253, 448

Density, optical, 502 Detailed structure analyses, 448ff, 469ff Diad, 36, 95 Difference-Fourier synthesis, 314

and correctness of structure analysis, 433 and least-squares refinement, 418 see also Fourier synthesis

Diffraction by atoms, 227 by holes, 227 by regular arrays of scattering centers,

131ff of visible light, 130 and wavelength, 130 see also x-ray scattering (diffraction) by

crystals Diffraction grating, 130 Diffraction pattern

of atoms, 227; see also Atomic scattering factor

of holes, 227 of crystal, 235ff

Diffractometer, optical, 232 Diffractometer, single-crystal, 139, 172ff

data collection by, 175 Diiodo-(N,N,N' ,N' -tetramethylethylene

diamine) zinc(II), 103, 104 Direct lattices, 73; see also Bravais

lattices

575

Direct methods of phase determination, 367ff examples of use of, 378ff, 388ff experience with, 404ff figures of merit in, 400 magic integers in, 396 multisolution procedure in, 394 see also E statistics; E-Factor; Signs of re

flections in centro symmetric crystals; Structure analysis

Direction cosines, 534 Directions, form of, 59 Disorder in single crystals, 437ff Dispersion, anomalous, 199, 335ff Displacement method for density measure-

ment,253 Duax, W. L., 428

E maps, 381 calculation of, 381 ff examples of, 382, 393, 478ff "sharp" nature of, 381

E statistics, 368ff, 473, 531ff; see also Direct methods of phase determination

Electron density distribution, 267ff, 273ff ball-and-stick model for, 276 computation and display of, 274ff contour map of, 2, 275, 461ff, 478ff and criteria for structure analysis correct-

ness, 433 determined from partial structures, 308ff equations for, 270 as Fourier series, 265ff Fourier transform of, 270 and hydrogen atom positions, 274, 314 interpretation of, 273 in large-molecule analysis, 333 nonnegativity of, 383 and Patterson function, 278ff peak heights and weights of, 274 periodicity of, 265 projections of, 268, 276 pseudosymmetry in, 312 resolution of, 274 standard deviation of, 433 and structure factors, 267ff and successive Fourier refinement, 313 in unit cell, 186 see also Contour map of electron density

Enantiomorph, 28 Enantiomorph selection, 392, 533ff Enhanced intensity averages, 529

576

Enzymes, 317 ribonuclease, 322

Epsilon (E) factor, 368, 369, 528ff Equi-inciination techniques, 162ff Equivalent positions

general, 83ff, 513 special, 84ff see also General equivalent positions; Spe-

cial equivalent positions Errors, superposition of, 432 Estimated standard deviation (esd), 432, 433 Eulerian angles, 172ff Euphenyl iodoacetate

contour map of electron density of, I, 2 crystal data, 483 molecular model and structural formula, 2,

3,276 Patterson section for, 315ff, 484 Weissenberg photograph for, 181

Evans-Sutherland picture system, 439ff Ewald sphere, 143, 145, 152ff Ewald's construction, 152ff Extinction of x-rays

primary, 499 secondary, 500

Extinction parameter (for x-rays), 500 Extinction positions (for polarized light), 120

and crystal system, 120, 127 for monoclinic crystals, 126 for orthorhombic crystals, 125 for tetragonal, trigonal and hexagonal crys

tals, 119ff for triclinic crystals, 127 E-factor, 368, 369, 528ff; see also Direct

methods of phase determination

Face-centered (F) unit cell, 63, 64 Family of planes, 72, 73 Figures of merit, 400 Filtered x-radiation, 497 Flotation method for density measurement,

253,448 Form

of direction;, 60 of planes, i8, 40, 41

Fourier analy;is, 263, 265 simple one ·dimensional example of, 268 and x-ray ciffraction, 273

Fourier map, 275, 461, 465, 478ff Fourier series, 260ff

termination errors, 263, 264, 483 weighting of coefficients, 310ff

Fourier summation tab!.es, 360, 362 Fourier synthesis, 264, 265

partial, 308ff and structure analysis, 273 see also Difference-Fourier synthesis

Fourier transform, 225ff, 265 of electron density, 267ff of molecule, 231 practical example of, 239 representation of, 239 and sign relationships, 247ff structure factors as, 270

Fractional coordinates of centered sites, 64 in space group diagrams, 82

Friedel pairs, 335ff

INDEX

Friedel's Law, 140, 152, 199ff, 213, 335 and absorption edges, 335

General form of planes, 40 for crystallographic point groups, 43ff

General positions, 40 equivalent, 40, 83ff, 513 molecules in, 255 see also Equivalent positions; Special

positions Geometric structure factor, 207ff Glass (silica) structure, 3, 4 Glide line, 81, 85 Glide plane, 98ff

and limiting conditions, 212ff translational component of, 99, 100

Goniometers, 16, 17, 449, 450 Graphic syrr;~ols

change of hand, 33, 34 diad axis in plane of diagram, 95 glide line, 85 glide plane, 100 inverse monad (center of symmetry), 36 inversion axis, 32, 36 mirror line, 30 mirror plane, 36 pole, 19, 21 representative point, 33, 34, 41 rotation axes, 3, 36 rotation points, 28, 32 screw axis, 97ff table of, 36 see also Stereogram

Great circle, 21

Habit of crystal, 124, 126 Half-translation rule, 113,512

INDEX

Hamilton R ratio, 336, 337 Harker and non-Harker regions of the Patter

son function, 286ff Hauptman, H., 375 Heavy-atom method, 245ff, 308ff

examples of, 290ff, 459ff and Fourier transform, 245 limitations of, 315 and Patterson function, 246 see also Isomorphous replacement; Patterson

selection; Structure analysis Hermann-Mauguin notation, 38ff, 54, 490ff

and Schoenflies notation, 490ff Hexarl. 16 Hexagonal crystal system

crystallographic axes in, 12, 37 crystals in, 14 lattice in, 70 triply primitive unit cell, 70, 71 see also Crystal systems

Hexagonal two-dimensional system, 32 Hexamethylbenzene, 373 Hierarchy for considering limiting conditions,

216,221 Hydrogen atoms, 436, 462

and difference-Fourier technique, 274, 314 and electron density distribution, 274 scattering of x-rays by, 274, 462 see also Light atoms

Ideal intensity, 198 Identity element, 28 Implication diagram, 326 Index of refraction, 122, 125 Indexing an oscillation photograph, 152, 155ff Indistinguishability, 27ff Integrated reflection, 50 I Intensity data, scaling of, 508 Intensity distributions, 527ff

statistics, 527ff Intensity of scattered x-rays; see x-ray scatter

ing (diffraction) by crystals Interaxial angles, 16, 22

for Bravais lattices, 68 for crystallographic axes, 9, 37

Interfacial angles, law of constant, 17 Interference figures, 128 Interference of x-rays

constructive, 132 due to finite atom size, 194 partially destructive, 136

Intermolecular contact distance, 425

Internuclear distances, 437 Interplanar spacings, 72, 77

577

Inverse diad, hexad, monad, tetrad, and triad, 33

Inversion axes, 31ff Ionic radii, table of, 434 Isogyres, 129 Isomorphous pairs, 318 Isomorphous replacement, 317ff, 327ff

for alums, 360 for proteins, 317ff multiple, 327ff single, 317ff, 327ff see also Heavy-atom method

Isotropic thermal vibrations, 195, 207 Isotropy, optical, of crystals, 119

Karle, I. L., 378 Karle, J., 375, 378 Karle-Hauptman inequalities, 383

Ladd, M. F. c., 444, 492 Lattice, 59ff

direct, 73 representative portion of, 69 rotational symmetry of, 79 three-dimensional, 59, 63ff two-dimensional (net), 35, 60ff see also Bravais lattices; Reciprocal lattice

Lattice points, symmetry at, 62, 68 Laue equations, 133 Laue group, 43ff

and point groups, 45, 140 projection symmetry of, 49

Laue method, 45, 130, 139ff experimental arrangement for, 130, 139 and synchrotron radiation, 142ff see also Laue x-ray photograph

Laue projection symmetry, 49 Laue treatment of x-ray diffraction, 131ff

equivalence with Bragg treatment, 137 Laue x-ray photograph, 130, 139, 148, 451

symmetry of, 45, 140 and uniaxial crystals, 141 see also Laue method

Layer lines, 148, 154 screens for, 162 spacings between, 147ff see also Oscillation method

Least-squares method, 415ff and esd, 421 and light atoms, 436

578

Least-squares method (Cont.)

and parameter refinement, 416ff refinement by and secondary extinction, 500 strategy in, 422 and unit-cell dimensions, 416 and weights, 420

Light atoms, 436; see also Hydrogen atom Limiting conditions on x-ray reflections, 88ff,

203ff for body-centered unit cell, 203 and geometric structure factors, 207 and glide-plane symmetry, 212ff hierarchical order of considering, 216 nonindependent (redundant), 98, 217 and screw-axis symmetry, 209 and systematic absences, 203 for various unit-cell types, 203ff see also Reflections, x-ray; Space groups,

by symbol, two-dimensional, threedimensional

Limiting sphere, 158, 277 Line, equation of, 5

intercept form of, 6 Liquids and x-ray diffraction, 436 Lonsdale, K., 373 Lorentz correction factor, 198, 506

Magic integers, 396 Matrix representation of symmetry operations,

108ff Mean planes, 431, 433, 469 Melatopes, 129 Miller-Bravais indices, 12 Miller indices, IOff

in stereograms, 24ff transformations of, 89, 509ff

Minimum function, 296ff Mirror line, 28ff Mirror plane, 31 ff Mirror symmetry: see Reflection symmetry Molecular geometry equations, 425ff Molecular symmetry, 255, 256 Molybdenum x-radiation, 498 Monad,36 Monochromatic x-radiation, 497 Monoclinic crystal system, 64

lattices of, 64ff limiting conditions in, 219, 220 optical behavior of crystals of, 126 orthogonal coordinates for, 464 reciprocal lattices for, 73ff space groups of, 92ff, 514

Monocliniccrystal system (Cont.)

unit cells of, 63, 64ff

INDEX

x-ray diffraction patterns of crystals in, 218ff

see also Crystal systems MULTAN, 394ff

example of, 402ff Multiplicities of reflection data, 40, 306 Multisolution procedure, 394ff

Napthalene, 254ff Net, 60ff; see also Lattice, two-dimensional Neutron diffraction, 358ff

complementary to x-ray diffraction, 350 location of light atoms by, 351 refinement of light atoms by, 352

Noncrystallographic point groups, 48 Nonindependent limiting conditions, 98, 217 Normal distribution, 530 Notation, xxi ff

Oblique coordinate axes, 5 Oblique extinction, 127 Oblique two-dimensional system, 60, 62; see

also Space Groups Optic axis

and crystallographic axes, 125 of uniaxial crystal, 122

Optical classification of crystals, 119ff Optical density, 502 Optical diffractometer, 232 Optical methods of crystal examination, 117ff Optically anisotropic crystals, 119ff; see also

Biaxial crystals; Uniaxial crystals Order of diffraction, 132, 137 Origin, change of, 104ff, 235 Origin-fixing reflections, 370ff, 379, 389, 473 Orthogonal lattice, direct 'and reciprocal, 76 Orthorhombic crystal system

lattices of, 63, 69 limiting conditions in, 22lff optical behavior of crystals in, 125 space groups of, 104ff, 51 Iff unit cells of, 63, 69 x-ray diffraction patterns of, 221 see also Crystal systems

Oscillation method, 147ff disadvantage of, 162 example of use of, 449, 450 experimental arrangement for, 147, 148 flat-plate technique in, 158ff indexing photograph by, 152, 155ff

INDEX

Oscillation method (ConI.)

symmetry indications from, lSI zero layer in, 148, 149, 154 see also Layer lines

PI, 283 PI, 283

amplitude and phase symmetry for, 377 E statistics for, 369 general positions in, 287 origin-fixing reflections for, 378

P2, amplitude and phase symmetry for, 377 diagram for, 98 general equivalent positions, 99, 207 geometric structure factors for, 207 and pseudosymmetry introduced by heavy

atom method, 312, 313 special positions in, absence of, 99 systematic absences in, 208

P2,/c

amplitude and phase symmetry for, 377, 473 analysis of symbol for, 101, \02, 107 general equivalent positions in, 114,212 geometric structure factors for, 213 limiting conditions for, 114,213 origin-fixing reflections of, 445 special position sets in, 114

Packing, 34, 35, 80, 425 Palmer, R. A., 370, 444 Papaverine hydrochloride

analysis of by Patterson functions, 298ff crystal data, 254 molecular structure, 255 and sharpened Patterson function, 286

Parametral line, 6 Parametral plane, 10, II

for crystal systems, 36 for cubic crystals, 18 for hexagonal crystals, 12

Parity, 208 Parity group, 371 Partial-structure phasing, 308ff

effective power of, 309 for protein molecule, 317ff see also Phase determination

Path difference, 184ff in Bragg diffraction, 135, 136 for constructive interference, 131 for parallel planes, 183

Pattern motif, 80 and asymmetric unit, 81ff

Patterson function, 278ff centrosymmetry of, 282, 284 as Fourier series, 282 and Fourier transform, 246 and Laue symmetry, 286 as map of interatomic vectors, 282 one-dimensional, 279ff oversharpening of, 286 partial results of, 298ff, 304 projection of, 303ff

579

search methods, 406ff sharpened, 285 sharpened, using normalized structure fac-

tors, 368 and solution of phase problem, 278ff symmetry of for Pm, 286ff and symmetry-related and symmetry-

independent atoms, 283 three-dimensional, 282 and vector interactions, 284, 286ff see also Peaks of Patterson function

Patterson sections, 291, 460ff for euphenyl iodacetate, 315ff, 484 for papaverine hydrochloride, 300, 301

Patterson selection, 315 Patterson space, 282 Patterson space group, 283 Patterson superposition, 292ff Patterson unit cell, 282ff Pc

equivalence to Pn. liS general equivalent positions in, 211 geometric structure factors for, 211 limiting conditions for, 211 reciprocal net for, 214 systematic absences in, 211, 224

Peak heights and weights for electron density, 274

Peaks of Patterson function, 282 arbitrariness in location of, 299 cross-vector, 325 Harker lines and sections, 287 implication diagram for, 326 non-origin, 284 positions of, 283ff spurious, 286 and symmetry-related atoms, 283, 286ff weights of, 283, 289 see also Patterson function

Phase angle, variance of, 385, 386 by anomalous scattering, 335ff direct methods of determining, 367ff

580

Phase angle, variance of (Cont.)

heavy-atom method of determining, 308ff by Patterson function, 278ff in space group PI, 377, 378ff in space group P2" 388ff see also Detailed structure analyses; Partial

structure phasing; Structure factor Phase angles in centrosymmetric crystals, 202 Phase determination, 199,273, 277ff

by magic integers, 396 Phase of structure factor, 198, 199; see also

Structure factor Phase of wave, 186ff

of resultant wave, 187ff Phase probability methods, 367ff; see also

Direct methods Phase symmetry, 377 Phasing, partial-structure; see Partial-structure

phasing Plane, mirror, 31, 100 Plane groups, 8lff

the 17 patterns, 86, 87 Planes

family of, 72 form of, 18,40,41 intercept form of equation for, 10, 12

Platinum derivative of ribonuclease, 322ff Point atoms, 285 Point groups, 27ff

centrosymmetric, 36, 44 and Laue group, 45, 140 matrix representation of, 108ff noncrystallographic, 48 and space groups, 80 recognition scheme for, 490ff stereograms for, 30ff, 41ff symbols, meaning of, 32, 39 and systems, 32, 38 tables of, 38, 39 three-dimensional, 31 ff

by Hermann-Mauguin symbol, 38ff, 46, 490ff

by Schoenflies symbol, 43ff, 492 noncrystallographic, by symbol, 48

table of, 39 two-dimensional, 28ff

by Hermann-Mauguin symbol, 28ff, 46 table of, 32 unit cells in plane lattices, 60, 61 unit cells of Bravais lattices, 63ff see also Crystallographic point groups;

Stereograms

Polarization correction, 198, 506 Polarized light, I17ff

and structure analysis, I17ff Polarizers, 118 Polarizing microscope, 118ff

examples of use of, 448, 470 Polaroid, 118

INDEX

Pole, in stereographic projection, 19 Positions: see Extinction, General, Special

positions Potassium dihydrogen phosphate (KH2P04 ),

119ff, 179,351 Potassium hexachloroplatinate, (56), 322ff,

354 Potassium 2-hydroxy-3,4-dioxocyclobut-l

ene-I-olate monohydrate (squaric acid), detailed structure analysis of, 469ff

E statistics for, 473 k2 listing for, 473, 474

Potassium dimercury (KHg2)' structure analysis of, 299ff

Powder diffraction, 139 Precession method, 166ff, 538ff

crystal setting for, 538ff Precision of calculations, 421, 431; see also

Errors, superposition of Primary extinction of x-rays, 499 Primitive, 19 Primitive circle, 19,20 Primitive plane, 19 Primitive unit cell, 62ff Principal symmetry axis, 38, 490 Probability of triple-product sign relationship,

375, 376 Projections

of electron density, 268, 276ff Patterson, 303ff spherical, 20 stereographic, 18ff see also Space groups; Stereograms

Proteins, 317ff, 438ff heavy-atom location in, 323ff noncentrosymmetric nature of, 321 phasing by anomalous scattering, 344ff phasing by multiple wavelength anomalous

dispersion technique (MAD) and SR, 345ff

sign determination for centric reflections of, 321, 322

structure analysis of, 312, 327 Pseudosymmetry, 312, 327

in trial structures, 312

INDEX

Pyridoxal phosphate oxime dihydrate, 370, 378ff

symbolic-addition procedure in, 379ff Pyrite, 268

Quartz structure, 3, 4

R factor, 312; see also Reliability (R) factor Reciprocal lattice, 73ff, 514ff

diffraction pattern as weighted, 199, 214 points of, in limiting sphere, 277 and sphere of reflection, 143, 152ff symmetry of, 73 unit cell of, 73 unit cell size in, 73, 155 units for, 75 vector components in, 75, 514ff weighted points, diagram of, 199,214 see also Lattice; Unit cell, two-dimensional

Reciprocal space, vector algebra treatment of, 514ff

Rectangular axes, 5 Rectangular two-dimensional system, 60ff

two-dimensional space groups in, 82ff Reduced structure factor equation, 204 Redundant limiting conditions, 98, 216 Reference axes, 5ff; see also Crystallographic

axes Refinement by least squares, 415ff, 500

strategy in, 422ff Reflecting goniometer, 17 Reflecting power, 500, 501 Reflection (mirror) line, 28, 30 Reflection (mirror) plane, 31 ff Reflection, sphere of, 143, 152ff Reflection symmetry, 28, 31

of a square-wave function, 261, 264 in three dimensions, 31 ff in two dimensions, 28ff see also Symmetry

Reflections, x-ray, 134 integrated, 500 intensity of, theory of, 183ff, 500, 501 number in data set, 277 origin-fixing, 370ff, 379, 389, 473 "unobserved", 306 see also Limiting conditions on x-ray reflec

tions; Signs of reflections in centrosymmetric crystals; Structure analysis; Systematic absences; X-ray scattering (diffraction) by crystals

Refractive indices, 122, 125 Reliability (R) factor, 312

and correctness of structure analysis, 433 and parameter refinement, 416ff, 422

Repeat period of a function, 253 Repeat vector, 59 Repetition, 100

581

Resolution of x-ray technique, 226, 240, 381, 439; see also Reflections, number in data set

Resolution of Patterson function, 284 Resultant phase, 186ff Resultant wave, 186ff

for unit cell, 196ff Rhombic dodecahedron, 18, 19 Rhombohedral unit cell, 70 Ribonuclease, 322

platinum derivative of, 322ff Rotation axes, 31 Rotation points, 28 Rotational symmetry, 28ff, 31 ff Row, 60

Sampling interval, 263 Sayre, D., 373 Scale factor, 198, 305ff, 416 Scaling of intensity data, 508 Scattering: see x-ray scattering (diffraction) Scattering, anomalous: see Anomalous

scattering Scattering ratio, 309 Schoenflies notation, 39, 42ff, 492ff

for crystallographic point groups, 43ff and Hermann-Mauguin notation, compared,

493 Scintillation counter, 172 Screw axis, 97, 209 Secondary extinction of x-rays, 500 Series termination errors, 245, 264, 433 Sigma-two (~2) examples of, 377ff, 473ff Sigma-two (~2) formula, 375ff

and symbolic-addition procedure, 379 listing, 377 see also Signs of reflections in centrosym

metric crystals Sign determination of reflections in centrosym

metric crystals, 372ff symbolic addition, 476ff see also Centro symmetric crystals; Direct

methods of phase determination; Reflections, x-ray; Sigma-two (~2) formula; Triple-product sign relationship

582

Sign relationships and Fourier transforms, 247ff

Signs of reflections in centrosymmetric crys-tals, 202

Silica glass, 3, 4 Single-crystal diffraction, 139ff Small circle, 22 Sodium chloride, I Space-filling patterns, 35, 80 Space groups, 80ff

"additional" symmetry elements of, 83 ambiguity in determination of, 219 center of symmetry in, 100 and crystals, 80ff fractional coordinates, 83 and geometric structure factor, 207 limiting conditions for reflection in, 219ff matrix representation of, 111 ff origin shift for, 105 pattern of, 59, 81ff and point groups, 80 practical determination of, 218ff as repetition of point-group pattern by

Bravais lattice, 80, 81 standard diagrams for, 82ff three-dimensional, 92ff three-dimensional, by symbol

PI, 283, 368, 378 PI, 283, 287, 368, 370, 378ff,445 P2, 92ff, 211, 220, 287ff, 356 P2 1, 98, 208, 209, 211, 220, 355, 32lff,

445,484 C2, 92ff, 219, 220, 224, 289 C2 1,98 Pm, 283, 286ff Pa, 115,356 Pc, 211,220,224,257 Pn, 115 Cm, 219, 220, 289 Ca, 115 Ce, 115,220 Cn, 115 P21m, 220, 283, 288ff, 356, 528 P2la, 356 P2/e, 115, 220, 258, 514, 528 P21/m, 220, 356, 528 P2 1/a, 224 P2 1/e, lOOff, 107, 212ff, 353, 298, 377,

445, 451ff, 473ff, 514, 528 P21/n, 224 C21m, 219, 220, 289 C2/e, 104, 220, 445

Space groups (Cont.)

three-dimensional, by symbol (Cont.)

A21a, 224 P2 12121, 104, 105,221,370,446 P2 122,224 P222 1,356 B2 122 1,224 Pma2,214ff Pbe21,221 Pea2 1, 115 Pee2,224 Im2a, 299 12ma, 299 Pbma, 115 Pban, 114 Pman, 218, 224 Pbem, 221 Pbea, 221 Pbnm, 355 Pnma, 106, 354 Pnna, 512, 513 Pmma, 299, 512, 513 Cmem, 115, 358 Immm, 224 Ibca, 107 Imam, 224 Imma, 299, 302, 512 P4 1212,107 P42 1e, 115 P63/m, 116 P6322, 116 Pa3, 116, 360 F43e, 107

two-dimensional, 82f diagrams for 17 groups, 86, 87

two-dimensional, by symbol pl,86 p2, 82ff, 86, 94, 99

INDEX

pm. pml, plm. plml, p11m. 84ff, 86, 88, 92, 94ff

pg. pgl, pig. plgl, p11g. 88ff, 99 em. eml, elm. elml, c1lm. 84ff, 94 pmm. p2mm. 86 pmg.pgm.p2mg.p2gm. 86,106,114 pgg. p2gg. 86, 90ff, 101, 100ff e2mm. 87, 106, 114 p4, 87 p4m. p4mm. 87 p4gm. p4mg. 87 p3, 87 p3ml,87 p31m.87

INDEX

Space groups (Cant.)

two-dimensional, by symbol (Cant.)

p6, 87 p6m, p6mm, 87 see also Crystal systems; Oblique two-

dimensional system; Projections Special intensity distributions, 530 Special form of planes, 40 Special positions, 40, 84ff

equivalent, 83ff, 88, 92 molecules in, 256ff see also Equivalent positions; General

positions Sphere of reflection, 143, 152ff Spherical projection, 20 Spherical trigonometry, 24 S pot integration, 503 Square two-dimensional system, 32, 62 Square wave, 261

as Fourier series, 26lff termination errors for, 264

Squaric acid, 470 SR: see Synchrotron radiation Standard deviation of electron density, 433 Stereograms, 19ff

assigning Miller indices in, 24ff for crystallographic point groups, 41 ff fundamental property of, 21, 536 notation for, 19. 33 practical construction of, 19ff for three-dimensional point groups, 3lff for two-dimensional point groups, 30, 31 uses of, 24 see also Graphic symbols; Point groups;

Projections; Stereographic projections Stereographic projections, 18ff; see also

Stereograms Stereoviewer, 487, 488 Stereoviews, 33, 65, 93, 103, 155,258, 327 Straight extinction, 124 Structural data, references for, 435ff, 444 Structure analysis

accuracy of; see Precision computer use in, 273, 276 criteria for correctness of, 273, 276 errors in trial structure, 311 ff limitations of, 436 as overdetermined problem, 278, 415 of papaverine hydrochloride, 298ff, 300,

301 phase problem in, 198,271, 277ff for potassium dimercury, 299, 302ff

Structure analysis (Cant.)

for potassium 2-hydroxy-3,4-dioxocyclobut -I-ene-I-olate monohydrate (squaric acid), 469ff

precision of, 421, 431 preliminary stages of, 117, 199ff, 218ff

examples of, 253ff, 447ff for proteins, 317ff published results on, 483 refinement of, 415ff, 467, 479, 500

583

with results of neutron diffraction, 348ff and symmetry analysis, 255, 258, 259, 290,

303 by x-ray techniques, 129ff see also Direct methods of phase determina

tion; Heavy-atom method; Reflections, x-ray; X-ray scattering (diffraction) by crystals

Structure factor, 91, 196ff absolute scale of, 305ff amplitude of, 196, 197 amplitude, symmetry of, 377 applications of equation for, 199ff calculated, 314 for centrosymmetric crystals, 20lff defined, 196 as Fourier transform of electron density, 270 generalized form of, 268 geometric, 207ff invariance under change of origin, 370 local average value of, 306 normalized, 367ff observed, 314, 503ff and parity group, 371 phase of, 196, 198 phase symmetry of, 377 plotted on Argand diagram, 197 reduced equation for, 204 sign-determining formula for, 372ff and special position sets, 258 and symmetry elements, 206ff symmetry of, 377 see also Phase determination; Phase of

structure factor Structure invariants, 372, 391 Subgroup, 43 Superposition technique, 292ff Symbolic-addition procedure, 378ff

advantages and disadvantages of, 392ff Symbolic phases, 379, 389 Symbolic signs of reflections, 379ff, 389ff,

476ff

584

Symmetric extinction, 126 Symmetry, 26ff

characteristic, 36, 37 of crystals, 26ff cylindrical, 48 examples of molecular, 48, 255ff and indistinguishability, 27ff inversion (axis), 31 ff of Laue photographs, 45, 140ff mirror: see Reflection symmetry onefold,28 and physical properties, 27 rotational, 28, 31 and structure analysis, 255ff of x-ray diffraction pattern, 45, 140, 151

Symmetry axis, 31, 36 alternating, 492 principal, 33, 490

Symmetry element, 27 intersecting, 38ff

Symmetry operations, 27 combination of, 29, 38ff matric (operator), 108ff matrix representations of, 108ff symbols for, 31, 36 in three dimensions, 31ff in two dimensions, 28ff

Symmetry-equivalent points, 33, 40 Symmetry-independent atoms, 206 Symmetry operations, 27, 34; see also Sym-

metry element Symmetry plane (mirror plane), 31 Symmetry point, 27 Symmetry-related atoms, 82ff, 201 ff Synchrotron radiation, 142ff, 541 ff Systematic absences, 88ff, 203ff, 218ff, 235,

528 for body-centered (I) unit cell, 203 and geometric structure factor, 207 and limiting conditions, 203ff and m plane, 289, 530 and translational symmetry, 219 see also Absences in x-ray spectra; Limiting

conditions on x-ray absences; Reflections, x-ray

Systems three-dimensional, 31ff; see also Crystal

systems two-dimensional, 29, 32, 62; see also Crys

tal systems

Temperature factor correction, 195, 274, 418 factor (overall) and scale, 305ff

INDEX

Termination errors for Fourier series, 245, 264,433

Tetrad, 36 Tetragonal crystal system

model of a crystal of, 488 optical behavior of crystals in, 119 symmetry of, 69 unit cells of, 63, 69 see also Crystal systems

Thermal vibrations of atoms, 195ff anisotropic, 207, 418 and smearing of electron density,

Torsion angles, 427 Translation, 59 Translation vector, 59, 98ff Triad, 36 Triclinic crystal system

lattice and unit cell of, 63, 64 optical behavior of crystals in, 124ff see also Crystal systems

Trigonal lattice, 63, 70; see also Crystal systems

Triple phase relationship, 382ff Triple-product sign relationship, 373ff

22 formula, 375ff see also Signs of reflections in centrosym

metric crystals Triply primitive hexagonal unit cell, 63, 70,

71 Tubercidin, 388ff Tungsten Lx-radiation, 452 Twinning, 176

Uniaxial crystals, 119ff defined, 119 idealized cross section of, 125 idealized interference figure for, 128, 129 and Laue photograph, 136, 140 optic axis of, 123 see also Anisotropy, optical; Crystal; Op

tically anisotropic crystals Unit cell, three-dimensional, 11, (61), 62

body-centered, 62, 63, 66, 68, 203ff of Bravais lattices, 62, 63, 68 contents of, determination of, 253ff conventional choice of, 61, 62 dimensions, determination of, I 48ff, 158ff,

163ff, 168, 416 face-centered (F), 63, 64 limiting conditions for type of, 203ff notation and data for, 64 primitive, 60, 62 reciprocity of F and /, 521

INDEX

Unit cell, three-dimensional (Cont.)

scattering of x-rays by, 189, 194ff symbols for, 62, 64, 203ff transformations of, 66ff, 509ff translations associated with each type of,

205 triply primitive hexagonal, 70, 71 type, 62, 64, 205 for various crystal systems, 62ff volume of, 76, 518; see also Reciprocal lat

tice; Unit cell Unit cell, two-dimensional, 61

centered, 61 conventional choice of, 61 edges and angles of, 62 symbols for, 60, 62: see also Unit cell

Van der Waals forces, 469 Vector algebra and the reciprocal lattice, 514ff Vector interactions, 286ff; see also Vectors Vector triplet, 373, 374 Vectors

complex, 191, 192 interatomic, in Patterson function, 280ff repeat, 59 scalar product of, 68 translation, 59 see also Vector interactions

Vector verification method, 406ff example of, 409ff optimisation of searches in, 410 orientation search in, 407, 408 translation search in, 409

Volume reciprocal unit cell, 76, 518 U11it cell, 76, 518

Volumes, reciprocity of direct and reciprocal unit cells, 519

Waves amplitude of, 186ff and Argand diagram, 191ff combinations of, 186ff energy associated with, 198 phase of, 186ff resultant, 187, 189ff

Weighted reciprocal lattice, 214, 527ff Weiss zone law, 16 Weissenberg method, 162ff

chart for, 455 examples of use of, 165, 452ff integrated photograph by, 503ff photograph by, 164, 181, 452ff, 505

White radiation (x-ray), 130, 139, 494ff Wilson, A. J. c., 305 Wilson plot, 305ff, 415, 456 Wilson's method, 305ff Wulff net, 23 Wyckoff, R. W. G., 483 Wyckoff notation (in space groups), 84ff

X-radiation copper, 496 dependence on wavelength, 495, 496 filtered, 497 molybdenum, 498 monochromatic, 497 tungsten, 452, 453 white, 130, 139, 494ff see also x-rays

585

X-ray diffraction and reciprocal lattice, 522ff; see also Reciprocal lattice

X-ray diffraction by liquids, 278; see also X-ray diffraction pattern

X-ray diffraction pattern centro symmetric nature of, 45, 140, 199 and Friedel's law, 140, 199 and geometric structure of crystal, 207 intensity of spots in, 130, 502ff position of spots in, 130, 140ff symmetry of, 45, 49, 140ff, 151, 152 and symmetry of crystals, 140ff, 151 ff,

166 as weighted reciprocal lattice, 199ff, 214 see a/so X-ray scattering (diffraction) by

crystals; X-ray diffraction by liquids; X-ray diffraction photograph

X-ray diffraction photograph, 131, I 39ff, 450ff

important features of, 130 indexing of, 155ff, 165 Laue, 130, 139ff, 148 measurements of intensity of reflection on,

502ff measurements of position of reflections on,

142ff, 155ff by oscillation method, 147ff for powder, 139 by precession method, 166ff, 538ff for single crystal, 139ff

X-ray reflections: see Reflections, x-ray X-ray scattering (diffraction) by crystals, 129ff

anomalous scattering, 335ff Bragg treatment of, 134ff as a Fourier analysis, 273ff and indices of planes, 88ff

586

X-ray scattering (diffraction) by crystals (Cont.)

intensity, measurement of, 502 corrections to, 503ff by diffractometer, 172ff examples of, 456, 471 by film, 502

intensity, theory of, 183ff, 198, 498ff ideal, 199

by lattice, 132, 133 Laue treatment, equivalence with Bragg

treatment of, 137 Laue treatment of, 131 ff for monoclinic crystals, 219ff order of, 132, 137 for orthorhombic crystals, 221 for single crystal, 139ff and space-group determination, 219ff symmetry of Patterson function, 286 from unit cell, 189, 194ff see also Diffraction; Reflections; X-ray dif

fraction pattern; X-ray diffraction photograph; X-ray structure analysis

INDEX

X-ray structure analysis: see Structure analysis X-ray tube, 495 X-rays

absorption of, 493, 507 characteristic, 496ff extinction of, 499ff generation of, 494, 495 nonfocusing property of, 199 properties of, 130, 494ff wavelengths of, 130, 494, 496 see also X-radiation

Zero layer, 148, 149, 152, 164 Zircon crystal, 15 Zone axis, 15 Zone circle, 21 Zones, 14ff

centrosymmetric, 211 Weiss law of, 16

Appendix - Springer978-1-4615-1793-1/1.pdf · Appendix A 1 Stereoviews and Crystal Models A 1.1 Stereoviews The representation of crystal and molecular structures by stereoscopic

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