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 from most shops that retail optical instruments or drawing materials. 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.

3. An inexpensive stereoviewer can be constructed with comparative 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 A.l and glued together with the lenses in position represents the simplest construction. This basic stereo viewer can be refined in various ways.

335

336

I I E u

<0

E u

"<t

Cut 3

Q .- - - - - - 6.4em - - - -J

""0 1 "0 IL l

A

- - - - - - - - '1 em - - - - - - - - -

FIGURE A.l. 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 together 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.)

A.1.2 Model of a Tetragonal Crystal

APPENDIX

The crystal model illustrated in Figure 1.30 can be constructed easily. This particular model has been chosen because it exhibits a 4 axis, 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 A.2 and then cut out

A.I STEREOVIEWS AND CRYSTAL MODELS

Qr-~------------~

K

FIGURE A.2. Construction of a tetragonal crystal of point group 42m:

NO = AD = BD = BC = DE = CE = CF = KM = lOem;

AB =CD=EF=GJ=5 em;

AP=PO=FL =KL =2em;

AO = DN = CM = FK = FG = FH = EJ = 1 em.

337

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 EFHI is glued externally. The resultant model belongs to crystal class 42m.

338 APPENDIX

A.2 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:

1,3 (IV) m and I both present:

- 4 6 21m, mmm, 3m, 41m, m mm, 61m, mmm, m3, m3m

The further systematic identification is illustrated by means of the block diagram in Figure A.3. Here R refers to the maximum degree of rotational symmetry in a crystal, or crystal model, and N is the number of such 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 A.3 to a flow diagram. Indeed, this scheme is ideally suited to a computer-aided self-study enhancement of a lecture course on crystal symmetry, and some success with the method has been obtained.*

A.3 Schoenflies' Symmetry Notation

Theoretical chemists and spectroscopists use the Schoenflies notation for describing point-group symmetry, which is a little unfortunate, because although the crystallographic (Hermann-Mauguin) and Schoen flies nota­tions are adequate for point groups, only the Hermann-Mauguin system is satisfactory for space groups.

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

(I)

(II)

(I

II)

No

l

FIG

UR

E A

.3.

Flow

dia

gram

for p

oint

-gro

up re

cogn

ition

.

(IV

)

~

mm

m

,IR

-2

)or

Bm

m

m

(R-4

.6)

>

.....

en

()

X

0 tr1

Z fl @

CI]

en ~ tr1 ;l -< z 0 ..., >

:l

0 z '" '" '"

340 APPENDIX

(a) (b)

FIGURE A.4. Stereograms of point groups: (a) S2, (b) S4.

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.

A.3.1 Alternating Axis of Symmetry

A crystal is said to have an alternating axis of symmetry Sn of degree n, if it can be brought into apparent self-coincidence by the combined opera­tion of rotation through (360jn) degrees and reflection across a plane normal to the axis. It must be stressed that this plane is not necessarily a mirror plane. * Operations Sn are nonperlormable. Figure A.4 shows stereograms of S2 and S4; we recognize them as I and 4, respectively. The reader should consider which point groups are obtained if, additionally, the plane of the diagram is a mirror plane.

A.3.2 Notation

Rotation axes are symbolized by em 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. 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 usual Schoenflies symbol for 6 is C3h (3/m). The reason that 3/m 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.

A.4 GENERATION AND PROPERTIES OF X-RAYS 341

TABLEA.1. Schoenfties and Hermann-Mauguin Point-Group Symbols

Schoenflies Hermann-Mauguina Schoenflies Hermann-Mauguina

C, 1 D4 422 C2 2 Do 622 C3 3 DZh mmm C4 4 D3h 6m2 Co 6 C;,52 T D4h

4 -mm m C,,5, m (2)

S6 3 DOh

6 S4 4 -mm m C3h 6 C2h 2/m D2d 42m C4h 4/m D3d 3m C6h 6/m T 23 C2v mm2 Th m3 C3v 3m 0 432 C4v 4mm Td 43m C6v 6mm Oh m3m D2 222 Cxov 00

D3 32 DOOh oo/m (00)

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

the special symbols T and 0. Table A.I compares the Schoenfties and Hermann-Mauguin symmetry notations.

A.4 Generation and Properties of X-Rays

A.4.1 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 e V electron-volts. If this energy were converted entirely into a quantum hI' of X-rays, the wavelength A would be given by

A = hc/eV (A. I)

where h is Planck's constant, c is the velocity of light, and e is the charge on

342 APPENDIX

the electron. Substitution of numerical values in (A. 1) leads to the equation

A = 12.4/V (A. 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, (A.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 A.5 is a schematic diagram of an X-ray tube, and Figure A.6 shows typical intensity vs. 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.

A.4.2 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

o •

n-__ ~\w".-x _-----..jl l ·1 ~ll

'1 - C \UJ ! I we.-. - - - - ~ e - - - - - ~ A :

'1------ .' 'I iL---__ -, 1 B

E x

FIGURE A.5. 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.

A.4 GENERATION AND PROPERTIES OF X-RAYS

4

.~ 3 x III

~ 'iii c: Q)

.S 2 Q) > +' III Qi a::

20kV

0.4

FIGURE A.6. Variation of X-ray intensity with wavelength A.

343

be expelled from the atoms. Then, other electrons, from 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 !lE, we may write

A =hc/!lE (A.3)

This wavelength is characteristic of the target material. The white radiation distribution now has sharp lines of very high intensity superimposed on it (Figure A.7). 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 K{3 (A = 1.392 A); Ka and K{3 are always produced together.

A.4.3 Absorption of X-Rays

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

1 = 10 exp(-J-Lt) (A.4)

where 1 and 10 are, respectively, the transmitted and incident intensities, J-L is the linear absorption coefficient, and t is the path length through the

344

4 ~

., 'x 3 '" ~ 'in c Q)

.~ 2-Q) > .. '" 0; a::

o

Ka.

K{J

0.4

A axis. A

FIGURE A.7, Characteristic K spectrum super­posed on the "white" radiation continuum,

APPENDIX

material. The absorption of X-rays increases with increase in the atomic number of the elements in the material.

The variation of IL with'\ is represented by the curve of Figure A.8; IL

decreases approximately as,\3. 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.

A.4.4 Filtered Radiation

If we superimpose Figures A. 7 and A.8, we see that the absorption edge of nickel lies between the Ka and KfJ characteristic lines of copper (Figure A.9). Thus, the effect of passing X-rays from a copper target through a thin (0.018 mm) nickel foil is that the KfJ 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

A.4 GENERATION AND PROPERTIES OF X-RAYS

400

300

I

E '-'

'" .~ 200

I I I I I I I

:t

100 V o 0.4 0.8 1.2 1.6

o A axis, A

FIGURE A.S. Variation of IL(Ni) with wavelength A of X-radiation.

4

'" 3 'x '" l: 'iii c: ~ 2 -

Q)

> '';::

'" a:; a:

o 0.4

A axis, A

400

300

I

E '-'

200 vi 'x

100

'" :t

FIGURE A.9. Superposition of Figures A.7 and A.S to show diagrammatically the production of "filtered" radiation.

345

346 APPENDIX

the Ka line; we speak of filtered radiation, to indicate the production of effectively monochromatic radiation by this process. The copper Ka line (~= 1.542 A) actually consists of a doublet, at (,\ = 1.5405 A) and a2 (,\ = 1.5443 A); the doublet is resolved on photographs at high (J values, but we shall not be concerned here with that feature. The value of 1.542 A is a weighted mean (2at +a2)/3, the weights being derived from the relative intensities (2: 1) of the at and a2 lines.

The absorption effect is important also in considering the radiation to be used for different materials. We have mentioned that eu 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 re-emitted as characteristic Fe K spectrum. In this case, Mo Ka (,\ = 0.7107 A) is a satisfactory alternative.

A.5 Crystal Perfection and Intensity Measurement

A.5.1 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 stacked side by side in a completely regular manner. Few, if any, crystals exhibit this high degree of perfection. Figure A.l 0 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 Be is in the correct

FIGURE A. 10. Primary extinction: The phase changes on reflec­tion at B and C are each '1r/2, so that between the directions BE and CD there is a total phase difference of '1r. Hence, some attenuation of the intensity occurs for the beam incident upon planes deeper in the crystal.

A.5 CRYSTAL PERFECTION AND INTENSITY MEASUREMENT

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

347

position for a second reflection CD, and so on. Since there is a phase change of 7T/2 on reflection, * the doubly reflected ray has 7T phase difference with respect to the incident ray (BE). In general, rays reflected nand n - 2 times differ in phase by 7T, 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 ex:: IFI.

Most crystals, however, are composed of an array of slightly misoriented crystal blocks (mosaic character) (Figure All). The ranges of geometric perfection are quite small. Even crystals that show some primary extinction exhibit mosaic character to some degree, and we may write

(AS)

Generally, the mosaic blocks are small, 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 by sUbjecting the crystals to the thermal shock of dipping them in

* This 71"/2 phase change is usually neglected since it arises for all reflections.

348 APPENDIX

liquid air. The effect of secondary extinction on the intensity of a reflection can be brought into the least-squares refinement (page 289) as an additional variable, the extinction parameter (. The quantity minimized in the refine­ment of the atomic parameters is then

(A6)

A.S.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 'If(hkl) as the crystal, which is completely bathed in an X-ray beam of incident intensity 10 , passes through the reflecting range.

At a given angle fJ, let the power of the reflected beam be d'lf(hkl)/dt. The greater the value of 10 , the greater the power. Hence,

d'lf(hkl)/dt = R(fJ)/o (A7)

where R(fJ) is the reflecting power. Figure Al2 shows a typical curve of R(fJ) against fJ. The area under the curve is called the integrated reflection J(hkl):

('580

J(hkl) = t'i80

R(fJ) dfJ (A.8)

Using (A7), we obtain

('580

J(hkl) = (1/10 ) Lli80

[d 'If(hkl)/dt] dfJ (A9)

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

J(hkl) = w~(hkl)/ 10 (A 10)

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

A.5 CRYSTAL PERFECTION AND INTENSITY MEASUREMENT

FIGURE A.12. Variation of reflecting power R(8) with 8 arising from "mosaic" character: 80 is the ideal Bragg angle, and ±88o represent the limits of reflection.

energy per unit time, we have

't(hkl) = Io(hkl)t

and, from (4.50), we obtain

349

(All)

(A12)

where C(hkl) includes correcting factors for absorption and extinction, and for the Lorentz and polarization effects (page 351). Because of the propor­tionality between energy and intensity (All), although we are actually measuring the energy of the diffracted beam, we usually speak of the corresponding intensity.

A.5.3 Intensity Measurements

X-ray intensities are measured either from the blackening of photo­graphic film emulsion or by direct quantum counting.

350 APPENDIX

Film Measurements

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

D = log(io/ i) (A. 13)

where io is the intensity of light hitting the spot and i 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 grey 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 A.13 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 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 A. 14a) is allowed to strike the film successively over a grid of points (Figure A.14b). 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 A.15 illustrates, diagram­matically, the building up of the plateau, and Figure A.16 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 5%. The general

. . . . . . ,.

FIGURE A.13. Sketch of a crystal-intensity scale.

A.5 CRYSTAL PERFECTION AND INTENSITY MEASUREMENT

• • • • •

• • • • •

• • • • •

• • • • •

• • • • • (a) (b)

FIGURE A.14. Spot integration: (a) typical diffraction spot, (b) 5 x 5 grid of points.

351

subject of accuracy in photographic measurements has been discussed exhaustively by Jeffery.*

Diffractometer Measurements

The principle of a four-circle, single-crystal diffractometer is shown in Figure A17. The incident and diffracted beams are maintained in the horizontal plane, and the scintillation counter C, which counts the diffracted photons, rotates about an axis normal to this plane. The crystal is brought into the desired reflecting position by the correct setting of the angles of the circles designated by X, ¢ and O.

The intensity of a reflection peak (Figure A12) is measured by allowing the crystal to move on the 0 circle by amounts ±80o while following this movement with the counter moving along the 20 circle at twice the rate of the crystal rotation. The best recorded precision of diffractometer intensity measurements lies in the range of 1-3%. Routine work may be carried out with speed at a 5-6% level.

A.5.4 Intensity Corrections

From (All) and (A12), we see that certain corrections are necessary in order to convert measured intensities into values of IFI2. We shall write

(A14)

* See Bibliography, Chapter 3.

352

(a)

(b)

(e)

e axis

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

APPENDIX

A.5 CRYSTAL PERFECTION AND INTENSITY MEASUREMENT 353

--

(a)

- --

---

(b)

FIGURE A.16. Weissenbergphotographs: (a) normal, (b) integrated.

354

and

FIGURE A.17. Typical four-circle diffractometer; the counter rotates about the 28 axis in one plane, and the crystal is oriented about the three axes </J, x, and n so that the incident and reflected beams lie in the horizontal (28) plane. (Reproduced from An Introduction to X-ray Crystallography by M. M. Woolfson, with the permission of the Cambridge University Press, London.)

IFo(hkl)1 = KIF(hkl)lrel

APPENDIX

(A. 15)

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 IFI values onto an absolute scale, represented by IFol; it includes, implicitly, the proportionality constant of (A.12). Absorp­tion corrections may often be neglected, especially with small, approxi­mately equidimensional crystals containing light atoms, and we shall not consider it further. 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 l/sin 2().

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:

(A. 16)

A.6 TRANSFORMATIONS 355

Application of the Land p factors, where absorption and secondary extinction are negligible, is essential in order to bring the 1F12 data onto a correct relative scale. The scale factor K can be determined approximately by Wilson's method (page 245) and refined as a parameter in a least-squares analysis.

A.6 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 A18, 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,

A=a-b

B=a+b

a= A/2+B/2

b=-A/2+B/2

b

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

R - --1'

a

A __ ----------------------------_.

FIGURE A.18. Unit-cell transformations.

(AI7)

(AI8)

(AI9)

(A20)

356 APPENDIX

We have encountered this type of transformation before, in our study of lattices (page 79).

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 (A17) and (A18), we obtain

whence

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

x=X+Y

y=-X+Y

Similarly, it may be shown that

X=x/2-y/2

Y=x/2+y/2

(A21)

(A22)

(A23)

(A24)

(A25)

(A26)

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

d*(hk) = ha* + kb* (A27)

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

d*(HK) = Ha* + Kb* (A28)

The scalar d* • R is invariant with respect to unit cell transformation, since it represents the path difference between that point and the origint (see page 144). Hence, evaluating d*' R with respect to both unit cells and using the properties of the reciprocal lattice discussed on pages 68-72, we obtain

hx+ky=HX+KY (A29)

tThe full significance of this statement can be appreciated after studying Chapter 4.

A.7 ORTHORHOMBIC AND MONOCLINIC SPACE GROUPS

Substituting for x and y from (A23) and (A24), we find

Hence,

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

H=h-k

K=h+k

357

(A30)

(A31)

(A32)

which is the same form of transformation as that for the unit cell, given by (AI7) and (AI8). Generalization of this treatment to three dimensions and oblique unit cells is straightforward, if a little time consuming.

A.7 Comments on Some Orthorhombic and Monoclinic Space Groups

A.7.1 Orthorhombic Space Groups

In Chapter 2, we looked briefly at the problem of choosing the positions

of the symmetry planes in the space groups of class mmm (;;, ;;, ;;,) with

respect to a center of symmetry at the origin of the unit 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 +bI2+ c). We disregard the whole translations a and c, because they refer to neighboring unit cells; thus, T becomes b12, and the center of symmetry is displaced by T12, or b14, 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 = a12, and the center of symmetry is displaced by al4 from mma.

358 APPENDIX

Space group Imma may be formed from Pmma by introducing the body-centering translation!. !. !- (Fig. 6.18b). Alternatively, the half­translation rule may be applied to the complete space-group symbol. In all, Imma contains the translations (a +b +c)/2 and a/2, 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 i (half the I translation) from a Pmma center of symmetry. This alternative setting is given in the International Tables for X-Ray Crystallography, Vol. 1*; it corresponds to that in Figure 6.18b with the origin shifted tp the center of symmetry at t t i. Space groups based on A, B, C, and F uriit 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 2.36.

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 a/2 component, then 2x (parallel to a) == 21> otherwise 2x == 2. Similarly for 2y and 2zo with reference to the b/2 and c/2 components. Thus, in Pnna, T= b/2, and so 2x =2, 2y = 2}, and 2z == 2. In Pmma, T = a/2; hence, 2x == 2" 2y == 2, and 2z == 2.

The location of each twofold axis may be obtained from the symbol 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, :1-, n 21 along [:1-, y, n and another 2 along [i, 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 2.36.

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 ,,(:1-, y, z), my" (x, 0, z), a" (x, y, 0)

• N. F. M. Henry and K. Lonsdale (Editors), International Tables for X-Ray Crystallography, Vol. I, Birmingham, Kynoch Press.

A.7 ORTHORHOMBIC AND MONOCLINIC SPACE GROUPS 359

Taking a point x, y, z across the three symmetry planes in turn, we have (from Figure 2.33)

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

my _

~ x,y,z

a, _ ~ Z+X,y,z

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 2, (consider, for example, Figure 4.13), although we have not discussed specifically these space groups in this book.

A.7.2 Monoclinic Space Groups

In the monoclinic space groups of class 21m, a 2, axis, with a transla­tional component of b 12, shifts the center of symmetry by b 14 with respect to the point of intersection of 2, with m (Figure S.13b). In P2/e, the center of symmetry is shifted by e/4 with respect to 2/e, and in P2,le the corresponding shift is (b + e)1 4 (Figure 2.32).

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)[Ill]. (d)[llO].

1.4. (523); (523) and (523) are parallel. It should be noted that a similar situation exists in Problem 1.3, but for coincident zone symbols. [UVW] and [UVW].

1.5. (a) see Figure S.1. (b) c/a=cot29.37°=1.7771. (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

.100

0010 F -+-... -+----t-~:JE =*d-~ ·010

al00

FIGURE S.I

361

362 SOLUTIONS

-0 0-

+0 0+

(I) (b)

FIGURE S.2

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, OIl, 02J, and 010 (letter symbols indicate faces actually present). Hence, the zone symbols and poles may be deduced. Confirm the assignments of indices by means of the Weiss zone law.

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

1.7. See Figure S.2. (a) mmm (h) 2/m

1.S.

2/m 42m m3

{OlO}

2 4 6

m.m.m "" 1. 2.m"" I.

{1lO}

4 4

12

{1I3}

4 8

24

1.9. (a) 1. (b) m. (c) 2. (d) m. (e) 1. (f) 2. (g) 6. (h) 6mm. (i) 3. (j) 2mm.

1.10. (a) Hexachloroplatinate ion m3m Oh {lOO} (b) Carbonate ion 6m2 D3h (loIO} or {01 IO}

(c) Benzene .2. mm m D6h

(d) Methane 43m Td {lll} or {I I I}

(e) cis-1,2-Dichloroethylene mm2 C2v

(f) Naphthalene mmm D2h

(g) Sulfate ion 43m Td (h) Tetrakismethylthiomethane 4 S4 {1I1} or {ll I}

(tetramethylorthothiocarbonate) (i) Cyanuric triazide 6 C3h

(j) Bromochloroftuoromethane 1 C1

1.11. 2

Chapter 2

1.1. (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 FOR CHAPTER 2 363

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, e' = -a/2 + e/2. The new dimensions are c'=5.763A and fI'=139°lT; a' and b' remain as a and b, respectively. Vc(C cell) = Vc(F cell)/2.

2.3. (a) The symmetry is no longer tetragonal. (b) The tetragonal symmetry is apparently restored, but the unit cell no longer represents a lattice because the points do not all 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, e' = e.

2.4. 28.74 A (F cell); 28.64 A.

2.5. It is not a new system because the symmetry of the unit cell is not higher than 1. It represents a special case of the tricIinic system with 'Y = 90°.

2.6.

2.7

(a) Plane group c2mm.

(0, O;~, ~)+ Limiting conditions

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

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

(b) Plane group p2mg. See Figures S.3 and SA. 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.

See Figures S.5 and S.6.

4 (e) X,Y,z; x, y, i; x,!-y,!+z; hkl: None X, !+y, !-z hOI:/=2n

OkO: k =2n 2 (d) I !, o,!; !, t, 0

} 2 (c) I 0, O,~; o,!, 0 As above + 2 (b) I !, 0, 0; 11 1 hkl: k +1 = 2n 2",2,2" 2 (a) I 0,0,0; O,!, !

(100) p2gg b'=b, c'=c (010) p2 a'=a,c'=c/2 (001) p2gm a'= a, b'= b

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.

364 SOLUTIONS

o 0 0 0

_0 9 ____ • __________ *-___ ~ 0

I 0 0 I

o o 0 o 0

FIGURE S.3

I I d. t d+ p

I P I I I

i < I I I I

I I I Ib

1: qf q' I

I

I :> I I

I

d.p I d l

'p I

FIGURE S.4

FIGURE S.5

~c(110 ~

FIGURE S.6

SOLUTIONS FOR CHAPTER 2 365

2.S. Each pair of positions forms two vectors, between the origin and the points ±{(Xl - X2), (Yl - Y2), (z 1 - Z2)}: one vector at each of the locations

2.9.

2x,2y,2z; 2x,2y,2i; 2x,2y,2z; 2x,2y,2i

and two vectors at each of the locations

2x, U+2z; O,!+2y, t 2x, U-2z; OJ-2y, ~

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

x, y, ~ ~ 2p-x, -~+y, Z

,!.-I 1 {

x,y,i -a

-1 (or O)+2p-x, 2q -y,2r- z ~ -~+2p-x, 2q +~-y, z

The points X, y, i and 2p - x, 2q - y, 2r - z are one and the same; hence, by comparing coordinates, p = q = r = O.

*2.10. See Figure S.7. General equivalent positions:

x, Y,z; x,y,z; 1 1 _ z-X,z-Y,Z

1+X,Y,x; 1+X, y, i; X, !+y, z; xJ+y,i

Centers of symmetry:

tto; t~, 0; ~,t 0; ~,to 1 1 1 1 3 1 ;, .! !. 3 3 1 4,4,2; 4,4,2"; 4,4,2, 4,4, "2

Change of origin: (i) subtract i, i, 0 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; x,y,z; ~+x,~-y,i; ~-x,~+y,i)

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

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

~-- -t"----- ----t--------- .1. ___ .....

:-(])+ 0 : 0 i-(])+ I _ f.T\+1 I I WI I

~---~---------T---------~---?

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

: :-CD+ : ~---r---------L---------~---~

1m I 1m 1-\1/+ 1-\1/+ I I I I

\j \j

FIGURE S.?

366 SOLUTIONS

2.11. Two unit cells of space group Pn are shown on the (010) plane (see Figure S.8). In the transformation to Pc, only the c axis changed:

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

FIGURE S.8

Hence, Pn =Pe. 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 On, from the centering of the unit cell, Ca~ Ce, although Ce == Cn. We have Ca == On, and the usual symbol for this space group is Cm.1f the X and Z axes are interchanged in Ce, the equivalent symboi is Aa.

2.12. P21e (a) 21m; monoclinic. (b) Primitive unit cell, e-glide plane .lb, twofold axis lib. (c) hOI: 1= 2n.

Pea 21 (a) mm2; orthorhombic. (b) Primitive unit cell, c-glide plane .la, a-glide plane .lb, 21 axis lie.

Cmcm (a) mmm; orthorhombic. (b) C-face-centered unit cell, m plane .la, e-glide plane .lb, m plane .le. (c) hkl: h+k =2n;hO/: 1=2n.

P421e (a) 42m; tetragonal. (b) Primitive unit cell, 4 axis lie, 21 axes lIa and b, c-glide planes .1[110] and [110]. (c) hhl: 1= 2n; hOO: h = 2n; OkO: (k = 2n).

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) 000/: 1= 2n.

Pa3 (a) m3; cubic. (b) Primitive unit cell, a-glide plane .lc, b-glide plane .la, e-glide plane .lb (the glide planes are equivalent under the cubic symmetry), threefold axes II [111], [111], [Ill], and [Ill]. (c) Okl: k = 2n; hOI: (I = 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".

SOLUTIONS FOR CHAPTER 3

Chapter 3

3.1 (a) Tetragonal crystal; optic axis parallel to the needle axis (e) of the crystal. (b) Section extinguished for any rotation in the ab plane.

367

(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) 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. (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, e = 5.00 A. a": = 0.167, b* =0.250, c* =0.300 RU. 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 = 4.322, c = 7.506 A. (b) n max =4. (c) No symmetry in (i). Horizontal m line in (ii). (d) The photographs would be identical because of the fourfold axis of oscillation. (See Figure S.9)

I '

I '. / /

-:/ True cell-------------~' --+---e

FIGURE S.9

Apparent cell

3.6. Remembering that the {3 angle is, conventionally, oblique, and that in the monoclinic system {3 = 1800 -{3*, {3* = 86° and {3 = 94°.

368 SOLUTIONS

Chapter 4

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 + go), F(O, 5) = 2(gp - go)

F(5, 5) = 2(-gp - go), F(5,1O)=2(-gp+go)

For gp = 2go, </J(O, 5) = 0 and </J(5, 0) = </J(5, 5) = </J(5, 10) = 7T.

4.2. The structure is centrosymmetric.

A (hkl) = 4 cos 27T[ky +(h +k +/)/4] cos 27T(h +k)/4

iFc(020)1 iFc(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) P2b P2dm. (b) Pa, PZ/a. (c) Ce, C2/c. (d) P2, Pm, P2/m.

4.5. (a) P21212. (b) Pbm2, Pbmm. (c) Ibm2 (Icm2); Ib2m (Ic2m); Ibmm (Icmm)

hkl: h +k+1 = 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/: (l = 2n).

4.6. (a) (i) hOI: h = 2n ;OkO: k = 2n. No other independent conditions. (ii) hOI: 1= 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: 1= 2n. No other independent conditions. (vi) hkl: h +k +1 = 2n;hO/: h = 2n. No other independent conditions.

Space groups with the same conditions: (i) None. (ii) P2/c. (iii) Cm, C2/m. (iv) None. (v) Pccm. (vi) lma2, I2am. (b) hkl: None; hOI: h +1 =2n;OkO: k =2n. (c) C2/c, C222.

Chapter 5

5.1. A (hk/) =4 cos 27T[0.2h +0.11 +(k +/)/4] cos 27T(lj4). Systematically absent reflections are hkl for I odd. The c dimension appertaining to P21 / c should be halved, because the true cell contains two atoms in space group P2 1• 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 S.10.

SOLUTIONS FOR CHAPTER 5

, .0,..:--:-___ ...... ,...... ___ = ..

/ Z 8)(15

FIGURE S.lO

FIGURE S.l1

FIGURE S.12

.x axis

True repeal Space group P2,

0 Ah at .. -1

0 Rh at y ::;;;:

Ie o al )' .. lJ (0 Baly=jl

369

*5.2. There are eight Rh atoms in the unit cell. The separation of atoms related across any m plane is ~- 2y, which is less than b/2 and thus, prohibited. The Rh atoms must therefore lie in two sets of special positions, with either I or m symmetry. The positions on I may be eliminated, again by spatial considerations. Hence, we have (see Figures S.ll and S.12) t

4 Rh(l): ±{XI> t Zl; i+ XI> t i- Zl}

4 Rh(2): ±{X2, t Z2; i+ X2, ~J- Z2}

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

370

CH3

~~ CI- H / I

/1 H2 /1

,I

I CH3

FIGURE S.13.

-0 0- -8 0+ +0 0+

-8 0-- -8 0+ +8 0+

FIGURE S.13b

Origin at I ; unique axis b

4 fix, y, z; x, y, i; X, ~+y, z; x, ~-y, z.

2 e m x,l, z; - 3 -x, 4", Z

2 d I ~,O,~; 1 1 1 2,"2,2'

2 c I O,o,t O,!. ~. 2 b I ~,0,0; ~,~, O. 2 a I 0,0,0; O,~,O.

Symmetry of special projections

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

o

SOLUTIONS

Limiting conditions hkl: None hOI: None OkO: k=2n

1 Asabove+ hkl: k =2n

(010) p2; c' = c, a' = a

S.3. Space group P21/m. Molecular symmetry cannot be 1, so it must 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 CH3 that have their Con m planes) on m planes. This arrangement is shown schematically in Figure S.13a. The groups CH3 , HI, and H2lie above and below the m plane. (The alternative space group, P2j, was considered, but the structure analysist confirmed the assumption of P2 t! m. The diagram of space group P21/ m shown in Figure S.13b 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.)

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

SOLUTIONS FOR CHAPTER 5 371

F,

F,

B

H,

H,

FIGURE S.14a

FIGURE S.14b

5.4

XCI=0.23 XCI=0.24

hhh !Fa I gp, gK gCI IFcl Kt!Fol IFcl K2!Fol

111 491 73.5 17.5 15.5 341 315 317 329 222 223 66.5 14.5 13.0 152 143 160 150 333 281 59.5 12.0 10.5 145 180 191 189

K t R t K2 R2 0.641 0.11 0.671 0.036

372 SOLtmONS

Qearly, Xc = 0.24 is the preferred value. Pt-Q = 2.34 A. For sketch and point group, see Problem (and Solution) 1.10(a).

5.5. AuChk/) =4 cos 21T[hxu -(h+k)/4] cos 21T[kYu+(h +k +/)/4]. (mean of 1, fi, and 1).

xu=l. yu=0.20

5.6. Since Z = 2, the molecules lie either on t or m. Chemical knowledge eliminates t. The m planes are at ±(x, 1, z), and the C, N, and B atoms must lie on these planes. Since the shortest distance betweenm planes is 3.64 A. Flo B, N, C, and HI (see Figure S.14) 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 S.14 shows a stereoscopic pair of packing diagrams for CH~2' BF3 •

Flo B, N, C, and HI lie on a mirror plane; the F2 , F3, ~,Hs, and H 2, H3 , atom pairs are related across the same m plane.

Chapter 6

6.1. (a) IF(hk/)1 = IF(hkl)1 IF(Ok/)1 = IF(Okl)1 IF(hO/)1 = IF(hOl)1

(b) IF(hkl)1 = IF(hH)1 = IF(hk/)1 = IF(hkl)1 IF(Okl)1 = IF(Okl)1 = IF(Okl)1 = IF(Okl)1 IF(hOl)1 = IF(hOl)1

(c) IF(hkl)1 = IF(hkl)1 = IF(hkl)1 = IF(hhl)1 = IF(hkl)1 = IF(hkl)1 = IF(hkl)1 = IF(hkl)1 IF(Okl)1 = IF(OH)I = IF(Okl)1 = IF(Okl)1 IF(hOl)1 = IF(hOl)1 = IF(hOl)1 = IF(hOf)1

6.2. (a) Pa; [!. v, 0]. P2/ a : li, v, 0] and (u, 0, w). P2221 ; (0, v, w), (u, 0, w), and (u, v, ~).

(b) (u,O, 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 and passing through the origin. Since the crystal is noncentrosymmetric, and assuming no other concentra­tions of peaks, 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.

6.3. (a) P21/n (a nonstandard setting of P2t/c; see Problem 2.11 for a similar relationship between Pc and Pn). (b) Vectors: 1: ±{!,!+2y,!}

2: -±{!+2xJJ+2z} 3: ±{2x, 2y, 2z} 4: ±{2x, 2;, 2z}

Section v =~: type 2 vector. Section v = 0.092: type 1 vector. Section v = 0.408: type 3 or 4 vector.

double weight double weight single weight single weight

4 S: ±{0.182, 0.204, 0.226} and ±{0.682, 0.296, 0.726}

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

SOLUTIONS FOR CHAPTER 6 373

z-axis

-+ x-axis

.0.20

.0.70

0.30 • 0.80 •

FIGURE S.15

You may have selected one of the other seven centers of symmetry, in which case the coordinates determined may be transformed accordingly. The positions are plotted in Figure S.15. Differences in the third decimal places of the coordinates determined from the maps in Problems 6.3 and 6.4 are not significant.

6.4. (a) The sulfur atom x and z coordinates are S(0.264, 0.141), S'(-0.264, -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 S.16a). On another tracing, carry out the same procedure with respect to -S' (Figure S.16b). Superimpose the two tracings (Figure S.16c). Atoms are located where both maps have positive areas.

6.5. (a) P(v) shows three nonorigin peaks. If the highest is assumed to arise from Hf atoms at ±{O, YHi, ~}, then YHi = 0.11. The other two peaks may be Hf-Si vectors; the difference in their height is due partly to the proximity of the peak of lesser height to the low vector density around the origin peak-an example of poor resolution-and partly due to the 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.lD7, which is a better value for YHh 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.

374 SOLUTIONS

(a' Superposition on - S

0 0

0 0

0 0 0

0 0 0 0

0

0

0 0 0

0

0 0 0

0 0

0 0

Original peak~O 0 0 displaced to - S

FIGURE S.16a

0 0 ... Origin Peak ~ displaced to -S'

o 0 0

0 0 0

0 0

0

0

0 0 0

0 0

0 0

0

FlGURE S.16b

SOLUTIONS FOR CHAPTER 6 375

s

b •

c

FIGURE S.16c

*6.6. Since the sites of the replaceable atoms are the same in each derivative, and the space group is centrosymmetric, we may write F(M1) = F(M2 ) +4(fMl -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 i represent K and AI, respectively. The peak at 0.35 is due, presumably, to the S atom. (c) The effect of the isomorphous replacement can be noted first from the increases in IF(555)1 and IF(666)1 and the decrease in IF(333)1. These changes are not in accord with the findings in (b). Comparison of the electron density plots shows that xs/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.

376 SOLUTIONS

Chapter 7

7.1. 705,617,814. 426 is a structure invariant, 203 is linearly related to 814 and 617, and 432 has a low lEI value. Alternative sets are 705, 203, 814 and 705, 203, 617. A vector triplet exists between 814, 426, and 432.

7.2. IF(hk/)1 = IF(hkT)1 = IF(hk/)1 = /P(hkl)1 k =2n: t/J(hk/) = -t/J(hkT) =-t/J(hkl) =t/J(hkl) k = 2n + 1: t/J(hk/) = -t/J(hkT) = 7T-t/J(hkl) = 7T+t/J(hkl)

7.3. Set (b) would be chosen. There is a redundancy in set (a) among 041, 162, and 123, because F(041) =F(041) in this space group. In space group C2/c, h +k must be even, Hence, reflections 012 and 162 would not be found. The origin could be fixed by 223 and 137 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±0.3A2. (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

(u 2 ) = 0.28 A.

7.5. The shortest distance is between points like t y, z and t y, z. Hence, from (7,41), d 2(O··· CI)=a 2/4+4y 2b 2, or d(O··· O)=4.64A. Usinl;l (7.44), [2du(d)f= [2au(a)/4]2 + [8y 2bu(b)]2 + [8yb 2u(y)]2, whence u(d) = 0.026 A.

7.6. IF(OIO)1 = 149, IF(010)1 = 145, t/J(01O) = 55°, t/J(010) = -49°

Chapter 8

8.1. The I-I vector lies at 2x,!, 2z. Hence, by measurement, x =0.423 and z =0.145, with respect to the origin O.

hkl (sin (J)/A 2fl cos 27T[(0.423h) +(0.146/)] FI IFol

001 0.0261 105 0.608 64 40 0014 0.365 66 0.962 63 37 300 0.207 82 -0.119 -10 35 106 0.176 86 -0.303 -26 33

The signs of 001, 0014, and 106 are probably +, +, and -, respectively. The magnitude of FI (300) is a small fraction of /Po (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 fl.

SOLUTIONS FOR CHAPTER 8

8.2. A simplified ~2 listing follows:

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

0018 081 0817 9.5 001 024 035 3.0

026 035 0.3 021 038 059 3.6

0310 059 3.2 024 035 059 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 P2t!a, s(hkl)=s(hkl)=(-I)h+ks(hkl), which means that s(hk/)= (-I)h+ks(hkl). 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(Ol1, 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(0018)= -

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

Note that s (026) = A has a low probability compared with those of the other conclusions. To obtain most nearly equal numbers of plus and minus signs, A = + would be chosen. Hence, finally, s(035) = + and s(024) = -; s(026) is +, but with low probability.

377

Index

Absences in X-ray spectra acciden tal, 160 see also Systematic absences

Absorption correction, 155, 351 Absorption edge, 344 Absorption of X-rays, 343-344 Acciden tal absences, 160 Accuracy of calculations, 291-292; see

also Errors Acetanilide, 13 9 Alternating axis of symmetry, 340 a-Aluminum oxide, 113 Alums, 267-269 Amorphous state, 3 Amplitude symmetry and phase symmetry,

281 Amplitude of wave, 146, 147 Analyzer (of polarized light), 102 Anatase, 50-51 Angles, see Bonds, In teraxial, Interfacial,

Phase angles Anisotropic thermal vibration, 164, 289 Anisotropy, optical, 102, 103, 106;see

also Biaxial crystals, Uniaxial crystals Anomalous dispersion, 157 Anomalous scattering, 293-295

and atomic scattering factor, 293 and phasing reflections, 295 see also Scattering, anomalous

Argand diagram, 148-149 Arndt- Wonnacott camera, 134 Asymmetric unit of pattern, 75-77, 79 Atom, finite size of, 152 Atomic scattering factor, 151-152

and anomalous scattering, 293 and electron density, 209 factors affecting, 152-153 for chlorine, 199

Atomic scattering factor (cant 'd)

for iodine, 333 for platinum, 199 for potassium, 199 and spherical symmetry of atoms, 157 tern pera ture-corrected, 153 variation with (sin e)/?-.., 152

Averaging function, 220, 221 Axes: see Coordinate, Crystallographic,

Inversion, Optic, Rotation, Screw axes

Axial angles; see Interaxial angles Axial ratios, 14 Azaroff, L. V., 138 Azidopurine monohydrate, 216

Baenziger, N. C., 242, 245 Barrett, A. N., 273, 282 Beevers, C. A., 266 Bell, R. J., 4 Benzene, 53 Bertrand lens, 111 Biaxial crystals, 103, 108; see also Aniso­

tropy, optical; Optically anisotropic crystals; Crystal

idealized interference figure for, 111, 112 optical behavior, 108-111 refractive index, 109

Biphenyl, 99 Birefringence, 106-111 Birefringent crystals, 102, 106 Biscyclopentadienyl ruthenium, 47, 48 Bisdiphenylmethyldiselenide, 229-238

Patterson studies of, 230-234 Body-centered (I) unit cell, 58-60, 160-

162 Bonds, lengths and angles, 289-291 Boron trifluoride, 200

379

380

Bragg, W. L., 48, 190, 197 Bragg angle, 117 Bragg diffraction, 79,116-121

and Laue diffraction, 120-121 see also Bragg reflection

Bragg equation, 119 Bragg reflection, 117, 118; see also Bragg

diffraction order of, 119, 120

Bravais lattices, 58; see also Lattice and crystal system, 58, 59 interaxial angles of unit cells in, 66 and space groups, 73 table of, 66 unit cells of, 58, 59, 66

2-Bromobenzo(b] indeno(3,2-e) pyran (BBIP), detailed structure analysis of, 299-321

Bromochlorofluoromethane, 54 Buerger, M. J., 136, 138, 234, 262 Bull, R. J" 321

Carbon monoxide, molecular symmetry, 47 Carbonate ion, 53 Carlisle, C. H., 258 Center of symmetry, 29, 33, 338

alternative origins on, 274-276 and X-ray diffraction pattern, 45-46,

123, 156 Centered unit cell,

in two dimensions, 56-57 in three dimensions, 59-66

Centric reflections; see Signs of reflections Centrosymmetric crystal

structure factor for, 158-160 Ileealso Signs of reflections in centrosym­

metric crystals, Crystal Centrosymmetric point group, 33, 45

and X-ray patterns, 45-46, 123, 156 Centrosymmetric zones, 168-169 Change of hand, 30 Change of origin, 96-97 Characteristic symmetry, 34, 35 Characteristic X-rays, 342-343, 344 Classification of crystals

optically, 102-111 by symmetry; see Crystal class, Crystal

systems· Complex conjugate, 150 Complex plane, 148, 149

INDEX

Contact goniometer, 16, 17 Contour map of electron density, 1,2,

215,216 Coordinate axes, 5-9; see olIO Crystal-

lographic axes Copper pyrites crystal, 39 Copper X-radiation, 343 Corrections to measured intensities, 351-

355 absorption, 155,354 Lorentz, 155, 354 polarization, 155, 354 temperature factor, 152, 215, 245-247

Cruickshank, D. W. J., 187 Crystal, 5

as a stack of unit cells, 146 density measurement, 183,300 external symmetry of, 25 habit, 108, 109 ideally perfect, ideally imperfect, 347 internal symmetry of, 55 law of constantin terfacial angles of, 16 mosaic character of imperfect, 347 optical classification of, 102-111 perfection of, 346-348 symmetry and physical properties of, 25 unit cell of, 11 see also Biaxial crystals, Centrosym­

metric crystals, Uniaxial crystals Crystal class

and €-factor, 272, 273 and point group, 30 restrictions on symmetry of, 30 see also Point group, Classification of

crystals Crystal morphology, 17, 18 Crystal systems

and Bravais lattices, 58, 59 and characteristic symmetry, 34, 35 and idealized cross sections, 108 crystallographic axes of, 34 and Laue groups, 46 and optical behavior, 103, 109-110 and point group scheme, 35, 36 and symmetry in Laue photograph, 46,

123-124 see al:ro Cubic, Hexagonal, Monoclinic,

Orthorhombic, Tetragonal, Tri­clinic, Trigonal crystal systems; Sys­tems

INDEX

Crystalline substance, 2, 5, 55 Crystallographic axes, 8-9, 34

conventional, 34, 35, 38,40-44 for cubic system, 18 for hexagonal system, 12, 29 interaxial angles for, 9, 34, 66 and optic axis, 106, 109 see also Coordinate axes; Reference axes

Crystallographic point groups, 33, 35 of Bravais lattice u·nit cell, 66 classification for recognition, 338 and crystal systems, 35 and Laue groups, 46 and noncrystallographic point groups,

47-48 notation for, 35, 36, 39,40-44,339-341 restrictions on, 30 and space groups, 73 and special position sites of space groups,

186 stereograms for, 31, 33, 37-39,40-44 study and recognition scheme for, 338-339 tables of, 35, 36, 341 see also Point groups

Cubic crystal system Bravais lattices for, 59, 64 crystallographic axes for, 18 see also Crystal systems

Cyanuric triazide, 54

Dean, P., 4 Debye, P., 219 De Moivre's theorem, 148 Density of crystal

and contents of unit cell, 183 measurement of, 183, 300

Density, optical, 350 Detailed structure analyses, 299-321,321-333 Diad, 33, 86 cis-1,2-Dichloroethylene,53 Difference-Fourier synthesis, 252-253

and correctness of structure analysis, 292 and least-squares refinement, 289 see also Fourier synthesis

Diffraction by regular arrays of scattering centers,

114-116 of visible light, 112 and wavelength, 112 see also X-ray diffraction by crystals

381

Diffraction grating, 113 Diffractometer, single-crystal, 121, 351,

354 Diiodo-(N,N,N',N'-tetramethylethylenedi-

amine)zinc (II), 93, 94 Diphenyl sulfoxide, 263 Direct lattice, 68; see also Bravais lattice Direct methods of phase determination,

271-287 examples of use of, 282-285, 299, 321-

333 see also lEI statistics, E-Factor, Signs of

reflections in centro symmetric crystals, Structure analysis

Directions, form of, 55 Dirhodium boron, 197; see also Rh,B Dispersion, anomalous, 157 Displacement method for density measure­

ment,183 Duwell, E. J., 242, 245

E maps, 285 calculation of, 285-287 examples of, 286, 328-329 "sharp" nature of, 286

lEI statistics, 273-274, 323; see also Direct methods of phase determination

Eisenberg, D., 263 Electron density distribution, 208-209

ball-and-stick model for, 215, 217 computation and display of, 215-217 contour map of, 1,2,215-216 and criteria for structure analysis correct-

ness, 292 determined from partial structures, 250 as Fourier series, 211-214 Fourier transform of, 211 and hydrogen atom positions, 214 interpretation of, 214 and Patterson function, 219-223 peak heights and weights of, 214 periodicity of, 209 projections of, 209, 215-217 pseudosymmetry in, 251, 252 resolution of, 282 and structure factors, 208-211 and successive Fourier refinemen t, 251 in unit cell, 146 see also Contour map of electron density

Enantiomorph, 27

382

Enzymes, 254 ribonuclease, 258

Epsilon (e) factor, 272-273 Equi-incIination techniques, 137 Equivalent positions

general, 76-77, 358-359 special,77 see also General (equivalent) positions,

Special (equivalent) positions Errors, superposition of, 291; see also

Accuracy Estimated standard deviation (esd), 291 Euphenyl iodoacetate

contour map of electron density of, 1, 2 crystal data, 333 molecular model and structural formula,

3,217 Patterson section for, 334 Weissenberg photograph for, 140, 141

Ewald, P. P., 48 Ewald's construction, 128-131 Ewald sphere, 128-130 Extinction of X-rays

primary, 346, 347 secondary, 347, 348

Extinction parameter (for X-rays), 348 Extinction positions (for polarized light),

105 and crystal system, 111 for monoclinic crystals, 110 for orthorhombic crystals, 109 for tetragonal crystals, 108 see also Extinction of polarized light

e-factor, 272-273; see also Direct methods of phase determination

Face-centered (F) unit cell, 58, 60 Family of planes, 66, 67 Filtered X-radiation, 344-346 Flotation method for density measure-

ment, 183 Form

of directions, 55 of planes, 18, 37-38

Fourier analysis, 204, 208 simple example of, 213 and X-ray diffraction, 213

Fourier map, 216 Fourier series, 201-217

termination errors, 204, 205, 292 weighting of coefficients, 249-250

Fourier summation tables, 266 Fourier synthesis, 205, 208

partial, 248-252 and structure analysis, 213

INDEX

see also Difference-Fourier synthesis Fourier transform, 208

of electron density, 209-211 structure factors as, 211

Fractional coordinates of centered sites, 60 in space group diagrams, 75

Friedel's law, 123, 128, 156-158, 170-171 and absorption edges, 293

Friedel pairs, 293, 295

Gay, P., 138 General form of planes, 37-38

for crystallographic point groups, 40-44 General positions, 37

equivalent, 77, 358-359 molecules in, 185 see also Special positions, Equivalent

positions Geometric structure factor, 164-166 Glass (silica) structure, 3,4 Glide line, 75, 78 Glide plane, 91

and limiting conditions, 169-175 translational component of, 91

Glusker, J. P., 216 Goniometers 16, 17, 300 Graphic symbols

change of hand, 30 diad axis in plane of diagram, 86 glide line, 78 glide plane, 91 inversion axis, 31, 33 inverse monad (center), 33 mirror line, 27 mirror plane, 33 pole, 19,21 representative point, 30, 31, 38 rotation axes, 33, 86 rotation points, 27 screw axis, 89, 95 table of, 33 see also Stereogram

Great circle, 20

Habit of crystal, 108, 109 Hafnium disilicide, 265

INDEX

Half-translation rule, 357-358 Hamilton, W. C., 179, 293 Harker, D., 225, 228 Harker and non-Harker regions of the

Patterson function, 225-228 Hartshorne, N. H., 138 Hauptman, H., 279 Heavy-atom method, 248-252

example of, 299-321 limitations of, 253-254 see also Isomorphous replacement,

Structure analysis Henry, N. F. M., 49, 98,138,179,197,

281, 358 Herman-Mauguin notation, 39,52, 338,

339,340,341 and Schoenflies notation, 338-341

Hexachloroplatinate ion, 53 Hexad,33 Hexagonal crystal system

crystallographic axes in, 12, 34 crystals of, 14 lattices of 64 triply primitive unit cell, 64, 65 see also Crystal systems

Hexagonal two-dimensional system, 29 Hexamethylbenzene, 279 Hierarchy for considering limiting con­

ditions, 173, 178 Htoon, S., 94 Hydrogen atom, 295

and difference-Fourier technique, 214, 253

and electron density distribution, 214 scattering of X-rays by, 317 see also Light atoms

Ibers, James A., 179 Ideal intensity, 155 Identity element, 26 Implication diagram, 261, Index of refraction, 106, 109 Indexing an oscillation photograph,

128, 131-134 Integrated reflection, 348 Intensity of scattered X-rays; see X-ray

diffraction by crystals Interaxial angles

for Bravais lattices, 66 for crystallographic axes, 9, 34

Interfacial angles, law of constant, 16

Interference figures, 111, 112 Interference of X-rays

constructive, 114 due to finite atom size, 152, 153 partially destructive, 119

383

Intermolecular contact distance, 289, 290 International Tables for X-ray Crystal-

lography, 281 Internuclear distances, 296 Interplanar spacings, 66-67, 71 Inverse diad, hexad, monad, tetrad, and

triad, 33 Inversion axes, 29, 30, 33 Isogyres, 111 Isomorphous pairs, 254 Isomorphous replacement, 254-262

for alums, 267-269 for proteins, 254-26 2

see also Heavy atom method Isotropic thermal vibrations, 153, 164 Isotropy, optical, of crystals, 102, 103

Jeffery, J. W., 138, 351 Jensen, L. H., 179, 262, 296

Karle, I. L., 282 Karle, J., 279, 282, 287 Keeling, R. 0., 188 Kennard, 0., 333

Ladd, M. F. C., 22, 52, 94, 266, 299, 333 Lattice, 55, 58-66

d!rect,69 rotational symmetry of, 72-73 three-dimensional, 55, 58-66 two-dimensional (net), 32, 55, 56-57 see also Bravais lattices, Reciprocal lattice

Laue equations, 114 Laue group, 40-44

defined,45 and point groups, 46, 123 projection symmetry of, 46

Laue method, 45,113-114,122-124 experimental arrangement for, 113 see also Laue X-ray photograph

Laue treatment of X-ray diffraction, 114-116

and Bragg treatment, 120-121 Laue X-ray photograph, 113-114,122,303

symmetry of, 46, 123 and uniaxial crystals, 123 see also Laue method

384

Layer lines, 124, 130 screens for, 134-135 spacings between, 124, 125, 126 see also Oscillation method

Least-squares method, 287-289 and esd, 291 and light atoms, 295 and p.arameter refinement, 288-289 rermement by and secondary extinction,

348 Light atoms, 295; see also Hydrogen atom Limiting conditions on X-ray reflections,

79,82-83, 160-162 for body centered unit cell, 161 and geometric structure factors, 165 and glide-plane symmetry, 169-174 hierarchical order of considering, 173 nonindependent (redundant), 89, 173 and screw-axis symmetry, 166-167 and systematic absences, 162 for various unit-cell types, 162 see also Space groups, by symbol, two­

dimensional, three-dimensional, Re­flections, X-ray

Limiting sphere, 134,218 Lipson, H., 138, 262, 266 Liquids and X-ray diffraction, 295 Lonsdale, K.,49, 98,179,197,279,

281,358 Lorentz correction, 155, 354 Lorentz factor, 155

Melatopes, 112 Methane, 53 Methylamine, 200 Miller-Bravais indices, 12,29 Miller indices, 10-12

in stereograms, 22-24 transformations of, 82, 355-357

Minimum function, 234-235 Mirror line, 27, 29 Mirror plane, 29, 33 Mirror symmetry; see Reflection symmetry Molecular geometry equations, 289-291 Molecular symmetry, 185-186 Molybdenum X-radiation, 346 Monad,33 Monochromatic X-radiation, 346 Monoclinic crystal system, 61

lattices of, 61-63 limiting conditions in, 176-177

INDEX

Monoclinic crystal system (cont'd) optical behavior of crystals of, 110 orthogonal coordinates for, 318 reciprocal lattices for, 68, 69 space groups of, 85-89, 359 unit cells of, 61, 62 X-ray diffraction patterns of crystals in,

175-177 see also Crystal systems

Multiplicities of reflection data, 246

Napthalene, 53,185-187 Net, 56-58 Neutron diffraction, 295, 296 Nickel tungstate, 187-188, 189 Noncrystallographic point groups, 47-48 Nonindependent limiting conditions, 89,

173

Oblique coordinate axes, 5-8 Oblique extinction, 110 Oblique two-dimensional system, 56, 58;

see also Space groups Optic axis

and crystallographic axes, 109 of uniaxial crystal, 106

Optical classification of crystals, 102-111 Optical density, 350 Optic.'ll methods of crystal examination,

101-111 Optically anisotropic crystals, 102, 103,

106; see also Bia.xial crystals, Uni­axial crystals

Order of diffraction, 115, 119, 120 Origin, change of, 96-97 Origin-fIxing reflections, 274-276

examples of, 283, 323, 325 Orthogonal lattice, direct and reciprocal,70 Orthorhombic crystal system

lattices of, 63-64 limiting conditions in, 177-179 optical behavior of crystals in, 109 space groups of, 94-97, 357-359 unit cells of, 63 X-ray diffraction patterns of, 175, 177-

179 see also Crystal systems

Oscillation method, 124-128 experimental arrangement for, 124, 125 disadvantage of, 134 example of use of, 301, 302

INDEX

Oscillation method (eont'd) indexing photograph by, 128, 131-134 symmetry indications from, 127 zero-layer in, 124 see also Layer lines

pI, 224 amplitude and phase symmetry of, 281 lEI statistics for, 273-274 general positions in, 228 origin-fixing reflections for, 296

P2 1

amplitude and phase symmetry for, 297 diagram for, 90 general equivalent positions, 164 geometric structure factors for, 164 and pseudosymmetry introduced by

heavy-atom method, 251, 252 special positions in, 89 systematic absences in, 165

P2l/e amplitude and phase symmetry for, 281,

323 analysis of symbol for, 96 general equivalent positions in, 98, 169 geometric structure factors for, 169 limiting conditions for, 98, 170 origin-fIxing reflections of, 297 special position sets in, 98, 186

Packing, 32, 73, 289 Palmer, H.T., 235, 274 Palmer, R.A., 235, 273, 282 Papaverine hydrochloride

analysis of by Patterson functions, 235-239, 240-241

crystal data, 184 molecular structure, 184 and partial-structure phasing, 249 and sharpened Patterson function, 225

Parametralline, 7 Parametral plane, 10, 11

for crystal systems, 34, 35 for cubic systems, 18 for hexagonal crystals, 12 preferred choice of, 12

Parity, 165 Parity group, 275 Partial-structure phasing, 248-251

effective power of, 249 for protein molec1lle, 254 see also Phase determination

Path difference in Bragg diffraction, 118, 119 for constructive interference, 114 for parallel planes, 144-146

Pattern unit, 74 and asymmetric unit, 75, 76-77,79

Patterson, A.L., 219 Patterson function, 218-229

centrosymmetry of, 222, 225 as Fourier series, 223 and Laue symmetry, 227 as map of interatomic vectors, 222 one-dimensional, 219-222, 266 oversharpening of, 226 partial results of, 244-245 projection of, 243, 244, 245 sharpened, 225-226

385

sharpened, using normalized structure factors, 272

and solution of phase problem, 218-245 symmetry of for Pm, 226-228 and symmetry-related and symmetry-

independent atoms, 224 three-dimensional, 223 and vector interactions, 226, 228-229 see also Peaks of Pa tterson function

Patterson sections, 231, 313-315 for euphenyl iodacetate, 334 for papaverine hydrochloride, 240-241

Patterson space, 223 Patterson space group, 224 Patterson superposition, 233, 234, 244 Patterson unit cell, 223, 224, 225 Pc

equivalence to Pn, 99 general equivalent positions in, 168 geometric structure factors for, 168 limiting conditions for, 181 reciprocal net for, 170, 171 systematic absences in, 169

Peak heights and weights for electron density, 214

Peaks of Patterson function, 222 arbitrariness in location of, 239 cross-vector, 261 Harker, 228 non-origin, 225 positions of, 223-225 spurious, 226 and symmetry-related atoms, 228 weights of, 223, 224, 228

386

Peaks of Patterson function (cont'd) flee alflo Patterson function

Pento-uloside sugar, 274 Phase angles in centrosymmetric crystals,

159 Phase determination, 156,213,217-262

by anomalous scattering, 295 direct methods of, 271-287 heavy-atom method of, 248-252, 253-

254 by Patterson function, 218-245 flee also Partial-structure phasing; Struc­

ture factor; Detailed structure analyses

Phase of structure factor, 155; see also Structure factor

Phase of wave, 146 of resultant wave, 147, 151, 154

Phase probability methods, 271 Phasing, partial-structure; flee Partial-struc-

ture phasing Phillips, F.C., 48 Phillips, D.C., 263 Plane, mirror, 29, 33 Plane groups, 75-85

the seventeen patterns, 80-81 Planes

family of, 66, 67 form of, 18,37-38 intercept form of equation for, 10, 12

Platinum derivative of ribonuclease, 259-262

Point groups centrosymmetric, 33, 45 and Laue group, 46,123 noncrystallographic, 47-48 and space groups, 73 stereograms for, 27, 28, 30, 31, 37, 38,

40-44 and systems, 29, 35 tables of, 29, 35, 36 three-dimensional, 30-47 three-dimensional, by Hermann-

Mauguin symbol

1- 35,36,40,46,66,87,90,95,96, 273, 338-339

1- 35, 36, 40, 45, 46, 52, 96, 273, 338-339,340

2- 31,35,36,40,46,86,87,273,338-339,340

INDEX

Point groups (coot'd) three dimensional, by Hermann­

Mauguin symbol (coot'd) m-33, 34, 35, 36,40,46,52,86,96,

273,340 2/m-35, 36,40,46,52,66,86,91,97,

273, 338-339 222- 35, 36,40,46, 273, 338-339 mm2- 35,36,41,46, 273,338-339 mmm - 35,36,41,46,52,66,96,97,

127,273,338-339,357 3-35,36,43,46,52,338-339 3- 35, 36,43,46, 338-339 32- 35, 36, 44, 45, 46, 338-339 3m-35, 36, 44, 46, 52,338-339 3m - 35, 36,44,45,46,66, 338-339 4- 35,36,41,46,338-339 ,r- 31,35,36,41,46,338-339,340 4/m- 35,36,41,45,46,338-339 422- 35, 36,41,46,52,97 4mm - 35, 36, 37, 42, 46, 97 42m-35, 36, 39,42,45,46,52,337,

338-339

i mm - 35,36,42,45,46,47,66,338-m339 6- 35,36,42,46,52,338-339 '6 - 35, 36,42,46, 338-339 6/m-35, 36,42,46,338-339 622- 35, 36,43,45,46, 338 6mm - 35, 36,43,46, 339 6m2- 35, 36,43,45,46, 52, 338-339

.!.mm -35, 36,43,45,46,66,338-339 m 23 - 35, 36, 44, 46, 52, 340 m3- 35,36,44,46,52,338-339 432- 35,36,44,45,46,52,338-339 43m-35, 36, 44, 45, 46, 97, 338-339 m3m - 35, 36, 44,45, 46, 66, 338-339

three-dimensional, by Schoenflies symbol

CI -40, 341 C2 -40, 341 C.-43,341 C4 -41,341 C.-42,341 Ci -40, 341 Cs-40, 341 8 1 -40,341 8 2 -40,340,341 84 -41,340,341 8. - 41, 341

INDEX

Point groups (cant 'd) three dimensional, by

Schoenflies symbol (cont'd)

C2 h - 41, 340,341 C3h - 42,340 C.h - 41, 341 C6h - 42,341 C2V - 41,341 C3V - 44,341 C.v -42,341 C6V - 43,341 D2 - 40,341 D3 - 44,341 D. - 41,341 D6 - 43,341 D2h - 40,341 D,h-43,341 D.h - 42,341 D6h - 43,341 D 2d- 42,341 D 3d- 44 ,341 T- 44,341 Th - 44,341 0-44,341 Td - 44,341 0h - 44, 341

three-dimensional, non-crystal-lographic, by symbol

1-m-47 m 10m2-47,48 ~-341

~/m-341

~-341

Dsh- 48 C~v-341

D~h-341 two-dimensional, 25-29 two-dimensional, by Herman-

Mauguin symbol 1-26-29,46,77,78,84 2-26-29,46,58,75-77,84 3-26-29,46 4-26-29,46 6-26-29,46 m-26-29, 46,77,78 2mm-28-29, 46, 58,77,83,127 3m-28-29,46,113 4mm - 28-29,46,58 6mm - 28-29,46,58

unit cells in plane lattices, 58

387

unit cells of Bravais lattices, 66 see also Crystallographic point groups,

Stereograms

Polarization correction, 155, 354 Polarized light, 102

and structure analysis, 102-111 Polarizers, 102 Polarizing microscope, 102, 103-111

examples of use of, 300, 321 Polaroid, 102 Pole, in spherical projection, 19, 20 Positions, see Extinction; General, Special

positions Potassium dihydrogen phosphate

(KH 2PO.), 103, 108, 139 Potassium hexachloroplatinate, 198, 258 Potassium 2-hydroxy-3,4-dioxocyclobut-l-

ene-l-olate monohydrate detailed structure analysis of, 321-332 lEI statistics for, 323 ~ 2 listing for, 323, 324

Potassium dimercury (KHg 2), structure analysis of, 239, 242-245

Potassium tetrathionate, 33, 34 Povey, D.C., 299 Powder diffraction, 121 Precession method, 136-13 7 Primary extinction of X-rays, 346, 347 Primitive, 19 Primitive circle, 19, 20 Primitive plane, 19 Primitive unit cell, 56, 58 Principal symmetry axis, 35 Probability of triple-product sign relation­

ship, 279, 280 Projections

of electron density, 209, 215-217 Patterson, 243-245 spherical, 19 stereographic, 18-24 see also Space groups; Stereograms

Proteins, 254 heavy-atom location in, 258-262 non-centrosymmetric nature of, 257 sign-determination for centric reflections

of, 257, 258 structure analysis of, 254-262, 295

Pyridoxal phosphate oxime dihydrate, 273, 274, 282-286

symbolic-addition procedure of, 282-285

388

Pyrite, 190, 209, 211 structure analysis of, 194-197

Pseudosymmetry in trial structures, 25 I, 252

Quartz structure, 3,4

R factor, 193; gee also Reliability (R) factor Reciprocal lattice, 68-72

diffraction pattern as weighted, 156 points of in limiting sphere, 218 and sphere of reflection, 128-130 symmetry of, 68 unit cell of, 68 unit cell size in, 68, 131 units for, 68, 69 vector components in, 70 weighted points diagram of, 170, 171 gee algo Lattice; Unit cell, two-dimen-

sional Rectangular two-dimensional system, 56-

57,58 two-dimensional space groups in, 77-79

Reduced structure factor equation, 161,162 Redundant limiting conditions, 89, 173,174 Reference axes, 5-8; gee also Crystal-

lographic axes Reference line, 7 Ref"mement by least squares, 288-289,348 Reflecting goniometer, 16, 17 Reflecting power, 348, 349 Reflection (mirror) line, 27, 29 Reflection (mirror) plane, 29, 33 Reflection, sphere of, 128, 129 Reflection symmetry, 25, 27

of a square-wave function, 204, 206 in three dimensions, 29 in two dimensions, 27 gee algO Symmetry

Reflections, X-ray, 117 integrated, 348 intensity of, theory of, 143 ff, 348-349 number in data set, 218 0rigin-f"JXing,274-276 "unobserved", 246 see also Limiting conditions, Signs of

reflections in centrosymmetric crystals, Structure analysis, Sys­tematic absences, X-ray diffraction by crystals

Refractive indices, 106, 109

INDEX

Reliability (R) factor, 193 and correctness of structure analysis, 292 and parameter refinement, 288-289

Repeat period of a function, 201 Repeat vector, 55 Resolution of electron density map, 282 Resolution of Patterson function, 225 Remltantphase, 147,151,154 Resultantwave,147,149-151

for unit cell, 153-155 Reynolds, C.D., 184 Rh2B,197 Rhombic dodecahedron, 18, 19 Rhom bohedral unit cell, 65 Ribonuclease, 258

platinum derivative of, 259-262 Rotation axes, 29 Rotation points, 26 Rotational symmetry, 25, 26, 29 Row, gymbol for, 55

Sampling interval, 204 Sayre, D., 277 Sayre formula, 277 Scale factor, ISS

absolute (for IFol), 245-247 for sodium chloride, 193

Scattering; see X-ray diffraction Scattering, anomalous; gee Anomalous

scattering Schoenflies notation, 39, 52, 338, 339- 341

for crystallographic point groups, 40- 44 and Hermann-Mauguin notation com-

pared,338 Scintillation counter, 351 Screw axis, 88, 166-167 Secondary extinction of X-rays, 347, 348 Sigma,.two (1:J formula, 279-281, 282 Signs of reflections in centrosymmetric

crystals, 159 Sign determination ofreflections in centro­

symmetric crystals, 274-281 gymbolic, 282-285, 325-327 gee also Triple-product sign relation­

ship; 1:2 formula, Centrosymmetric crystal; Direct methods of phase determination; Reflections, X-ray

Sim, G.A., 250 . Single-crystal diffraction, 121, 122!! Single-crystal diffractometer, 121, 351,

354

INDEX

Small circle, 20 Sodium chloride, 1

structure analysis of, 190-193 Space-ruling patterns, 32, 73 Space groups, 73-98

"additional" symmetry elements of, 75 ambiguity in determination of, 176 centre of symmetry in, 91 and crystals, 74-75 fractional coordinates in, 75 and geometric structure factor, 164 limiting conditions for reflection in,

176-178 origin shift in diagrams for, 94 pattern of, 55, 74 and point groups, 73 practical determination of, 175 as repetition of point-group pattern by

Bravais lattice, 73, 74 standard diagrams for, 75ff three-dimensional, 85-98 three-dimensional, by symbol

P1- 224, 273, 282 Pl-224, 228, 273, 281, 282, 298; see

also pI P2-86, 87, 88, 90, 167, 177, 228,

229,263,272 P2,-90, 164, 165, 168, 177,200,257,

258, 298, 333; see also P2, C2-86, 87, 88, 89, 90, 176, 177, 181,

230,232 C2,-90 Pm - 127, 224, 226-228, 229 Pa-100 Pe-100, 168, 169, 177, 181, 188; see

also Pc Pn-IOO Cm-176, 177,230,232 Ca-IOO Ce-IOO,I77 Cn-IOO P2lm -177, 224, 228, 229 P2le-IOO, 177, 188, 359 P2,1m-I77, 200, 251, 257 P2,/a-181, 263, 234 P2,1e-91, 92, 93, 97, 98, 99,169-

171, 176, 177, 184, 185, 186, 197, 238,281,298,301,308,322,323, 359;see also P2,/e

P2,/n-181 C2Im-176,177,230

Space groups (cant 'd) three-dimensional, by symbol (eont'd) C21e- 93,177,298 A21a-181 P2,2,2,-94, 95,178,274 P2,22-181 P222,- 263 B2,22,-181 Pma2-171-173 Pbe2,-178 Pea2,-IOO Pee2-181 Im2a-242 I2ma-242 Pbam-IOO Pban -100 Pman -173-174 Pbem-178 Pbea-178 Pbnm-199 Pnna-357,358 Pmma-357,358 Cmem - 100, 265 Immm-178 Ibea-97,98 Imam-181 Imma-239, 242, 243, 358 P4,2,2-97,98 P42,e-IOO P6 322-100 Pa3 -100, 194,267 F23-190 Fm3-190 F432-190 F43m-190 F43e-97,98 Fm3m-190

tW<rdimensional, 75-83 tW<rdimensional, by symbol

pl-80 p2-75-77, 80, 87, 90 pm, pm I, plm, plml, pllm -77-79, 8~ 83,85, 87,88,8~ 171

pg, pgl, pig, plgl, pllg-79, 80, 83, 90

em, em I, elm, elm 1, ellm, eg-77-79, 81, 82, 87, 88

pmm, p2mm - 80 pmg, pgm, p2mg, p2gm - 80,96, 99,

121

389

pgg, p2gg-80, 83, 84, 85, 90, 94, 95,96

390

Space groups (cont'd) two-dimensional, by symbol (cont 'd)

c2mm-81, 96, 99 p4-81 p4m, p4mm - 81 p4gm, p4mg- 81 p3-81 p3ml-81 p31m-81 p6-81 p6m, p6mm - 81

see also Crystal systems, Oblique two­dimensional system, Projections

Special forms, 37, 38 Special positions, 28, 37, 77, 186

equivalent, 77, 83, 86 molecules in, 186-187 see also Equivalent positions, General

positions Sphere of reflection, 128-130 Spherical projection, 19 Spot integration, 351, 352-353 Square two-dimensional system, 29, 58 Square wave, 202

as Fourier series, 202-205, 206-207 termination errors for, 205

Squaric acid, 321 Standard deviation of electron density, 292 Stereogram, 19-24

assigning Miller indices in, 22-24 for crystallographic point groups, 38, 39,

4~4

description of, 17-24 fundamental property of, 20 practical construction of, 21-22 for three-dimensional point groups, 30,

31, 38,4~4 for two-dimensional point groups, 27,

28 uses of, 23 see also Graphic symbols, Point groups,

Projections, Stereographic projec­tions

Stereographic projections, 17-24; see also Stereograms

Stereoviewer, 335, 336 Stereoviews, 30, 61, 86, 93, 187,262 Stout, G.H., 179,262,297 Str~textinction,107 Structural data, references for, 333

Structure analysis accuracy of, 291-292

INDEX

of 2-bromobenzo[b] indeno[3,2-e] pyran, 299-321

computer use in, 214 criteria for correctness of, 292 errors in tria1 structure, 251, 252-253 limitations of, 295-296 as overdetermined problem, 218 of papaverine hydrochloride, 235, 238-

239,240-241 phase problem in, 156, 213, 217-262 for potassium dimercury, 239, 242-245 for potassium 2-hydroxy-3,4-dioxo-

cylobut-l-ene-l-olate monohydrate, 321-332

preliminary stages of, 101, 156, 175-179 preliminary stages of, examples of, 183-

197,300-302,321 for proteins, 254-262, 295 published results on, 333 for pyrite, 194-197 refmement of, 288-289, 317-318,

329-332, 348 with results of neutron diffraction, 296 for sodium chloride, 190-193 and symmetry analysis, 185, 189, 230 by X-ray techniques, 111-137 see also Direct methods of phase deter­

mination; Heavy-atom methods; Reflections, X-ray; X-ray diffraction by crystals

Structure factor, 83, 153-155 absolute scale of, 245 amplitude of, 153, 154 amplitude, symmetry of, 281 applications of equation for, 156-157 calculated,253 for centrosymmetric crystals, 158-160 defmed, 153 as Fourier transform of electron density,

211 generalized form of, 210 geometric, 164-166 invariance under change of origin, 275 local average value of, 246 normalized,271-275 observed, 193,253 and parity group, 275-276 phase symmetry of, 281

INDEX

Structure factor (cont'd) phase of, 155 plotted on Argand diagram, 154 reduced equation for, 161, 162 sign-determining formula for, 276-281 for sodium chloride, 191-192 and special position sets, 187 symmetry of, 281 and symmetry elements, 163-166 see also Phase determination; Phase

of structure factor Structure invariants, 276 Stuart, A., 138 Subgroup, 45 Sucrose, 139 Sulfate ion, 54 Superposition technique, see Patterson Sutton, L.E., 263, 296 Symbolic-addition procedure, 282-285 Symbolic phases, 283 Symbolic signs of reflections, 282-285,

325-327 Symmetric extinction, 109 Symmetry, 25

characteristic, 34, 35 of crystals, 25 cylindrical, 47 examples of molecular, 47,185-186 of Laue photographs, 46, 123-124 mirror, see Reflection symmetry onefold,26 and physical properties, 25 rotational, 25, 26, 29 and self-coincidence, 25 and structure analysis, 185, 189, 230 of X-ray diffraction pattern, 45

Symmetry axis, 29 alternating, 340 principal, 35

Symmetry element, 25 interacting, 25 intersecting, 38 symbols for, 27,33,78,89,91,95 in three dimensions, 29 in two dimensions, 25

Symmetry-equivalent points, 31, 37 Symmetry-independent atoms, 163 Symmetry operations, 25, 33;

see also Symmetry element Symmetry plane (mirror plane), 29, 33

Symmetry point, 26 Symmetry-related atoms, 76-163 Systematic absences, 79, 82, 158, 175

for body-centered (l) unit cell, 162 and geometric structure factor, 165 and limiting conditions, 162 and m plane, 228 and translational symmetry, 175 see also Absences in X-ray spectra;

Limiting conditions on X-ray absences; Reflections, X-ray

Systems

391

three-dimensional, see Crystal systems two-dimensional, 29, 56-57, 58; see

also Crystal systems ~2 formula, 279-281

and symbolic-addition procedure, 282 listing, 282 see also Signs of reflections in centro­

symmetric crystals ~2 examples of, 283-285, 323, 324

Temperature factor correction, 153, 215, 354

factor (overall) and scale, 245-247 Termination errors for Fourier series, 204,

205,292 Tetrad,33 Tetragonal crystal system

model of a crystal of, 336-337 optical behaviour of crystals in,

103-108 symmetry of, 64 unit cells of, 59, 60, 64 see also Crystal systems

Tetrakismethylthiomethane (tetramethyl­orthothiocarbonate), 54

Thermal vibrations of atoms, 152-153 anisotropic, 164, 289 and smearing of electron density, 2

Translation vectors, 55 Triad,33 Tric1inic crystal system

lattice and unit cell of, 58, 59, 60 optical behaviour of crystals in, 111 see also Crystal systems

Trigonal crystal system, 64-66; see also Crystal systems

392

Trimethylammonium chloride, 198 Triplo-product sign relationship, 277-279

Sayre formula, 277 1:2 formula, 279 lee oIso Signs of reflections in centro­

symmetric crystals Triply primitive hexagonal unit cell, 64,

65 Tungsten LX-radiation, 304

Uniaxial crystals, 103-108 def"med,103 idealized cross section of, 108 idealized interference f"lgure for, 111,

112 and Laue photograph, 113, 123 optic axis of, 106 aee also Anisotropy, optical; Optically

anisotropic crystals; Crystal Unit cell, threo-dimensional, II, 59

body-centered, 58, 60, 160-162 of Bravais lattices, 58, 59, 66 contents of determination of, 183-189 conventional choice of, 58 dimensions, determination of, 126, 288 faco-centered (F), 58, 60 limiting conditions for type of, 162 nomenclature and data for, 60 point groups of, 66 primitive, 56, 58 scattering of X-rays by, 146, 151-153,

155 symbols for, 58, 60, 162 transformations of, 61-63, 355-357 translations associated with each type,

162 triply primitive hexagonal, 64, 65 type, 60, 66, 162 for various crystal systems, 58-59, 61-

62,63,64,65,66 volume of, 72 lee also Reciprocal lattice; Unit cell,

two-dimensional Unit ceO, two-dimensional, 56-57

centered, 56 conventional choice of, 57 edges and angles of, 58 point groups of, 58 symbols for, 56, 58 see oIso Unit cell, three-dimensional

INDEX

Van der Waals' forces, 321 Vector interactions, 226, 228-229

lee also Vectors Vector triplet, 277, 278 Vectors

complex, 148, 149 interatomic in Patterson function, 222 repeat, 55 scalar product of, 63 translation, 55 aee also Vector interactions

Waves amplitude of, 146, 147 and Argand diagram, 148-149 combinations of, 146-151 energy associated with, 155 phase of, 146 resultant, 147, 149-151, 153-155

Weighted reciprocal lattice levels, 170, 171 Weiss zone law, 16 Weissenberg method, 134-135, 137

chart for, 307 examples of use of, 301-302, 304-307 integrated photograph by, 350, 353 photograph by, 141

White radiation (X-ray), 113,342,344 Wilson, A.J.C., 245, 246 Wilson's method, 245-246 Wilson plot, 246-247, 287, 309 Woolfson, M.M., 138, 179, 249, 262, 287,

296,354 Wooster, W.A., 138 Wulff net, 22, 23 Wyckoff, R.W.G., 333 Wyckoff notation (in space groups), 76, 77

X-radiation copper, 343 dependence on wavelength, 342 mtered, 344-346 molybdenum, 346 monochromatic, 346 tungsten, 304 white, 1l3, 342, 344 lee also X-rays

X-ray scattering (diffraction) by crystals, 111-137

anomalous scattering, 293-295 Bragg treatment of, 116-120

INDEX

X-ray scattering (diffraction) (cont'd) as a Fourier analysis, 213 and indices of planes, 79 intensity, measurement of, 349-351

corrections for, 351-355 by diffractometer, 351 examples of, 302, 308, 322-323 by film, 350-351

intensity, theory of, 143f[, 348-349 ideal, 155

by lattice, 115-116 Laue treatment of, 114-116 Laue treatment equivalence to Bragg

treatment of, 120-121 for monoclinic crystals, 175-177 order of, 115, 119, 120 for orthorhombic crystals, 175, 177-179 for single-crystal, 121, 122-128 and space-group determination, 175-179 symmetry of and Patterson function, 227 from unit cell, 146, 151-153, 155 see also Diffraction, Reflections, X-ray

structure analysis, X-ray diffraction pattern, X-ray diffraction photograph

X-ray diffraction by fluids, 219, 295; see also X-ray diffraction pattern

X-ray diffraction pattern centrosymmetric nature of, 45-46,123,

156 and Friedel's law, 123 and geometric structure of crystal, 164 intensity of spots in, 114,350-351 position of spots in, 114, 122-136 symmetry of, 45, 123 and symmetry of crystals, 123-124,

127-128, 137 as weighted reciprocal lattice, 156 see also X-ray diffraction by crystals,

X-ray diffraction by fluids, X-ray diffraction photograph

393

X-ray diffraction photograph, 113, 121-137

important features of, 113-114 Laue, 113, 114, 122 measurements of intensity of reflection

on, 349-351 measurements of position of reflections

on, 131-137 measurements of unit-cell from, 124-126 by oscillation method, 124-128 for powder, 121 by precession method, 16, 136-137 for single-crystal, 121 Weissenberg, 134-135,350,353 see also X-ray diffraction by crystals;

X-ray diffraction pattern X-ray reflections; see Reflections, X-ray X-ray scattering; see X-ray diffraction by

crystals X-ray structure analysis; see Structure

analysis X-ray tube, 342 X-rays

absorption of, 343-344 characteristic, 342-34 3, 344 extinction of, 346, 347, 348 generation of, 341-342 non-focusing property of, 156 properties of, 113, 342-346 wavelengths of, 112,343 see also X-radiation

Zero-layer, 124, 125, 129 Zircon crystal, 15, 16 Zone axis, 15 Zone circle, 21, 22 Zones, 15, 16

centrosymmetric, 168 Weiss law of, 16