12

Projections of Cubic Crystals

by

Ian O. Angell and Moreton Moore

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only.

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Copyright in this electronic edition (i)2001 International l.Jnion of Crystallography

P u b l i s h e d for the I n t e r n a t i o n a l U n i o n o f C r y s t a l l o g r a p h y

b y U n i v e r s i t y Col lege Cardi f f Press

Cardiff , Wales

O 1984. by the International Union of Crystallography. All rights reserved.

Published by the University College Cardiff Press for the International Union of Crystallography with the financial assistance of Unesco Contract No. SC/R.P 250.271

This pamphlet is one of a series prepared by the Commission on Crystallographic Teaching of the International Union of Crystallography, under the General Editorship of Professor C. A. Taylor. Copies of this pamphlet and other pamphlets in the series may be ordered direct from the University College Cardiff Press, P.O. Box 78, Cardiff CFI IXL, U.K.

ISBN 0 906449 16 2

Printed by J. W. Arrowsmith Ltd., Bristol

Series Preface

The long-term aim of the Commission on Crystallographic Teaching in establishing this pamphle t p rogramme is to produce a large collection of short statements each dealing with a specific topic at a specific level. The emphasis is on a particular teaching approach and there may well, in time, be pamphlets giving alternative teaching approaches to the same topic. It is not the function of the Commission to decide on the 'best ' approach but to make all available so that teachers can make their own selection. Similarly, in due course, we hope that the same topics will be covered at more than one level.

The first set of ten pamphlets, published in 1981, and this second set of nine represent a sample of the various levels and approaches and it is hoped that they will stimulate many more people to contribute to this scheme. It does not take very long to write a short pamphlet , but its value to someone teaching a topic for the first time can be very great.

Each pamphlet is prefaced by a statement of aims, level, necessary background, etc.

C. A. Taylor Editor for the Commission

The financial assistance of UNESCO, ICSU and of the International Union of Crystallogra- phy in publishing the pamphlets is gratefully acknowledged.

Teaching Aims

To present numerous views of cubic crystals as an aid to the understanding of the relations between symmetry and morphology in three dimensions.

Level

This would be suitable early in an undergraduate study of crystallography or mineralogy.

Background

Elementary knowledge of symmetry operations: (centre of symmetry, axes of rotational symmetry and mirror planes): and of the unit cell is assumed.

Practical Resources

Models (made of cardboard, wood or plaster) of the simpler shapes would

be helpful.

Time Required for Teaching

This could form two or three lectures in an introductory course, together with time to study the diagrams (and to make models).

Projections of Cubic Crystals

Ian O. Angel l and M o r e t o n M o o r e

Royal Holloway College, University of London, Egham, Surrey, TW20 0EX, England

1. Introduction

Crystals are three-dimensional objects and are represented on paper by suitable projections. The use to which the resulting picture is to be put determines the choice of projection. Clinographic, orthographic and per- spective projections are briefly described here, with examples taken from the cubic crystal system.

2. Clinographic, Orthographic and Perspective Projections

Imagine one wishes to represent a cube on paper. There are six square faces but one cannot see them all at once (unless the cube is transparent!). The hidden edges may be included in the drawing to indicate their positions as if they were visible.

Only one face will be seen if the cube is viewed centrally and perpendicular to this face: not a very informative view if one wishes to obtain an overall impression of the object (which in this case might be a square prism). It is usual to arrange the viewpoint so that as many faces as possible are visible, or equivalently the object is turned so that this is the case.

In the clinographic projection the cube is turned through an angle (0) about a vertical axis, making both the front and right hand faces visible. The cube is then projected on to a vertical plane by parallel straight lines, which are inclined to the horizontal so that the top face is brought into view, (Fig. ia).

The orthographic projection is also a parallel one, but here the projection lines meet the (vertical) plane at right angles. The cube is tilted forwards through an angle 4~ before projection to show the top face, (Fig. ib). If the angle of tilt 4~ equals the angle that the projection lines make with the horizontal in the clinographic projection, then these clinographic and ortho- graphic views are closely similar, (Fig. ii), but differ as follows. The vertical dimensions in the clinographic projection are magnified by the factor see ~b compared with the orthographic: or in other words, the height h of the crystal will be preserved in the clinographic projection, whereas in this orthographic projection it will appear as h cos 6.

In the standard setting of the crystal, ~b is usually chosen to be 9028 ' so that its tangent is ~, for ease of drawing, see 4~ is then (x/3-7)/6 -- 1.0138, and

(a) CLINOGRAPHIC

h. f

(b) ORTHOGRAPHIC lJ

1-

----.._1 i Picture ptane

I Fig. L

cos q5 = 0.9864. The fact that these figures are very close to unity shows that these two views will be nearly the same. (In early books on mineralogy, tan q5 was taken to be ~.) The angle 0 about the vertical is usually chosen to be 18°26 ' so that tan 0 = ½. An orthographic projection which is closely similar to this clinographic standard is one projected along the direction which (in the cubic crystal system) has zone-axis symbol [621], (and appear- ing in the diagrams as 6.0 2.0 1.0: these are the components, referred to the cube, of a vector in this direction).

CuOJ:

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IL

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Clinographic and orthographic projections are used widely in crystal- lography because parallel lines in the crystal project to parallel lines in the projections, and thus zonal relationships are preserved visually. The eye however sees things differently. A perspective view is a conical projection, with the apex of the cone at the eye. The picture plane is placed between the observer and the object; and parallel lines in space project to lines which converge to vanishing points on the picture plane] The size of the picture will be determined not only by the size of the object but also by the relative distances of eye to picture plane and eye to object. Rear faces of the crystal will appear smaller than front faces because they are further away. In Fig. iic the viewpoint has coordinates (36, 12, 6) on an arbitrary scale, that is, the components are in the ratio 6:2: l, and so one is looking along the same direction as the [621] orthographic view. Perspective projec- tions with slightly differing viewpoints, corresponding to the left and the right eye, may be used to construct stereoscopic pairs.

3. Cubic Crystals

The variety of crystal shapes is so great that only crystals built up from cubes are considered in this pamphlet: other structural units have been omitted. Three mutually perpendicular crystallographic axes may be chosen parallel to the edges of the cubic unit cells. These are the familiar Cartesian axes Ox, Oy and Oz of coordinate geometry, where O is the origin. Their directions are represented by [100], [010] and [001]. Directions related by symmetry, such as these three, are written for brevity as (100). Other directions in the crystal may be referred to these axes; as in the earlier remark that the [621] direction has components in the ratios 6:2:1 along these three axes respectively.

A set of crystal faces, related to one another by symmetry, is called a form; the symmetry being more obvious in specimens in which the faces are equally developed. For example, the cube, in the most symmetrical class, has three tetrad axes coinciding with the Ox, Oy and Oz axes, and three mirror planes perpendicular to these axes; four triad axes coinciding with the body diagonals (111}; six diad axes parallel to the face diagonals

Fig. iii. Rhombic dodecahedron made from cubes.

4

(110), and six mirror planes (making nine in all) perpendicular to these diad axes. The cube has a centre of symmetry. The underlying symmetry of crystals is much less obvious in specimens for which the faces of individual forms are unequally developed; but may still be deduced by considering the perpendiculars drawn to the faces from a point within the crystal, and their distribution in space. This is best done using a stereographic projection: the subject of another pamphlet-'.

In the cubic crystal system there are five distinct classes or point groups: that is, five different combinations of symmetry elements, but all have in common the presence of four triad axes of symmetry. These are listed in Table 1. An axis of symmetry X perpendicular to a mirror plane m is written X --m (or X/m). The maximum symmetry of a cube, as described above, is

4 - 2 represented by the point group symbol - - 3 - - (or 4/m33/m, often abbreviated to m3 m). m m

The orientation of a plane in crystallography is described in terms of its Miller indices. These are the reciprocals of the relative intercepts, in terms of unit cell edges, that the plane makes with the three crystallographic axes, expressed in whole numbers. In general, if the intercepts are a/h, b/k, c/l, where a, b and c are the cell edges, then the Miller indices of the plane are h, k, l, usually written (hkl). In the cubic crystal system, the unit cells are cubes, so a = b = c. (In the drawings here a = 1.) The face of the cube, perpendicular to the [100] direction (i.e. to the Ox Cartesian axis) has Miller indices (100), since the intercepts on the three axes are a / l , a/0, a/0: the latter two intercepts (on Oy and Oz), being at infinity. (For cubic crystals,

Table 1. Cubic crystal classes or point groups

Symmetry elements along the directions Full Abbreviated

symbol symbol (100) (111) (110)

4 _ 2 4-fold axes 3-fold rotation- 2-fold axes - - 3 - - r n 3 m m rn perpendicular to inversion axes perpendicular to

mirror planes mirror planes

4 3 2 4 3 4-fold axes 3-fold axes 2-fold axes

~, 3 rn ~, 3 m 4-fold rotation- 3-fold axes mirror planes inversion axes perpendicular

to these directions 9

m

23

rn 3 2-fold axes 3-fold rotation- perpendicular inversion axes to mirror planes

2 3 2-fold axes 3-fold axes

the [hkl] direction is perpendicular to the (hkl) plane, but this is not so for crystals of other crystal systems.) The faces of an octahedron make equal intercepts, positive and negative, on all three axes andtherefore have Miller indices (111), ( ] l l ) ,~0TI) , (111), (111), (111), (111) and (1-]1). The minus signs arewritten above the symbols for compactness. All eight symmetrically related faces of the form are written as {11 I}.

4. The Diagrams r

In each computer drawn diagram, the Miller indices of the face(s) defining the form(s) are shown under the letters H, K, L (printed by the computer as capital letters). The distance of this face from the centre, on an arbitrary scale, is shown under D. The point group symmetry (shown at the top of each diagram tinder the word CUBIC) then operates on the specified face (or faces) to give the complete solid. For example, m 3 m symmetry generates all six faces of the cube from the single face (100). For the solids thus generated from a single face, four orthographic views are given with hidden edges omitted: a 'general' view along the [621] axis, closely similar to a standard clinographic projection; and views along the [100], [110] and [I 11] directions. For each cubic crystal viewed along the [111] direction the three-fold symmetry is evident. There are three other equivalent three-fold axes along [111], [111] and [1-i-1]. One can also see immediately whether the crystal possesses a four-fold axis along [100], a diad axis along [110], or mirror planes bisecting any of the projections along [ 100], [ 110], and [ 111 ].

As far as possible, similarly shaped solids are placed together in the pages which follow. In some cases two different settings ('positive' and 'negative') are given for the same crystal: (3 &4, 9 & 10, 15 & 16, 19 &20, 25 &26); and some crystals are mirror images (enantiomorphs) of one another: (22 & 23, 27 & 2g).

Table 2 lists the names of the crystal forms and table 3 shows their distribution amongst the five cubic point groups, or crystal classes.

The cube (no. 1) appears in all five columns because any one of the cubic point group symmetries operating on the (100) plane will generate all six faces of the cube. The rhombic dodecahedron (5) also appears five times for the same reason; whilst the octahedron (2) appears three times, and the tetrahedron (3) twice. {210}, {310} and {320} are particular examples of {hk0}; {211} and {311} of {hll} with h>l; and {221} is an example of {h h l} with h > L {321} is a particular case of the general form {hkl} in which all the indices are different and non-zero. In table 3, it will be seen for example, that the same crystal form (15) will be generated ei ther by

3 m or by 23 operating on (211). Very often crystals exhibit faces of more than one form together. Some

examples of combinations of two forms are shown here. The overall shape

Table 2. Cubic crystal forms: key to the figure numbers

(Maximum) No. of Fig. No. Form Symmetry Name faces

1 {100} m 3 m 2 {111} m 3 m 3 {111} 4 3 m

• 4 { f l u z~3 ,n 5 {110} m 3 m 6 {210}1 7 {310} 7 rn 3 m 8 {32o}j 9 {210})

lo {12o}(. I 1 { 3 1 0 } f m 3 12 {320}) 13 {211} m 3 m 14 {311} m 3 m 15 {211}] 16 {211} ~, 43 m 17 {311}J 18 {221} m 3 m 19 {221} 4-3 m 20 {221} 43 m 21 {321} m 3 m 22 {321} 43 23 {312} 43 24 {321} 43 m 25 {321} m 3 26 {312} m 3 27 {321} 23 28 {312} 23

Cube 6 Octahedron 8 Tetrahedron (positive) 4 Tetrahedron (negative) 4 Rhombic dodecahedron 12

Tetrahexahedra 24

Pentagonal dodecahedra (or pyritohedra) 12

Icositetrahedra (or trapezohedra) 24

Tristetrahedra 12

Trisoctahedron 24 Deltoid dodecahedron (or deltohedron) 12

Hexoctahedron 48 Pentagonal icositetrahedra (or gyroids)

24 Hexatetrahedron 24 Didodecahedron (or diploid)

24 Tetrahedral pentagonal dodecahedra (or tetartoids) 12

(The figures continue unnumbered for the combinations of forms.)-

Table 3. Distribution o f crystal forms amongst the five cubic classes

The numbers are figure numbers. See Table 2 for the names of the forms

m 3 m 43 /43m m3 23

{ I 0 0 } 1 1 I I 1 {110} 5 5 5 s 5 {111} 2 2 3 2 3 {hkO} 6 6 6 9 9

{hll} h > l 13 13 15 13 15 {hhl} h > l 18 18 19 18 19

{h k 1} 21 22 24 25 27

7

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of a crystal is very much determined by the relative distances from the centre of the two kinds of face: one dominant form perhaps being only slightly modified by another. The drawings are still of ideal crystals, in which all faces of each form have the same size and shape. Real crystals seldom have the faces of forms so uniformly developed.

Acknowledgements

The authors would like to thank Dr A. F. Seager and Dr. E. J. W. Whittaker for helpfu! comments on the manuscript.

Further Reading 1. White, Gwen, Perspective: A Cruide for Artists, Architects and Designers, London, Batsford

Ltd. (1982). (Also available in German, Japanese and Portuguese.) 2. Whittaker, E. J. W., The Stereographic Projection. I.U.Cr. Pamphlet, Cardiff, University

College Cardiff Press (1983). 3. Angell, I. O., A Practical Introduction to Computer Graphics, London; Macmillan (1981). 4. Phillips, F. C., An Introduction to Crystallography, London, Longmans (1964). 5. Whittaker, E. J. W., Crystallography: An Introduction for Earth Science (and other Solid

State) Students, Oxford, Pergamon Press (1981).

28

International Union of Crystallography Commission on Crystallographic Teaching

FIRST SERIES PAMPHLETS (1981) 1. A non-mathematical introduction to X-ray diffraction. C.A. Taylor 2. An introduction to the scope, potential and applications

of X-ray analysis.

3. Introduction to the Calculation of Structure Factors. 4. The Reciprocal Lattice.

5. Close-packed structures.

6. Pourquoi les groupes de Symetrie en Cristallographie.

7. Solving the phase problem when heavy atom.s are in special positions.

8. Anomolous Dispersion of X-rays in Crystallography.

9. Rotation Matrices and Translation Vectors in Crystallography. S. Hovm6ller

I0. Metric Tensor and Symmetry operations in Crystallography. G. Rigault

SECOND SERIES PAMPHLETS (1984) 11. The Stereographic Projection.

12. Projections of Cubic Crystals.

13. Symmetry.

14. Space Group Patterns.

15. Elementary X-Ray Diffraction for Biologists.

16. The Study of Metals and Alloys by X-ray Powder Diffraction Methods. H. Lipson

17. An Introduction to Direct Methods. The Most Important Phase Relationships and their Application in Solving the Phase Problem. H. Schenk

18. An Introduction to Crystal Physics. Ervin Hartmann

19. Introduction to Neutron Powder Diffractometry. E. Arzi

This selection of booklets represents a sample of teaching approaches at various levels (undergraduate and postgraduate) and in various styles. The Commission on Crystallographic Teaching of the International Union of Crystallography hopes to develop the series into a large collection from which teachers can make selections appropriate to their needs and has particularly borne in mind the needs of developing countries and of the problem of teaching crystallography to students of other disciplines such as chemistry, biology, etc. and priced as follows: 95p each.

Available from: University College Cardiff Press P O Box 78 Cardiff CF1 1XL Wales, U.K. Cheques should be made payable to University College Cardiff.

M. Laing

S. C. Wallwork

A. Authier

P. Krishna and D. Pandey D. Weigel

L. Hohne and L. Kutchabsky

S. Caticha-Ellis

E. J. W. Whittaker

Ian O. Angell and Moreton Moore

L. S. Dent Glasser

W. M. Meier

Jenny P. Glusker