Hammond C. The Basics of Crystallography and Diffraction

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334 Appendix 1

Ca. R. Ine Crystallography, version 3.1 and 4.0 by Cyrille Boudias and Daniel Mon-

ceau, CIRIMAT Laboratory, CNRS INPT UPS, ENSIACET, 31077 Toulouse, France,

distributed by C Sellier, Centre de Transfert UTC BP 50149, 60200 Compiegne. Tel. 03

44 23 45 30, e-mail: christine.sellier@utc.fr

CARINE CRYSTALLOGRAPHY software is distributed in two versions (both on the

same disk). The new version (4.0) is focused on 3D modelling of crystals, including

surfaces and interfaces. The 3.1 version is used for simple rendering and for simulation

and analysis. CaRIne 3.1 exposes in a clear way the relationships between different geo-

metrical representations of crystals and brings them together in a user friendly interface:

stereographic projections, X-ray diffraction on powders, reciprocal lattices in 3D and

2D and (hkl) plane-based calculation lists. All the unit cells created with the 4.0 version

can be sent directly to the 3.1 version via the [Send to CaRIne 3.1] command. Daniel

Monceau also edits the ‘E-materials’ website: http://e-materials.ensiacet.fr/.

Crystallographica by A R Renshaw, Oxford Cryosystems, 3 Blenheim Ofﬁce

Park, Long Hanborough, Oxford OX29 8LN, UK, e-mail: AlexRenshaw@

oxfordcryosystems.co.uk.

CRYSTALLOGRAPHICA runs under any version of Windows including 3.1, 95, 98, NT,

2000, XP and Vista and is available free from www.crystallographica.com. The package

includes structure and reciprocal lattice displays, powder simulation, graphing (includ-

ing common X-ray data formats) and a unique crystallographic scripting language, all

supported by comprehensive help and numerous example scripts and structures. See

also Crystallographica—a software toolkit for crystallography, J. Appl. Cryst., 30, 418

(1997).

CrystalMaker by David C Palmer, CrystalMaker Software Ltd., Centre for Innovation

and Enterprise, Begbroke Science Park, Sandy Lane, Yarnton, Oxford, OX5 1PF, UK,

e-mail: info@crystalmaker.com.

CRYSTALMAKER is a program for building, displaying and manipulating crystal and

molecular structures, including real-time photo-realistic graphics and ‘out-of-the-screen’

three-dimensional displays. CrystalMaker allows visualization of massive structures

(millions of atoms), with fast bond and polyhedra generation and the display of thermal

ellipsoids. New features include the ability to seamlessly browse multiple structures

in the same window and generate animations and movies (e.g. showing structural

changes resulting from temperature or pressure). The program supports a wide range

of data formats, exports high-resolution graphics in a wide range of formats, plus Quick-

Time movies and VR objects—and works with the ‘SingleCrystal’and ‘CrystalDiffract’

diffraction software.

CrystalMaker is available in two versions: for Mac (a ‘universal binary’ application

that runs natively on the new Intel-based Macs, as well as the older PowerPC-based

Macs), and for Windows computers running XP or Vista. The software is applicable

in the areas of materials science, solid-state physics, chemistry, earth sciences and

crystallography.

Crystals by David Watkin, Chemical Crystallography Department, University of

Oxford, Oxford, OX1 3TA, UK, e-mail: david.watkin@chem.ox.ac.uk.

Appendix 1 335

CRYSTALS is available without charge for non-commercial use from the web server

xtl.chem.ox.ac.uk. It provides all the basic and advanced facilities needed for crystal

structure determination, plus a visualization window which continuously displays the

structure and a graphics module for manipulating and drawing the structure. A ‘guide’

leads theuser through allthestages of anormal crystal structureanalysis and the manualis

included as Postscript ﬁles and as browsable html. The distribution kit includes versions

of SHELXS, SIR92 and Superﬂip (with the author’s permissions). Tutorials and data

are included for a number of demonstration structures. A scripting interface enables the

program to be customized as a teaching aid, and two extensible examples are included.

Introduction to Crystallography and Diffraction by Anne Fretwell and Peter Good-

hew, University of Liverpool, Liverpool, L69 3BX, UK, e-mail: goodhew@liv.ac.uk

INTRODUCTION TO CRYSTALLOGRAPHY is part of (one module of 18) a CD-ROM

which covers the basics of materials science and related subjects at undergraduate level,

viz Crystals, 2-D and 3-D Crystallography, Crystal Structures and Indexing Directions

and Planes. Embedded multiple choice questions and interactive simulations and a

comprehensive glossary are included.

DIFFRACTION runs live on the web from the MATTER site and includes geometry,

the Ewald sphere, structure factors, electron approximations, etc. The best URL for both

programs is the root MATTER site at www.matter.org.uk.

!Polynet, !Polysymm and !Stellate by Kate Crennell, ‘Fortran Friends’, PO Box 64,

Didcot, Oxon, OX11 OTH. Website: http://fortran.orpheusweb.co.uk/poly/prodp.htm

THESE POLYHEDRA PLOTTING PROGRAMS can run on RISCOS systems or on PCs

under the Virtual Acorn emulator. They show the construction and symmetries of the

Platonic andArchimedean solids and their duals and some of their stellations. For details

see the website above, from which demonstration copies of the software may be freely

downloaded. Free copies of plans to cut out and construct 3D models are also available

to make space-ﬁlling models and examples of each of the crystal classes.

A1.2 Crystallography models, model-building and suppliers

The use of crystal models in the teaching or exposition of crystallography has a long

history, the ﬁrst perhaps being the terracotta models of Rome de L’Isle (1783). Later,

comprehensive sets in wood were made by geological suppliers, particularly by the

German ﬁrm of Krantz. Glass, aluminium, paper and card have also been used. Some of

the strangest and most intriguing crystal models were those made byWilliam Barlow. His

workshop is said to have resembled a toy factory with thousands of balls of various sizes

and colours and plaster dolls’ hands which he arranged in frameworks to demonstrate

enantiomorphism and space group symmetry. Unfortunately, toy suppliers, unlike their

Victorian forebears, are today unwilling or unable to provide dolls’ hands for such

crystallographic purposes.

Cardboard/paper models of polyhedra are very easily constructed. ‘Ready-made’ tem-

plates are available on the BCA website and also in several Tarquin Publications (99

336 Appendix 1

Hatﬁeld Road, St Albans, AL1 4JL, UK; www.tarquinbooks.com), e.g. Make Shapes,

books 1 and 2 by Jenkins and Wild; Paper Polyhedra in Colour by Jenkins and Bear;

Geodesic Domes by van Loon and the classic text Mathematical Models by Cundy and

Rollett. By making collections of polyhedra and glueing them together you can see,

in a very direct way, those which are space-ﬁlling and those which are not. You can

also represent twinned and multiply-twinned crystals. Figure A1.1 shows an interesting

example. Each tetrahedron represents a ccp crystal, the faces of which correspond to

the {111} twin planes. The twinning angle across each face is 70.53

◦

and the total angle

for the ﬁve twins is 70.53

◦

× 5 = 352.65

◦

, i.e. very close to 360

◦

. Such a multiply-

twinned ccp crystal shows ‘pseudo-pentagonal’ symmetry—not to be confused with the

pentagonal symmetry of quasicrystals (see Section 4.9).

Fig. A1.1. (a) The faces of the tetrahedron correspond to the {111} twin planes in a ccp crystal;

(b) a singly-twinned crystal; (c) a multiply-twinned crystal (ﬁve twins) showing pseudo-pentagonal

symmetry. By a small distortion of the model the pentagonal symmetry of a quasicrystal (d) is obtained.

Plastic (polystyrene) sphere models of the ccp and hcp metal structures may be made

by glueing spheres together to form close-packed hexagonal layers and then stacking

them together in theABCABC… and ABAB… sequences. The insertion of smaller (e.g.

plasticine) spheres shows the sizes and distribution of the interstitial sites. The more

complex stacking sequences (e.g. in SiC (Fig. 1.27) or of the oxygen atoms in Al

)

can also be demonstrated. The bcc metal structure is best made by making layers of the

closest-packed {110} planes (Fig. 1.9(b)) and stacking them in the ABAB… sequence,

atoms in each layer being located at the ‘saddle-points’ of the layer below. Notice that

the layer-separation is reduced if the layers are slipped into the recesses each side of the

saddle-points, giving a more closely packed but no longer cubic structure.

Childrens’ wooden building bricks come in a variety of shapes—cubes, tetragonal,

hexagonal and orthorhombic prisms—and (apart from building castles), may be used

to represent the 32 point groups (monoclinic and triclinic models must be home-made).

Appendix 1 337

Fig. A1.2. Models showing the symmetries of the three orthorhombic point groups: mmm, mm2 and

222. Note the two enantiomorphic forms of point group 222.

Fig. A1.3. Models showing the symmetries of the ﬁve cubic point groups m

3m,

43m, m

3, 432 and 23.

Note the two enantiomorphic forms of point groups 432 and 23.

The plain models represent the holosymmetric point groups; those of lower symmetry

may be made by cutting-off appropriate corners and edges (as shown in Fig. A1.2 for

the three orthorhombic point groups) or by marking the faces (as shown in Fig. A1.3 for

the ﬁve cubic point groups). Bricks may also be cut across a non-mirror plane and the

two halves then rotated 180

◦

to show the geometry of twinning. Clearly, many beautiful

multiply-twinned crystals can be created in this way.

Model-building kits and complete models are obtainable from a number of suppliers of

which the following is a selection.

Cochranes of Oxford Ltd., 29 Groves Yard, Shipton Road, Milton under Wychwood,

Chipping Norton, Oxford, OX7 6JP, UK; http://www.cochranes.co.uk makes Orbit™

and Minit™ molecular model kits based on plastic atom centres colour-coded according

338 Appendix 1

to the element which are joined together with plastic tubes of the appropriate lengths.

The scale of the Orbit™ system is 30 mm=1Åwhen used for organic and inorganic

chemistry and typically 12–25 mm =1 Å for crystal structures. For the smaller-scale

Minit™ system the scale is typically 10–15 mm = 1 Å. Also available are larger-scale

Unit™ molecular models in which the atoms are joined together with rigid aluminium

or ﬂexible vinyl tubes.

Miramodus Limited (formerly Beevers Miniature Models Unit), c/o School of Chem-

istry, West Mains Road, Edinburgh, EH9 3JJ; http://www.miramodus.com produces

crystal models for display and teaching on the scale of 1 cm=1Å.Coloured PMMA

balls are individually drilled and connected with stainless steel rods to give accurate and

attractive models of any crystal or molecular structure. Models can be made to indi-

vidual requirements, including mounting for gifts or presentations and are suitable for

museum-quality displays. Rods and balls can also be supplied for self-assembly.

Spiring Enterprises Ltd., Billingshurst, West Sussex, RH14 9EZ, UK; http://www.

molymod.com makes the Molymod™ molecular model system. The colour-coded atom

centres are joined together with ﬂexible mushroom-shaped links. A wide range of sets

for building organic and inorganic molecular models is available as well as DNA (and

RNA) model kits in which abstract-shaped parts are designed to represent the bases,

sugar and phosphate groups of the double helix.

3-D Molecular Designs, 2223 North 72nd Street, Wauwatoza, WI 53213, USA;

http://3DMolecularDesigns.com make molecular and crystal models in which the atoms

or molecules are held together by internal magnets. These models simulate, for example,

the hydrogen bonding and polarity of water molecules in the structures of liquid water

and ice.

Appendix 2

Polyhedra in crystallography

In this Appendix the crystallographic and geometrical properties of the polyhedra which

are introduced and brieﬂy described in this book are summarized. The word comes from

the Greek meaning many (poly) faces or planes (hedra). The (simpler) polyhedra are

simply described by their number of faces as indicated below. It is curious that only the

hexahedron has a ‘simple’ name—the cube.

Polyhedron names

For the simpler polyhedra the preﬁx indicates the number of faces. Those of importance

in crystallography are: tetra (4), penta (5), hexa (6), octa (8), deca (10), dodeca (12),

icosa (20), icosidodeca (32). Further preﬁxes indicate the shapes of the faces them-

selves and/or their symmetry, e.g. pentagonal or rhombic dodecahedra (polyhedra with

twelve pentagonal or rhombic (diamond shaped) faces respectively), or trapezorhombic

dodecahedra (polyhedra in which the twelve rhombic faces are arranged in hexagonal

symmetry). Further preﬁxes such as triakis or pentakis indicate that plane triangular or

pentagonal faces are replaced by a group of three or ﬁve faces respectively. Thus a triakis

tetrahedron has twelve (4 × 3) faces, and so on.

Truncation

Many polyhedra are described as ‘truncated’which simply means ‘cutting off the corners’

of a polyhedron—and the truncated polyhedron thus created depends on ‘how much’ of

the corners are cut off. As shown in Fig.A.2.1(a)–(c) the truncated cube, cubeoctahedron

and truncated octahedron are created by cutting increasing amounts off the cube corners.

Alternatively, we could ‘start off’ with an octahedron and obtain the same polyhedra by

again cutting off the corners as shown in Fig.A.2.1(d)–(f). Truncation can also be applied

to the newly-created corners and edges: the possibilities are almost endless and a whole

series of polyhedra, with many faces of different shapes and sizes (and with rather long

descriptive names) can be created. The process has its analogy in the beautiful patterns

of faces which can develop in crystal growth.

Duality

Polyhedra of identical symmetry are said to be dual if the faces of one correspond to

the corners (or vertices) of the other and vice versa. Thus the cube and octahedron are

duals: the eight vertices of the cube correspond to the eight faces of the octahedron and,

340 Appendix 2

Fig. A2.1. Alternative truncation procedures to create a truncated cube, cubeoctahedron and truncated

octahedron. (a) → (b) → (c) successive truncation of a cube; (d) → (e) → (f) successive truncation

of an octahedron. (From Polyhedra by Peter R. Cromwell, Cambridge University Press, Cambridge,

1999.)

conversely, the six vertices of the octahedron correspond to the six faces of the cube.

Similarly, the pentagonal dodecahedron and icosahedron are duals. The tetrahedron is

self-dual: the four vertices of one correspond to the four faces of the other and vice versa.

Classiﬁcation of polyhedra

There is an enormous number of polyhedra, including beautiful star-shapes with re-

entrant faces. However, those of most importance in crystallography are the Platonic and

Archimedean polyhedra, deltahedra, coordination and space-ﬁlling (Voronoi) polyhedra

and Brillouin zones. These groupings are of course not mutually exclusive.

(A) The ﬁve perfect (regular) polyhedra or Platonic solids

These polyhedra (Fig. A.2.2) each have identical faces and edges of equal length. The

vertices and faces of each circumscribe and inscribe respectively the surface of a sphere.

The tetrahedron has cubic point group symmetry

43m and the octahedron and cube

have the full (holosymmetric) cubic point group symmetry

≡ m

3m. The icosa-

hedron and pentagonal dodecahedron both have point group symmetry

5—the

symmetry possessed by quasicrystals, buckminsterfullerene and some viruses. Of all

the Platonic polyhedra, only the cube is space-ﬁlling.

These solids were known to the Ancient Greeks and were clearly identiﬁed and

described by Theaetatus of Athens (d. 369 bc). They were considered by Plato to

constitute the fundamental building-blocks of matter: ﬁre (tetrahedron), solid (cube),

air (octahedron), liquid (icosahedron) and the universe or cosmos itself (pentagonal

Appendix 2 341

Fig. A2.2. The ﬁve perfect polyhedra or Platonic solids: tetrahedron, octahedron, cube, icosahedron,

pentagonal dodecahedron.

dodecahedron). Hence, they are attributed to him, instead of (more properly) to

Theaetatus.

The notion that the Platonic solids had a cosmic signiﬁcance was taken up by Johannes

Kepler in his ﬁrst book Mysterium Cosmographicum, 1596, in which he attempted to

show that the ratios of the diameters of the (circular) orbits of the then-known six planets

corresponded with the ratios of the inscribed and exscribed diameters of the spheres of

the ﬁve perfect solids. Thus the ratios of the diameters of the inscribed and exscribed

spheres for an octahedron correspond to the orbits of Mercury and Venus; that for an

icosahedron the orbits of Venus and Earth, and so on. In short he attempted to explain

or model the orbits of the planets in purely geometrical terms. The model failed of

course, even in Kepler’s time, by observation and calculations of greater precision on

the ellipticity of the planetary orbits.

(B) The thirteen semi-regular or Archimedean Polyhedra

In these polyhedra, the discovery of which is attributed to Archimedes, every face is a

regular polygon as in the Platonic polyhedra but the faces by be of different kinds—two

kinds in the case of ten of the polyhedra and three kinds in the remaining three. Around

each vertex the faces are arranged in the same order, the edges are of equal length and

all the polyhedra are inscribable in a sphere. A complete list is given in Table A.2.1 and

those of particular crystallographic interest are shown in Fig. A.2.3.

We owe the (rather long) names of these polyhedra to Kepler, and as indicated, they

may be obtained by truncation of the corresponding Platonic solids—either a simple

truncation of the vertices as shown for example in Fig. A.2.1, or a more complex trunca-

tion of the newly created vertices or edges. For example the truncated cubeoctahedron

(also known as the great rhombicubeoctahedron) and the (small) rhombicubeoctahe-

dron are obtained by truncating the cubeoctahedron in two different ways. The last two

polyhedra, described by the curious word ‘snub’, are not derived simply by a process

342 Appendix 2

Table A2.1 The Archimedean polyhedra

No. Name No. and shapes of faces Point group symmetry

1 Truncated tetrahedron 4 hexag, 4 triang

43m

2 Truncated cube 8 triang, 6 oct m

3 Truncated octahedron 6 square, 8 hexag m

4 Truncated dodecahedron 20 triang, 12 decag

5 Truncated icosahedron 12 pentag, 20 hexag

6 Cubeoctahedron 8 triang, 6 square m

7 Icosidodecahedron 20 triang, 12 pentag

8 Truncated cubeoctahedron

(great rhombicubeoctahedron)

12 square, 8 hexag, 6 oct m

9 (small) rhombicubeoctahedron 8 triang, 18 square m

10 Truncated icosidodecahedron 30 square, 20 hexag, 12 decag

11 Rhombicosidodecahedron 20 triang, 30 square, 12 pentag

12 Snub cubeoctahedron 32 triang, 6 square 432

13 Snub icosidodecahedron 80 triang, 12 pentag 235

Fig. A2.3. Six of the thirteen Archimedean polyhedra, Nos. 1, 5, 8, 9, 11, 12 (see Table A.2.1). Figure

A.2.1 shows Nos. 2, 3, 6. (From Mathematical Models by H.M. Cundy and A.P. Rollett, 2nd edn,

Clarendon Press, Oxford, 1961 reprinted by Tarquin Publications, St Albans, 2006.)

of truncation, but also involve a ‘twist’ or rotation, which destroys mirror planes and

centres of symmetry. Hence, they occur in two enantiomorphic forms. It should be noted

that the point group symmetry 432 of the snub cubeoctahedron is not known to occur in

any inorganic crystal (see pp. 38 and 103).

Appendix 2 343

Of these thirteen polyhedra, only the truncated octahedron (the Voronoi polyhedron

for the cubic I lattice) is space-ﬁlling (Fig. 3.9). The pattern of faces in the truncated

icosahedron correspondsto thatof C

(see Section1.11.6). The dualsof theArchimedean

polyhedra have the same symmetry. They are perhaps of less interest, except for the

dual of the cubeoctahedron which is a rhombic dodecahedron (the space-ﬁlling Voronoi

polyhedron for the cubic F lattice (Fig. 3.8(a))) and that of the icosidodecahedron which

is a rhombic triacontahedron (Fig. A.2.4). The ratios of the lengths of the long and short

diagonals of the rhombic or diamond-shaped faces are not the same: for the rhombic

dodecahedron itis

√

2 : 1 and for therhombic triacontahedron it is

√

5+1 : 2 or1.618 : 1—

the Golden ratio (see Section 2.9).

(a) (b)

Fig. A2.4. (a) Rhombic dodecahedron and (b) rhombic triacontahedron. (a) is the space-ﬁlling

polyhedron for the cubic F lattice and also the coordination polyhedron for the cubic I lattice.

In thesepolyhedra allthefaces areequilateral triangles (Greek). Three(the tetrahedron,

octahedron and icosahedron) are also Platonic polyhedra. The remaining ﬁve have 6, 10,

12, 14 and 16 faces. Two, the triangular and pentagonal dipyramids with 6 and 10 faces,

are shown in Fig. A.2.5. Their duals are triangular and pentagonal prisms respectively.

Fig. A2.5. Two of the eight deltahedra: the triangular and pentagonal dipyramids.