Christopher Hammond

Print publication date: 2015

Print ISBN-13: 9780198738671

Published to Oxford Scholarship Online: August 2015

DOI: 10.1093/acprof:oso/9780198738671.001.0001

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(p.469) Appendix 7 Group theory in crystallography

Source:
The Basics of Crystallography and Diffraction
Publisher:
Oxford University Press

From our reading of Chapter 4 in this book, we understand the term point group of a crystal simply to denote the ‘collection and arrangement’ of symmetry elements that together describe its symmetry and distinguish it from all the other 31 point groups. Similarly, we understand the term space group of a crystal to denote the distribution and arrangements of symmetry elements in space that describe its three-dimensional pattern and distinguish it from all the other 229 space groups.

But the word group has a precise and much deeper meaning. We now need to make clear the distinction between the terms symmetry element and symmetry operation. A symmetry operation is a movement of body (a molecule or crystal) in which, after the operation has been carried out, is in an equivalent configuration: that is to say that it is indistinguishable from the original configuration, though not necessarily identical to it. For example, rotation of a pencil by $60∘$ about the pencil-lead direction or axis results in an equivalent configuration—and of course, repeating the process a total of six times returns us to the original configuration. The symmetry element (the hexad axis) is the geometrical line about which these symmetry operations are carried out. The same applies to all the other symmetry elements that we have learned about: their existence is demonstrated (as with the pencil) by showing that the appropriate symmetry operations exist.

Group theory in crystallography is concerned with the number and distribution of symmetry operations in a crystal and the relationships between them. By its means, the 32 point groups and the 230 space groups can be formally identified and enumerated. In the words of Vainshtein (1981) ‘Group theory … enables one to reveal the deeper meaning and regularities of the symmetry concept itself’. It also has important applications in the analysis of the symmetry properties of atoms and molecules, as well as crystals. However, largely because of the mathematical formalism and abstract terms in which group theory is inevitably expressed, the subject is generally by-passed in most books on crystallography—either because of the ‘difficulty’ of the subject or because it is only needed in such particular applications as those mentioned earlier. Another practical hindrance for the crystallographer is that the Schönflies’ notation for point groups and space groups is generally used rather than the Hermann–Mauguin notation (see Note and Table A7.4).

Here, we will describe group theory as it applies to crystallographic point group symmetry in the very simplest terms, with only a brief mention of space group symmetry. Readers with a sufficient mathematical background and who wish to pursue the subject further will find the books listed in Further Reading useful.

(p.470) A7.1 Group theory: axioms and multiplication tables

We begin by reconsidering the symmetry operations between the pair of right and left hands shown in Fig. 4.5. We see that a reflection (symbol $m)$ in Fig. 4.5(b) plus a rotation of 180° (a diad axis, symbol 2, perpendicular to the mirror plane) is equivalent to a centre of symmetry or inversion monad axis, symbol $1ˉ$ as shown in Fig. 4.5(a). Writing this algebraically we may put m.2 = $1ˉ$ where the dot means the product or combination of one symmetry operation (m) followed by another (2) giving rise to a centre of symmetry ($1ˉ)$. Conversely, we may write $1ˉ$.2 = m. Furthermore, the order in which the symmetry operations are carried out gives the same result, e.g. m.2 = 2.$m=1ˉ$, etc. (This is only the case for the so-called Abelian groups, named after the Norwegian mathematician, Niels Henrik Abel 1802–1829.) Finally, if we find the product or combination of a mirror symmetry operation with itself, we ‘get back where we started’; e.g. the operation of a reflection m (right to left hand) followed by another reflection m in the same plane returns us back to a right hand. This combination of symmetry operations is equivalent to the operation of a monad axis of symmetry, symbol 1 (see Section 2.3, p. 59), i.e. a rotation of $360∘$ (about any axis) that brings the crystal into coincidence with itself. Writing this algebraically: m.m = 1 and similarly 2.2 = 1 and $1ˉ.1ˉ$ = 1.

In group theory the symmetry operations m, 2, $1ˉ$, and 1 are called a set of elements of the group and they fulfil the first requirement or first axiom, of group theory, namely that the combination or product of any two elements (including an element with itself) is also an element of the set. As we see, these four symmetry operations do indeed accord with this axiom. They are also in accord with the second axiom of group theory, namely the combination of elements is associative. This means that if we combine three elements, the sequence of combination gives the same result, e.g. for m, 2 and $1ˉ$, (m.2).$1ˉ=1ˉ.1ˉ=1$, and m.(2.$1ˉ)=m.m=1$.

The third axiom introduces a new idea. It may be stated as follows: the set of elements contains a single element known as the identity, which has the property that, when it is combined with any other element, the result is that same element. Here, the identity is the monad axis, symbol 11. Clearly, in the previous example, 1.$m=m$ etc. and thus the third axiom is fulfilled. Finally, the fourth axiom may be stated: for each element in the set there exists another unique element called its inverse, such that the combination of the element and its inverse is the identity. In this example, each of the symmetry elements in the set is its own inverse: we may consider the combination m.m = 1 to mean, as indicated earlier ‘reflection one way followed by reflection back again’, or 2.2 = 1 to mean ‘a rotation of 180° clockwise followed by a rotation of 180° anti-clockwise’ (which is equivalent of course to two clockwise or anti-clockwise rotations).

Fig. A7.1. (a) The multiplication table for monoclinic point group 2/m and (b) the corresponding stereographic projection. Note that, as with space group symbols, the symbols ○ and are related by a reflection or a centre of symmetry.

The combinations, or products, of a set of elements in a group can be set out in the form of a matrix array, known as a multiplication table or ‘Cayley’s Square’ (after the English mathematician Arthur Cayley 1821–1895) as shown in Fig. A7.1(a).

(p.471) Notice that all the combinations of symmetry elements in the table contain the original four symmetry elements, i.e. the first axiom is fulfilled. This multiplication table (said to be of order 4 since there are four elements in the set), represents monoclinic point group symmetry 2/m that is also represented by a stereogram (see Section 12.5.1) in Fig. A7.1(b).

Fig. A7.2. Multiplication tables for (a) point group 2 and (b) point group m.

We can now write down the multiplication tables of monoclinic point group 2 (elements 1 and 2) and monoclinic point group m (elements 1 and m), both are of order 2.

These two groups are called subgroups of group 2/m; conversely group 2/m is the supergroup of these subgroups. A considerable part of the application of group theory to crystallography consists of first devising simple groups and then finding supergroups based upon them by adding the appropriate symmetry operations. In this way, complicated groups are constructed from simple ones. We can proceed in this way to write down the multiplication tables of all the 32 point groups—which are of higher order of course—the greater the number of the symmetry operations present. However, we shall only do this for four further examples, orthorhombic point groups 222 and mm2, trigonal point group 32, and tetragonal point group 4.

Fig. A7.3. (a) The multiplication table for orthorhombic point group 222 and (b) the corresponding stereographic projection.

For point group 222 the symmetry operations are 1, 2x, 2y, 2z; the subscripts referring to the crystallographic axis along which the corresponding diad axis lies.

(p.472) Notice that a rotation of 180° about x followed by a rotation of 180° about y equals a rotation of 180° about z, i.e. 2x.2y = 2z, and so on.

Fig. A7.4. (a) The multiplication table for orthorhombic point group mm2 and (b) the corresponding stereographic projection.

For point group mm2 the symmetry operations are 1, mx, my, and 2z.

Notice that a reflection in a plane perpendicular to x followed by a reflection in a plane perpendicular to y equals a rotation of 180° about z, i.e. mx.my = 2z. Indeed, we encountered this equivalence in our derivation of the ten plane point groups (Section 2.3) (p.473) where we found (Fig. 2.3(3)) that the operation of two perpendicular mirror lines ‘automatically’ generates a diad axis of symmetry along their intersection.

Notice also the similarity between the multiplication tables for point groups 222 and mm2, i.e. 1$≡$1, $mx≡$2x, $my≡$2y, $2z≡$2z. These two groups are said to be isomorphic (meaning ‘equal form’) since there is a unique one-to-one correspondence between the elements of the two groups. The groups in Fig. A7.2 are also isomorphic.

Fig. A7.5. (a) The multiplication table for trigonal point group 32 and (b) the corresponding stereographic projection.

For point group 32 (quartz) the symmetry operations are 1, 3, 32, 2x, 2y, 2u. These are indicated on the stereographic projection in Fig. A7.5(b). Here, we have to distinguish a symmetry operation and its inverse: 3 represents an anti-clockwise rotation of 120° and 32 a clockwise rotation of 120° (equivalent to an anti-clockwise rotation of 240), i.e. 3.32 = 1 (fourth axiom) and 3.3 = 32. (The symbol 32 should not be confused with the symbol 32 that denotes a screw triad axis.) Also, unlike our previous examples, the order of combination is important (non-Abelian). For example, 3.2x = 2y and 2x.3 = 2u; i.e. depending on the order of combination, the operation of an anti-clockwise rotation of 120° about the z-axis combined with a rotation of 180° about the x-axis gives a rotation of 180° either about the y-axis or the u-axis. The multiplication table is given in Fig. A7.5(a), the order of combination being ‘column times row’.

Fig. A7.6. (a) The multiplication table for tetragonal point group 4 and (b) the corresponding stereographic projection.

For point group 4, the symmetry operations (about the z-axis) are 1, 4 (90° anti-clockwise), 42 (180° anti-or-clockwise = 2), and 43 (270° anti-clockwise or 90° clockwise). These operations are shown in the multiplication table and stereographic projection in Fig. A7.6(a) and (b).

Notice that for each symmetry operation there is an inverse, i.e. 4$−1$ = 43, (4$2)−1$ = 42 and (4$3)−1$ = 4. We may check that the second axiom (the associative law) holds, e.g. 4.(42.43) = 4.4 = 42 and (4.42).43 = 43.43 = 42

(p.474) A note on nomenclature for symmetry operations

The operations of rotation about a symmetry axis, both rotation axes and screw axes, are sometimes called proper operations, or operations of the first sort because they do not result in a change of hand of the asymmetrical unit. Conversely, the operations of inversion, reflection, and glide-reflection, are called improper operations, or operations of the second sort since they do result in a change of hand. This distinction is important in the classification of point groups (see Table 3.1) and space groups.

A7.2 Matrix representation of symmetry operations: character tables

Matrices were used (see Section 5.8) to describe the transformations of plane indices and zone axis symbols from one axis system to another. Here, we use matrices to describe symmetry operations and their combinations: in effect we substitute the algebraic process of matrix multiplication for the geometrical process of successively applying symmetry operations. This in turn leads to a new (matrix) way of representing point groups. Note that in what follows we refer the matrices to axes x, y, and z.

The identity (monad symmetry operation), symbol 1 or E

Suppose we have a point with coordinates x, y, z. The identity transforms the point ‘into itself’, i.e. $x→x$, $y→y$, $z→z$. In matrix notation this is:

$Display mathematics$

(p.475) Reflection mx (i.e. in the y–z plane, perpendicular to the x-axis)

Here, a point (x, y, z) becomes ($xˉ$, y, z). (p.476) In matrix notation this is:

$Display mathematics$

Inversion, symbol $1ˉ$

Here, a point (x, y, z) becomes ($xˉ$, $yˉ$, $zˉ)$. In matrix notation this is:

$Display mathematics$

Rotation operations

Fig. A7.7. Coordinate transformation of a point following an anticlockwise rotation, angle [Greek] phi.

First, consider a general rotation of $ϕ∘$ anti-clockwise about the z-axis (Fig. A7.7). A point (x1, y1, $z1)$ becomes a point (x2, y2, z2) (or vector $r1$ becomes vector $r2)$. These are related by the equations:

(I leave it to the reader to work out these equations. Hint: consider the x1 and y1 components of vector $r1$ separately, rotate both anti-clockwise and find x2 and y2 from the sums.)

In matrix notation this is:

$Display mathematics$

and:

$Display mathematics$

Notice (a) that these matrices are the transposes of each other (i.e. rows and columns interchanged) and (b) that the inverse of each matrix (see Section 5.10) is simply equal to the transpose. They are said to be conjugate.

For $ϕ$ = 180° the anti-clockwise and clockwise matrices are identical.

$Display mathematics$

For $ϕ$ = 90° the matrices are:

$Display mathematics$

$Display mathematics$

Combinations of symmetry operations; here we give two examples:

$Display mathematics$

Matrix representation of point groups

Consider for example point group $2/m$ (Fig. A7.1). We now represent the symmetry operations 1, 2z, mz, and $1ˉ$ by the matrices:

(p.477)

$Display mathematics$

and for point group mm2 (Fig. A7.4), we now represent the symmetry operations 1, mx, my, and 2z by the matrices:

$Display mathematics$

Each matrix corresponds to a single operation in the group, which may be summarized by considering their operations on a point $xyz$. For example, for point group mm2 above:

$Display mathematics$

mm2

1 (or $E)$

mx

my

2z

x

1

$−1$

1

$−1$

y

1

1

$−1$

$−1$

z

1

1

1

1

xy

1

$−1$

$−1$

1

Table A7.1 Character table for point group mm22.

mm2

1($E)$

mx

my

2z

A1

1

1

1

1

z

$x2$, $y2$, $z2$

A2

1

$−1$

$−1$

1

Rz

xy

B1

1

$−1$

1

$−1$

x, Ry

xz

B2

1

1

$−1$

$−1$

y, Rx

yz

The results may be set out in the form of a table.

Also included in the table is the effect of the symmetry operations on the product function xy (simply multiply x and y in each of the columns here). As can be seen, xy transforms differently to x, y, and z. If we do the same for the product functions xz, yz, $x2$, $y2$, and $z2$, we find that we simply repeat the combinations already in the table. For example, yz transforms like y and xz transforms like x. Hence, in summary, (p.478) there are four ways in which the functions transform in point group mm2. They are tabulated in the form of a character table (not to be confused with a multiplication table—Section A7.1).

The four rows A1, A2, B1, and B2 are called representations of the point group mm2 and Rz, Ry, and Rx are axial vectors. However, at this point our introduction to character tables must stop, leaving a reader to pursue these topics in the books listed in Further Reading.

A7.3 Symmorphic and enantiomorphous space groups

The 230 space groups are derived by combining the 32 point group symmetries with the appropriate Bravais lattices and then taking into account the additional space groups arising from the existence of the translational symmetry elements—glide planes and screw axes. It is, in essence, the same procedure by which we derived the 17 plane groups (Sections 2.4 and 2.5) but is obviously more complicated. In group theory, the procedure is to combine the multiplication table of each point group with the appropriate lattice translations. Consider the general symbol for a direction (Section 5.7):

$Display mathematics$

Table A7.2 The 73 symmorphic space groups (based on Burns and Glazer, 2013).

Space groups—based on point groups:

System

Bravais lattice

Non-centrosymmetric enantiomorphous (total 24)

Non-centrosymmetric non-enantiomorphous (total 25)

Centrosymmetric non-enantiomorphous (total 24)

Cubic

P

P23, P432

$P4ˉ$3m

Pm $3ˉ$, Pm$3ˉ$m

I

I23, I432

$I4ˉ$3m

Im $3ˉ$, Im $3ˉ$m

F

F23, F432

$F4ˉ$3m

Fm $3ˉ$, Fm $3ˉ$m

Tetragonal

P

P4, P422

$P4ˉ$, P4mm, $P4ˉ$2m, $P4ˉm$2

P4/m, P4/mmm

I

I4, I422

$I4ˉ$, I4mm, $I4ˉ$2m, $I4ˉm$2

I4/m, I4/mmm

Orthorhombic

P

P222

Pmm2

Pmmm

I

I222

Imm2

Immm

$C(A)$

C222

Cmm2, Amm2

Cmmm

F

F222

Fmm2

Fmmm

Trigonal

R

R3, R32

R3m

$R3ˉ$, $R3ˉm$

P

P3, P312, P321

P3m1, P31m

$P3ˉ$, $P3ˉ$1m, $P3ˉ$ m1

Hexagonal

P

P6, P622

$P6ˉ$, P6mm, $P6ˉm$2, $P6ˉ$2m

P6/m, P6/mmm

Monoclinic

P

P2

Pm

P2/m

C

C2

Cm

C2/m

Triclinic

P

P1

$P1ˉ$

For an ‘infinite’ crystal the integers u, v, w take any value from -$∞$ to +$∞$ and specify the coordinates of the origins of every unit cell. Each set of such integers u, v, w constitute a group, i.e. they are in accord with the four axioms stated here except (1) the rule of combination is addition and (2) the identity is zero. From this, multiplication tables may be constructed but the mathematical procedures involved (which begin so simply!) are too sophisticated to be covered in detail in this book. Instead we derive the 73 symmorphic space groups by combining the appropriate point groups and Bravais lattices. For example, the five cubic point groups (23, 432, $4ˉ$3m, $m3ˉ$, and $m3ˉm)$ combined with the three (P, I, and $F)$ Bravais lattices give rise to 15 symmorphic space groups. In other systems there is more than one way in which the point group and Bravais lattice can be combined. For example, the trigonal point group 3m combined with the hexagonal P lattice gives rise to two space groups, P3m1 and P31m (recall the two possible plane groups, p31m and p3m1 in Table 2.6). In the orthorhombic system point group mm2 may be combined with the C-centred lattice with the diad axis either along the C-axis (space group Cmm2) or along the a- (equivalently b-) axis (space group Amm2, equivalently Bmm2). The 73 symmorphic space groups are listed in Table A7.2. As can be seen, they are almost equally divided between the three groups of point groups (see Table 3.1).

The 24 symmorphic space groups based on the 11 centrosymmetric point groups correspond to the 24 Patterson space groups (Section 13.3.2) and include the eleven Laue groups (Table 9.1) with the appropriate/permissible cell centrings.

The space groups may also be divided into those based on:

1. 1. The eleven non-centrosymmetric enantiomorphous point groups. These space groups, total 65, which were first worked out by Sohncke in 1867 (Section 4.6), (p.479) are sometimes called the axial space groups and are important in that proteins and nucleic acids (which do not possess mirror planes or inversion axes of symmetry) crystallize in these space groups. They include the eleven enantiomorphous pairs (Table A7.3).

2. 2. The ten non-centrosymmetric non-enantiomorphous point groups, total 73 space groups; the same number (coincidence?) as the symmorphic space groups.

3. 3. The eleven centrosymmetric (non-enantiomorphous) point groups, total 92 space groups.

Table A7.3 The eleven pairs of enantiomorphous space groups.

 P31 and P32 P41 and P43 P61 and P65 P62 and P64 P3112 and P3212 P3121 and P3221 P4122 and P4322 P41212 and P43212 P6122 and P6522 P6222 and P6422 P4132 and P4332

(p.480) A note on Schönflies’ notation for symmetry elements, point groups, and space groups

Schönflies treated the crystallographic symmetries as mathematical ‘groups’ of operations. He called a group containing one rotation operation and its powers a cyclic group, symbol C. For example, the operations of a four-fold axis, symbol C4 are the rotations $1,π2,π,3π2$ which are all ‘powers’ of the fundamental rotation $π2$. A mirror (reflection) operation, symbol σ‎, normal to the rotation axis he symbolized with a superscript h. A mirror operation parallel to the rotation axis he symbolized with the superscript v.

As noted in Section 4.3 (see p. 104), Schönflies used rotation-reflection or alternating axes (rather than rotation-inversion (or simply inversion/axes)) symbolized by S (from Sphenoidisch). Next, we have the groups with normal diad axe(s) symbolized by D (from Diëdergruppe). Those ‘D’ groups that also include a plane of symmetry normal to the rotation axis are symbolized, as before, by the superscript h and those with a diagonal plane of symmetry are symbolized by the superscript d. In the cubic system the symbols T and O are used to represent tetrahedral and octahedral symmetry, respectively.

The Schönflies symbol may also include the letter i denoting a centre of symmetry, so Ci is a synonym for S2 (=$1ˉ)$ and $C3i$ is a synonym for S6 (= $3ˉ)$. We may also note that the $5ˉ$ inversion axis that describes icosahedral (non-crystallographic) symmetry (Section 4.9) corresponds to Schönflies’ $S10$ rotation-reflection axis.

Table A7.4 Hermann–Mauguin and Schönflies point group symbols.

Non-centrosymmetric point groups

Centrosymmetric point groups (non-enantiomorphous)

System

Enantiomorphous

Non-enantiomorphous

Cubic

T(23), O (432)

$Td(4ˉ$3$m)$

$Th(m$3), $Oh(m3ˉm)$

Tetragonal

C4(4), D4(422)

$S4(4ˉ)$, $C4v$(4mm), $D2d(4ˉ$2$m)$

$C4h4m$, $D4h4mmm$

Orthorhombic

D2(222)

$C2v$(mm2)

$D2h$(mmm)

Trigonal

C3(3), D3(32)

$C3v$ (3m)

$S6(3ˉ)$, $D3d(3ˉ$m)

Hexagonal

C6(6), D6(622)

$C3h(6ˉ)$, $C6v$(6mm), $D3h(6ˉm$ 2)

$C6h(6m)$, $D6h(6mmm)$

Monoclinic

C2(2)

Cs, $C1h(m)$

$C2h(2m)$

Triclinic

C1 (1)

$Ci(1ˉ)$

The corresponding Schönflies and Hermann–Mauguin point group symbols are set out in Table A7.4 (arranged in the same form as Table 3.1).

Clearly, the Hermann–Mauguin notation is to be far preferred since it contains the essential point group symmetry information. This is even more the case for Schönflies’ space group notation in which the space groups, isomorphous with each point group, are simply numbered sequentially (presumably in the order in which he derived them). For example, the five space groups isomorphous with point group 23 (Nos. 195–199 in the International Tables) are denoted T1(P23), T2(F23), T3(I23), T4(P213), and T5(I213).

Notes:

(1) The identity is also given the symbol E (German: Einheit)

(2) This is the character table for the water molecule. Its use in describing the atomic vibrations of the water molecule may be found (e.g.) in Hargittai & Hargittai (1995) and Ladd (2014).