Ted Janssen, Gervais Chapuis, and Marc de Boissieu

Print publication date: 2007

Print ISBN-13: 9780198567776

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198567776.001.0001

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(p.414) APPENDIX A

Source:
Aperiodic Crystals
Publisher:
Oxford University Press

HIGHER-DIMENSIONAL SPACE GROUPS

A.1 Crystallographic operations in n dimensions

Physical systems are zero-, one-, two- or three-dimensional (a point, a line, a surface, bulk). Furthermore, as we have seen, aperiodic crystals of finite rank may be embedded as periodic structures in still higher dimensions. This makes it convenient to consider the crystallography from a more general point of view. All positive dimensions have crystallographic properties in common. Of course, in lower dimensions more details and complete classifications are known than in higher dimensions, but the general concepts are to a large extent the same. Therefore, we want to give a brief overview of crystallography in spaces of arbitrary dimension.

Crystallographic operations are elements of the Euclidean group in n dimensions, the group of all distance-preserving motions in Euclidean space. All Euclidean transformations are pairs of an orthogonal transformation and a translation. What types of orthogonal transformations may one encounter?

A real orthogonal transformation in n dimensions is a linear transformation leaving all distances invariant. It may be decomposed as the direct sum of two-dimensional rotations, and one-dimensional identity (+1) or inversion (−1) operations. Its determinant is ±1. If the determinant of the transformation is +1, the number of inverses is even (2p), and these can be viewed as p two-fold rotations in a plane. If the determinant is −1, there is one unpaired inversion. As example, consider the four-fold rotation in three dimensions, and its related roto-inversion $4 ¯$. They can be written in block form:

$Display mathematics$
The first is the sum of a four-fold rotation and a one-dimensional identity, the second a four-fold rotation and a one-dimensional inversion. Each N-fold rotation in the plane is a rotation over 2πmN, where 0 < m < N and N and m are co-prime. Such a planar rotation then may be denoted by N m, and an arbitrary rotation in n dimensions can be denoted by a string $N 1 m 1 N 2 m 2 .. ,$ whereas an orthogonal transformation which is not a rotation (if its determinant is −1) has a similar string with a $1 ¯$ at the end. The two examples: the four-fold rotation in three dimensions becomes the string 41, and the roto-inversion $4 ¯$ becomes $4 3 1 ¯ .$. The inverse of the latter is $4 1 ¯$. The eigenvalues of the two-dimensional rotation N m are exp(±2πimN). Because the rotations are real, an eigenvalue (p.415) exp(2πimN) comes along with an eigenvalue exp(−2πim)⧏N). In the notation the values of m can therefore be limited to the range 0 < mN⧏2.

Not all these rotations leave a lattice invariant. A theorem from number theory leads to the statement that such an invariant lattice exists if and only if the rotation N is accompanied by its conjugates (the rotations N m with m co-prime to N), and the set of all two-dimensional rotations may be divided into complete sets of conjugates. The number of conjugates for the eigenvalue exp(2πiN) is the Euler function of N: Φ(N) is the number of positive integers smaller than N which are co-prime with N. The eigenvalues exp(2πimN) and exp(2πi(Nm)⧏N) then combine to the rotation N m. Examples of the value of the Euler function are Φ(4) = 2 (1,3), Φ(5) = 4 (1,2,3,4), Φ(6) =2 (1,5), Φ(7) = 6, etc. Therefore a four-fold rotation is crystallographic (leaves a lattice invariant) in two dimensions, a five-fold rotation in four dimensions, a three- or six-fold rotation in two dimensions, and a seven-fold rotation in six dimensions. That is the reason why one has to go to four dimensions to get a crystallographic embedding of the Penrose tiling.

Because in crystallography the crystallographic rotations are relevant, the set of conjugated rotations is indicated by [N]. If N = 8 the symbol [8] stands for the direct sum of the rotations 8 and 83. It is a four-dimensional rotation. Crystallographic transformations therefore are indicated by strings of symbols [N], N, $[ N ] , N , 1 ¯$, and 1, where the numbers in square brackets correspond to four- or higher-dimensional rotations, without brackets to three-, four-, or six-fold rotations in two dimensions (N = 3, 4, or 6), 2s to 2-fold rotations and $1 ¯$ appears when the transformation has determinant −1. As an example $[ 8 ] 62 1 ¯$ is a transformation with determinant −1 in nine dimensions: an eight-fold rotation in four dimensions, a six-fold rotation in two dimensions, a two-fold rotation in two dimensions and an inversion in one dimension.

For aperiodic crystals there is an embedding in an n-dimensional space with an invariant d-dimensional physical space (d = 1,2,3). The invariant physical space consists of one or more of the invariant subspaces of the orthogonal transformation. The action of the transformation in the physical space may be given separately, and the action in the internal space then is put in parentheses. A symmetry operation for a three-dimensional octagonal aperiodic structure may act as a rotation 81 in physical space, and, because 8 needs its conjugate 83, as 83 in internal space. The operator [8]1 in five dimensions is then denoted as 81(831) for the aperiodic crystal. Very often, one denotes this transformation by 81, because for a crystallographic transformation the component 83 follows necessarily. Strictly speaking, 81 is a non-crystallographic eight-fold rotation in three dimensions, but the shorter symbol is certainly less heavy. In the context of the crystallography of quasiperiodic crystals, there will be no confusion, and then the shorter notation is more convenient.

(p.416) A.2 Lattices

A lattice translation group in n dimensions is a group of translations spanned by n linearly independent basis vectors. That means that the lattice translations span the whole space. If the translations do not span the whole space, or if the basis vectors are not linearly independent, it is better to use the expression Z-module or Bravais module. In both cases any translation may be written as a sum of basis translations:

$Display mathematics$

If n is larger than the dimension of the space, or if the basis vectors a j do not span the whole space, the vectors form a Z module, not a lattice. As a Euclidean transformation a translation is written as {E|a}. The lattice is characterized by its metric tensor, a symmetric tensor with elements

$Display mathematics$
This implies only the mutual relations between the basis vectors, but not their absolute orientation. Rotating a lattice does not change its metric tensor.

The dual of a lattice is the reciprocal lattice with basis vectors $a j ∗$ defined by

$Display mathematics$
Its metric tensor is g * with elements
$Display mathematics$

The holohedry of a lattice is the group of all orthogonal transformations leaving the lattice invariant. With respect to a lattice basis the (arithmetic) holohedry is a group of integer matrices. It is the group of all integer matrices S that satisfy

(A.1)
$Display mathematics$
as one may verify easily.

Two lattices are equivalent if their arithmetic holohedries are arithmetically equivalent. This means that there are bases for the two lattices such that the holohedries are the same groups of matrices. All mutually equivalent lattices form the equivalence class, called a Bravais class. In three dimensions there are fourteen Bravais classes.

A crystallographic point group leaves a lattice invariant, and actually a whole set of lattices. The maximal point group that leaves all the lattices invariant that are invariant under the point group K is called the system group of K.

(p.417) A crystallographic point group K that leaves a lattice with metric tensor g invariant corresponds to a group of integer matrices D(K) after the choice of a lattice basis. Then

(A.2)
$Display mathematics$
All tensors g satisfying Eq. A.2 form a set G. A group that includes the arithmetic group D(K) is the group of matrices D(S) satisfying
$Display mathematics$
Clearly the elements of D(K) belong to this group, but it may be larger. It is called the Bravais group B(D(K)) of D(K).

For example, in three dimensions the point group 3 leaves a family of lattices invariant that belong either to the rhombohedral Bravais class or to the hexagonal Bravais class. The holohedries are $3 ¯ m$ and 6/mmm, respectively. The first is a subgroup of the latter. So the system group of 3 is the point group $3 ¯ m$. The arithmetic holohedries of the two Bravais classes are $3 ¯ mR .$ and 6/mmmP, respectively. With respect to a lattice basis the point group 3 is either of the type 3R or of the type 3P. The Bravais group for 3R is the arithmetic holohedry $3 ¯ mR .$, that for 3P is 6/mmmP.

A.3 Crystal classes

Point groups are subgroups of the orthogonal group O(n) in n dimensions. On an orthogonal basis they are represented by orthogonal matrices, for which the product with their transpose is the identity matrix. If the point group leaves a lattice invariant, it is called a crystallographic point group. On the basis of an invariant lattice the point group is represented by integer matrices. This implies that the trace of a crystallographic point group transformation is an integer. This is called the crystallographic restriction. For example, in three dimensions the trace of an orthogonal transformation with determinant +1 is 1 + 2cos(Φ), and this is only an integer if Φ/2π is 1⧏2, 1⧏3, 1⧏4, 1⧏6, or 0. This is the reason why there are ‘forbidden’ symmetries in two dimensions and three dimensions. In higher dimensions the crystallographic restriction means that an N-fold rotation in n dimensions is non-crystallographic if n is smaller than the Euler function Φ(N).

A point group may leave a subspace invariant. Because it is a group of orthogonal transformations, the complement of the invariant subspace is also an invariant subspace. For example, the tetragonal group 4/mmm leaves both the unique axis and the plane perpendicular to it invariant. This means that there is a basis with respect to which the matrices of the point group are the direct sum of matrices corresponding to the action in the subspace and its complement. In general, these matrices are real and non-integer. That means there is, in general, not a basis for an invariant lattice with vectors only in the two invariant subspaces. If there is an invariant subspace, the point group is said to (p.418) be R-reducible, otherwise R-irreducible. Here R stands for the real numbers. If there is an invariant lattice having a basis with vectors in the invariant space and its complement, the point group is Z-reducible, otherwise Z-irreducible (Z is the group of integers). If there is a sublattice with vectors in the two subspaces, the point group is Q-reducible (Q for the rational numbers). This means that there is a sublattice for which the invariant lattice is a centring.

If the point group is R-reducible with an n 1-dimensional and an n 2-dimensional invariant subspace, the n-dimensional matrices are the direct sum of an n 1-dimensional and an n 2-dimensional matrix. The former form a point group K 1 in n 1 dimensions, and the latter a point group K 2 in n 2 dimensions. If K 1 and K 1 are both subgroups of the point group K, then one says that K is the external product of the two subgroups. This is denoted as K = K 1K 2. For example, the tetragonal group 4/mmm has two complementary invariant subspaces. In the one-dimensional subspace along the unique axis it acts as K 2 = m, in the two-dimensional perpendicular subspace as K 1 = 4mm. Then 4/mmm = 4mm⊥m. If K 1 and K 2 are not subgroups, the group K is a subdirect product of the two groups. Then the group K consists of pairs (R 1, R 2), the R 1s forming K 1, and the R 2s forming K 2. An example is the group 422. The unique axis and the plane perpendicular to it are, also here, invariant subspaces. The elements of 422 act as 4mm in the two-dimensional plane, and as the group m in the complementary one-dimensional subspace. The generators now are pairs 4z = (4,1) and 2y = (mx, mz). 422 is the subdirect product of 4mm and m.

Two point groups are considered to be the same, if there is an orthogonal transformation transforming one into the other:K = SKS −1 for some orthogonal transformation S. Equivalently one may say that there are two bases such that K with respect to one gives the same matrices as K with respect to the other. All point groups equivalent to each other according to this definition form an equivalence class, called a geometric crystal class. All point groups belonging to one geometric crystal class can be seen as the same group of transformations, but with a different orientation. In three dimensions there are 32 geometric crystal classes of crystallographic point groups.

With respect to a basis of an an invariant lattice, a crystallographic point group corresponds to a group of integer matrices M(K). With respect to another basis, obtained from the former by a basis transformation S, the matrices form the group SM(K)S −1. Two such groups of integer matrices are called arithmetically equivalent. The equivalence classes, the sets of all groups of integer matrices arithmetically equivalent to each other, form an arithmetic crystal class. Two arithmetically equivalent point groups are also geometrically equivalent, but the inverse is not true. In three dimensions there are 73 arithmetic crystal classes. Each of the 32 geometric crystal classes is composed of one or more complete arithmetic crystal classes.

If two groups of integer matrices D 1(K) and D 2(K) are geometrically equivalent, but not arithmetically equivalent, then there is a matrix S with rational coefficients such that (p.419)

$Display mathematics$
The matrix S (or its inverse) describes the transformation of a lattice basis on which the point group K corresponds to the matrix group D 1(K) to the basis of a sublattice. Then in the unit cell of this sublattice there are lattice points of the other lattice. One lattice is a so-called centring of the other.

For example, in two dimensions the point group mm leaves two types of lattices invariant, a primitive rectangular lattice with as basis vectors (a,0) and (0, b), and a centred lattice with basis vectors (a⧏2, b⧏2) and (−a⧏2, b⧏2). The point groups D 1(mm) and D 2(mm) are related by a rational matrix S:

$Display mathematics$

In the notation of the International Tables for Crystallography geometrically equivalent, but arithmetically non-equivalent point groups are distinguished by a symbol, which indicates a centring of the lattice. For example, the geometric crystal class 432 contains the three arithmetic crystal classes 432P, 432I, and 432F, corresponding to the three types of centring of the cubic lattice. The symbol for an arithmetic crystal class is the symbol for the geometric crystal class with a postfix indicating the centring.

A.4 Space groups

Space groups are groups of Euclidean transformations that contain a translation subgroup, the intersection with the translation group, which is generated by n independent lattice translations. The elements are denoted, in the Seitz notation, by {R | t}, where R is an orthogonal transformation in n dimensions. The product of two elements is given by

$Display mathematics$
The translations are all elements {E | a}, with the identity transformation E. They form an invariant subgroup, because
$Display mathematics$
Then each element of the space group can be decomposed into the product of a lattice translation and a representative of the coset to which the element belongs:
$Display mathematics$
where the translation t R is determined up to a lattice vector. To each element R of the point group one may choose a translation t R. The elements t R form the so called system of non-primitive translations or the vector system. The translations satisfy
$Display mathematics$
These are called the Frobenius congruences.

(p.420) Euclidean transformations are the product of an orthogonal transformation with a centre O and a translation. Choosing another origin for the orthogonal transformations, the associated translations t change to (1−R)t. Therefore, the vector system t R is determined up to a lattice translation and an origin shift: t R and t R + a + (1−R)v are equivalent for every lattice vector a and origin shift v.

The point groups may be generated by a (finite) number of generators Rj, which satisfy a number of ‘defining relations’. These relations fix the isomorphism class of the group. For example, the point group mm2 has two generators, R 1 = m x and R 2 = m y, which satisfy $R 1 2 = R 2 2 = R 1 R 2 R 1 R 2 = E$. The defining relations are of the form of words in or strings with the generators, which are equal to the identity: Wj(R 1,…, Rm) = E. For a space group, with a vector system t R, the defining relations of the point group imply that

$Display mathematics$
for lattice translations b j. Any solution of these equations gives a vector system that determines a space group. For example, for the point group generated by m x and m y the vector system $t 1 = ( 1 2 , 1 2 ) , t 2 = ( 1 2 , 1 2 )$ satisfies
$Display mathematics$
Therefore, t R is a solution to the equations. The space group here is pgg. For known point group matrices, this is a way to determine all space groups. However, in this way one finds an infinite number of space groups. We shall have to say when space groups can be considered to be the same. This will be discussed in the next section.

A space group then is specified by the point group K, and the vector system t R (RK). For a basis of the lattice the group K corresponds to a group of matrices D(K) with integer coefficients. On the same basis the vectors t R can be specified. A non-primitive translation t R may be decomposed in the sum of a vector that may be transformed to 0 by an origin shift, and an intrinsic part that cannot be transformed away. The symbol for a symmorphic space group then consists of the symbol for the point group K, and a prefix indicating the centring (together with symbol for K this fixes the arithmetic crystal class). For a non-symmorphic space group a variation on the symbol for the symmorphic space group is used. The variation consists in a change of the symbol for a generator of K to a symbol that indicates the intrinsic part of the associated non-primitive translation. The non-primitive translations in two dimensions are translations along a mirror line, a glide. In three dimensions they are translations in a mirror plane or a translation along the axis of a rotation. The combinations are called glide and screw, respectively.

(p.421) For example, the arithmetic crystal class 2/mP in three dimensions has a symmorphic space group with symbol P2/m. If the two-fold rotation has a non-primitive translation along the rotation axis, the combined operation is a screw operation. If this is the only non-primitive translation the non-symmorphic space group has symbol P21⧏m, indicating that the intrinsic part of the non-primitive translation is half the lattice constant in the direction of the axis. If there is a glide plane, instead of a mirror, with intrinsic part of the non-primitive translation along the a-axis, the symbol of the non-symmorphic space group is P2/a.

Another way to fix a space group type or isomorphism class is by using a set of generators and defining relations. Suppose that the point group K has m generators (R 1,…, Rm) and k defining relations Wi(R 1,…, Rm) = E (i = 1,…, k). Furthermore, the translation group has a basis with n vectors a j (j = 1,…, n). Then the defining relations of the space group are (1) relations between the generating translations

$Display mathematics$
(2) relations between the elements {Rj|t j}
$Display mathematics$
and (3) relations between these elements and the translations
$Display mathematics$
with i = 1,…, m and p = 1,…, n. The translation group is Abelian, which is seen from the first relation. And, the action of the point group on a lattice basis is given by the matrices D(Ri). Notice that in these relations only the integer matrices D(Ri) and the lattice translations b i occur. These data can all be given by integers, in contrast to the other approach where, generally, non-primitive translations appear. The reason is that the structure of a space group bears some similarity to that of a direct product of two groups. Here we have a combination of the point group K and the translation group A. The group A is an invariant subgroup of the space group G, and the factor group G⧏A is isomorphic to K. In mathematical terms G is an extension of K by A. And the direct product A × K is the simplest example of such an extension. However, space groups are not the direct product of the translation group and the point group. Symmorphic space groups are semi-direct products of A and K, but the structure of a non-symmorphic space group is a more general extension.

A.5 Classification

The answer to the question of which space groups should be considered as being the same depends on what one wants to do with it. For use in physical applications one may require that two groups are equivalent if they are connected (p.422) by a change in coordinate system. In mathematical terms this is called affine conjugated. Bieberbach has shown, in 1911, that this requirement is equivalent to the requirement of isomorphism between the two. In two dimensions one finds in this way 17, and in three dimensions 219 different space groups. Equivalent groups belong to one equivalence class. So there are 219 equivalence classes in three dimensions. An equivalence class here is also called a space group type. For some physical applications it is useful to have a stronger equivalence relation. Then the coordinate transformation between two equivalent space groups should be orientation preserving. This is stronger, because there is an additional requirement. With this definition of equivalence there are 230 space group types in three dimensions. In higher dimensions physical properties do not play an immediately obvious role. Therefore, one might use isomorphism as equivalence relation. However, for incommensurate modulated phases, for example, it makes sense to use a stronger equivalence relation. It is obvious, that the number of equivalence classes depends on the equivalence relation. A stronger relation leads to a larger number of classes. For incommensurate phases this will be discussed in Section A.6. The role of enantiomorphism in superspace is unclear, as long as no experiments are performed where mirror operations in internal space play a role.

A space group has a lattice of translations and a point group. The operations of the point group with respect to a basis of the translations give a group of n×n integer matrices, the arithmetic point group. All space groups of one space group type have arithmetic point groups in the same arithmetic crystal class. So one can get all space group types, by taking one representative of each arithmetic crystal class, determining all solutions to the equations above, and eliminating the equivalent ones.

The translation subgroup of a space group, which is the intersection of the space group with the translation subgroup, is a group isomorphic with the group Z n, the abelian group of n-tuples of integers, vectors (n 1 ,…, nn). Each n-tuple corresponds to a lattice point. Moreover, the basis vectors span the whole n-dimensional space.

If one wants to be more precise, one should distinguish the translation group with translations {E|a} and the lattice points one obtains by the action of the translation group on an origin. The latter is a set of vectors a in the n-dimensional space. We shall not make this distinction here. Then one may say that for a space group the translations span the whole n-dimensional space if the coefficients are real. With integer coefficients one obtains the discrete point lattice.

The point group of all orthogonal transformations leaving the lattice invariant is the holohedry of the lattice. It is a finite subgroup of O(n). On the basis of the invariant lattice the holohedry gives a group of integer matrices, the arithmetic holohedry. For another basis the holohedry gives an arithmetic holohedry in (p.423) the same arithmetic crystal class. All lattices with arithmetic holohedries in the same arithmetic crystal class belong to the same Bravais class. Bravais classes are equivalence classes for lattices.

Fig. A.1. Relation between the various crystallographic notions.

Because a space group determines uniquely an arithmetic crystal class (that of its point group on the basis of the lattice) and a geometric crystal class, one may assign a space group to an arithmetic crystal class and a geometric crystal class. Each geometric crystal class contains complete arithmetic crystal classes, and the latter complete space group types.

If the metric tensor of the lattice is g, defined by

$Display mathematics$
the elements of the arithmetic point group D(K) with respect to the same basis satisfy
$Display mathematics$
All non-singular integer matrices S satisfying S T gS = g then form the Bravais group of D(K). This is an arithmetic holohedry. Therefore, a space group may also be assigned to a Bravais class. On the other hand, the subgroup of O(n) that leaves all lattices invariant that are left invariant by K forms the system group of K. All lattices with a holohedry in the same geometric class as this system group form a system. So each space group may be assigned to an arithmetic point group, which may be assigned to a Bravais class. Similarly, each space group may be assigned to a geometrical crystal class and a point group system. Finally these two lines may be combined in still larger classes, the families. A family is (p.424) the smallest union of arithmetic crystal and geometric crystal classes such that with each crystallographic point group all point groups belonging to the same system or the same Bravais class are contained in the family. Schematically this can be presented as in Fig. A.1

As an example, consider the hexagonal family in three dimensions. It contains two systems, the hexagonal system with holohedry P6/mmm, and the rhombohedral system with holohedry $3 ¯ m$. The first contains geometric crystal classes 6, $6 mm, 6 ¯ , 622 , 6 ¯ 2 m$, and 6/mmm. The second contains $3 , 3 m, 32 , 3 ¯$, and $3 ¯ m$. The geometric crystal classes of the first series contain only a single arithmetic crystal class, those of the second series contain each two arithmetic crystal classes, one in the Bravais class 6/mmmP and one in the rhombohedral Bravais class $3 ¯ mR .$. Then each arithmetic crystal class contains one or more space group types. Strictly speaking, space group types are equivalence classes of space groups, (arithmetic and geometric) crystal classes are equivalence classes of point groups, and Bravais classes and systems are classes of lattices. But, because each space group uniquely determines the other classes, one may consider them also as equivalence classes of space groups.

The number of space group types and that of crystal classes increase rapidly with increasing dimension. For the lower dimensions these numbers are given in the following table (Janssen et al., 1999).

Dimension

1

2

3

4

5

6

Families

1

4

6

23

32

91

Bravais classes

1

5

14

64

189

841

Geometric classes

2

10

32

227

955

7,104

Arithmetic classes

2

13

73

710

6,079

85,311

Space group types

2

17

219

4,783

222,018

28,927,922

Space groups may be calculated starting from representatives of the arithmetic crystal classes. They are what is known as group extensions of a point group K with a translation group A. The first is a finite subgroup of O(n) that leaves the n-dimensional lattice invariant. The latter is isomorphic to the group of n-tuples of integers (Z n) corresponding to the lattice nodes. The representatives of the cosets of A in G are r(R), the elements of A are a. Then an arbitrary element may be written either as a pair (R, a) or as a product a r(R). One introduces an action of K on A by Ra = r(R) a r(R)−1. Finally the product of r(R) and r(S) is, generally, not r(RS) but it is in the same coset. So, there is a translation vector ω(R, S) such that r(R)r(S) = ω(R, S)r(RS). Then the product of two elements of G is

$Display mathematics$
The elements ω(R, S) are lattice translations, elements of Z n. So the product may be written as
(A.3)
$Display mathematics$
(p.425) The pairs (R, a) consist of an orthogonal transformation R and a lattice translation a. In the Seitz notation the elements are {R|t}, pairs of R and a not necessarily primitive translation. The relation is t = t R+a, with r(R) = {R|t R}. The mapping ω from pairs of elements of K to a lattice translation is a 2-cocycle. These satisfy certain relations, and every solution of these equations gives an extension. The extensions correspond to what is called the second cohomology group $H φ 2 ( A , ℤ n )$, where Φ is the mapping from K to the automorphisms of Z n described by an integer matrix group.

The vector system t R in the space group elements {R|t R} forms a 1-cocycle, for which we have the relations

$Display mathematics$
The various extensions now correspond to the first cohomology group $H φ 1 ( A , ℤ n )$. But for finite point groups these two groups are isomorphic. Therefore, for space groups in Euclidean space, where the point groups are finite, both approaches are equivalent. Derivation of the list of non-equivalent space groups in n dimensions is based on the calculation of the first or second cohomology group (Burckhardt, 1947; Zassenhaus, 1947; Fast and Janssen, 1971).

A.6 Space groups for aperiodic crystals

Space groups for aperiodic crystals are space groups in n dimensions, but not every n-dimensional space group occurs as such. In superspace there is a physical subspace (of d = 1,2, or 3 dimensions) which is invariant under the point group. Therefore, the point groups of superspace groups are R-reducible into a d- and an (nd)-dimensional component. This eliminates a large number of n-dimensional space groups as candidates for aperiodic crystal symmetries. But the remaining space groups do not all allow an invariant space without periodicity. For example, in n = 2 the point group mm2 has two invariant subspaces, but both contain translations from an invariant lattice. However, when one wants to treat commensurate phases on the same footing as the incommensurate, there is a lattice in physical space, and it makes sense to call such an n-dimensional space group a superspace group if its point group leaves a d-dimensional subspace invariant. The condition that the point group of a superspace group leaves a subspace invariant restricts the number of space groups which are superspace groups.

The equivalence relation has to be reconsidered. Because of the special role of the physical space two superspace groups are called equivalent if they are conjugated in the n-dimensional affine group by an affine transformation {S|a} such that the physical space is left invariant by S. The additional constraint may be the reason why two groups, that are equivalent as space groups, are non-equivalent as superspace groups. Therefore, on one hand the number of space groups that are superspace groups is limited by the condition that the point group leaves a d-dimensional subspace invariant, and on the other hand, the (p.426) stronger equivalence condition (the affine transformation leaves the subspace invariant) increases this number. Therefore, the number of superspace groups is, generally, different than the number of space groups in the same dimension. In addition, similar to three dimensions, a finer classification may be introduced, by limiting the affine transformations to those for which the restriction to the physical subspace has a linear part with positive determinant. Of course, a definition of equivalence is important. Without such a definition a statement about the number of (super)space groups or equivalence classes is meaningless.

Let us consider a number of examples.

• 1. The hypercubic group in four dimensions is R-irreducible. Therefore, the corresponding symmorphic space group is not a superspace group.

• 2. The superspace groups $P 2 ( 1 ¯ )$ and $P 1 ¯ ( 1 )$ are isomorphic as four-dimensional space groups, but the connecting affine transformation interchanges the rotation axis in the physical space with the one-dimensional internal space. Therefore, they are non-equivalent as superspace groups.

• 3. The five primitive icosahedral six-dimensional superspace groups with as generators (besides three- and two-fold elements)

$Display mathematics$
fall into two equivalence classes. The first is symmorphic (p = 0). The groups with p = 1 and p = 4, and those with p = 2 and p = 3 are equivalent by a mirror, just as the screw operations 41 and 43 in three dimensions. The groups with p = 1 and p = 2 are equivalent because they are related by a centralizer element of the point group, which leaves the physical space invariant. Therefore, they are equivalent as superspace groups, and, a fortiori, also as six-dimensional space groups. This means that there are two non-equivalent superspace groups for this arithmetic crystal class.

As for the general space groups, the number of superspace groups grows rapidly with the dimension, but generally there are fewer superspace groups than space groups, because of the reducibility condition. The number of superspace groups for modulated phases is smaller than (or equal to) the number of general superspace groups of the same dimension, because the point groups for the former should be isomorphic to a d-dimensional crystallographic group, because the physical component of the point group should leave the basis structure invariant. For example, the (2+2)-dimensional space group p8m(83 m) does not occur as symmetry group for a modulated structure, but it is the symmetry group of the Ammann-Beenker tiling. For lower dimensions the numbers are given in the following table. For higher dimensions there are only partial lists.

(p.427)

Dimension

2+1

2+2

2+2

3+1

(modulated)

(modulated)

(general)

(modulated)

Bravais classes

4

17

20

24

Geometric crystal classes

5

13

21

30

Superspace group types

22

73

83

777

A.7 Notation

The notation for superspace groups for quasiperiodic crystals is based on the notation for two-dimensional and three-dimensional plane and space groups as used in the International Tables for Crystallography and for higher-dimensional groups as recommended in (Janssen et al., 1999; Janssen et al., 2002). The extension to higher dimensions requires new symbols, and the fact that the physical space plays a special role in the space of the superspace group leads to additional information one should give in the symbol. The symbol is a compromise between a number as the number of a space group in ITC, and full specification. The symbol contains symbols for the geometric crystal class of the space group, the arithmetic crystal class and on the vector system of non-primitive translations.

A.7.1 Superspace groups for incommensurate phases

For superspace groups, the fact that here one may distinguish main reflections and satellites in the diffraction pattern is taken into account. In this case one may always choose a basis of the n-dimensional reciprocal space with d basis vectors in the physical space. These are the basis vectors of the reciprocal lattice of the basis structure. Then the basis of the direct lattice has nd basis vectors in the internal space. With respect to such a basis the point group elements D(R) may be written as

$Display mathematics$

The elements {RE|t E} form in this case a d-dimensional space group, for which a symbol can be given according to the International Tables of Crystallography. The internal components of the point group elements are determined by the action on the basic satellites. For example, if the physical part of the super-space group is the three-dimensional space group Pcmn, and the basic satellite is (0,0,γ) = γc *, then the point group generators are (m x,1), (m y,1), and $( m z , 1 ¯ )$. Therefore, indicating the physical part of the point group and the basic satellites fixes the higher-dimensional point group. A second, alternative, possibility is indicating the groups K E and K I. This is the more general approach, which is also applicable for quasiperiodic structures that are not modulated phases. So, the space group Pmmm(0, 0,γ) can also be denoted by Pmmm $( 11 1 ¯ )$. It is a matter of convenience what to choose.

The notation for the non-primitive translations (the vector system) may be differentiated as well. A non-primitive translation consists of an intrinsic (or rational) and a non-intrinsic (or irrational) part: (p.428)

$Display mathematics$
The latter may be changed, or even eliminated, by another choice of origin, and corresponds to the components of the general solutions which have continuous solutions. For example, in two dimensions, the mirror mx in a rectangular lattice has as general solutions in the case of mm2P: t = (α, β) with $1 2$. The intrinsic (or rational) part is here (0,β). The intrinsic part has rational components with respect to a lattice basis. The irrational, not intrinsic part is (α,0). For modulated phases this means that
$Display mathematics$
where −Δt E is the internal component associated with t E:
$Display mathematics$
So the intrinsic part of the non-primitive translation has an internal component stemming from the component in physical space and the rational components fj for j = d + 1,…, n.

In the case that nd = 1 (‘co-dimension one’) the fractions f 4 have denominators equal to 1, 2, 3, 4, or 6, because the maximal order for a point group element which is a symmetry for a modulated structure is six, the point group being crystallographic in d dimensions. These values are denoted by 0, s, t, q, and h, respectively. Then the symbol Pcmn(00γ>)1ss means that the mirror m x has internal component 1 and non-primitive translation $10 mm . 10 22 , 10 ¯ 2 m ( 1 2 c, − Δ 1 2 c ) = ( 1 2 c, − γ / 2 )$. The mirror m y has a non-primitive translation $5 ¯ m . 1 2 a 4 = ( 0 , 1 2 )$, and the mirror m z has an internal component −1 (because m z(00γ) = (00−γ)), and non-primitive translation $10 mm, 10 22 , 5 ¯ m, 1 2 ( a 1 + a 2 ) + 1 2 a 4 = ( 1 2 ( a + b ) , 1 2 )$. Notice that Δa = Δb = 0, because a = q.a = (0, 0,γ).(1,0,0) = 0. Here q is the modulation wave vector in the symbol Pcmn(00γ)1ss.

For higher co-dimension the principle remains the same. The internal component of a non-primitive translation is the sum of the mapping Δ applied to the physical component and the component along the last nd basis vectors. This means that the same letters (1, s, t, q, and h) may be used to indicate the last nd components. Now for each generator of the point group one has to give an m-tuple of the letters (m = nd). As an example, the two-dimensional system of rank four with point group mm2(mm2) and two modulation wave vectors (α, 0) and (0,β), has several superspace groups. One of them has symbol pgg(α,0;0β)m dm c. It has a point group with two generators

$Display mathematics$
(p.429) and corresponding superspace group generators $10 ¯ 2 m { ( m x , m z ) | 1 2 ( a 2 + a 4 ) }$ , and $8 2 ¯ m { ( m y , m u ) | 1 2 ( a 1 + a 3 ) }$. Here x, y, z, u are the Cartesian coordinates in superspace. The physical space has the coordinates x and y, the internal space z and u. In six dimensions the Cartesian coordinates are usually x, y, z, u, v, w.

Earlier symbols for the superspace groups for modulated phases used two-line symbols. For instance, Pcmn(00γ)0s0 and $( m x ¯ y , 1 ) P P cmn 0 s 1 ¯$ denote the same group. This convention has been abandoned for typographical reasons.

A.7.2 General space groups

For composites and quasicrystals there is no obvious reciprocal lattice of main reflections. In the case of composites it is still possible to choose the first three reciprocal basis vectors in the physical space, although there is not always an obvious choice. In general, all basis vectors of the n-dimensional reciprocal lattice have internal components. In this case the notation may be based on the property that still holds, namely that there is a distinguished physical subspace in the n-dimensional superspace. This means that the physical space is invariant under the point group. This implies that the point group K may be written as (K E, KI) with elements (RE, RI). The non-primitive translations may be indicated as subindices to the generator symbols. Because the symbol for a point group element now has two components, the symbols for the non-primitive translations may be attached, as subindex, to the external part.

For example, the four-dimensional group Pcmn(00γ)1ss according to this principle gets the symbol $P cm d n d ( 11 1 ¯ )$. One keeps the notation for the physical components. So, now the first generator (m x,1) has non-primitive translation $( 1 2 c, − γ / 2 )$, the second generator (m y,1) has non-primitive translation $( 0 , 1 2 )$, and the third generator $( m z , 1 ¯ )$ has non-primitive translation $( 1 2 ( a + b ) , 1 2 )$.

The four-dimensional group pgg(α, 0; 0, β)m dm c would get, in this context, the notation pb da c(mm).

A number of examples for quasicrystals with icosahedral point group are given in Table A.8.

A.8 Extinction rules

Intrinsic non-primitive translations in elements of a non-symmorphic space group lead to extinction rules. If a vector k of the Fourier module is invariant under the homogeneous part R of the superspace group element {(R, RI)|(v, v I)}, then the intensity of the diffraction at k is zero if exp(2πi(k.v + k I. v I)) ≠ 1. If the indices of k with respect to a basis of the Fourier module are integers h 1,…, hn and the coordinates of (v, v I) are ξ1,…,ξn, then the intensity can only be non-zero if These are the reflection conditions. The extinction rules are the complement of this: the intensity is zero if

(A.4)
$Display mathematics$
These are the reflection conditions. The extinction rules are the complement of this: the intensity is zero if ∑i hiξ ≠ 0 modulo integers. The non-primitive (p.430) translations (v, v I) can be changed by an origin shift, but, of course, this does not affect the extinction rules. Therefore, one may use the intrinsic part of the non-primitive translation in Eq. A.4.

Extinction rules (or their complement, the reflections conditions) also occur because of centring. If the translations (v, v I) are given with respect to the basis of a conventional cell, a unit cell of a sublattice, there are lattice translations with non-integer, but rational, coordinates. This means that the non-zero intensities are found on positions in reciprocal space which form a superlattice of the conventional lattice (the lattice with the conventional cell as unit cell). Therefore, there are additional extinction rules. For example, in two dimensions the rectangular lattice may be primitive or centred. The primitive cell has basis (a,0) and (0, b), and the centred cell has basis (a⧏2, b⧏2) and (−a⧏2, b⧏2). The reciprocal basis for the p-lattice is (1⧏a,0) and (0,1⧏b). Then the points of the reciprocal c-lattice are ((h 1h 2)⧏a,((h 1 + h 2)⧏b). With respect to the reciprocal p-lattice the coordinates are (h 1h 2) and (h 1 + h 2) which are both even or both odd: their sum is even. Then the reflection condition given in terms of the indices with respect to the p-lattice H 1 and H 2 is: H 1 + H 2 = even.

A.9 Tables

A.9.1 Introduction

In this section a number of examples are given of point and space groups, and the related character tables for quasiperiodic crystals. Their total number is very large. Therefore, it does not make sense to incorporate them here. As illustration, and as an aid to what has been said in the chapters of this book, we just give examples here. We start with the representations of point groups. Notice that the n-dimensional crystallographic point groups for superspace groups for aperiodic crystals are isomorphic to three-dimensional point groups, in general non-crystallographic. We can analyse the point groups in terms of irreducible representations.

For the two-dimensional group 5, one reads from Table A.1 that the matrices of the point group form a two-dimensional reducible representation with two one-dimensional irreducible complex components, D2 and D5. The characters of these two representations are complex conjugates of each other, and their sum is R-irreducible. The character of this representation is +(A n) = 2 cos(2nπ/5). It is not a crystallographic group in two dimensions, but it becomes crystallographic in four dimensions. Then the four-dimensional point group is the sum of the four non-trivial irreducible representations. For this sum all characters are integers: χ(E) = 4, χ(A n) = −1 (n ≠ 0).

The two-dimensional group 5m forms the irreducible representation D3, which is not crystallographic. The group becomes crystallographic in four dimensions, and then has two irreducible components: D3 and D4, with integer character. In three dimensions the point groups 5m and 52 are reducible representations (see Table A.2). A is the five-fold rotation, B the mirror or the two-fold rotation, respectively. The first has components D3 and D1, the second D3 and D2.

(p.431) In three dimensions there are four geometric crystal classes with point groups isomorphic to the dihedral group of order 20. They are $10 mm , 10 22 , 10 ¯ 2 m ,$, and $5 ¯ m,$. Their decomposition into irreducible components (cf. Table A.3) shows the difference:

$Display mathematics$
A similar situation occurs for the cyclic groups of order 8 and 12, and for the dihedral groups of order 16 and 24.

The group 10 mm is isomorphic to the direct product of the group 5m and the group with two elements. The irreducible representations of such a direct product may simply be obtained from those of the subgroup of index two (5m in this case). The irreducible representations of the direct product K × Z 2 are obtained from the irreps D1,…, D m as

$Display mathematics$
for all elements R in K, when the group of order two consists of E and A. The character table of K × Z 2 then looks like

(R, E)

(R, A)

Dj+

χj(R)

χj(R)

Dj−

χj(R)

−χj(R)

For the group 10mm both notations are given.

The three-dimensional icosahedral group is an irreducible representation of the isomorphism class. It corresponds to representation D3(Table A.7). In three dimensions it is a non-crystallographic point group. The corresponding Z-irreducible representation is D3 + D4. It is a six-dimensional Z-irreducible integer representation. There are other integer representations, for example, in four dimensions, but these are not R-reducible. There is no invariant three-dimensional space that can be identified with the physical space.

In this subsection we have presented examples of character tables for some of the point groups which are relevant for quasicrystals. A full list may be found in the International Tables for Crystallography, Vol.D.

A.9.2 Tables for irreducible representation of point groups: point groups of 5-, 8-, 10-, 12-fold and icosahedral symmetry

(p.432)

Table A.1 Character table for the point group 5

Class:

E

A

A2

A3

A4

Order:

1

5

5

5

5

No.:

1

1

1

1

1

D1

1

1

1

1

1

D2

1

exp(2Φi⧏5)

exp(4Φi⧏5)

exp(6Φi⧏5)

exp(8Φi⧏5)

D3

1

exp(4Φi⧏5)

exp(8Φi⧏5)

exp(2Φi⧏5)

exp(6Φi⧏5)

D4

1

exp(6Φi⧏5)

exp(2Φi⧏5)

exp(8Φi⧏5)

exp(4Φi⧏5)

D5

1

exp(8Φi⧏5)

exp(6Φi⧏5)

exp(4Φi⧏5)

exp(2Φi⧏5)

Table A.2 Character table for the point groups 5m and 52

Class:

E

A

A2

B

Order:

1

5

5

2

No.:

1

2

2

5

D1

1

1

1

1

D2

1

1

1

−1

D3

2

Φ

−τ

0

D4

2

−τ

Φ

0

$τ = ( 5 + 1 ) / 2 , Φ = ( 5 − 1 ) / 2$

Table A.3 Character table for the point groups 10mm, 10 22, $5 ¯ m$, and $10 ¯ 2 m$, the representations dj(a) are the representations of table A.2 with a = A 2.

Class:

E

A

A2

A3

A4

A5

B

AB

Order:

1

10

5

10

5

2

2

2

No.:

1

2

2

2

2

1

2

2

D1

d1+

1

1

1

1

1

1

1

1

D2

d2+

1

1

1

1

1

1

−1

−1

D3

d1

1

−1

1

−1

1

−1

1

−1

D4

d2

1

−1

1

−1

1

−1

−1

1

D5

d3+

2

τ

Φ

−Φ

−τ

−2

0

0

D6

d4+

2

Φ

−τ

−τ

Φ

2

0

0

D7

d4

2

−Φ

−τ

τ

Φ

−2

0

0

D8

d3

2

−τ

Φ

Φ

−τ

2

0

0

(p.433)

Table A.4 Character table for the point group 8

Class:

E

A

A2

A3

A4

A5

A6

A7

Order:

1

8

4

8

2

8

4

8

No.:

1

1

1

1

1

1

1

1

D1

1

1

1

1

1

1

1

1

D2

1

α

i

β

−1

−α

i

−β

D3

1

i

−1

i

1

i

−1

i

D4

1

β

i

α

−1

−β

i

−α

D5

1

−1

1

−1

1

−1

1

−1

D6

1

−α

i

−β

−1

α

i

β

D7

1

i

−1

i

1

i

−1

i

D8

1

−β

i

−α

−1

β

i

α

$α = exp ⁡ ( π i / 4 ) = 1 2 2 ( 1 + i ) , β = exp ⁡ ( 3 ⁢ π / 4 ) = 1 2 2 ( 1 − i )$

Table A.5 Character table for the point groups 8mm, 8 22, and $8 2 ¯ m$

Class:

E

A

A2

A3

A4

B

AB

Order:

1

8

4

8

2

2

2

No.:

1

2

2

2

1

4

4

D1

1

1

1

1

1

1

1

D2

1

1

1

1

1

−1

−1

D3

1

−1

1

−1

1

1

−1

D4

1

−1

1

−1

1

−1

1

D5

2

$2$

0

$− 2$

−2

0

0

D6

2

0

−2

0

2

0

0

D7

2

$− 2$

0

$2$

−2

0

0

(p.434)

Table A.6 Character tables for the point groups 23 and 432

Class:

E

A

A2

B

Order:

1

3

3

2

No.:

1

4

4

3

D1

1

1

1

1

D2

1

ω

ω2

1

D3

1

ω2

ω

1

D4

3

0

0

−1

ω=exp(2πi⧏3)

Class:

E

B

A2

A

AB

Order:

1

3

2

4

2

No.:

1

8

3

6

6

D1

1

1

1

1

1

D2

1

1

1

−1

−1

D3

2

−1

2

0

0

D4

3

0

−1

1

−1

D5

3

0

−1

−1

1

Table A.7 Character table for the point group 532

Class:

E

A

A2

B

AB

Order:

1

5

5

3

2

No.:

1

12

12

20

15

D1

1

1

1

1

1

D2

3

τ

0

−1

D3

3

τ

0

−1

D4

4

−1

−1

1

0

D5

5

0

0

−1

1

$τ = ( 5 + 1 ) / 2 , Φ = ( 5 − 1 ) / 2$

(p.435) A.9.3Examples of superspace groups for modulated phases

The full list of all superspace groups in (3 + 1) dimensions may be found in the International Tables for Crystallography, Vol. C(ITCC). Here, we give examples mentioned in the text.

Example 1. The four-dimensional superspace groups Pcmn(00γ) and Pcmn(00γ)0s0 have numbers 62.5 and 62.6 in ITCC. Their basic space group is Pcmn, number 62 in ITCA, where the standard setting is Pnma. The point group has generators mx, my, and mz, and because mz(00γ) = (00−γ) it has an internal component – 1. Therefore, the four-dimensional generators are (mx, 1), (my, 1), and (mx, −1). The three generators of the superspace group in four dimensions act as follows on a point with lattice coordinates x, y, z, u:

$Display mathematics$
and
$Display mathematics$
for the two superspace groups. The (special) reflection conditions are, respectively Group 1: 0klm: l = 2n hk00: h + k = 2n Group 2: 0klm: l = 2n h0lm: m = 2n hk00: h + k = 2n. This is immediately seen if one takes into account, that if [hklm] is invariant under a point transformation with non-primitive translation (x, y, z, u), then the reflection condition is hx + ky + lz + mu = 0 modulo integers.

Example 2. The four-dimensional superspace group I41 ⧏acd(00γ)s0s0 has basic space group I41 ⧏acd. This has three generators for the point group, 4z, mx, and mz. The latter changes the sign of the modulation wave vectorγc *. Therefore, the four-dimensional operation (mz, −1) belongs to the point group. The three-dimensional group is the group 142 in ITCA. The action of the three generators in the four-dimensional superspace group transforms (x, y, z, u) into

$Display mathematics$
There is a centring condition h + k + l = 2n. Moreover, one has
$Display mathematics$
because of the existence of the operations (4z, 1), (mx, 1), (mz, −1, and $( m x ¯ y , 1 )$, together with their non-primitive translations.

Example 3. The group Abm2($Abm 2 ( 1 2 0 ⁢ γ )$) has three-dimensional basic space group Abm2, nr. 39 in ITCA. The centring A causes centring conditions for the main reflections (hkl0: k + l = 2n). In this case, the modulation wave vector has a rational component a*⧏2. This implies that there is a further centring in four

(p.436) dimensions. A general reflection is (h+ m⧏2)a* + kb* + (l + m)c*. Introducing a conventional cell such that a $a c ∗ = a ∗ / 2 , b c ∗ = b ∗$, and $c c ∗ = c ∗$, the new indices are H = 2h + m, K = k, L = l and the modulation wave vector is written as [H, K, L, m]. Because k + l = 2n, one has the reflection conditions K + L = even and H + m = even. These are the centring conditions. In addition, there is the reflection condition 0KLm: K = 2n, because of the existence of a non-primitive translation associated with (mx, 1).

A.9.4 Superspace groups for quasiperiodic structures with five-, eight-, ten-, twelve-fold, or icosahedral symmetry

The list of superspace groups occurring in quasicrystals found so far, is rather limited. However, the list of possibly relevant space groups for the three-dimensional quasicrystals with point groups with five-, eight-, ten- or twelve-fold rotation symmetries (including the icosahedral systems) is not too long. We give here a condensed form of the information concerning these space groups in five and six dimensions.

Again, the generators of the space group are the n basis translations and a number of elements {R|v }. A position r in the unit cell is given by the coordinates with respect to the lattice basis: the first n of the letters x, y, z, u, v, w. Then the lattice translation a1 is indicated by the transformed point x + 1, y, z, u, v, …, and the element {R|v } by the transformed point r′ with coordinates x, y, z, …. For non-symmorphic groups or groups with centred lattices there are extinction rules or reflection conditions.

Superspace groups for the pentagonal, octagonal, decagonal, and dodecagonal quasicrystals are listed in (Janssen, 1988)

(p.437)

Table A.8 Superspace groups: icosahedral family

Symbol

Short

Rank Generators

Reflection Dimension

P532(5232)

P532

6

(x, z, u, v, w, y)

$( w , x , u , v , z ¯ , u ¯ , y )$

P5132(5232)

P5132

6

$( x + 1 5 , z , u + 1 5 , v , w , y − 1 5 ) ( w , x , u , v , z ¯ , u ¯ , y )$

h2 = h3 = h4 = h5 = h6:

h1 = 5m

I532(5232)

I532

6

(x, z, u, v, w, y)

$( w , x , u , v , z ¯ , u ¯ , y )$

Σi hi = even

I5132(5232)

I1532

6

$( x + 1 5 , z + 1 5 , u − 1 5 , v + 2 5 , w − 1 5 , y − 1 5 ) ( w , x , u , v , z ¯ , u ¯ , y )$

h2 = h3 = h4 = h5 = h6:

hP1 = 5m

Σi hi = even

F532(5232)

F532

6

(x, z, u, v, w, y)

$( w , x , u , v , z ¯ , u ¯ , y )$

hi + hj = even (all i, j)

F532(5232)

F532

6

$( x − 3 10 , z + 1 10 , u + 1 5 , v , w − 1 5 , y − 1 5 ) ( w , x , u , v , z ¯ , u ¯ , y )$

hi + hj = even (all i, j)

$P 53 ¯ m ( 5 2 3 m ) P 53 ¯ m$

6

$( x , z , u , v , w , y ) ( w , x , u , v , z ¯ , u ¯ , y ) ( x ¯ , y ¯ , z ¯ , u ¯ , v ¯ , w ¯ )$

$P 53 ¯ m 34 ( 53 ¯ m ) P 53 ¯ ( cd )$

6

$( x , z + 1 2 , u + 1 2 , v + 1 2 , w + 1 2 , y ) ( w , x , u , v , z ¯ , u ¯ , y ) ( x ¯ , y ¯ , z ¯ , u ¯ , v ¯ , w ¯ )$

h5 = −h2, h6 = −h1:

h3 + h4= even

$I 53 ¯ m ( 5 2 32 ) I 53 ¯ m$

6

$( x , z , u , v , w , y ) ( w , x , u , v , z ¯ , u ¯ , y ) ( x ¯ , y ¯ , z ¯ , u ¯ , v ¯ , w ¯ )$

Σi hi = even

$F 53 ¯ m ( 5 2 32 ) ( F 53 ¯ m )$

6

$( x , z , u , v , w , y ) ( w , x , u , v , z ¯ , u ¯ , y ) ( x ¯ , y ¯ , z ¯ , u ¯ , v ¯ , w ¯ )$

hi + hj = even (all i, j)

$F 53 ¯ a ( 5 2 32 ) F 53 ¯ a$

6

$( x , z , u , v , w , y ) ( w , x , u , v , z ¯ , u ¯ , y ) ( x ¯ , y ¯ , z ¯ , u ¯ , v ¯ , w ¯ )$

h6 = h1, h5 = h2, h4 = −h3:

h1 even

hi + hj = even (all i, j)