(p.414) APPENDIX A
(p.414) APPENDIX A
A.1 Crystallographic operations in n dimensions
Physical systems are zero, one, two or threedimensional (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 distancepreserving 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 twodimensional rotations, and onedimensional 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 twofold rotations in a plane. If the determinant is −1, there is one unpaired inversion. As example, consider the fourfold rotation in three dimensions, and its related rotoinversion $\overline{4}$. They can be written in block form:
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 coprime to N), and the set of all twodimensional rotations may be divided into complete sets of conjugates. The number of conjugates for the eigenvalue exp(2πi⧏N) is the Euler function of N: Φ(N) is the number of positive integers smaller than N which are coprime with N. The eigenvalues exp(2πim⧏N) and exp(2πi(N − m)⧏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 fourfold rotation is crystallographic (leaves a lattice invariant) in two dimensions, a fivefold rotation in four dimensions, a three or sixfold rotation in two dimensions, and a sevenfold 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 8^{3}. It is a fourdimensional rotation. Crystallographic transformations therefore are indicated by strings of symbols [N], N, $\left[N\right],\text{}N,\text{}\overline{1}$, and 1, where the numbers in square brackets correspond to four or higherdimensional rotations, without brackets to three, four, or sixfold rotations in two dimensions (N = 3, 4, or 6), 2s to 2fold rotations and $\overline{1}$ appears when the transformation has determinant −1. As an example $\left[8\right]62\overline{1}$ is a transformation with determinant −1 in nine dimensions: an eightfold rotation in four dimensions, a sixfold rotation in two dimensions, a twofold rotation in two dimensions and an inversion in one dimension.
For aperiodic crystals there is an embedding in an ndimensional space with an invariant ddimensional 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 threedimensional octagonal aperiodic structure may act as a rotation 81 in physical space, and, because 8 needs its conjugate 8^{3}, as 8^{3} in internal space. The operator [8]1 in five dimensions is then denoted as 81(8^{3}1) for the aperiodic crystal. Very often, one denotes this transformation by 81, because for a crystallographic transformation the component 8^{3} follows necessarily. Strictly speaking, 81 is a noncrystallographic eightfold 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 Zmodule or Bravais module. In both cases any translation may be written as a sum of basis translations:
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 {Ea}. The lattice is characterized by its metric tensor, a symmetric tensor with elements
The dual of a lattice is the reciprocal lattice with basis vectors ${a}_{j}^{\ast}$ defined by
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
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
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 $\overline{3}\text{m}$ and 6/mmm, respectively. The first is a subgroup of the latter. So the system group of 3 is the point group $\overline{3}\text{m}$. The arithmetic holohedries of the two Bravais classes are $\overline{3}\text{mR}.$ 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 $\overline{3}\text{mR}.$, 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 Nfold rotation in n dimensions is noncrystallographic 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 noninteger. 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 Rreducible, otherwise Rirreducible. 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 Zreducible, otherwise Zirreducible (Z is the group of integers). If there is a sublattice with vectors in the two subspaces, the point group is Qreducible (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 Rreducible with an n _{1}dimensional and an n _{2}dimensional invariant subspace, the ndimensional 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 _{1}⊥K _{2}. For example, the tetragonal group 4/mmm has two complementary invariant subspaces. In the onedimensional subspace along the unique axis it acts as K _{2} = m, in the twodimensional 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 _{1}s forming K _{1}, and the R _{2}s 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 twodimensional plane, and as the group m in the complementary onedimensional subspace. The generators now are pairs 4_{z} = (4,1) and 2_{y} = (m_{x}, m_{z}). 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)
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:
In the notation of the International Tables for Crystallography geometrically equivalent, but arithmetically nonequivalent 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
(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 R_{j}, 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: W_{j}(R _{1},…, R_{m}) = E. For a space group, with a vector system t _{R}, the defining relations of the point group imply that
A space group then is specified by the point group K, and the vector system t _{R} (R ∈ K). 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 nonprimitive 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 nonsymmorphic 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 nonprimitive translation. The nonprimitive 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 twofold rotation has a nonprimitive translation along the rotation axis, the combined operation is a screw operation. If this is the only nonprimitive translation the nonsymmorphic space group has symbol P2_{1}⧏m, indicating that the intrinsic part of the nonprimitive 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 nonprimitive translation along the aaxis, the symbol of the nonsymmorphic 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},…, R_{m}) and k defining relations W_{i}(R _{1},…, R_{m}) = 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
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 ntuples of integers, vectors (n _{1} ,…, n_{n}). Each ntuple corresponds to a lattice point. Moreover, the basis vectors span the whole ndimensional space.
♯ If one wants to be more precise, one should distinguish the translation group with translations {Ea} 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 ndimensional space. We shall not make this distinction here. Then one may say that for a space group the translations span the whole ndimensional 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.
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
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 $\overline{3}\text{m}$. The first contains geometric crystal classes 6, $6\text{mm,}\text{}\overline{6},\text{}622,\text{}\overline{6}2\text{m}$, and 6/mmm. The second contains $3,\text{}3\text{m,}\text{}32,\text{}\overline{3}$, and $\overline{3}\text{m}$. 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 $\overline{3}\text{mR}.$. 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 ndimensional lattice invariant. The latter is isomorphic to the group of ntuples 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
The vector system t _{R} in the space group elements {Rt _{R}} forms a 1cocycle, for which we have the relations
A.6 Space groups for aperiodic crystals
Space groups for aperiodic crystals are space groups in n dimensions, but not every ndimensional 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 Rreducible into a d and an (n − d)dimensional component. This eliminates a large number of ndimensional 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 ndimensional space group a superspace group if its point group leaves a ddimensional 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 ndimensional affine group by an affine transformation {Sa} 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 nonequivalent 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 ddimensional 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 Rirreducible. Therefore, the corresponding symmorphic space group is not a superspace group.

2. The superspace groups $\text{P}2\left(\overline{1}\right)$ and $\text{P}\overline{1}\left(1\right)$ are isomorphic as fourdimensional space groups, but the connecting affine transformation interchanges the rotation axis in the physical space with the onedimensional internal space. Therefore, they are nonequivalent as superspace groups.

3. The five primitive icosahedral sixdimensional superspace groups with as generators (besides three and twofold elements)
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 ddimensional 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(8^{3} m) does not occur as symmetry group for a modulated structure, but it is the symmetry group of the AmmannBeenker tiling. For lower dimensions the numbers are given in the following table. For higher dimensions there are only partial lists.
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 twodimensional and threedimensional plane and space groups as used in the International Tables for Crystallography and for higherdimensional 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 nonprimitive 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 ndimensional 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 n − d basis vectors in the internal space. With respect to such a basis the point group elements D(R) may be written as
The elements {R_{E}t _{E}} form in this case a ddimensional 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 superspace group is the threedimensional 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 $\left({\text{m}}_{\text{z}},\overline{1}\right)$. Therefore, indicating the physical part of the point group and the basic satellites fixes the higherdimensional 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 $\left(11\overline{1}\right)$. It is a matter of convenience what to choose.
The notation for the nonprimitive translations (the vector system) may be differentiated as well. A nonprimitive translation consists of an intrinsic (or rational) and a nonintrinsic (or irrational) part: (p.428)
In the case that n − d = 1 (‘codimension 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 nonprimitive translation $10\text{mm}.10\text{}22,\overline{10}2\text{m}\left(\frac{1}{2}\text{c,}\Delta \frac{1}{2}\text{c}\right)=\left(\frac{1}{2}\text{c,}\text{}\gamma /2\right)$. The mirror m _{y} has a nonprimitive translation $\overline{5}\text{m}.\frac{1}{2}{a}_{4}=\left(0,\frac{1}{2}\right)$, and the mirror m _{z} has an internal component −1 (because m _{z}(00γ) = (00−γ)), and nonprimitive translation $10\text{mm,}\text{}10\text{}22,\text{}\overline{5}\text{m,}\frac{1}{2}\left({a}_{1}+{a}_{2}\right)+\frac{1}{2}{a}_{4}=\left(\frac{1}{2}\left(a+b\right),\frac{1}{2}\right)$. 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 codimension the principle remains the same. The internal component of a nonprimitive translation is the sum of the mapping Δ applied to the physical component and the component along the last n − d basis vectors. This means that the same letters (1, s, t, q, and h) may be used to indicate the last n − d components. Now for each generator of the point group one has to give an mtuple of the letters (m = n − d). As an example, the twodimensional 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 _{d}m _{c}. It has a point group with two generators
Earlier symbols for the superspace groups for modulated phases used twoline symbols. For instance, Pcmn(00γ)0s0 and $\left({m}_{\overline{x}y},1\right)\text{P}\text{}\begin{array}{c}{\text{P}}_{\text{cmn}}\\ 0\text{s}\text{}\overline{1}\end{array}$ 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 ndimensional 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 ndimensional 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}, K_{I}) with elements (R_{E}, R_{I}). The nonprimitive 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 nonprimitive translations may be attached, as subindex, to the external part.
For example, the fourdimensional group Pcmn(00γ)1ss according to this principle gets the symbol ${\text{P}}_{{\text{cm}}_{\text{d}}{n}_{\text{d}}}\left(11\overline{1}\right)$. One keeps the notation for the physical components. So, now the first generator (m _{x},1) has nonprimitive translation $\left(\frac{1}{2}\text{c,}\gamma /2\right)$, the second generator (m _{y},1) has nonprimitive translation $\left(0,\frac{1}{2}\right)$, and the third generator $\left({\text{m}}_{\text{z}},\overline{1}\right)$ has nonprimitive translation $\left(\frac{1}{2}\left(a+b\right),\frac{1}{2}\right)$.
The fourdimensional group pgg(α, 0; 0, β)m _{d}m _{c} would get, in this context, the notation pb _{d}a _{c}(mm).
A number of examples for quasicrystals with icosahedral point group are given in Table A.8.
A.8 Extinction rules
Intrinsic nonprimitive translations in elements of a nonsymmorphic 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, R_{I})(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},…, h_{n} and the coordinates of (v, v _{I}) are ξ_{1},…,ξ_{n}, then the intensity can only be nonzero if These are the reflection conditions. The extinction rules are the complement of this: the intensity is zero if
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 noninteger, but rational, coordinates. This means that the nonzero 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 plattice is (1⧏a,0) and (0,1⧏b). Then the points of the reciprocal clattice are ((h _{1} − h _{2})⧏a,((h _{1} + h _{2})⧏b). With respect to the reciprocal plattice the coordinates are (h _{1} − h _{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 plattice 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 ndimensional crystallographic point groups for superspace groups for aperiodic crystals are isomorphic to threedimensional point groups, in general noncrystallographic. We can analyse the point groups in terms of irreducible representations.
For the twodimensional group 5, one reads from Table A.1 that the matrices of the point group form a twodimensional reducible representation with two onedimensional irreducible complex components, D_{2} and D_{5}. The characters of these two representations are complex conjugates of each other, and their sum is Rirreducible. 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 fourdimensional point group is the sum of the four nontrivial irreducible representations. For this sum all characters are integers: χ(E) = 4, χ(A ^{n}) = −1 (n ≠ 0).
The twodimensional group 5m forms the irreducible representation D_{3}, which is not crystallographic. The group becomes crystallographic in four dimensions, and then has two irreducible components: D_{3} and D_{4}, with integer character. In three dimensions the point groups 5m and 52 are reducible representations (see Table A.2). A is the fivefold rotation, B the mirror or the twofold rotation, respectively. The first has components D_{3} and D_{1}, the second D_{3} and D_{2}.
(p.431) In three dimensions there are four geometric crystal classes with point groups isomorphic to the dihedral group of order 20. They are $10\text{mm},\text{}10\text{}22,\text{}\overline{10}2\text{m},$, and $\overline{5}\text{m,}$. Their decomposition into irreducible components (cf. Table A.3) shows the difference:
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 D_{1},…, D _{m} as
(R, E) 
(R, A) 


D_{j+} 
χ_{j}(R) 
χ_{j}(R) 
D_{j−} 
χ_{j}(R) 
−χ_{j}(R) 
The threedimensional icosahedral group is an irreducible representation of the isomorphism class. It corresponds to representation D_{3}(Table A.7). In three dimensions it is a noncrystallographic point group. The corresponding Zirreducible representation is D_{3} + D_{4}. It is a sixdimensional Zirreducible integer representation. There are other integer representations, for example, in four dimensions, but these are not Rreducible. There is no invariant threedimensional 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, 12fold and icosahedral symmetry
Table A.1 Character table for the point group 5
Class: 
E 
A 
A^{2} 
A^{3} 
A^{4} 

Order: 
1 
5 
5 
5 
5 
No.: 
1 
1 
1 
1 
1 
D_{1} 
1 
1 
1 
1 
1 
D_{2} 
1 
exp(2Φi⧏5) 
exp(4Φi⧏5) 
exp(6Φi⧏5) 
exp(8Φi⧏5) 
D_{3} 
1 
exp(4Φi⧏5) 
exp(8Φi⧏5) 
exp(2Φi⧏5) 
exp(6Φi⧏5) 
D_{4} 
1 
exp(6Φi⧏5) 
exp(2Φi⧏5) 
exp(8Φi⧏5) 
exp(4Φi⧏5) 
D_{5} 
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 
A^{2} 
B 

Order: 
1 
5 
5 
2 
No.: 
1 
2 
2 
5 
D_{1} 
1 
1 
1 
1 
D_{2} 
1 
1 
1 
−1 
D_{3} 
2 
Φ 
−τ 
0 
D_{4} 
2 
−τ 
Φ 
0 
$\tau =\left(\sqrt{5}+1\right)/2,\text{}\text{}\text{}\text{}\text{}\Phi =\left(\sqrt{5}1\right)/2$ 
Table A.3 Character table for the point groups 10mm, 10 22, $\overline{5}\text{m}$, and $\overline{10}\text{}2\text{m}$, the representations d_{j}(a) are the representations of table A.2 with a = A ^{2}.
Class: 
E 
A 
A^{2} 
A^{3} 
A^{4} 
A^{5} 
B 
AB 


Order: 
1 
10 
5 
10 
5 
2 
2 
2 

No.: 
1 
2 
2 
2 
2 
1 
2 
2 

D_{1} 
d_{1+} 
1 
1 
1 
1 
1 
1 
1 
1 
D_{2} 
d_{2+} 
1 
1 
1 
1 
1 
1 
−1 
−1 
D_{3} 
d_{1} _{−} 
1 
−1 
1 
−1 
1 
−1 
1 
−1 
D_{4} 
d_{2} _{−} 
1 
−1 
1 
−1 
1 
−1 
−1 
1 
D_{5} 
d_{3+} 
2 
τ 
Φ 
−Φ 
−τ 
−2 
0 
0 
D_{6} 
d_{4+} 
2 
Φ 
−τ 
−τ 
Φ 
2 
0 
0 
D_{7} 
d_{4} _{−} 
2 
−Φ 
−τ 
τ 
Φ 
−2 
0 
0 
D_{8} 
d_{3} _{−} 
2 
−τ 
Φ 
Φ 
−τ 
2 
0 
0 

Table A.4 Character table for the point group 8
Class: 
E 
A 
A^{2} 
A^{3} 
A^{4} 
A^{5} 
A^{6} 
A^{7} 

Order: 
1 
8 
4 
8 
2 
8 
4 
8 
No.: 
1 
1 
1 
1 
1 
1 
1 
1 
D_{1} 
1 
1 
1 
1 
1 
1 
1 
1 
D_{2} 
1 
α 
i 
β 
−1 
−α 
−i 
−β 
D_{3} 
1 
i 
−1 
−i 
1 
i 
−1 
−i 
D_{4} 
1 
β 
−i 
α 
−1 
−β 
i 
−α 
D_{5} 
1 
−1 
1 
−1 
1 
−1 
1 
−1 
D_{6} 
1 
−α 
i 
−β 
−1 
α 
−i 
β 
D_{7} 
1 
−i 
−1 
i 
1 
−i 
−1 
i 
D_{8} 
1 
−β 
−i 
−α 
−1 
β 
i 
α 
$\alpha =\mathrm{exp}\left(\pi i/4\right)=\frac{1}{2}\sqrt{2}\left(1+i\right),\text{}\text{}\text{}\text{}\beta =\mathrm{exp}\left(3\pi /4\right)=\frac{1}{2}\sqrt{2}\left(1i\right)$ 
Table A.5 Character table for the point groups 8mm, 8 22, and $8\overline{2}\text{m}$
Class: 
E 
A 
A_{2} 
A_{3} 
A_{4} 
B 
AB 

Order: 
1 
8 
4 
8 
2 
2 
2 
No.: 
1 
2 
2 
2 
1 
4 
4 
D_{1} 
1 
1 
1 
1 
1 
1 
1 
D_{2} 
1 
1 
1 
1 
1 
−1 
−1 
D_{3} 
1 
−1 
1 
−1 
1 
1 
−1 
D_{4} 
1 
−1 
1 
−1 
1 
−1 
1 
D_{5} 
2 
$\sqrt{2}$ 
0 
$\sqrt{2}$ 
−2 
0 
0 
D_{6} 
2 
0 
−2 
0 
2 
0 
0 
D_{7} 
2 
$\sqrt{2}$ 
0 
$\sqrt{2}$ 
−2 
0 
0 
Table A.6 Character tables for the point groups 23 and 432
Class: 
E 
A 
A_{2} 
B 

Order: 
1 
3 
3 
2 
No.: 
1 
4 
4 
3 
D_{1} 
1 
1 
1 
1 
D_{2} 
1 
ω 
ω^{2} 
1 
D_{3} 
1 
ω^{2} 
ω 
1 
D_{4} 
3 
0 
0 
−1 
ω=exp(2πi⧏3) 
Class: 
E 
B 
A_{2} 
A 
AB 

Order: 
1 
3 
2 
4 
2 
No.: 
1 
8 
3 
6 
6 
D_{1} 
1 
1 
1 
1 
1 
D_{2} 
1 
1 
1 
−1 
−1 
D_{3} 
2 
−1 
2 
0 
0 
D_{4} 
3 
0 
−1 
1 
−1 
D_{5} 
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 
Φ 
τ 
0 
−1 
D3 
3 
τ 
Φ 
0 
−1 
D4 
4 
−1 
−1 
1 
0 
D5 
5 
0 
0 
−1 
1 
$\tau =\left(\sqrt{5}+1\right)/2,\text{}\text{}\text{}\text{}\text{}\Phi =\left(\sqrt{5}1\right)/2$ 
(p.435) A.9.3 Examples 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 fourdimensional superspace groups Pcmn(00_{γ}) and Pcmn(00_{γ})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 m_{x}, m_{y}, and m_{z}, and because m_{z}(00_{γ}) = (00−_{γ}) it has an internal component – 1. Therefore, the fourdimensional generators are (m_{x}, 1), (m_{y}, 1), and (m_{x}, −1). The three generators of the superspace group in four dimensions act as follows on a point with lattice coordinates x, y, z, u:
Example 2. The fourdimensional superspace group I4_{1} ⧏acd(00_{γ})s0s0 has basic space group I4_{1} ⧏acd. This has three generators for the point group, 4_{z}, m_{x}, and m_{z}. The latter changes the sign of the modulation wave vectorγc ^{*}. Therefore, the fourdimensional operation (m_{z}, −1) belongs to the point group. The threedimensional group is the group 142 in ITCA. The action of the three generators in the fourdimensional superspace group transforms (x, y, z, u) into
Example 3. The group Abm2($\text{Abm}2\left(\frac{1}{2}0\gamma \right)$) has threedimensional 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 ${a}_{c}^{\ast}={a}^{\ast}/2,\text{}{b}_{\text{c}}^{\ast}={b}^{\ast}$, and ${\text{c}}_{c}^{\ast}={\text{c}}^{\ast}$, 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 nonprimitive translation associated with (m_{x}, 1).
A.9.4 Superspace groups for quasiperiodic structures with five, eight, ten, twelvefold, 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 threedimensional quasicrystals with point groups with five, eight, ten or twelvefold 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 {Rv }. 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 a_{1} is indicated by the transformed point x + 1, y, z, u, v, …, and the element {Rv } by the transformed point r′ with coordinates x′, y′, z′, …. For nonsymmorphic 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)
Table A.8 Superspace groups: icosahedral family
Symbol 
Short 
Rank Generators 
Reflection Dimension 


P532(5^{2}32) 
P532 
6 
(x, z, u, v, w, y) 

$\left(w,x,u,v,\overline{z},\overline{u},y\right)$ 

P5132(5232) 
P5132 
6 
$\begin{array}{l}\left(x+\frac{1}{5},\text{}z\text{},u+\frac{1}{5},\text{}v,w,y\frac{1}{5}\right)\\ \left(w,x,u,v,\overline{z},\overline{u},y\right)\end{array}$ 
h2 = h3 = h4 = h5 = h6: h1 = 5m 
I532(5232) 
I532 
6 
(x, z, u, v, w, y) 

$\left(w,x,u,v,\overline{z},\overline{u},y\right)$ 

Σ_{i} hi = even 

I5132(5232) 
I1532 
6 
$\begin{array}{l}\left(x+\frac{1}{5},z+\frac{1}{5},u\frac{1}{5},v+\frac{2}{5},w\frac{1}{5},y\frac{1}{5}\right)\\ \left(w,x,u,v,\overline{z},\overline{u},y\right)\end{array}$ 
h2 = h3 = h4 = h5 = h6: hP1 = 5m 
Σ_{i} hi = even 

F532(5232) 
F532 
6 
(x, z, u, v, w, y) 

$\left(w,x,u,v,\overline{z},\overline{u},y\right)$ 

hi + hj = even (all i, j) 

F532(5232) 
F532 
6 
$\begin{array}{l}\left(x\frac{3}{10},z+\frac{1}{10},u+\frac{1}{5},v,w\frac{1}{5},y\frac{1}{5}\right)\\ \left(w,x,u,v,\overline{z},\overline{u},y\right)\end{array}$ 
hi + hj = even (all i, j) 
$\text{P}\overline{53}\text{m}\left({5}^{2}3\text{m}\right)\text{}\text{}\text{}\text{}\text{}\text{P}\overline{53}\text{m}$ 
6 
$\begin{array}{l}\left(x,z,u,v,w,y\right)\\ \left(w,x,u,v,\overline{z},\overline{u},y\right)\\ \left(\overline{x},\overline{y},\overline{z},\overline{u},\overline{v},\overline{w}\right)\end{array}$ 

$\text{P}\overline{53}{\text{m}}_{34}\left(\overline{53}\text{m}\right)\text{}\text{}\text{}\text{}\text{P}\overline{53}\left(\text{cd}\right)$ 
6 
$\begin{array}{l}\left(x,z+\frac{1}{2},u+\frac{1}{2},v+\frac{1}{2},w+\frac{1}{2},y\right)\\ \left(w,x,u,v,\overline{z},\overline{u},y\right)\\ \left(\overline{x},\overline{y},\overline{z},\overline{u},\overline{v},\overline{w}\right)\end{array}$ 
h5 = −h2, h6 = −h1: h3 + h4= even 

$\text{I}\overline{53}\text{m}\left({5}^{2}32\right)\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{I}\overline{53}\text{m}$ 
6 
$\begin{array}{l}\left(x,z,u,v,w,y\right)\\ \left(w,x,u,v,\overline{z},\overline{u},y\right)\\ \left(\overline{x},\overline{y},\overline{z},\overline{u},\overline{v},\overline{w}\right)\end{array}$ 
Σ_{i} hi = even 

$\text{F}\overline{53}\text{m}\left({5}^{2}32\right)\text{}\text{}\text{}\text{}\text{}\text{}\text{}\left(\text{F}\overline{53}\text{m}\right)$ 
6 
$\begin{array}{l}\left(x,z,u,v,w,y\right)\\ \left(w,x,u,v,\overline{z},\overline{u},y\right)\\ \left(\overline{x},\overline{y},\overline{z},\overline{u},\overline{v},\overline{w}\right)\end{array}$ 
hi + hj = even (all i, j) 

$\text{F}\overline{53}a\left({5}^{2}32\right)\text{}\text{}\text{}\text{}\text{}\text{}\text{F}\overline{53}a$ 
6 
$\begin{array}{l}\left(x,z,u,v,w,y\right)\\ \left(w,x,u,v,\overline{z},\overline{u},y\right)\\ \left(\overline{x},\overline{y},\overline{z},\overline{u},\overline{v},\overline{w}\right)\end{array}$ 
h6 = h1, h5 = h2, h4 = −h3: h1 even hi + hj = even (all i, j) 