InternationalMathematical, physical and chemical tablesTables for Crystallography Volume C Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C, ch. 9.8, pp. 937-945
## Section 9.8.4. Theoretical foundation |

A periodic crystal structure is defined in a three-dimensional Euclidean space *V* and is invariant with respect to translations **n** which are integral linear combinations of three fundamental ones : These translations are linearly independent and span a lattice Λ. The *dimension* of Λ is the dimension of the space spanned by and the *rank* is the (smallest) number of free generators of those integral linear combinations. In the present case, both are equal to three. Accordingly, The elements of are triples of integers that correspond to the coordinates of the lattice points. The Bragg reflection peaks of such a crystal structure are at the positions of a reciprocal lattice Λ*, also of dimension and rank equal to three. Furthermore, the Fourier wavevectors **H** belong to Λ* (after identification of lattice vectors with lattice points): where is the reciprocal basis The two corresponding metric tensors *g* and , are positive definite and dual: We now consider crystal structures defined in the same three-dimensional Euclidean space *V* with Fourier wavevectors that are integral linear combinations of *n* = (3 + *d*) fundamental ones , , : The components are the indices labelling the corresponding Bragg reflection peaks.

A crystal is *incommensurate* when *d* > 0 and the vectors linearly independent over the rational numbers. In that case, the crystal does not have lattice periodicity and is said to be *aperiodic*. The above description can still be convenient, even in the case that the vectors are not independent over the rationals: one or more of them is then expressed as rational linear combinations of the others. A typical example is that of a superstructure arising from the (commensurate) modulation of a basic structure with lattice periodicity.

Let us denote by *M** the set of all integral linear combinations of the vectors . These are said to form a *basis*. It is a set of free Abelian generators, therefore the *rank* of *M** is *n*. The *dimension* of *M** is the dimension of the Euclidean space spanned by *M** The elements of are precisely the set of indices introduced above. Mathematically speaking, *M** has the structure of a (free Abelian) module. Its elements are vectors. So we call *M** a *vector module*. This nomenclature is intended as a *generic* characterization. When a series of structures is considered with different values of the components of the last *d* vectors with respect to the first three, the generic values of these components are irrational, but accidentally they may become rational as well. This situation typically arises when considering crystal structures under continuous variation of parameters like temperature, pressure or chemical composition. In the case of an ordinary crystal, rank and dimension are equal, the crystal structure is *periodic*, and the vector module becomes a (reciprocal) lattice.

Lattices and vector modules are, mathematically speaking, free *Z* modules. For such a module, there exists a dual one that is also free and of the same rank. In the periodic crystal case, that duality can be expressed by a scalar product, but for an aperiodic crystal this is no longer possible. It is possible to keep the metrical duality by enlarging the space and considering the vector module *M** as the projection of an *n*-dimensional (reciprocal) lattice in an *n*-dimensional Euclidean space . with the orthogonal projection of onto *V* defined by This corresponds to attaching to the diffraction peak with indices the point of an *n*-dimensional reciprocal lattice having the same set of coordinates. The orthocomplement of *V* in is called internal space and denoted by . The embedding is uniquely defined by the relations where is a basis of and a basis of *M**. The vectors span .

The crystal density ρ in *V* can also be embedded as in by identifying the Fourier coefficients at points of *M** and of having correspondingly the same components. Then is invariant with respect to translations of the lattice with basis dual to (9.8.4.9). In the commensurate case, this correspondence requires that the given superstructure be considered as the limit of an incommensurate crystal [for which the embedding (9.8.4.10) is a one-to-one relation].

As discussed below, point-group symmetries *R* of the diffraction pattern, when expressed in terms of transformation of the set of indices, define *n*-dimensional integral matrices that can be considered as being *n*-dimensional orthogonal transformations in , leaving invariant the Euclidean metric tensors: The crystal classes considered in the tables suppose the existence of main reflections defining a three-dimensional reciprocal lattice. For that case, the embedding can be specialized by making the choice and, correspondingly, with and . These are called *standard lattice bases*.

*Definition 1.* The *Laue point group* of the diffraction pattern is the point group in three dimensions that transforms every diffraction peak into a peak of the same intensity.^{2}

Because all diffraction vectors are of the form (9.8.4.5), the action of an element *R* of the Laue group is given by The (3 + *d*) × (3 + *d*) matrices form a finite group of integral matrices for *K* equal to or to one of its subgroups. A well known theorem in algebra states that then there is a basis in 3 + *d* dimensions such that the matrices on that basis are orthogonal and represent (3 + *d*)-dimensional orthogonal transformations . The corresponding group is a (3 + *d*)-dimensional crystallographic group denoted by . Because *R* is already an orthogonal transformation on *V*, is reducible and can be expressed as a pair of orthogonal transformations, in 3 and *d* dimensions, respectively. The basis on which acts according to is denoted by . It spans a lattice that is the reciprocal of the lattice with basis elements . The pairs , sometimes also noted , leave invariant: where Γ(*R*) is the transpose of .

In many cases, one can distinguish a lattice of main reflections, the remaining reflections being called satellites. The main reflections are generally more intense. Therefore, main reflections are transformed into main reflections by elements of the Laue group. On a standard lattice basis (9.8.4.13), the matrices Γ(*R*) take the special form The transformation of main reflections and satellites is then given by as in (9.8.4.15), the relation with Γ(*R*) being (as already said) where the tilde indicates transposition. Accordingly, on a standard basis one has The set of matrices for *R* elements of *K* forms a crystallographic point group in three dimensions, denoted , having elements *R* of *O*(3), and the corresponding set of matrices forms one in *d* dimensions denoted by with elements of *O*(*d*).

For a modulated crystal, one can choose the (*i* = 1, 2, 3) of a standard basis. These span the (reciprocal) lattice of the basic structure. One can then express the additional vectors (which are modulation wavevectors) in terms of the basis of the lattice of main reflections: The three components of the *j*th row of the (*d* × 3)-dimensional matrix σ are just the three components of the *j*th modulation wavevector with respect to the basis . It is easy to show that the internal components (*i* = 1, 2, 3) of the corresponding dual standard basis can be expressed as This follows directly from (9.8.4.19) and the definition of the reciprocal standard basis (9.8.4.13). From (9.8.4.16) and (9.8.4.17), a simple relation can be deduced between σ and the three constituents , , and of the matrix Γ(*R*): Notice that the elements of are integers, whereas σ has, in general, irrational entries. This requires that the irrational part of σ gives zero when inserted in the left-hand side of equation (9.8.4.21). It is therefore possible to decompose formally σ into parts and as follows. where the sum is over all elements of the Laue group of order *N*. It follows from this definition that This implies The matrix has rational entries and is called the rational part of σ. The part is called the irrational (or invariant) part.

The above equations simplify for the case *d* = 1. The elements are the three components of the wavevector **q**, the row matrix has the components of and = = ±1 since, for *d* = 1, **q** can only be transformed into ±**q**. One has the corresponding relations and The reciprocal-lattice vector that gives the difference between and has as components the elements of the row matrix .

According to the previous section, in the case of modulated structures a standard basis can be chosen (for *M** and correspondingly for ). According to equation (9.8.4.15), for each three-dimensional point-group operation *R* that leaves the diffraction pattern invariant, there is a point-group transformation in the external space (the physical one, so that ) and a point-group transformation in the internal space, such that the pair is a (3 + *d*)-dimensional orthogonal transformation leaving a (3 + *d*)-dimensional lattice invariant. For incommensurate crystals, this internal transformation is unique and follows from the transformation by *R* of the modulation wavevectors [see equations (9.8.4.15) and (9.8.4.18) for the basis vectors]: there is exactly one for each *R*. This is so because in the incommensurate case the correspondence between *M** and is uniquely fixed by the embedding rule (9.8.4.10) (see Subsection 9.8.4.1). Because the matrices Γ(*R*) and the corresponding transformations in the (3 + *d*)-dimensional space form a group, this implies that there is a mapping from the group of elements to the group of elements that transforms products into products, *i.e.* is a group homomorphism. A point group of the (3 + *d*)-dimensional lattice constructed for an incommensurate crystal, therefore, consists of a three-dimensional crystallographic point group , a *d*-dimensional crystallographic point group , and a homomorphism from to .

*Definition 2.* Two (3 + *d*)-dimensional point groups and are *geometrically equivalent* if they are connected by a pair of orthogonal transformations in and , respectively, such that for every from the first group there is an element of the second group such that and .

A point group determines a set of groups of matrices, one for each standard basis of each lattice left invariant.

*Definition 3.* Two groups of matrices are *arithmetically equivalent* if they are obtained from each other by a transformation from one standard basis to another standard basis.

The arithmetic equivalence class of a (3 + *d*)-dimensional point group is fully determined by a three-dimensional point group and a standard basis for the vector module *M** because of relation (9.8.4.15).

In three dimensions, there are 32 geometrically non-equivalent point groups and 73 arithmetically non-equivalent point groups. In one dimension, these numbers are both equal to two. Therefore, one finds all (3 + 1)-dimensional point groups of incommensurately modulated structures by considering all triples of one of the 32 (or 73) point groups, for each one of the two one-dimensional point groups and all homomorphisms from the first to the second.

Analogously, in (3 + *d*) dimensions, one takes one of the 32 (73) groups, one of the *d*-dimensional groups, and all homomorphisms from the first to the second. If one takes all triples of a three-dimensional group, a *d*-dimensional group, and a homomorphism from the first to the second, one finds, in general, groups that are equivalent. The equivalent ones still have to be eliminated in order to arrive at a list of non-equivalent groups.

The Laue group of the diffraction pattern is a three-dimensional point group that leaves the positions (and the intensities)^{3} of the diffraction spots as a set invariant, thus the vector module *M** also. As discussed in Subsection 9.8.4.2, each of the elements of the Laue group can be combined with an orthogonal transformation in the internal space. The resulting point group in 3 + *d* dimensions leaves the lattice * invariant for which the vector module *M** is the projection. Conversely, if one has a point group that leaves the (3 + *d*)-dimensional lattice invariant, its three-dimensional (external) part with elements *R _{E}* =

*R*leaves the vector module invariant.

*Definition 4.* The holohedry of the lattice * is the subgroup of the direct product *O*(3) × *O*(*d*), *i.e.* the group of all pairs of orthogonal transformations that leave the lattice invariant.

This choice is possible because the point groups are reducible, *i.e.* leave the subspaces *V* and of the direct sum space invariant. In the case of an incommensurate crystal, the projection of * on *M** is one-to-one as one can see as follows. The vector of * is projected on . The vectors projected on the null vector satisfy, therefore, the relation . For an incommensurate phase, the basis vectors are rationally independent, which means that and for any *i* and *j*. Consequently, precisely one vector of * is projected on each given vector of *M**.

Suppose now *R* = 1. This transformation leaves the component of every vector belonging to in *V* invariant. If is the corresponding orthogonal transformation in of an element of the point group, a vector with component is transformed into a vector with component . Since a given **H** is the component of only one vector of *, this implies . Consequently, is also the identity transformation. Therefore, for incommensurate modulated phases, there are no point-group elements with and . For commensurate crystal structures embedded in the superspace, this is different: point-group elements with internal component different from the identity associated with an external component equal to unity can occur.

For modulated crystal structures, the holohedral point group can be expressed with respect to a lattice basis of standard form (9.8.4.13). It is then faithfully represented by integral matrices that are of the form indicated in (9.8.4.17) and (9.8.4.18).

*Definition 5.* A *crystallographic system* is a set of lattices having geometrically equivalent holohedral point groups.

In this way, a given holohedral point group (and even each crystallographic point group) belongs to exactly one system. Two lattices belong to the same system if there are orthonormal bases in *V* and in , respectively, such that the holohedral point groups of the two lattices are represented by the same set of matrices.

The elements of a (3 + *d*)-dimensional superspace group are pairs of Euclidean transformations in 3 and *d* dimensions, respectively: *i.e.* are elements of the direct product of the corresponding Euclidean groups. The elements form a three-dimensional space group, but the same does not hold for the elements of . This is because the internal translations also contain the `compensating' transformations associated with the corresponding translation **v** in *V* [see (9.8.4.32)]. In other words, a basis of the lattice does not simply split into one basis for *V* and one for .

As for elements of a three-dimensional space group, the translational component of the element can be decomposed into an intrinsic part and an origin-dependent part : with where *n* denotes the order of the element *R*. In particular, for *d* = 1 the intrinsic part of is equal to if = = +1 and vanishes if = −1. The latter means that for *d* = 1 there is always an origin in the internal space such that the internal shift can be chosen to be zero for an element with = −1.

The internal part of the intrinsic translation can itself be decomposed into two parts. One part stems from the presence of a translation in the external space. The lattice of the (3 + *d*)-dimensional space group has basis vectors The internal part of the first three basis vectors is according to equation (9.8.4.20). The three-dimensional translation then entails a *d*-dimensional translation −Δ**v** in given by These are the so-called compensating translations. Hence, the internal translation can be decomposed as where .

This decomposition, however, does still depend on the origin. Consider the case *d* = 1. Then an origin shift **s** in the three-dimensional space changes the translation **v** to **v** + (1 − *R*)**s** and its internal part −Δ**v** = to . This implies that for the case that = 1 the part δ changes to , because is invariant under *R*. Therefore, δ changes, in general. The internal translation however, is invariant under an origin shift in *V*.

*Definition 7.* *Equivalent superspace groups.* Two superspace groups are equivalent if they are isomorphic and have point groups that are arithmetically equivalent.

Another definition leading to the same partition of equivalent superspace groups considers equivalency with respect to affine transformations among bases of standard form.

This means that two equivalent superspace groups admit standard bases such that the two space groups are represented by the same set of (4 + *d*)-dimensional affine transformation matrices. We recall that an *n*-dimensional Euclidean transformation if referred to a basis of the space can be represented isomorphically by an (*n* + 1)-dimensional matrix, of the form with an *n* × *n* matrix and an *n*-dimensional column matrix, all with real entries.

A (3 + *d*)-dimensional space group that leaves a function invariant maps points in (3 + *d*)-space to points where the function has the same value. The atomic positions of a modulated crystal represent such a pattern, and the superspace group leaving the crystal invariant leads to a partition into equivalent atomic positions. These relations can be formulated either in (3 + *d*)-dimensional space or, equally well, in three-dimensional space. As a simple case, we first consider a crystal with a one-dimensional occupation modulation: this implies *d* = 1. Again, as in §9.8.1.3.2, we omit to indicate the basis vectors and and give only the corresponding components.

An element of the (3 + 1)-dimensional superspace group is a pair of Euclidean transformations in *V* and , respectively. This element maps a point located at to one at . Suppose the probability for the position to be occupied by an atom of species *A* is given by where . By , the position is transformed to the equivalent position = *R***n** + *R***r**_{j} + **v**. As the crystal is left invariant by the superspace group, the occupation probability on equivalent points has to be the same: This implies that for the structure in the three-dimensional space one has the relation In terms of the modulation function this means In the same way, one derives the following property of the modulation function: Analogously, for a displacive modulation, the position with displacement , where , is transformed to with displacement To be invariant, the displacement function has to satisfy the relation The expressions for are straightforward generalizations of these.

The scattering from a set of atoms at positions is described in the kinematic approximation by the structure factor: where is the atomic scattering factor. For an incommensurate crystal phase, this structure factor is equal to the structure factor of the crystal structure embedded in 3 + *d* dimensions, where **H** is the projection of on . This structure factor is expressed by a sum of the products of atomic scattering factors and phase factors over all particles in the unit cell of the higher-dimensional lattice. For an incommensurate phase, the number of particles in such a unit cell is infinite: for a given atom in space, the embedded positions form a dense set on lines or hypersurfaces of the higher-dimensional space. Disregarding pathological cases, the sum may be replaced by an integral. Including the possibility of an occupation modulation, the structure factor becomes (up to a normalization factor) where the first sum is over the different species, the second over the positions in the unit cell of the basic structure, the integral over a unit cell of the lattice spanned by in ; is the atomic scattering factor of species *A*, is the probability of atom *j* being of species *A* when the internal position is **t**.

In particular, for a given atomic species, without occupational modulation and a sinusoidal one-dimensional displacive modulation According to (9.8.4.45), the structure factor is For a diffraction vector **H** = **K** + *m***q**, this reduces to For a general one-dimensional modulation with occupation modulation function and displacement function , the structure factor becomes Because of the periodicity of and , one can expand the Fourier series: and consequently the structure factor becomes The diffraction from incommensurate crystal structures has been treated by de Wolff (1974), Yamamoto (1982*a*,*b*), Paciorek & Kucharczyk (1985), Petricek, Coppens & Becker (1985), Petříček & Coppens (1988), Perez-Mato *et al.* (1986, 1987), and Steurer (1987).

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