Incommensurate and commensurate modulated structures

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 907-955
https://doi.org/10.1107/97809553602060000624

Chapter 9.8. Incommensurate and commensurate modulated structures

T. Janssen,^a A. Janner,^a A. Looijenga-Vos^b and P. M. de Wolff^c ^‡

^a Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,^bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and ^cMeermanstraat 126, 2614 AM, Delft, The Netherlands

This chapter discusses incommensurate and commensurate modulated structures. After an introduction to the concepts used to describe modulated structures, including the basic ideas of higher-dimensional crystallography, four-dimensional space groups and displacive and occupational modulation, superspace-group determination is outlined. Tables of Bravais lattices, geometric and arithmetic crystal classes, and superspace groups (with reflection conditions) are presented and their use is illustrated with different examples.

Keywords: arithmetic crystal classes; basic structural features; Bravais classes; centring reflection conditions; composite crystal structures; crystallographic systems; higher-dimensional crystallography; holohedry; incommensurate modulated structures; lattices; Laue classes; modulated crystal structures; modulations; occupation modulations; point groups; structure factors; superspace groups.

9.8.1. Introduction

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9.8.1.1. Modulated crystal structures

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Lattice periodicity is a fundamental concept in crystallography. This property is widely considered as essential for the characterization of the concept of a crystal. In recent decades, however, more and more long-range-ordered condensed-matter phases have been observed in nature that do not have lattice periodicity, but nevertheless only differ from normal crystal phases in very subtle and not easy to observe structural and physical properties.

It is convenient to extend the concept of a crystal in such a way that it includes these ordered phases as well. If properly applied, crystallographic symmetry concepts are then still valid and of relevance for structural (and physical) investigation. Instead of limiting the validity of crystallographic notions, these developments show how rich the crystallographic ordering can be.

Incommensurate crystals are characterized by Bragg reflection peaks in the diffraction pattern that are well separated but do not belong to a lattice and cannot be indexed by three integer indices. This means that the description in terms of three-dimensional point and space groups, as given in IT A (2005 ), breaks down. It is possible, however, to generalize the crystallographic concepts on which the usual description is based. The ensuing symmetry groups and their equivalence classes are different from those of three-dimensional crystallography. The main properties of these new groups and the specific problems posed by their application to incommensurate crystal structures form the subject of the following sections.

The tables presented here, and their explanation, have been restricted to cover mainly the simplest case: modulated crystal structures (in particular, those with a single modulation wave). Nevertheless, most of the basic concepts also apply for the other classes of incommensurate crystal structures.

A modulation is here considered to be a periodic deformation of a `basic structure' having space-group symmetry. If the periodicity of the modulation does not belong to the periodicities of the basic structure, the modulated crystal structure is called incommensurate.

That description is based on the observation in the diffraction pattern of modulated structures of main reflections , situated on a reciprocal lattice, and additional reflections, generally of weaker intensity, called satellites.

As far as we know, the first indications that complex crystal structures could exist not having normal lattice periodicity came, at the end of the 19th century, from studies of the morphology of the mineral calaverite. It was found that the faces of this compound do not obey the law of rational indices (Smith, 1903 ; Goldschmidt, Palache & Peacock, 1931 ). Donnay (1935 ) showed that faces that could not be indexed correspond to additional reflections in the X-ray diffraction pattern. From the experience with manufacturing of optical gratings, it was already known that periodic perturbation gives rise to additional diffraction spots. Dehlinger (1927 ) used this theory of `Gittergeister' (lattice ghosts) to explain line broadening in Debye–Scherrer diagrams of metals and alloys. Preston (1938 ) called the `Gittergeister' he found in diffraction from an aluminium–copper alloy `satellites'. Periodic displacement (modulation) waves were considered by Daniel & Lipson (1943 ) as the origin of satellites in a copper–iron–nickel alloy. Complex magnetic ordering giving rise to satellites was found in magnetic crystals (Herpin, Meriel & Villain, 1960 ; Koehler, Cable, Wollan & Wilkinson, 1962 ). In the early 1970's, the importance of the irrationality was realized and the term `incommensurate phase' was introduced for modulated crystal phases with wavevectors that have irrational components. Such phases were found in insulators as well (Tanisaki, 1961 , 1963 ; Brouns, Visser & de Wolff, 1964 ). The remarkable morphological properties of calaverite could also be related to the existence of an incommensurate modulation (Dam, Janner & Donnay, 1985 ). Structures that are well ordered, show sharp incommensurate spots in the diffraction pattern, but cannot be described as a modulation of a lattice periodic system, were also discovered, indicating that incommensurability is not restricted to modulated crystals. Examples are the composite (or intergrowth) crystals (Jellinek, 1972 ) and, more recently, quasicrystals (Shechtman, Blech, Gratias & Cahn, 1984 ). The latter not only lack three-dimensional lattice periodicity, but also show non-crystallographic symmetry elements , such as five-, eight- or twelvefold axes. For a review, see Janssen & Janner (1987 ).

We denote a basis for the lattice of main reflections by a*, b*, c*. The satellites are separated from the main reflections by vectors that are integral linear combinations of some basic modulation vectors denoted by q_j $[(\,j=1,2,\ldots,d)]$ . Accordingly, the positions of the Bragg reflections are given by $[{\bf H} =h{\bf a}^*+k{\bf b}^*+ l{\bf c}^*+ m_1{\bf q}_1 + m_2{\bf q}_2 + \ldots + m_d{\bf q_d}. \eqno (9.8.1.1)]$ If the modulation is incommensurate, the set of all positions (9.8.1.1) does not form a lattice. Mathematically, it has the structure of a Z-module, and its elements are three-dimensional vectors (of the reciprocal space). Accordingly, we call it a (reciprocal) vector module.

The modulation that gives rise to the satellites can be:

(i) a displacive modulation , consisting of a periodic displacement from the atomic positions of the basic structure;
(ii) an occupation modulation , in which the atomic positions of the basic structure are occupied with a periodic probability function.

Mixed forms also occur. Other modulation phenomena found are, for example, out-of-phase structures and translation interface modulated structures. The satellite reflections of modulated structures for the various types of modulation were systematically studied by Korekawa (1967 ).

For an incommensurate structure, at least one component of a basic modulation wavevector q_j with respect to the lattice of main reflections is irrational. As argued later, even in the case of a commensurate modulation, i.e. when all components of these modulation wavevectors are rational, it is sometimes convenient to adopt the description of equation (9.8.1.1) and to apply the formalism developed for the incommensurate crystal case.

The modulated crystal case presented above is only the simplest one giving rise to incommensurate crystal structures. Another class is represented by the so-called composite or intergrowth crystal structures. Here the basic structure consists of two or more subsystems each with its own space-group symmetry (neglecting mutual interaction) such that, in general, the corresponding lattices are incommensurate. This means that these lattices are not sublattices of a common one. Actually, a mutual interaction is in general present giving rise to periodic deviations, i.e. to modulations. These crystals are different from the previous ones, in the sense that they show more than one set of main reflections. In the case of an intergrowth crystal, the Bragg reflections can be labelled by a finite set of integral indices as well and the Fourier wavevectors form a vector module as defined above. The difference from the previous case is that the basic structure itself is incommensurate, even if one disregards modulation.

A third class of incommensurate crystals found in nature is represented by the quasicrystals (examples of which are the icosahedral phases). In that case also, the basic structure is incommensurate, the difference from the previous case being that the splitting in subsystems of main reflections and/or in main reflections and satellites is not a natural one, even in the lowest approximation.

It is justified to assume that crystal phases allow for an even wider range of ordered structures than those mentioned above. Nevertheless, all the cases studied so far allow a common approach based on higher-dimensional crystallography, i.e. on crystallographic properties one obtains after embedding the crystal structure in a higher-dimensional Euclidean space.

To present that approach in its full generality would require a too abstract language. On the other hand, restriction to the simplest case of a modulated crystal would give rise to misleading views. It what follows, the basic ideas are formulated as simply as possible, but still within a general point of view. Their application and the illustrative examples are mainly restricted to the modulated crystal case. The tables presented here, even if applicable to more general cases, cover essentially the one-dimensional modulated case [d = 1 in equation (9.8.1.1)].

9.8.1.2. The basic ideas of higher-dimensional crystallography

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Incommensurate modulated crystals are systems that do not obey the classical requirements for crystals. Nevertheless, their long-range order is as perfect as that of ordinary crystals. In the diffraction pattern they also show sharp, well separated spots, and in the morphology flat faces. Dendritic crystallization with typical point-group symmetry is observed in both commensurate and incommensurate materials. Therefore, we shall consider both as crystalline phases and generalize for that reason the concept of a crystal. The positions of the Bragg diffraction peaks given in (9.8.1.1) are a special case. In general, they are elements of a vector module M* and can be written as $[{\bf H}=\textstyle\sum\limits^n_{i=1}h_i{\bf a}^*_i,\quad\hbox{integers $h_i$.} \eqno (9.8.1.2)]$ This leads to the following definition of crystal.

An ideal crystal is considered to be a matter distribution having Fourier wavevectors expressible as an integral linear combination of a finite number (say n) of them and such that its diffraction pattern is characterized by a discrete set of resolved Bragg peaks, which can be indexed accordingly by a set of n integers $[h_1,h_2,\ldots,h_n]$ .

Implicit in this definition is the possibility of neglecting diffraction intensities below a given threshold, allowing one to identify and to label individual Bragg peaks even when n is larger than the dimension m of the crystal, which is usually three. Actually, for incommensurate crystals, the unresolved Bragg peaks of arbitrarily small intensities form a dense set, because they may come arbitrarily close to each other.

Here some typical examples are indicated. In the normal crystal case, n = m = 3 and $[{\bf a}^*_1]$ , $[{\bf a}^*_2]$ , $[{\bf a}^*_3]$ are conventionally denoted by a*, b*, c* and the indices [h_1, h_2, h_3] by h, k, l.

In the case of a one-dimensionally modulated crystal, which can be described as a periodic plane wave deformation of a normal crystal (defining the basic structure), one has n = 4. Conventionally, $[{\bf a}^*_1]$ , $[{\bf a}^*_2]$ , $[{\bf a}^*_3]$ are chosen to be a*, b*, and c* generating the positions of the main reflections, whereas $[{\bf a}^*_4]$ = q is the wavevector of the modulation. The corresponding indices are usually denoted by h, k, l and m. Also, crystals having two- and three-dimensional modulation are known: in those cases, n = 5 and n = 6, respectively.

In the case of a composite crystal, one can identify two or more subsystems, each with its own space-group symmetry. As an example, consider the case where two subsystems share a* and b* of their reciprocal-lattice basis, whereas they differ in periodicity along the c axis and have, respectively, $[{\bf c}^*_1]$ and $[{\bf c}^*_2]$ as third basis vector. Then again, n = 4 and $[{\bf a}^*_1,\ldots, {\bf a}^*_4]$ can be chosen as $[{\bf a}*]$ , $[{\bf b}*]$ , $[{\bf c}^*_1]$ , and $[{\bf c}^*_2]$ . The indices can be denoted by h, k, l₁, l₂. In general, the subsystems interact, giving rise to modulations and possibly (but not necessarily) to a larger value of n. In addition to the main reflections from the undistorted subsystems, satellite reflections then occur.

In the case of a quasicrystal, the Bragg reflections require more than three indices, but they do not arise from structurally different subsystems having lattice periodicity. So, for the icosahedral phase of AlMn alloy, n = 6, and one can take for $[{\bf a}^*_1]$ , $[{\bf a}^*_2]$ , $[{\bf a}^*_3]$ a cubic basis a*, b*, and c* and for $[{\bf a}^*_4]$ , $[{\bf a}^*_5]$ , $[{\bf a}^*_6]$ then τa*, τb*, and τc*, respectively, with τ the golden number $[[1+\sqrt5]/2]$ .

The Laue point group [P_L] of a crystal is the point symmetry group of its diffraction pattern. This subgroup of the orthogonal group O(3) is of finite order. The finite order of the group follows from the discreteness of the (resolved) Bragg peak positions, implying a finite number only of peaks of the same intensity lying at a given distance from the origin.

Under a symmetry rotation R, the indices of each reflection are transformed into those of the reflection at the rotated position. Therefore, a symmetry rotation is represented by an $[n\times n]$ non-singular matrix Γ(R) with integral entries. Accordingly, a Laue point group admits an n-dimensional faithful integral representation $[\Gamma(P_L)]$ . Because any finite group of matrices is equivalent with a group of orthogonal matrices of the same dimension, it follows that:

(1) is isomorphic to an n-dimensional crystallographic point group;
(2) there exists a lattice basis $[a^*_{s1}, \ldots, a^*_{sn}]$ of a Euclidean n-dimensional (reciprocal) space, which projects on the Fourier wavevectors $[{\bf a}^*_1]$ , $[\ldots, {\bf a}^*_n]$ ;
(3) the three-dimensional Fourier components $[\hat\rho(h_1,\ldots, h_n)]$ of the crystal density function ρ(r) can be attached to corresponding points of an n-dimensional reciprocal lattice and considered as the Fourier components of a density function $[\rho_s(r_s)]$ having lattice periodicity in that higher-dimensional space ( is an n-dimensional position vector).

Such a procedure is called the superspace embedding of the crystal.

Note that this procedure only involves a reinterpretation of the structural data (expressed in terms of Fourier coefficients): the structural information in the n-dimensional space is exactly the same as that in the three-dimensional description.

The symmetry group of a crystal is then defined as the Euclidean symmetry group of the crystal structure embedded in the superspace.

Accordingly, the symmetry of a crystal whose diffraction pattern is labelled by n integral indices is an n-dimensional space group.

The equivalence relation of these symmetry groups follows from the requirement of invariance of the equivalence class with respect to the various possible choices of bases and embeddings. Because of that, the equivalence relation is not simply the one valid for n-dimensional crystallography.

In the case of modulated crystals, such an equivalence relation has been worked out explicitly, giving rise to the concept of the (3 + d)-dimensional superspace group.

The concepts of point group, lattice holohedry, Bravais classes, systems of non-primitive translations, and so on, then follow from the general properties of n-dimensional space groups together with the (appropriate) equivalence relations.

A glossary of symbols is given in Appendix 9.8.1 and a list of definitions in Appendix 9.8.2 .

9.8.1.3. The simple case of a displacively modulated crystal

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9.8.1.3.1. The diffraction pattern

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To introduce what follows, the simple case of a displacively modulated crystal structure is considered. The point-atom approximation is adopted and the modulation is supposed to be a sinusoidal plane wave.

This means that the structure can be described in terms of atomic positions of a basic structure with three-dimensional space-group symmetry, periodically displaced according to the modulation wave. Writing for the position of the jth particle in the unit cell of the basic structure given by the lattice vector n: $[{\bf r}_0({\bf n}, j) = {\bf n} + {\bf r}_j, \eqno (9.8.1.3)]$ the position of the same particle in the modulated structure is given by $[{\bf r}({\bf n}, j)={\bf n} + {\bf r}_j+{\bf U}_j\sin[2\pi{\bf q}\cdot({\bf n}+{\bf r}_j)+\varphi_j], \eqno (9.8.1.4)]$ where q is the wavevector of the modulation and $[{\bf U}_j]$ is the polarization vector for the jth particle's modulation. (This is not the most general sinusoidal modulation, because different components $[U_{j\alpha}]$ may have different phases $[\varphi_{j\alpha}, \ \alpha = 1,2,3]$ .) In general, the symmetry of the modulated structure is different from that of the basic structure and the only translations that leave the modulated structure invariant are those lattice translations m of the basic structure satisfying the condition $[{\bf q}\cdot{\bf m}]$ = integer . If the components α, β and γ of the wavevector q with respect to the basis a*, b*, c* are all rational numbers, there is a full lattice of such translations, the structure then is a superstructure of the original one and has space-group symmetry. If at least one of the α, β, and γ is irrational, the structure does not have three-dimensional lattice translation symmetry. Nevertheless, the crystal structure is by no means disordered: it is fully determined by the basic structure and the modulation wave(s).

The crystalline order is reflected in the structure factor, which is given by the expression $[\eqalignno{ S_{\bf H} &=\textstyle\sum\limits_{{\bf n},\,j}\,f_j\exp[2\pi i{\bf H}\cdot {\bf r}({\bf n},j)] \cr &=\textstyle\sum\limits_{{\bf n},\,j}\,f_j\exp[2\pi i {\bf H}\cdot ({\bf n}+{\bf r}_j)] \cr &\quad\times\exp\{2\pi i{\bf H}\cdot{\bf U}_j\sin[2\pi{\bf q}\cdot({\bf n}+{\bf r}_j)+\varphi_j]\}, &(9.8.1.5)}]$ where [f_j] is the atomic scattering factor (which still, in general, depends on H). Using the Jacobi–Anger relation, one can rewrite (9.8.1.5) as $[\eqalignno{ S_{\bf H} &=\textstyle\sum\limits_{\bf n}\textstyle\sum\limits_j\textstyle\sum\limits^\infty_{m=-\infty}\exp[2\pi i({\bf H}-m{\bf q})\cdot({\bf n}+{\bf r}_j)] \cr&\quad\times f_j\exp(-im\varphi_j)\, J_{-m}(2\pi{\bf H}\cdot{\bf U}_j), & (9.8.1.6)}]$ where [J_m(x)] is the mth-order Bessel function. The summation over n results in a sum of δ functions on the positions of the reciprocal lattice: $[\Delta({\bf H}-m{\bf q})=\textstyle\sum\limits_{h,k,l}\delta({\bf H}-m{\bf q}-h{\bf a}^*-k{\bf b}^* - l{\bf c}^*).]$ Consequently, the structure factor $[S_{\bf H}]$ vanishes unless there are integers h, k, l, and m such that $[{\bf H}=h{\bf a}^*+k{\bf b}^*+l{\bf c}^*+m {\bf q}. \eqno (9.8.1.7)]$ If q is incommensurate, i.e. if there is no integer N such that Nq belongs to the reciprocal lattice spanned by a*, b*, c*, one needs more than three integers (in the present case four) for indexing H. This is characteristic for incommensurate crystal phases.

In the diffraction pattern of such a modulated phase, one distinguishes between main reflections (for which m = 0) and satellites (for which $[m\ne 0)]$ . The intensities of the satellites fall off rapidly for large m so that the observed diffraction spots remain separated, although vectors of the form (9.8.1.7) may come arbitrarily close to each other.

In the commensurate case also, there are main reflections and satellites, but, since here there is an integer N such that Nq belongs to the reciprocal lattice, one may restrict the values of m in (9.8.1.7) to the range from 0 to N − 1.

9.8.1.3.2. The symmetry

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There is more than one way for expressing the long-range order present in an incommensurate crystal in terms of symmetry. One natural way is to adopt the point of view that the measuring process limits the precision in the determination of a modulation wavevector. Accordingly, one can try an approximation of the modulation wavevector q by a commensurate one: an irrational number can be approximated arbitrarily well by a rational one.

There are two main disadvantages in this approach. Firstly, a good approximation implies, in general, a large unit cell for the corresponding superstructure, which involves a large number of parameters. Secondly, the space group one finds may depend essentially on the rational approximation adopted. Consider, for example, an orthorhombic basic structure with space group [Pcm2_1] and modulation with wavevector $[{\bf q}=\gamma{\bf c}^*]$ , polarization along the b direction and positions of the particles given by $[{\bf r}({\bf n},j)={\bf n}+{\bf r}_j+{\bf U}\sin[2\pi{\bf q}\cdot({\bf n}+{\bf r}_j)], \eqno (9.8.1.8)]$ where for convenience the polarization has been taken as independent of j. Under the glide reflection $[\{m_x|(m+{1\over2}){\bf c}\}]$ (integer m) present in the basic structure, the transformed positions are $[\eqalignno{ {\bf r}'({\bf n}, j)= {\bf r}+{\bf U}\sin&\{2\pi[{\bf q}\cdot {\bf r}-\gamma(m+\textstyle{1\over2})]\}, \cr &\hbox{with }{\bf r} ={\bf n}+{\bf r}_j. & \ (9.8.1.9)}]$ Consider the rational approximation of γ given by P/Q, with P and Q relatively prime integers. Then such a glide transformation as given above is (for certain values of m) a symmetry of the modulated structure only if P is even and Q odd, because only in that case does the equation $[P(m+{1\over2})=lQ]$ have a solution for integers l and m. Analogously, the mirror [m_y] only occurs as a glide plane if P/Q is odd/even, whereas the screw axis along the z axis requires the case odd/odd. Hence, if, for example, one approximates the same irrational number successively by 5/12, 7/17 or 8/19, one finds different symmetry groups. Furthermore, in all the cases, the point group is monoclinic and the information contained in the orthorhombic point-group symmetry of the main reflections is lost.

These difficulties are avoided if one embeds the modulated crystal according to the basic ideas expressed in the previous section. Conceptually, it corresponds (de Wolff, 1974 , 1977 ; Janner & Janssen, 1977 ) to enlarging the class of symmetry transformations admitted, from Euclidean in three dimensions to Euclidean in n dimensions (here n = 4). A four-dimensional Euclidean operation transforms the three-dimensional structure but does not leave distances invariant in general. To illustrate this phenomenon, consider the effect of a lattice translation of the basic structure on the modulation: it gives rise to a shift of the phase. Combining such a translation with an appropriate (compensating) shift of the phase, one obtains a transformation that leaves the basic structure and the modulation invariant. Such a transformation corresponds precisely to a lattice translation in four dimensions for the embedded crystal structure. In the real space, however, a shift of the phase is not an admitted Euclidean transformation as it does not leave invariant the distance between the particles, so that the combination of lattice translation and (compensating) phase shift also is not admitted as such, even if for an incommensurate modulation it simply corresponds to a relabelling of the atoms leaving the global structure invariant.

From what has been said, it should be apparent that the four-dimensional embedding of the modulated structure according to the superspace approach can also be obtained directly by considering as additional coordinate the phase t of the modulation. The positions in three dimensions are then the intersection of the lines of the atomic positions (parametrized by t) with the hyperplane t = 0. For the modulated structure of (9.8.1.4), these lines are given by $[\eqalignno{ ({\bf n}+{\bf r}_j+{\bf U}_j\sin\{2\pi[{\bf q}\cdot({\bf n}+{\bf r}_j)+t]+\varphi_j\}&,t) \cr \hbox{ any real $t$}&, &(9.8.1.10a)}]$ and for a more general, not necessarily sinusoidal, modulation by $[({\bf n}+{\bf r}_j+{\bf u}_j\{{\bf q}\cdot[{\bf n}+{\bf r}_j]+t\}, t), \eqno(9.8.1.10b)]$ where $[{\bf u}_j(x)]$ is a periodic vector function: $[{\bf u}_j(x+1)={\bf u}_j(x)]$ . A general point in four-dimensional space is r_s = (r, t) with r = (x, y, z).

The notation adopted here for the (3 + 1)-dimensional embedding of the modulated structure is a shorthand notation intended to stress the double role of the variable t as a parameter in the three-dimensional description of the structure and as a coordinate in the (3 + 1)-dimensional embedding. In the latter case, it implicitly assumes the choice of a fourth basis vector d perpendicular to the physical space of the crystal, so that the following four-dimensional vector notations are considered to be equivalent: $[r_s=({\bf r, t})=({\bf r}, t{\bf d})=({\bf r}, t).]$

The pattern of lines (9.8.1.10a , b ) has lattice periodicity. Indeed, it is invariant under the shift $[t\rightarrow t+1]$ and, for every lattice translation n from the basic structure, there is a compensating phase shift: the pattern is left invariant under the combination of the translation n and the phase shift $[t\rightarrow t-{\bf q}\cdot {\bf n}]$ . Therefore, (9.8.1.10a , b ) is invariant under translations from a four-dimensional lattice with basis $[\eqalign{a_{s1}&=({\bf a, -q\cdot a}),\quad a_{s2}=({\bf b,-q\cdot b}), \cr a_{s3}&=({\bf c,-q\cdot c}),\quad a_{s4}=(0,1). }\eqno (9.8.1.11)]$ In accordance with the equivalent descriptions given for [r_s] , equivalent descriptions for $[a_{s4}]$ are $[a_{s4}]$ = (0, 1) = (0, d) with dual basis vector $[a^*_{s4}]$ = (q, 1) = (q, d*), where d* is reciprocal to d in the fourth direction. With respect to the axes (9.8.1.11), a general point in four dimensions can be written as $[r_s=\textstyle\sum\limits^4_{i=1}\,x_ia_{si}=({\bf r}, r_I),]$ with $[{\bf r}=x_1{\bf a}+x_2{\bf b}+x_3{\bf c},\quad \hbox{and}\quad r_I=t=x_4-{\bf q}\cdot {\bf r}. \eqno (9.8.1.12)]$ Some, or sometimes all, transformations of the space group of the basic structure give rise to a symmetry transformation for the modulated structure when combined with an appropriate phase shift, possibly together with an inversion of the phase. If $[\{ R|{\bf v}\}]$ is an element of the space group of the basic structure, Δ is a phase shift and ɛ = ±1, then the point (r, t) is transformed to (Rr + v, ɛt + Δ). The pattern (9.8.1.10b ) is left invariant by ({R|v}, {ɛ|Δ}) if $[\eqalignno{ &(R{\bf n}+R{\bf r}_j+{\bf v}+R{\bf u}_j[{\bf q}\cdot({\bf n}+{\bf r}_j)+\varepsilon t-\varepsilon\Delta], t) \cr &\quad=({\bf n}'+{\bf r}_{j\,'}+{\bf u}_{j\,'}[{\bf q}\cdot({\bf n}'+{\bf r}_{j\,'})+t], t). &(9.8.1.13)}]$ The position $[{\bf n}'+{\bf r}_{j\,'}]$ is the transform of $[{\bf n}+{\bf r}_j]$ in the basic structure and therefore also belongs to the basic structure.

One can conclude that the pattern has as symmetry transformations all elements ({R|v}, {ɛ|Δ}) satisfying (9.8.1.13). These form a space group in four dimensions.

The reciprocal to the basis (9.8.1.11) is $[a^*_{s1}=({\bf a}^*,0), \quad a^*_{s2}=({\bf b}^*,0), \atop a^*_{s3}=({\bf c}^*,0),\quad a^*_{s4}=({\bf q},1).{\phantom*} \eqno (9.8.1.14)]$ A general reciprocal-lattice vector is now $[H_s=\textstyle\sum\limits^4_{i=1}\,h_ia^*_{si}=(h_1{\bf a}^*+h_2{\bf b}^*+h_3{\bf c}^*+h_4{\bf q}, h_4).]$ The projection of this reciprocal-lattice vector is of the form (9.8.1.7), as it should be. Moreover, the projection of the four-dimensional Fourier transform on the hyperplane t = 0 is exactly the Fourier transform of the structure in this hyperplane. If, as a consequence of the four-dimensional space-group symmetry, the four-dimensional diffraction pattern shows systematic extinctions, the same extinctions are present in the diffraction pattern of the three-dimensional structure.

There are also other ways to describe the symmetry relations of incommensurate modulated structures. One is based on representation theory. This method has in particular been used when the modulation occurs as a consequence of a soft mode. Then the irreducible representation of the space group to which the soft mode belongs gives information on the modulation function as well. Heine & McConnell have treated the symmetry with a method related to the Landau theory of phase transitions (Heine & McConnell, 1981 ; McConnell & Heine, 1984 ). Perez-Mato et al. have given a formulation in terms of three-dimensional structures parametrized by what are here the additional coordinates (Perez-Mato, Madariaga & Tello, 1984 , 1986 ; Perez-Mato, Madariaga, Zuñiga & Garcia Arribas, 1987 ). Other treatments of the problem can be found in Koptsik (1978 ), Lifshitz (1996 ), and de Wolff (1984 ).

9.8.1.4. Basic symmetry considerations

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9.8.1.4.1. Bravais classes of vector modules

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For a modulated crystal structure with a one-dimensional modulation, the positions of the diffraction spots are given by vectors $[{\bf H}=\textstyle\sum\limits^3_{i=1}\, h_i{\bf a}^*_i+m{\bf q}. \eqno (9.8.1.15)]$ This set of vectors is a vector module M*. The vectors $[{\bf a}^*_1, {\bf a}^*_2,{\bf a}^*_3]$ form a basis of the reciprocal lattice Λ* of the basic structure and q is the modulation wavevector. The choice of the basis of Λ* has the usual freedom, the wavevector q is only determined up to a sign and up to a reciprocal-lattice vector of the basic structure.

A vector module M* has point-group symmetry K, which is the subgroup of all elements R of O(3) leaving it invariant.

In the case of an incommensurate one-dimensional modulation, M* is generated by the lattice Λ* of main reflections and the modulation wavevector q. It then follows that K is characterized by the following properties:

(1) It leaves Λ* invariant. (Only in this way are main reflections transformed into main reflections and satellites into satellites.)
(2) Any element R of K then transforms q into ±q (modulo reciprocal-lattice vectors of Λ*).

An element R of K then transforms the basic vectors $[{\bf a}^*_1]$ , $[{\bf a}^*_2]$ , $[{\bf a}^*_3]$ , q into ones of the form (9.8.1.15). If one denotes, as in (9.8.1.2), q by $[{\bf a}^*_4]$ , this implies $[R{\bf a}^*_i=\textstyle\sum\limits^4_{j=1}\,\Gamma^*(R)_{ji}{\bf a}^*_j,\quad i=1,\ldots,4, \eqno (9.8.1.16)]$ with Γ*(R) a 4 × 4 matrix with integral entries. In the case of an incommensurate modulated crystal structure, only two vectors with the same length as q are q and −q. As Λ* is left invariant, it follows that for a one-dimensionally modulated structure Γ*(R) has the form $[\Gamma^*(R)= \left(\matrix{ \Gamma^*_E(R)&\Gamma^*_M(R) \cr 0&\varepsilon(R)}\right), \quad\hbox{where }\kern3pt\varepsilon(R)=\pm1. \eqno (9.8.1.17)]$ This matrix represents the orthogonal transformation R when referred to the basis vectors $[{\bf a}^*_i]$ (i = 1, 2, 3, 4) of the vector module M*. As in the case of lattices, two vector modules of modulated crystals are equivalent if they have bases (i.e. a basis for the reciprocal lattice Λ* of the basic structure together with a modulation wavevector q) such that the set of matrices Γ*(K) representing their symmetry is the same for both vector modules. Equivalent vector modules form a Bravais class.

Again, as in the case of three-dimensional lattices, it is sometimes convenient to consider a vector module that includes as subset the one spanned by all diffraction spots as in (9.8.1.15). Within such a larger vector module, the actual diffraction peaks then obey centring conditions. For a vector module associated with a modulated structure, centring may involve main reflections (the basic structure then has a centred lattice), or satellites, or both. For example, if in a structure with primitive orthorhombic basic structure the modulation wavevector is given by $[\alpha {\bf a}^*_1+{1\over2}{\bf a}^*_2]$ , one may describe the diffraction spots by means of the non-primitive lattice basis $[{\bf a}^*_1]$ , $[{1\over2}{\bf a}^*_2]$ , $[{\bf a}^*_3]$ and by the modulation wavevector $[\alpha{\bf a}^*_1]$ .

Crystallographic point groups are denoted generally by the same letter K.

9.8.1.4.2. Description in four dimensions

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The matrices Γ*(R) form a faithful integral representation of the three-dimensional point group K. It is also possible to consider them as four-dimensional orthogonal transformations leaving a lattice with basis vectors (9.8.1.14) invariant. Indeed, one can consider the vectors (9.8.1.15) as projections of four-dimensional lattice vectors $[H_s=({\bf H},H_I)]$ , which can be written as $[H_s=\textstyle\sum\limits^4_{i=1}\,h_ia^*_{si}, \eqno (9.8.1.18)]$ where [cf. (9.8.1.14)] m has now been replaced by [h_4] and $[a^*_{si}=({\bf a}^*_i,0),\quad i=1,2,3\semi \quad a^*_{s4}=({\bf q}, 1). \eqno (9.8.1.19)]$

As will be explained in Section 9.8.4 , these vectors span the four-dimensional reciprocal lattice for a periodic structure having as three-dimensional intersection (say defined by the hyperplane t = 0) the modulated crystal structure (a specific example has been given in Subsection 9.8.1.3 ). In direct space, the point group [K_s] in four dimensions with elements [R_s] of O(4) then acts on the corresponding dual basis vectors (9.8.1.11) of the four-dimensional direct lattice as $[R_sa_{si}=\textstyle\sum\limits^4_{j=1}\,\Gamma(R)_{ji}a_{sj} \quad (i=1,2,3,4), \eqno(9.8.1.20a)]$ where Γ(R) is the transpose of the matrix $[\Gamma^*(R^{-1})]$ appearing in (9.8.1.17) and therefore for incommensurate one-dimensionally modulated structures it has the form $[\Gamma(R)=\left(\matrix{\Gamma_E(R)&0 \cr \Gamma_M(R)&\varepsilon(R)} \right). \eqno (9.8.1.20b)]$

9.8.1.4.3. Four-dimensional crystallography

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Let us summarize the results obtained in the previous paragraph. The matrices Γ(R) form a faithful integral representation of the three-dimensional point group K with a four-dimensional carrier space [V_s] . It is a reducible representation having as invariant subspaces the physical three-dimensional space, denoted by V (or sometimes also by [V_E] ), and the additional one-dimensional space, denoted by [V_I] . In V, the four-dimensional point-group transformation acts as R (sometimes also denoted by [R_E] ), in [V_I] it acts as one of the two one-dimensional point-group transformations: the identity or the inversion. Therefore, the space [V_s] can be made Euclidean with Γ(R) defining a four-dimensional point-group transformation [R_s] , which is an element of a crystallographic subgroup [K_s] of O(4). The four-dimensional point-group transformations are of the form (R, ɛ), with ɛ = ±1 and they act on the four-dimensional lattice basis as $[(R,\varepsilon)a_{si}= \textstyle\sum\limits^4_{j=1}\,\Gamma(R)_{ji}a_{sj}, \quad i=1,\ldots,4, \eqno (9.8.1.21)]$ where ɛ stands for ɛ(R) as in (9.8.1.20b ). So the point-group symmetry operations are crystallographic and given by pairs of a three-dimensional crystallographic point-group transformation and a one-dimensional ɛ = ±1, respectively. The case ɛ = −1 corresponds to an inversion of the phase of the modulation function.

As in the three-dimensional case, one can define equivalence classes among those four-dimensional point groups.

Two point groups [K_s] and [K'_s] belong to the same geometric crystal class if their three-dimensional (external) parts (forming the point group [K_E] and [K'_E] , respectively) are in the same three-dimensional crystal class [i.e. are conjugated subgroups of O(3)] and their one-dimensional internal parts (forming the point groups [K_I] and [K'_I] , respectively) are equal. The latter condition implies that corresponding point-group elements have the same value of ɛ.

Such a geometric crystal class can then be denoted by the symbol of the three-dimensional crystal class together with the values of ɛ that correspond to the generators.

Also, the notion of arithmetic equivalence can be generalized to these four-dimensional point groups, as they admit the same faithful integral representation Γ(K) given above. This means that two such groups are arithmetically equivalent if there is a basis transformation for the reciprocal-vector module, which transforms main reflections into main reflections and satellites into satellites and which transforms one of the matrix groups into the other. The arithmetic classes are determined by the arithmetic equivalence class of the three-dimensional group [K_E] [i.e. by $[\Gamma_E(K)]$ ] and by the components of the modulation wavevector with respect to the corresponding reciprocal-lattice basis. This is because the elements ɛ are fixed by the relation $[R{\bf q}\equiv\varepsilon{\bf q}\hbox{ (modulo reciprocal-lattice vectors}\atop \hbox{of the basic structure).}\eqno (9.8.1.22)]$ Note that these (3 + 1)-dimensional equivalence classes are not simply those one obtains in four-dimensional crystallography, as the relation between the higher-dimensional space [V_s] and the three-dimensional physical space V plays a fundamental role.

The embedded structures in four dimensions have lattice periodicity. So the symmetry groups are four-dimensional space groups, called superspace groups. The new name has been introduced because of the privileged role played by the three-dimensional subspace V. A superspace-group element [g_s] consists of a point-group transformation (R, ɛ) and a translation (v, Δ). The action of such an element on the four-dimensional space is then given by $[g_sr_s=\{(R,\varepsilon)|({\bf v},\Delta)\}({\bf r},t)=(R{\bf r}+{\bf v}, \varepsilon t+\Delta). \eqno (9.8.1.23)]$ It is important to realize that a superspace-group symmetry of an embedded crystal induces three-dimensional transformations leaving the original modulated structure invariant. Corresponding to (9.8.1.23), one obtains the following relations [cf. (9.8.1.13)]: $[{\bi u}_{j\,'}[{\bf q}\cdot({\bf n}'+{\bf r}_{j\,'})]=R{\bf u}_j[{\bf q}\cdot({\bf n}+{\bf r}_j)-\varepsilon\Delta] \eqno (9.8.1.24)]$ with $[{\bf n}' + {\bf r}_{j\,'}=R({\bf n}+{\bf r}_j)+{\bf v}.]$ These are purely three-dimensional symmetry relations, but of course not Euclidean ones.

In three-dimensional Euclidean space, the types of space-group transformation are translations, rotations, rotoinversions, reflections, central inversion, screw rotations, and glide planes. Only the latter two transformations have intrinsic non-primitive translations. For superspace groups, the types of transformation are determined by the point-group transformations. By an appropriate choice of the basis in [V_s] , each of the latter can be brought into the form $[\left(\matrix{ \cos\varphi&-\sin\varphi&0&0 \cr \sin\varphi&\cos\varphi&0&0 \cr 0&0&\delta&0 \cr 0&0&0&\varepsilon}\right)\semi\quad \varepsilon,\delta=\pm1. \eqno (9.8.1.25)]$ By a choice of origin, each translational part can be reduced to its intrinsic part, which in combination with the point-group element (R, ɛ) gives one of the transformations in V indicated above together with the inversion, or the identity, or a shift in [V_I] . So, for phase inversion (when ɛ = −1), the intrinsic shift in [V_I] vanishes. When ɛ = +1, the intrinsic shift in [V_I] is given by τ. It will be shown in Subsection 9.8.3.3 that the value of τ is one of $[0,{1\over2},{\pm1\over3}, {\pm1\over4},{\pm1\over6}. \eqno (9.8.1.26)]$ Therefore, a superspace-group element can be denoted by a symbol that consists of a symbol for the three-dimensional part following the conventions given in Volume A of International Tables for Crystallography, a symbol that determines ɛ, and one for the corresponding intrinsic internal translation τ.

9.8.1.4.4. Generalized nomenclature

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In Section 9.8.4 , the theory is extended to structures containing d modulations, with $[d \, \ge \, 1]$ . In this case, each point-group transformation in internal space is given by [R_I] and the associated internal translation by the (d-dimensional) vector $[{\bf v}_I]$ . Thus, $[g_s=\{(R,R_I)|({\bf v},{\bf v}_I)\}.]$ The transformations R and [R_I] are represented by the matrices $[\Gamma_E(R)]$ and $[\Gamma_I(R)]$ , respectively. In the following discussion, this nomenclature (but with v_I rather than v_I) is sometimes also applied for the (3 + 1)-dimensional case. The usual formulae are obtained by replacing R_I by ɛ and v_I by Δ.

9.8.1.4.5. Four-dimensional space groups

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Four-dimensional space groups were obtained in the (3 + 1)-reducible case by Fast & Janssen (1969 ) and in the general case by Brown, Bülow, Neubüser, Wondratschek & Zassenhaus (1978 ). The groups were determined on the basis of algorithms developed by Zassenhaus (1948 ), Janssen, Janner & Ascher (1969a ,b ), Brown (1969 ), and Fast & Janssen (1971 ). In the book by Brown, Bülow, Neubüser, Wondratschek & Zassenhaus, quoted above, a mathematical characterization of the basic crystallographic concepts is given together with corresponding tables for the dimensions one, two, three, and four. One finds there, in particular, a full list of four-dimensional space groups. The list by Fast & Janssen is restricted to space groups with (3 + 1)-reducible point groups. The four-dimensional groups in the work of Brown et al. are labelled by numbers. For these same groups, alternative symbols have been developed by Weigel, Phan, Veysseyre and Grebille generalizing the principles of the notation adopted by International Tables for Crystallography, Volume A , for the three-dimensional space groups (Weigel, Phan & Veysseyre, 1987 ; Veysseyre & Weigel, 1989 ; Grebille, Weigel, Veysseyre & Phan, 1990 ).

The difference in the listing of four-dimensional crystallographic groups one finds in Brown et al. and in Weigel et al. with respect to that in the present tables is not simply a matter of notation. In the first place, here only those groups appear that can occur as symmetry groups of one-dimensional incommensurate modulated phases (there are 371 such space groups). Furthermore, as already mentioned, a finer equivalence relation has been considered that reflects the freedom one has in embedding a three-dimensional modulated structure in a four-dimensional Euclidean space. Instead of 371, one then obtains 775 inequivalent groups for which the name superspace group has been introduced. A (3 + 1)-dimensional superspace group is thus a four-dimensional space group having some additional properties. In Section 9.8.4 , the precise definitions are given.

In the commensurate one-dimensionally modulated case, 3833 four-dimensional space groups may occur, out of which 320 already belong to the previous 371. The corresponding additional (3 + 1)-dimensional superspace groups are also present in the listing by Fast & Janssen (1969 ) and have been considered again (and applied to structure determination) by van Smaalen (1987 ). The Bravais classes for the commensurate (3 + 1)-dimensional case are given in Table 9.8.3.2(b).

The relation between modulated crystals and the superspace groups is treated in a textbook by Opechowski (1986 ). That between the superspace-group symbols of the present tables and those of Weigel et al. is discussed in Grebille et al. (1990 ).

Note that no new names have been introduced for the underlying crystallographic concepts like Bravais classes, geometric and arithmetic crystal classes , even if in those cases also the equivalence relation is not simply that of four-dimensional Euclidean crystallography, an explicit distinction always being possible by specifying the dimension as (3 + 1) instead of four.

9.8.1.5. Occupation modulation

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Another type of modulation, the occupation modulation, can be treated in a way similar to the displacive modulation. As an example consider an alloy where the positions of the basic structure have space-group symmetry, but are statistically occupied by either of two types of atoms. Suppose that the position r is occupied by an atom of type A with probability $[p({\bf r})]$ and by one of type B with probability $[1-p({\bf r})]$ and that p is periodic. The probability of finding an A atom at site $[{\bf n}+{\bf r}_j]$ is $[P_A({\bf n}+{\bf r}_j)=p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)], \eqno (9.8.1.27)]$ with [p_j(x)=p_j(x+1)] . In this case, the structure factor becomes $[\eqalignno{\qquad S_{\bf H} &=\textstyle\sum\limits_{\bf n}\textstyle\sum\limits_j\big[\big(\,f_A\,p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)]+f_B\{1-p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)]\}\big) \cr &\quad\times\exp[2\pi i{\bf H}\cdot({\bf n}+{\bf r}_j)]\big], & (9.8.1.28)}]$ where [f_A] and [f_B] are the atomic scattering factors. Because of the periodicity, one has $[p_j(x)=\textstyle\sum\limits_mw_{jm}\exp(2\pi imx). \eqno (9.8.1.29)]$ Hence, $[\eqalignno{ S_{\bf H} &=\textstyle\sum\limits_j\bigg\{f_B\Delta({\bf H})\exp(2\pi i{\bf H}\cdot{\bf r}_j)+(\,f_A-f_B)\textstyle\sum\limits_m\Delta({\bf H}+m{\bf q})w_{jm} \cr &\quad \times\exp[2\pi i({\bf H}+m{\bf q})\cdot{\bf r}_j]\bigg\}, & (9.8.1.30)}]$ where Δ(H) is the sum of δ functions over the reciprocal lattice of the basic structure: $[\Delta({\bf H})=\textstyle\sum\limits_{h_1h_2h_3}\delta\left({\bf H}-\textstyle\sum\limits^3_{i=1}h_i{\bf a}^*_i\right).]$ Consequently, the diffraction peaks occur at positions H given by (9.8.1.7). For a simple sinusoidal modulation [m = ±1 in (9.8.1.29)], there are only main reflections and first-order satellites (m = ±1). One may introduce an additional coordinate t and generalize (9.8.1.27) to $[P_A({\bf n}+{\bf r}_j, t) = p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)+t], \eqno (9.8.1.31)]$ which has (3 + 1)-dimensional space-group symmetry. Generalization to more complex modulation cases is then straightforward.

9.8.2. Outline for a superspace-group determination

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In the case of a modulated structure, the diffraction pattern consists of main reflections and satellites. The main reflections span a reciprocal lattice generated by $[{\bf a}^*_1]$ , $[{\bf a}^*_2]$ , $[{\bf a}^*_3]$ . Considerations are here restricted for simplicity to the one-dimensional modulated case, i.e. to the n = 4 case. Extension to the more general n = 3 + d case is conceptually not difficult and does not modify the general procedure outlined here.

(1) The first step is the determination of the Laue group [P_L] of the diffraction pattern: it is the point group in three dimensions that transforms every diffraction peak into a peak of the same intensity.¹

As [P_L] leaves invariant the subset of main reflections, this Laue group belongs to one of the 11 Laue symmetry classes. Accordingly, the Laue group determines a three-dimensional holohedral point group which determines a crystallographic system.

(2) The second step consists of choosing a basis according to the conventions of ITA for the main reflections and choosing a modulation wavevector.

From the centring extinctions, one can deduce to which Bravais class the main reflections belong. This is one of the 14 three-dimensional Bravais classes. Notice that the cubic Bravais classes do not occur because a one-dimensional (incommensurate) modulation is incompatible with cubic symmetry. For this same reason, only the nine non-cubic Laue-symmetry classes occur in the one-dimensional incommensurate case.

The main reflections are indexed by hkl0 and the satellite reflections by hklm. The Fourier wavevector of a general reflection hklm is given by $[{\bf H}=h{\bf a}^*+k{\bf b}^*+l{\bf c}^*+m{\bf q}. \eqno (9.8.2.1)]$ Note that this step involves a choice because the system of satellite reflections is only defined modulo the main reflections. When a satellite is in the vicinity of a main reflection, it is reasonable to assign it to that reflection. But one has, especially when deciding whether or not situations are equivalent, to be aware of the fact that each satellite may be assigned to an arbitrary main reflection. It is even possible to assign a satellite to an extinct main reflection. One takes by preference the q vector along a symmetry axis or in a mirror plane. According to equation (9.8.2.1), the fourth basis vector $[{\bf a}^*_4]$ is equal to the chosen q, the modulation wavevector.

(3) In the third step, one determines the space group of the average structure (from the main reflections).

The average structure is unique but possibly involves split atoms. The space group of the average structure is often the symmetry group of the undistorted phase. That helps to make a good choice for the basic structure and also gives an insight as to how the satellite reflections split from the main reflections at the phase transition.

(4) Step four is the identification of the (3 + 1)-dimensional Bravais lattice type. In superspace also, centring gives rise to centring extinctions, and that corresponds to making the choice of a conventional unit cell in (3 + 1) dimensions.

The previous three steps establish a*, b*, c*, the three-dimensional Bravais class and q = αa* + βb* + γc*, where the components α, β, and γ are given with respect to the three-dimensional conventional basis. $[\alpha={\bf q} \cdot {\bf a},\quad \beta={\bf q}\cdot {\bf b},\quad \gamma={\bf q}\cdot {\bf c}.\eqno (9.8.2.2)]$ The (3 + 1)-dimensional Bravais class is fixed by that three-dimensional Bravais class and the components α, β, γ of q.

Just as for three-dimensional lattices, a conventional cell can be chosen for (3 + 1)-dimensional lattices. To this end, the vector q must be considered. The vector q can be split into an invariant (or irrational) and a rational component according to: $[{\bf q}={\bf q}^i+{\bf q}^r \quad\hbox{ with }\quad {\bf q}^i={1\over N}\sum_R \varepsilon(R)(R{\bf q}),\eqno (9.8.2.3)]$ where summation is over the elements R of the Laue group [P_L] , N is the order of [P_L] , and ɛ = ±1 follows from the property (valid in the one-dimensional modulated case): $[R{\bf q}=\varepsilon{\bf q}\hbox{ (modulo main-reflection vectors).} \eqno (9.8.2.4)]$ If the splitting (9.8.2.3) results in $[{\bf q}^r=0]$ , there is no problem. For $[{\bf q}^r\neq0]$ we choose $[{\bf q}^i]$ as the new modulation wavevector. This may cause satellites to be assigned to extinct main reflections, or even to `main reflections' with non-integer indices. In this case, it is desirable to choose a new basis $[{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]$ such that all reflections can be written as $[H{\bf a}^*_c+K{\bf b}^*_c+L{\bf c}^*_c+m{\bf q}^i \eqno (9.8.2.5)]$ with H, K, L, and m integer numbers. The basis obtained is the conventional basis for the M* module and corresponds to the choice of a conventional cell for the (3 + 1)-dimensional Bravais lattice.

Observed (possible) centring conditions for the reflections can be applied to determine the (3 + 1)-dimensional Bravais class.

Example 1. As an example, consider a C-centred orthorhombic lattice with modulation wavevector $[{\bf q} = (\alpha0{1\over2}\,)]$ , or $[{\bf q}^i = (\alpha 00)]$ and $[{\bf q}^r=(00{1\over2}\,)]$ . The general reflection condition is: hklm, h + k = even. A reflection $[h{\bf a}^*+k{\bf b}^*+l{\bf c}^*+m(\alpha{\bf a}^*+\textstyle{1\over2}{\bf c}^*) \eqno (9.8.2.6)]$ can be described on a conventional basis with $[{\bf a}^*_c={\bf a}^*]$ , $[{\bf b}^*_c={\bf b}^*]$ , $[{\bf c}^*_c={\bf c}^*/2]$ as $[H{\bf a}^*_c+K{\bf b}^*_c+L{\bf c}^*_c+m\alpha{\bf a}^*_c. \eqno (9.8.2.7)]$ Because H = h, K = k, L/2 = l + m/2, and h, k and l are integers with h + k even, one has the conditions H + K = even and L + m = even. The adopted conventional basis has centring $[(\,{1\over2}{1\over2}00,00{1\over2}{1\over2}, {1\over2}{1\over2}{1\over2}{1\over2}\,)]$ . The conditions H + K = even and L + m = even become, after interchange of H and L, K + L = even and H + m = even, which according to Table 9.8.3.6 gives mmmA(½0γ). Changing back to the original setting gives the Bravais class mmmC(α0½).

Example 2. As a second example, consider a C-centred orthorhombic lattice of main reflections. Suppose that an arbitrary reflection is given by $[H{\bf a}^*+K{\bf b}^*+L{\bf c}^*+m\gamma{\bf c}^* \eqno (9.8.2.8)]$ with respect to the conventional basis a*, b*, c* of the C-centred lattice and that the general reflection condition is $[H+K+m={\rm even}. \eqno (9.8.2.9)]$ This is in agreement with the C-centring condition H + K = even for the main reflections. Equation (9.8.2.9) implies a centring $[(\,{1\over2}{1\over2}0{1\over2}\,)]$ of the (3 + 1)-dimensional lattice $[\Sigma]$ . Its conventional basis corresponding to (9.8.2.8) is $[({\bf a},0), ({\bf b},0), ({\bf c},-\gamma),(0,1). \eqno (9.8.2.10)]$ Because of the centring translation [(a + b)/2, 1/2], a primitive basis of $[\Sigma]$ is $[\left({{\bf a}+{\bf b}\over 2},-1/2\right),\left({{\bf a}-{\bf b}\over 2},-1/2\right), ({\bf c},-\gamma), (0,1), \eqno (9.8.2.11)]$ which corresponds, according to (9.8.1.11), to a choice q = a*+ γc*. In terms of this modulation vector, an arbitrary reflection (9.8.2.8) can be described as $[h{\bf a}^*+k{\bf b}^*+l{\bf c}^*+m({\bf a}^*+\gamma{\bf c}^*). \eqno (9.8.2.12)]$ Because H = h + m, K = k, and L = l, the reflection condition (9.8.2.9) now becomes h + k = even, again in agreement with the fact that the lattice of main reflections is C-centred.

(5) Step five: one lists all point groups (more precisely, the arithmetic crystal classes ) compatible with the external Bravais class (i.e. the three-dimensional one), the components of the modulation vector q with respect to the conventional basis of this class, and the observed general extinctions. These groups are listed in Table 9.8.3.3 .

The external part of the (3 + 1)-dimensional point group does not need to be the same as the point group of the average structure: it can never be larger, but is possibly a subgroup only.

(6) Step six consists of finding the superspace groups compatible with the previously derived results and with the special extinctions observed in the diffraction pattern.

For each arithmetic point group, the non-equivalent superspace groups are listed in Table 9.8.3.5 .

Example 3. Continue with example 2 of step 4. The external Bravais class of the modulated crystal is mmmC and q = (10γ). Consider the two cases of special reflection conditions.

(1) The observed condition is: 00Lm, m = 2n. Then there is only one superspace group that obeys this condition: No. 21.4 with symbol C222(10γ)00s in Table 9.8.3.5 . [See Subsection 9.8.3.3 for explanation of symbols; note that C222 denotes the external part of the (3 + 1)-dimensional space group.]
(2) There is no special reflection condition. Possible superspace groups are: $[{C222(10\gamma), \hbox{ No. 21.3},\hfill \atop C2mm(10\gamma), \hbox{ No. 38.3},} \quad\quad {Cmm2(10\gamma), \hbox{ No. 35.4},\hfill \atop Cmmm(10\gamma), \hbox{ No. 65.4}.}]$

In this second case, there are thus four possible superspace groups, three without and one with inversion symmetry. The difficulty stems from the fact that (in the absence of anomalous dispersion) the Laue group always has an inversion centre. Just as in three-dimensional crystallography, the ambiguity can only be solved by a complete determination of the structure, or by non-diffraction methods.

In the determination of modulated structures, one generally starts with the assumption that the external part of the (3 + 1)-dimensional point group is equal to the point group of the basic structure. This is, however, not necessarily true. The former may be a subgroup of the latter. Especially when the basic structure contains atoms at special positions, the point-group symmetry may be lowered by the modulation.

(7) Step seven concerns the restrictions imposed on/by the displacive or occupation modulation waves. For each occupied special Wyckoff position of the basic structure, one has to verify the validity of the restrictions imposed by the elements of the superspace group on the modulation as expressed in equation (9.8.1.24).

During the structure determination, one should check for each Wyckoff position which is occupied by an atom of the basic structure, whether the assumed displacive or occupation modulations obey the site symmetry of the Wyckoff position considered. An atom at a special position transforms into itself by the site symmetry of the position. For such a symmetry transformation, $[j\,'=j]$ , v = 0, and thus Δ(ɛ = 1) = τ, Δ(ɛ = −1) = 0 (cf. Subsection 9.8.3.3 ). In the modulated structure, the site symmetry is preserved only if, for each of its symmetry operations, the appropriate relation is obeyed by the modulations [cf. (9.8.1.24)].

For displacive modulations, the conditions are $[\matrix{ {\bf u}_j({\bf q\cdot r})=R{\bf u}_j({\bf q\cdot r}-\tau) &\hbox{ for $\varepsilon=1$},\hfill \cr {\bf u}_j({\bf q}\cdot{\bf r}) = R{\bf u}_j({\bf q}\cdot {\bf r})\hfill & \hbox{ for $\varepsilon=-1$}}\eqno (9.8.2.13)]$ and, for occupation modulation, $[\matrix{ p_j({\bf q}\cdot {\bf r})=p_j({\bf q}\cdot{\bf r}-\tau)&\hbox{ for $\varepsilon=1$},\hfill \cr p_j({\bf q}\cdot {\bf r})=p_j({\bf q}\cdot{\bf r})\hfill & \hbox{ for $\varepsilon=-1$}.} \eqno (9.8.2.14)]$

Example 4. Assume, as for the case discussed above, that the basic structure has the space group Cmmm. Can the superspace group be Cmmm(10γ)? In this superspace group, τ = 0 for all symmetry operations with ɛ = 1. Displacive modulations at special positions must thus obey $[{\bf u}_j({\bf q}\cdot{\bf r})=R{\bf u}_j({\bf q}\cdot {\bf r})]$ for the superspace group to be correct. For an atom at special position mmm, this is not possible for all site symmetry operations unless $[{\bf u}_j=0]$ . Suppose that the structure model contains a displacive modulation polarized along a for that atom. The allowed site symmetry is then lowered to 2mm, and as a consequence the superspace group is C2mm(10γ) rather than Cmmm(10γ).

9.8.3. Introduction to the tables

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In what follows, the tables dealing with the (3 + 1)-dimensional case will be presented. The explanations can easily be applied to the (2 + d)-dimensional case also [Tables 9.8.3.1 and 9.8.3.4 ].

9.8.3.1. Tables of Bravais lattices

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The (3 + 1)-dimensional lattice $[\Sigma^*]$ is determined by the three-dimensional vectors a*, b*, c* and the modulation vector q. The former three vectors give by duality a, b, and c, the external components of lattice basis vectors, and the products $[-{\bf q}\cdot{\bf a}=-\alpha]$ , $[-{\bf q}\cdot{\bf b}=-\beta]$ , and $[-{\bf q}\cdot{\bf c}=-\gamma]$ the corresponding internal components. Therefore, it is sufficient to give the arithmetic crystal class of the group $[\Gamma_E(K)]$ and the components σ_j (σ₁ = α, σ₂ = β, and σ₃ = γ) of the modulation vector q with respect to a conventional basis a*, b*, c*. The arithmetic crystal class is denoted by a modification of the symbol of the three-dimensional symmorphic space group of this class (see Chapter 1.4 ) plus an indication for the row matrix σ (having entries $[\sigma_j]$ ). In this way, one obtains the so-called one-line symbols used in Tables 9.8.3.1 and 9.8.3.2 .

Table 9.8.3.1| top | pdf |
(2 + 1)- and (2 + 2)-Dimensional Bravais classes for incommensurate structures

(a) (2 + 1)-Dimensional Bravais classes. The holohedral point group K_s is given in terms of its external and internal parts, K_E, and K_I, respectively. The reflections are given by ha* + kb* + mq where q is the modulation wavevector. If the rational part q^r is not zero, there is a corresponding centring translation in three-dimensional space. The conventional basis ( $[{\bf a}_c^*]$ , $[{\bf b}_c^*]$ , qⁱ) given for the vector module M* is shown such that q^r = 0. The basis vectors are given by components with respect to the conventional basis a*, b* of the lattice Λ* of main reflections.

No.	Symbol	K_E	K_I	q	Conventional basis	Centring
Oblique
1	2p(αβ)	2	$[\bar 1]$	(αβ)	(10), (01), (αβ)
Rectangular
2	mmp(0β)	mm	$[1\bar 1]$	(0β)	(10), (01), (0β)
3	mmp( $[{{1}\over{2}}]$ β)	mm	$[1\bar 1]$	( $[{{1}\over{2}}]$ β)	( $[{{1}\over{2}}]$ 0), (01), (0β)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$
4	mmc(0β)	mm	$[1\bar 1]$	(0β)	(10), (01), (0β)	$[{{1}\over{2}}]$ $[{{1}\over{2}}]$ 0

(b) (2 + 2)-Dimensional Bravais classes. The holohedral point group K_s is given in terms of its external and internal parts, K_E and K_I, respectively. The basis of the vector module M* contains two modulation wavevectors and the reflections are given by ha* + kb* + m₁q₁ + m₂q₂. If $[{\bf q}_1^r]$ or $[{\bf q}_2^r]$ are not zero, there are corresponding centring translations in four-dimensional space. The conventional basis ( $[{\bf a}_c^*]$ , $[{\bf b}_c^*]$ , $[{\bf q}_1^i]$ , $[{\bf q}_2^i]$ ) for the vector module M* is chosen such that $[{\bf q}_1^r = {\bf q}_2^r =0]$ . The basis vectors are indicated by their components with respect to the conventional basis a*, b* of the lattice Λ* of main reflections.

No.	Symbol	K_E	K_I	q₁	q₂	Conventional basis	Centring
Oblique
1	2p(αβ, λμ)	2	2	(αβ)	(λμ)	(10), (01), (αβ), (λμ)
Rectangular
2	mmp(0β, 0μ)	mm	12	(0β)	(0μ)	(10), (01), (0β), (0μ)
3	mmp( $[{{1}\over{2}}]$ β, 0μ)	mm	12	( $[{{1}\over{2}}]$ β)	(0μ)	( $[{{1}\over{2}}]$ 0), (01), (0β), (0μ)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0
4	mmp(α0, 0μ)	mm	mm	(α0)	(0μ)	(10), (01), (α0), (0μ)
5	mmp(α $[{{1}\over{2}}]$ , 0μ)	mm	mm	(α $[{{1}\over{2}}]$ )	(0μ)	(10), (0 $[{{1}\over{2}}]$ ), (α0), (0μ)	0 $[{{1}\over{2}}{{1}\over{2}}]$ 0
6	mmp(α $[{{1}\over{2}}]$ , $[{{1}\over{2}}]$ μ)	mm	mm	(α $[{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ μ)	( $[{{1}\over{2}}]$ 0), (0 $[{{1}\over{2}}]$ ), (α0), (0μ)	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$ , 0 $[{{1}\over{2}}{{1}\over{2}}]$ 0
7	mmp(αβ)	mm	mm	(αβ)	$[(\alpha{\bar\beta})]$	(10), (01), (α0), (0β)	00 $[{{1}\over{2}}{{1}\over{2}}]$
8	mmc(0β, 0μ)	mm	12	(0β)	(0μ)	(10), (01), (0β), (0μ)	$[{{1}\over{2}}{{1}\over{2}}]$ 00
9	mmc(α0, 0μ)	mm	mm	(α0)	(0μ)	(10), (01), (α0), (0μ)	$[{{1}\over{2}}{{1}\over{2}}]$ 00
10	mmc(αβ)	mm	mm	(αβ)	$[(\alpha{\bar\beta})]$	(10), (01), (α0), (0β)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, 00 $[{{1}\over{2}}{{1}\over{2}}]$
Square
11	4p(αβ)	4	4	(αβ)	$[(\bar \beta\alpha)]$	(10), (01), (αβ), $[(\bar\beta\alpha)]$
12	4mp(α0)	4m	4m	(α0)	(0α)	(10), (01), (α0), (0α)
13	4mp(α $[{{1}\over{2}}]$ )	4m	4m	(α $[{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ α)	( $[{{1}\over{2}}{{1}\over{2}}]$ ), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ ), (γγ), ( $[\delta\bar \delta]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 00, 00 $[{{1}\over{2}}{{1}\over{2}}]$
13	4mp(α $[{{1}\over{2}}]$ )	4m	4m	(α $[{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ α)	$[\gamma]$ = (2α + 1)/4, δ = (2α − 1)/4
14	4mp(αα)	4 $[\dot m]$	4 $[\ddot m]$	(αα)	( $[\bar \alpha\alpha]$ )	(10), (01), (αα), ( $[\bar \alpha\alpha]$ )
Hexagonal
15	6p(αβ)	6	6	(αβ)	( $[\bar \beta\alpha + \beta]$ )	(10), (01), (αβ), ( $[\bar \beta\alpha + \beta]$ )
16	6mp(α0)	6m	6m	(α0)	(0α)	(10), (01), (α0), (0α)
17	6mp(αα)	6 $[\dot m]$	6 $[\ddot m]$	(αα)	( $[\bar \alpha2\alpha]$ )	(10), (01), (αα), ( $[\bar \alpha2\alpha]$ )

Table 9.8.3.2| top | pdf |
(3 + 1)-Dimensional Bravais classes for incommensurate and commensurate structures

(a) (3 + 1)-Dimensional Bravais classes for incommensurate structures. The holohedral point group K_s is given in terms of its external and internal parts, K_E and K_I, respectively. The reflections are given by ha* + kb* + lc* + mq, where q is the modulation wavevector. If the rational part q^r is not zero, there is a corresponding centring translation in four-dimensional space. A conventional basis ( $[{\bf a}_c^*]$ , $[{\bf b}_c^*]$ , $[{\bf c}_c^*]$ , qⁱ) for the vector module M* is then chosen such that q^r = 0. The basis vectors are indicated by their components with respect to the conventional basis a*, b*, c* of the lattice Λ* of main reflections. The Bravais classes can also be found in Janssen (1969 ) and Brown et al. (1978 ). The notation of the Bravais classes there is here given in the columns Ref. a and Ref. b, respectively.

No.	Symbol	K_E	K_I	q	Conventional basis	Centring translation(s)	Ref. a	Ref. b
Triclinic
1	$[\bar 1]$ P(αβγ)	$[\bar 1]$	$[\bar 1]$	(αβγ)	(100), (010), (001), (αβγ)		I P	I/I
Monoclinic
2	2/mP(αβ0)	2/m	$[\bar 11]$	(αβ0)	(100), (010), (001), (αβ0)		II P	II/I
3	2/mP(αβ $[{{1}\over{2}}]$ )	2/m	$[\bar 11]$	(αβ $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ ), (αβ0)	00 $[{{1}\over{2}}{{1}\over{2}}]$	II I	II/II
4	2/mB(αβ0)	2/m	$[\bar 11]$	(αβ0)	(100), (010), (001), (αβ0)	$[{{1}\over{2}}0{{1}\over{2}}0]$	II I	II/II
5	2/mP(00γ)	2/m	$[1\bar 1]$	(00γ)	(100), (010), (001), (00γ)		III P	III/I
6	2/mP( $[{{1}\over{2}}]$ 0γ)	2/m	$[1\bar 1]$	( $[{{1}\over{2}}]$ 0γ)	( $[{{1}\over{2}}]$ 00), (010), (001), (00γ)	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$	III I	III/II
7	2/mB(00γ)	2/m	$[1\bar 1]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	III I	III/II
8	2/mB(0 $[{{1}\over{2}}]$ γ)	2/m	$[1\bar 1]$	(0 $[{{1}\over{2}}]$ γ)	(100), (0 $[{{1}\over{2}}]$ 0), (001), (00γ)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0, 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	III G	III/III
Orthorhombic
9	mmmP(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)		IV P	IV/I
10	mmmP(0 $[{{1}\over{2}}]$ γ)	mmm	$[11\bar 1]$	(0 $[{{1}\over{2}}]$ γ)	(100), (0 $[{{1}\over{2}}]$ 0), (001), (00γ)	0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV B	IV/III
11	mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	mmm	$[11\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (001), (00γ)	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$ , 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV F	IV/VI
12	mmmI(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	IV I	IV/IV
13	mmmC(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 00	IV C	IV/II
14	mmmC(10γ)	mmm	$[11\bar 1]$	(10γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV I	IV/IV
15	mmmA(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)	0 $[{{1}\over{2}}{{1}\over{2}}]$ 0	IV B	IV/III
16	mmmA( $[{{1}\over{2}}]$ 0γ)	mmm	$[11\bar 1]$	( $[{{1}\over{2}}]$ 0γ)	( $[{{1}\over{2}}]$ 00), (010), (001), (00γ)	0 $[{{1}\over{2}}{{1}\over{2}}]$ 0, $[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$	IV G	IV/V
17	mmmF(00γ)	mmm	$[11\bar 1]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	IV F	IV/VI
18	mmmF(10γ)	mmm	$[11\bar 1]$	(10γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}{{1}\over{2}}]$	IV G	IV/V
Tetragonal
19	4/mmmP(00γ)	4/mmm	$[1\bar 111]$	(00γ)	(100), (010), (001), (00γ)		VII P	VI/I
20	4/mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	4/mmm	$[1\bar 111]$	( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	VII I	VI/II
21	4/mmmI(00γ)	4/mmm	$[1\bar 111]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	VII I	VI/II
Trigonal
22	$[\bar 3]$ mR(00γ)	$[\bar 3m]$	$[\bar 11]$	(00γ)	(100), (010), (001), (00γ)	$[{{1}\over{3}}{{2}\over{3}}{{2}\over{3}}]$ 0	VI P	VII/I
23	$[\bar 3]$ 1mP( $[{{1}\over{3}}{{1}\over{3}}]$ γ)	$[\bar 31m]$	$[\bar 111]$	( $[{{1}\over{3}}{{1}\over{3}}]$ γ)	( $[{{1}\over{3}}{{1}\over{3}}]$ 0), ( $[{{\bar 1}\over{3}}{{2}\over{3}}]$ 0), (001), (00γ)	$[{{1}\over{3}}{{2}\over{3}}]$ 0 $[{{2}\over{3}}]$	VI P	VII/I
Hexagonal
24	6/mmmP(00γ)	6/mmm	$[1\bar 111]$	(00γ)	(100), (010), (001), (00γ)		V P	VII/II

(b) (3 + 1)-Dimensional Bravais classes for commensurate structures. The holohedral point group K_s is given in terms of its external and internal parts, K_E and K_I, respectively. The reflections are given by ha* + kb* + lc* + mq, where q is the modulation wavevector. Here q is a commensurate vector having rational components with respect to a*, b*, c*. The rank of the vector module M* is three. Therefore, there are three basis vectors for M*. They are given by their components with respect to the conventional basis a*, b*, c* of the lattice of main reflections. If they do not coincide with the primitive basis vectors of the lattice Λ* of main reflections, there is a centring in four-dimensional space. The notation of the Bravais classes in Janssen (1969 ) is here given in the column Ref. a. Notice that for a commensurate one-dimensional modulation cubic symmetry is also possible.

No.	Symbol	K_E	K_I	q	Conventional basis	Centring translation(s)	Ref. a
Triclinic
1	$[\bar 1]$ P(000)	$[\bar 11]$	$[1\bar 1]$	(000)	(100), (010), (001)		II P
2	$[\bar 1]$ P( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	$[\bar 11]$	$[1\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{\bar 1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ ), ( $[{{1}\over{2}}{{\bar 1}\over{2}}{{1}\over{2}}]$ ), $[{{1}\over{2}}{{1}\over{2}}{{\bar 1}\over{2}}]$	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	II I
Monoclinic
3	2/mP(000)	2/m1	$[11\bar 1]$	(000)	(100), (010), (001)		IV P
4	2/mP( $[{{1}\over{2}}{{1}\over{2}}]$ 0)	2/m1	$[11\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}]$ 0)	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV B
5	2/mP(00 $[{{1}\over{2}}]$ )	2/m1	$[11\bar 1]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$	IV C
6	2/mP( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	2/m1	$[11\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	IV F
7	2/mB(000)	2/m1	$[11\bar 1]$	(000)	(100), (010), (001)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	IV B
8	2/mB(100)	2/m1	$[11\bar 1]$	(100)	(100), (010), (001)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}{{1}\over{2}}]$	IV I
9	2/mB(0 $[{{1}\over{2}}]$ 0)	2/m1	$[11\bar 1]$	(0 $[{{1}\over{2}}]$ 0)	(100), (0 $[{{1}\over{2}}]$ 0), (001)	$[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0, 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	IV G
Orthorhombic
10	mmmP(000)	mmm1	$[111\bar 1]$	(000)	(100), (010), (001)		VIII P
11	mmmP(00 $[{{1}\over{2}}]$ )	mmm1	$[111\bar 1]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$	VIII A
12	mmmP(0 $[{{1}\over{2}}{{1}\over{2}}]$ )	mmm1	$[111\bar 1]$	(0 $[{{1}\over{2}}{{1}\over{2}}]$ )	(100), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	VIII F
13	mmmP( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	mmm1	$[111\bar 1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$ , 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	VIII S
14	mmmI(000)	mmm1	$[111\bar 1]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	VIII E
15	mmmI(111)	mmm1	$[111\bar 1]$	(111)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	VIII I
16	mmmI( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	mmm	$[{\bar 1}{\bar 1}{\bar 1}]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{4}}{{1}\over{4}}{{1}\over{4}}{{1}\over{4}}]$ , $[{{\bar 1}\over{4}}{{1}\over{4}}{{1}\over{4}}{{\bar 1}\over{4}}]$ , $[{{1}\over{4}}{{1}\over{4}}{{\bar 1}\over{4}}{{\bar 1}\over{4}}]$	VIII K
17	mmmF(000)	mmm1	$[111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	VIII F
18	mmmF(001)	mmm1	$[111{\bar 1}]$	(001)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}{{1}\over{2}}]$	VIII H
19	mmmC(000)	mmm1	$[111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 00	VIII A
20	mmmC(100)	mmm1	$[111{\bar 1}]$	(100)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	VIII E
21	mmmC(00 $[{{1}\over{2}}]$ )	mmm1	$[111{\bar 1}]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 00, 00 $[{{1}\over{2}}{{1}\over{2}}]$	VIII G
22	mmmC(10 $[{{1}\over{2}}]$ )	mmm1	$[111{\bar 1}]$	(10 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	VIII H
Tetragonal
23	4/mmmP(000)	4/mmm1	$[1111{\bar 1}]$	(000)	(100), (010), (001)		XII P
24	4/mmmP(00 $[{{1}\over{2}}]$ )	4/mmm1	$[1111{\bar 1}]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$	XII A
25	4/mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ 0)	4/mmm1	$[1111{\bar 1}]$	( $[{{1}\over{2}}{{1}\over{2}}]$ 0)	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$	XII E
26	4/mmmP( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	4/mmm1	$[1111{\bar 1}]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}{{1}\over{2}}]$ 0), ( $[{{\bar 1}\over{2}}{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	XII H
27	4/mmmI(000)	4/mmm1	$[1111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	XII E
28	4/mmmI(111)	4/mmm1	$[1111{\bar 1}]$	(111)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	XII I
29	4/mmmI( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	4/mmm	$[{\bar 1}{\bar 1}{\bar 1}1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	(100), (010), (001)		XII N
Trigonal
30	$[{\bar 3}]$ m1R(000)	$[{\bar 3}m1]$	$[11{\bar 1}]$	(000)	(100), (010), (001)	$[{{2}\over{3}}{{1}\over{3}}{{1}\over{3}}]$ 0	X R
31	$[{\bar 3}]$ m1R(00 $[{{1}\over{2}}]$ )	$[{\bar 3}m1]$	$[11{\bar 1}]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$ , $[{{2}\over{3}}{{1}\over{3}}{{1}\over{6}}{{1}\over{6}}]$	X RI
Hexagonal
32	6/mmmP(000)	6/mmm1	$[1111{\bar 1}]$	(000)	(100), (010), (001)		X P
33	6/mmmP(00 $[{{1}\over{2}}]$ )	6/mmm1	$[1111{\bar 1}]$	(00 $[{{1}\over{2}}]$ )	(100), (010), (00 $[{{1}\over{2}}]$ )	00 $[{{1}\over{2}}{{1}\over{2}}]$	X A
34	6/mmmP( $[{{1}\over{3}}{{1}\over{3}}]$ 0)	6/mmm	$[{\bar 1}11{\bar 1}]$	( $[{{1}\over{3}}{{1}\over{3}}]$ 0)	( $[{{1}\over{3}}{{1}\over{3}}]$ 0), ( $[{{\bar 1}\over{3}}{{2}\over{3}}]$ 0), (001)	$[{{1}\over{3}}{{2}\over{3}}]$ 0 $[{{2}\over{3}}]$	X R
35	6/mmmP( $[{{1}\over{3}}{{1}\over{3}}{{1}\over{2}}]$ )	6/mmm	$[{\bar 1}11{\bar 1}]$	( $[{{1}\over{3}}{{1}\over{3}}{{1}\over{2}}]$ )	( $[{{1}\over{3}}{{1}\over{3}}]$ 0), ( $[{{\bar 1}\over{3}}{{2}\over{3}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{3}}{{2}\over{3}}]$ 0 $[{{2}\over{3}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	X RI
Cubic
36	m3mP(000)	$[m{\bar 3}m1]$	$[111{\bar 1}]$	(000)	(100), (010), (001)		XIV P
37	m3mP( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	$[m{\bar 3}m1]$	$[111{\bar 1}]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{2}}]$ 00 $[{{1}\over{2}}]$ , 0 $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ , 00 $[{{1}\over{2}}{{1}\over{2}}]$	XIV S
38	m3mI(000)	$[m{\bar 3}m1]$	$[111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ 0	XIV V
39	m3mI(111)	$[m{\bar 3}m1]$	$[111{\bar 1}]$	(111)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$	XIV I
40	m3mI( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	$[m{\bar 3}m]$	$[{\bar 1}{\bar 1}1]$	( $[{{1}\over{2}}{{1}\over{2}}{{1}\over{2}}]$ )	( $[{{1}\over{2}}]$ 00), (0 $[{{1}\over{2}}]$ 0), (00 $[{{1}\over{2}}]$ )	$[{{1}\over{4}}{{1}\over{4}}{{1}\over{4}}{{1}\over{4}}]$ , $[{{\bar 1}\over{4}}{{1}\over{4}}{{1}\over{4}}{{\bar 1}\over{4}}]$ , $[{{1}\over{4}}{{1}\over{4}}{{\bar 1}\over{4}}{{\bar 1}\over{4}}]$	XIV K
41	m3mF(000)	$[m{\bar 3}m1]$	$[111{\bar 1}]$	(000)	(100), (010), (001)	$[{{1}\over{2}}{{1}\over{2}}]$ 00, $[{{1}\over{2}}]$ 0 $[{{1}\over{2}}]$ 0	XIV F

As an example, the symbol $[2/mB(0{1\over2}\gamma)]$ denotes a Bravais class for which the main reflections belong to a B-centred monoclinic lattice (unique axis c) and the satellite positions are generated by the point-group transforms of $[{1\over2}{\bf b}^*+\gamma{\bf c}^*]$ . Then the matrix σ becomes $[\sigma=(0{1\over2}\gamma)]$ . It has as irrational part $[\sigma^i=(00\gamma)]$ and as rational part $[\sigma^r=(0{1\over2}0)]$ . The external part of the (3 + 1)-dimensional point group of the Bravais lattice is 2/m. By use of the relation [cf. (9.8.2.4)] $[R{\bf q}^i=\varepsilon{\bf q}^i,\quad R{\bf q}^r\equiv\varepsilon{\bf q}^r\hbox{ (modulo {\bf b}}{^*}), \eqno (9.8.3.1)]$ we see that the operations 2 and m are associated with the internal space transformations ɛ = 1 and ɛ = −1, respectively. This is denoted by the one-line symbol $[(2/m,1\bar1)]$ for the (3 + 1)-dimensional point group of the Bravais lattice . In direct space, the symmetry operation {R, ɛ(R)} is represented by the matrix Γ(R) which transforms the components $[v_j, j=1,\ldots,4]$ , of a vector [v_s] to: $[v'_j=\textstyle\sum\limits^4_{k=1}\Gamma(R)_{jk}v_k.]$ The operations (2, 1) and $[(m,\bar1)]$ are represented by the matrices: $[\Gamma(2)=\left(\matrix{ -1&\hfill0&0&0 \cr \hfill0&-1&0&0 \cr \hfill0&\hfill0&1&0 \cr \hfill0&-1&0&1}\right)\semi \quad \Gamma(m)=\left(\matrix{ 1&0&\hfill0&\hfill0 \cr 0&1&\hfill0&\hfill0 \cr 0&0&-1&\hfill0 \cr 0&1&\hfill0&-1}\right). \eqno (9.8.3.2)]$ The 3 × 3 part $[\Gamma_E(R)]$ of each matrix is obtained by considering the action of R on the external part v of [v_s] . The 1 × 1 part $[\Gamma_I(R)]$ is the value of the ɛ associated with R and the remaining part $[\Gamma_M(R)]$ follows from the relation $[\Gamma_M(R)=-\Gamma_I(R)\sigma^r+\sigma^r\Gamma_E(R). \eqno (9.8.3.3)]$

Bravais classes can be denoted in an alternative way by two-line symbols . In the two-line symbol, the Bravais class is given by specifying the arithmetic crystal class of the external symmetry by the symbol of its symmorphic space group, the associated elements $[\Gamma_I(R)=\varepsilon]$ by putting their symbol under the corresponding symbols of $[\Gamma_E(R)]$ , and by the rational part $[\sigma^r]$ indicated by a prefix. In the following table, this prefix is given for the components of $[{\bf q}^r]$ that play a role in the classification. $[\let\normalbaselines\relax\openup4pt\matrix{ P \quad(000)\hfill& R\quad(\,{1\over3},{1\over3},0)\hfill \cr A\quad(\,{1\over2},0,0)\hfill& B\quad(0,{1\over2},0)\hfill& C\quad(0,0,{1\over2}\,)\hfill \cr L\quad(1,0,0)\hfill& M\quad(0,1,0)\hfill& N\quad(0,0,1)\hfill \cr U\quad (0,{1\over2},{1\over2}\,)\hfill& V\quad(\,{1\over2},0,{1\over2}\,)\hfill& W\quad(\,{1\over2},{1\over2},0).}]$ Note that the integers appearing here are not equivalent to zero because they express components with respect to a conventional lattice basis (and not a primitive one). For the Bravais class mentioned above, the two-line symbol is $[B^{2/mB}_{1\;\;\bar1}]$ . This symbol has the advantage that the internal transformation (the value of ɛ) is explicitly given for the corresponding generators. It has, however, certain typographical drawbacks. It is rare for the printer to put the symbol together in the correct manner: $[B^{2/mB}_{1\;\;\bar1}]$ .

In Tables 9.8.3.1 and 9.8.3.2 the symbols for the (2 + d)- and (3 + 1)-dimensional Bravais classes are given in the one-line form. It is, however, easy to derive from each one-line symbol the corresponding two-line symbol because the bottom line for the two-line symbol appears in the tables as the internal part of the point-group symbol.

The number of symbols in the bottom line of the two-line symbol should be equal to that of the generators given in the top line. A symbol `1' is used in the bottom line if the corresponding [R_I] is the unit transformation. If necessary, a mirror perpendicular to a crystal axis is indicated by $[\dot m]$ and one that is not by $[\ddot m]$ . This situation only occurs for $[d\ge2]$ . So the (2 + 2)-dimensional class $[P^{4mp}_{4m}]$ is actually $[P^{4 {\dot m} p}_{4\dot m}]$ and is different from the class $[P^{4{\dot m}p}_{4{\ddot m}}]$ . In a one-line symbol, their difference is apparent, the first being 4mp(α0), whereas the second is 4mp(αα).

9.8.3.2. Table for geometric and arithmetic crystal classes

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In Table 9.8.3.3 , the geometric and the arithmetic crystal classes of (3 + 1)-dimensional superspace are given.

Table 9.8.3.3| top | pdf |
(3 + 1)-Dimensional point groups and arithmetic crystal classes

The four-dimensional point group K_s has external part K_E, which belongs to a three-dimensional system. Depending on the Bravais class of the four-dimensional lattice left invariant by K_s, this point group gives rise to an integral 4 × 4 matrix group Γ(K) which belongs to one of the arithmetic crystal classes given in the last column.

System	Point group		External Bravais class	Arithmetic crystal class(es)
System	K_E	K_s	External Bravais class	Arithmetic crystal class(es)
Triclinic	1	(1, 1)	$[{\bar 1}]$ P	1P(αβγ)
Triclinic	$[{\bar 1}]$	( $[{\bar 1},{\bar 1}]$ )	$[{\bar 1}]$ P	$[{\bar 1}]$ P(αβγ)
Monoclinic	2	( $[2, {\bar 1}]$ )	2/mP	2P(αβ0), 2P(αβ $[{{1}\over{2}}]$ )
			2/mB	2B(αβ0)
		(2, 1)	2/mP	2P(00γ), 2P( $[{{1}\over{2}}]$ 0γ)
			2/mB	2B(00γ), 2B(0 $[{{1}\over{2}}]$ γ)
	m	(m, 1)	2/mP	mP(αβ0), mP(αβ $[{{1}\over{2}}]$ )
			2/mB	mB(αβ0)
		(m, $[{\bar 1}]$ )	2/mP	mP(00γ), mP( $[{{1}\over{2}}]$ 0γ)
			2/mB	mB(00γ), mB(0 $[{{1}\over{2}}]$ γ)
	2/m	(2/m, $[{\bar 1}1]$ )	2/mP	2/mP(αβ0), 2/mP(αβ $[{{1}\over{2}}]$ )
			2/mB	2/mB(αβ0)
		(2/m, $[1{\bar 1}]$ )	2/mP	2/mP(00γ), 2/mP( $[{{1}\over{2}}]$ 0γ)
			2/mB	2/mB(00γ), 2/mB(0 $[{{1}\over{2}}]$ γ)
Orthorhombic	222	(222, $[{\bar 1}{\bar 1}1]$ )	mmmP	222P(00γ), 222P(0 $[{{1}\over{2}}]$ γ), 222P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			mmmI	222I(00γ)
			mmmF	222F(00γ), 222F(10γ)
			mmmC	222C(00γ), 222C(10γ)
		(222, $[1{\bar 1}{\bar 1}]$ )	mmmC	222C(α00), 222C(α0 $[{{1}\over{2}}]$ )
	mm2	(mm2, 111)	mmmP	mm2P(00γ), mm2P(0 $[{{1}\over{2}}]$ γ), mm2P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			mmmI	mm2I(00γ)
			mmmF	mm2F(00γ), mm2F(10γ)
			mmmC	mm2C(00γ), mm2C(10γ)
		(2mm, 111)	mmmC	2mmC(α00), 2mmC(α0 $[{{1}\over{2}}]$ )
		(2mm, $[{\bar 1}1{\bar 1}]$ )	mmmP	2mmP(00γ), 2mmP(0 $[{{1}\over{2}}]$ γ), 2mmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			mmmI	2mmI(00γ)
			mmmF	2mmF(00γ), 2mmF(10γ)
			mmmC	2mmC(00γ), 2mmC(10γ)
		(mm2, $[{\bar 1}1{\bar 1}]$ )	mmmC	mm2C(α00), mm2C(α0 $[{{1}\over{2}}]$ )
		(m2m, $[1{\bar 1}{\bar 1}]$ )	mmmP	m2mP(0 $[{{1}\over{2}}]$ γ)
		(m2m, $[{\bar 1}{\bar 1}1]$ )	mmmC	m2mC(α00), m2mC(α0 $[{{1}\over{2}}]$ )
	mmm	(mmm, $[11{\bar 1}]$ )	mmmP	mmmP(00γ), mmmP(0 $[{{1}\over{2}}]$ γ), mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			mmmI	mmmI(00γ)
			mmmF	mmmF(00γ), mmmF(10γ)
			mmmC	mmmC(00γ), mmmC(10γ)
		(mmm, $[{\bar 1}11]$ )	mmmC	mmmC(α00), mmmC(α0 $[{{1}\over{2}}]$ )
Tetragonal	4	(4, 1)	4/mmmP	4P(00γ), 4P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	4I(00γ)
	$[{\bar 4}]$	( $[{\bar 4}]$ , $[{\bar 1}]$ )	4/mmmP	$[{\bar 4}]$ P(00γ), $[{\bar 4}]$ P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	$[{\bar 4}]$ I(00γ)
	4/m	(4/m, $[1{\bar 1}]$ )	4/mmmP	4/mP(00γ), 4/mP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	4/mI(00γ)
	422	(422, $[1{\bar 1}{\bar 1}]$ )	4/mmmP	422P(00γ), 422P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	422I(00γ)
	4mm	(4mm, 111)	4/mmmP	4mmP(00γ), 4mmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	4mmI(00γ)
	$[{\bar 4}]$ 2m	( $[{\bar 4}]$ 2m, $[{\bar 1}{\bar 1}1]$ )	4/mmmP	$[{\bar 4}]$ 2mP(00γ), $[{\bar 4}]$ 2mP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	$[{\bar 4}]$ 2mI(00γ)
	$[{\bar 4}]$ m2	( $[{\bar 4}]$ m2, $[{\bar 1}1{\bar 1}]$ )	4/mmmP	$[{\bar 4}]$ m2P(00γ), $[{\bar 4}]$ m2P( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	$[{\bar 4}]$ m2I(00γ)
	4/mmm	(4/mmm, $[1{\bar 1}11]$ )	4/mmmP	4/mmmP(00γ), 4/mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
			4/mmmI	4/mmmI(00γ)
Trigonal	3	(3, 1)	$[{\bar 3}]$ mR	3R(00γ)
			6/mmmP	3P(00γ), 3P( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
	$[{\bar 3}]$	( $[{\bar 3}]$ , $[{\bar 1}]$ )	$[{\bar 3}]$ mR	$[{\bar 3}]$ R(00γ)
			6/mmmP	$[{\bar 3}]$ P(00γ), $[{\bar 3}]$ P( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
	32	(32, $[1{\bar 1}]$ )	$[{\bar 3}]$ mR	32R(00γ)
			6/mmmP	312P(00γ), 312P( $[{{1}\over{3}}{{1}\over{3}}]$ γ), 321P(00γ)
	3m	(3m, 11)	$[{\bar 3}]$ mR	3mR(00γ)
			6/mmmP	3m1P(00γ), 31mP(00γ), 31mP( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
	$[{\bar 3}]$ m	( $[{\bar 3}m]$ , $[{\bar 1}1]$ )	$[{\bar 3}]$ mR	$[{\bar 3}]$ mR(00γ)
			6/mmmP	$[{\bar 3}]$ 1mP(00γ), $[{\bar 3}]$ 1mP( $[{{1}\over{3}}{{1}\over{3}}]$ γ), $[{\bar 3}]$ m1P(00γ)
Hexagonal	6	(6, 1)	6/mmmP	6P(00γ)
	$[{\bar 6}]$	( $[{\bar 6}]$ , $[{\bar 1}]$ )	6/mmmP	$[{\bar 6}]$ P(00γ)
	6/m	(6/m, $[1{\bar 1}]$ )	6/mmmP	6/mP(00γ)
	622	(622, $[1{\bar 1}{\bar 1}]$ )	6/mmmP	622P(00γ)
	6mm	(6mm, 111)	6/mmmP	6mmP(00γ)
	$[{\bar 6}]$ m2	( $[{\bar 6}m2]$ , $[{\bar 1}1{\bar 1}]$ )	6/mmmP	$[{\bar 6}]$ m2P(00γ)
	$[{\bar 6}2m]$	( $[{\bar 6}2m]$ , $[{\bar 1}{\bar 1}1]$ )	6/mmmP	$[{\bar 6}]$ 2mP(00γ)
	6/mmm	(6/mmm, $[1{\bar 1}11]$ )	6/mmmP	6/mmmP(00γ)

The symbols for geometric crystal classes indicate the pairs $[[R,\varepsilon(R)]]$ of the generators of the point group. This is done by giving the crystal class for the point group [K_E] and the symbols for the corresponding elements of [K_I] . So, for example, the geometric crystal class belonging to the holohedral point group of the Bravais class $[2/mB(0{1\over2}\gamma)]$ , mentioned above, is $[(2/m,1\bar1)]$ .

The notation for the arithmetic crystal classes is similar to that for the Bravais classes. In the tables, their one-line symbols are given. They consist of the (modified) symbol of the three-dimensional symmorphic space group and, in parentheses, the appropriate components of the modulation wavevector. The three arithmetic crystal classes implying a lattice belonging to the Bravais class $[2/mB(0{1\over2}\gamma)]$ are $[2B(0{1\over2}\gamma)]$ , $[mB(0{1\over2}\gamma)]$ , and $[2/mB(0{1\over2}\gamma)]$ . The corresponding geometric crystal classes are [(2,1)] , $[(m,\bar1)]$ , and $[(2/m,1\bar1)]$ .

9.8.3.3. Tables of superspace groups

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9.8.3.3.1. Symmetry elements

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The transformations [g_s] belonging to a (3 + 1)-dimensional superspace group consist of a point-group transformation [R_s] given by the integral matrix Γ(R) and of the associated translation. So the superspace group is determined by the arithmetic crystal class of its point group and the corresponding translational components. The symbol for the arithmetic crystal class has been discussed in Subsection 9.8.3.2 . Given a point-group transformation [R_s] , the associated translation is determined up to a lattice translation. As in three dimensions, the translational part generally depends on the choice of origin. To avoid this arbitrariness, one decomposes that translation into a component (called intrinsic) independent of the origin, and a remainder. The (3 + 1)-dimensional translation $[\upsilon_s]$ associated with the point-group transformation [R_s] is given by $[\upsilon_s=\textstyle\sum\limits^{3+1}_{i=1}\upsilon_i\,a_{is}. \eqno (9.8.3.4)]$ Its origin-invariant part $[\upsilon^o_s]$ is given by $[\upsilon^o_s=\textstyle\sum\limits^4_{j=1}\upsilon^o_j\,a_{sj}\quad\hbox{with}\quad \upsilon^o_j= \displaystyle{1\over n}\,\sum^n_{m=1}\,\sum^4_{k=1}\Gamma(R^m)_{jk}\upsilon_k, \eqno (9.8.3.5)]$ where n is now the order of the point-group transformation R so that [R^n] is the identity. As customary also in three-dimensional crystallography, one indicates in the space-group symbol the invariant components $[\upsilon^o_j]$ . Notice that this means that there is an origin for [R_s] in (3 + 1)-dimensional superspace such that the translation associated with [R_s] has these components. This origin, however, may not be the same for different transformations [R_s] , as is known in three-dimensional crystallography.

Written in components, the non-primitive translation $[\upsilon_s]$ associated with the point-group element [(R,R_I)] is $[({\bf v}, \upsilon_I)]$ , where $[\upsilon_I]$ can be written as $[\delta-{\bf q}\cdot {\bf v}]$ . In accordance with (9.8.1.12), δ is defined as $[\upsilon_4]$ . The origin-invariant part $[\upsilon^o_s]$ of $[\upsilon_s]$ is $[\upsilon^o_s=({\bf v}^o,\upsilon^o_I) = {1\over n}\, \sum^n_{m=1}\, (R^m{\bf v}, R^m_I\upsilon_I) = ({\bf v}^o, \tau-{\bf q}\cdot{\bf v}^o), \eqno (9.8.3.6)]$ where $[\tau=\upsilon^o_4=\upsilon^o_I+{\bf q}\cdot{\bf v}^o.]$ The internal transformation [R_I(R)] = ɛ(R) = ɛ is either +1 or −1. When ɛ = −1 it follows from (9.8.3.6) that $[\upsilon^o_I=0]$ . For $[\varepsilon=+1]$ , one has $[\upsilon^o_I=\upsilon_I]$ . Because in that case $[{\bf q}\cdot {\bf v}^o={1\over n}\,\sum^n_{m=1}\,{\bf q}\cdot R^m{\bf v}={\bf q}^i\cdot {\bf v}, \eqno (9.8.3.7)]$ it follows that $[\tau=\upsilon_I+{\bf q}\cdot{\bf v}^o=\delta-{\bf q}\cdot{\bf v}+{\bf q}\cdot{\bf v}^o=\delta-{\bf q}^r\cdot {\bf v}. \eqno (9.8.3.8)]$

For [R_s] of order n, [R^n_s] is the identity and the associated translation is a lattice translation. The ensuing values for τ are $[0,{1\over2}, \pm{1\over3}, \pm{1\over4}]$ or $[\pm{1\over6}]$ (modulo integers). This remains true also in the case of a centred basis. The symbol of the (3 + 1)-dimensional space-group element is determined by the invariant part of its three-dimensional translation and τ. Again, that information can be given in terms of either a one-line or a two-line symbol .

In the one-line symbol , one finds: the symbol according to International Tables for Crystallography, Volume A , for the space group generated by the elements {R|v}, in parentheses the components of the modulation vector q followed by the values of τ, one for each generator appearing in the three-dimensional space-group symbol. A letter symbolizes the value of τ according to $[\matrix{ \tau\hfill&0&{1\over2}&\pm{1\over3}&\pm{1\over4}&\pm{1\over6} \cr {\rm symbol}&0&s&t&q&h}. \eqno (9.8.3.9)]$ As an example, consider the superspace group $[P2_1/m(\alpha\beta0)0s.]$ The external components $[\{R|{\bf v}\}]$ of the elements of this group form the three-dimensional space group [P2_1/m] . The modulation wavevector is αa* + βb* with respect to a conventional basis of the monoclinic lattice with unique axis c. Therefore, the point group is $[(2/m,\bar11)]$ . The point-group element $[(2,\bar1)]$ has associated a non-primitive translation with invariant part $[(\,{1\over2}{\bf c},0)=(00{1\over2}0)]$ and the point-group generator (m, 1) one with $[({\bf 0},{1\over2}\,)=(000{1\over2}\,)]$ .

In the two-line symbol, one finds in the upper line the symbol for the three-dimensional space group , in the bottom line the value of τ for the case ɛ = +1 and the symbol ` $[\bar1]$ ' when ɛ = −1. The rational part of q is indicated by means of the appropriate prefix. In the case considered, q^r = 000. So the prefix is P and the same superspace group is denoted in a two-line symbol as $[P^{P2_1/m}_{\kern5pt{\bar1}\kern7pt s}.]$ In Table 9.8.3.5 , the (3 + 1)-dimensional space groups are given by one-line symbols. These are so-called short symbols . Sometimes, a full symbol is required. Then, for the example given above one has $[P112_1/m(\alpha\beta0)000s]$ and $[P^{P112_1/m}_{\kern5pt 11\bar1\kern6pt s}]$ , respectively. Note that in the short one-line symbol for τ = 0 superspace groups (where the non-primitive translations can be transformed to zero by a choice of the origin) the zeros for the translational part are omitted. Not so, of course, in the full symbol. For example, short symbol P2₁/m(αβ0) and full symbol P112₁/m(αβ0)0000. Table 9.8.3.5 is an adapted version of the tables given by de Wolff, Janssen & Janner (1981 ) and corrected by Yamamoto, Janssen, Janner & de Wolff (1985 ).

9.8.3.3.2. Reflection conditions

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The indexing of diffraction vectors is a matter of choice of basis. When the basis chosen is not a primitive one, the indices have to satisfy certain conditions known as centring conditions. This holds for the main reflections (centring in ordinary space) as well as for satellites (centring in superspace). These centring conditions for reflections have been discussed in Subsection 9.8.2.1 .

In addition to these general reflection conditions, there may be special reflection conditions related to the existence of non-primitive translations in the (3 + 1)-dimensional space group, just as is the case for glide planes and screw axes in three dimensions.

Special reflection conditions can be derived from transformation properties of the structure factor under symmetry operations. Transforming the geometric structure factor by an element $[g_s=(\{R|{\bf v}\}, \{R_I|\upsilon_I\})]$ , one obtains $[S_{\bf H}=S_{R^{-1}{\bf H}}\exp[-2\pi i ({\bf H}\cdot{\bf v}+H_I\cdot \upsilon_I)]. \eqno (9.8.3.10)]$ Therefore, if RH = H, the corresponding structure factor vanishes unless $[{\bf H}\cdot{\bf v}+H_I\cdot\upsilon_I]$ is an integer.

The form of such a reflection condition in terms of allowed or forbidden sets of indices depends on the basis chosen. When a lattice basis is chosen, one has $[H_s=({\bf H}, H_I)=\textstyle\sum\limits^4_{i=1}h_i{\bf a}^*_{si}, \eqno (9.8.3.11)]$ $[\upsilon_s=({\bf v}, \upsilon_I)+\textstyle\sum\limits^4_{i=1}\upsilon_i{\bf a}_{si}. \eqno (9.8.3.12)]$ Then the reflection condition becomes $[H_s\cdot\upsilon_s=\textstyle\sum\limits^4_{i=1}h_i\upsilon_i=\hbox{integer } \quad \hbox{ for }R{\bf H}={\bf H}. \eqno (9.8.3.13)]$

In terms of external and internal shift components, the reflection condition can be written as $[H_s\cdot\upsilon_s={\bf H}\cdot{\bf v}+H_I\cdot\upsilon_I={\bf H}\cdot{\bf v}+m\upsilon_I=\hbox{integer for }R{\bf H}={\bf H}. \eqno (9.8.3.14)]$ With $[{\bf H}={\bf K}+m{\bf q}]$ and $[\upsilon_I=\delta-\bf{q\cdot v}]$ , (9.8.3.14) gives $[{\bf K}\cdot{\bf v}+m\delta=\hbox{integer$\quad$ for }R{\bf H}={\bf H}. \eqno (9.8.3.15)]$ For $[{\bf v} =\upsilon_1{\bf a}+\upsilon_2{\bf b}+\upsilon_3{\bf c}]$ and $[{\bf K} =h{\bf a}^*+k{\bf b}^*+l{\bf c}^*]$ , (9.8.3.15) takes the form (9.8.3.13): $[h\upsilon _1+k \upsilon_2+l\upsilon_3+m\delta = \hbox{integer }\quad \hbox{ for }R{\bf H}={\bf H}. \eqno (9.8.3.16)]$ When the modulation wavevector has a rational part, one can choose another basis (Subsection 9.8.2.1 ) such that K′ = K + mq^r has integer coefficients: $[{\bf H}={\bf K}'+m{\bf q}^i=H{\bf a}^*_c + K{\bf b}^*_c + L{\bf c}^*_c+m{\bf q}^i.]$ Then, (9.8.3.15) with $[\tau=\delta-{\bf q}^r\cdot{\bf v}]$ becomes $[{\bf K}'\cdot{\bf v}+m\tau=\hbox{integer }\quad\hbox{ for }R{\bf H}={\bf H} \eqno (9.8.3.17)]$ and (9.8.3.16) transforms into $[H\upsilon'_1+K\upsilon'_2 +L\upsilon'_3 +m\tau=\hbox{integer }\quad\hbox{ for }R{\bf H}={\bf H}, \eqno (9.8.3.18)]$ in which $[\upsilon'_1,\upsilon'_2]$ , and $[\upsilon'_3]$ are the components of v with respect to the basis $[{\bf a}_c,{\bf b}_c]$ , and $[{\bf c}_c]$ .

As an example, consider a (3 + 1)-dimensional space-group transformation with R a mirror perpendicular to the x axis, $[\varepsilon=1,{\bf v}={1\over2}{\bf b}]$ , and $[\tau={1\over4}]$ with b orthogonal to a. The modulation wavevector is supposed to be $[(\,{1\over2}{1\over2}\gamma)]$ . Then $[\delta={1\over4}+{\bf q}^r\cdot{\bf v}={1\over2}]$ . The vectors H left invariant by R satisfy the relation 2h + m = 0. For such a vector, the reflection condition becomes $[{\bf K}\cdot{\bf v}+m\delta=\textstyle{1\over2}{\bf K}\cdot{\bf b}+{1\over2}m=\displaystyle{k+m\over2}={\rm integer, \quad or\ } k+m=2n.]$ For the basis $[{1\over2}({\bf a}^*+{\bf b}^*)]$ , $[{1\over2}({\bf a}^*-{\bf b}^*)]$ , c*, the rational part of the wavevector vanishes. The indices with respect to this basis are H = h + k + m, K = h − k, L = l and m. The condition now becomes $[{\bf K}'\cdot{\bf v}+m\tau={H-K+m\over4}= {\rm integer},]$ $[{\rm or}\quad H-K+m=4n, \quad {\rm for}\ K=-H.]$ Of course, both calculations give the same result: k + m = 2n for h, k, l, −2h and H − K + m = 4n for H, −H, L, m.

The special reflection conditions for the elements occurring in (3 + 1)-dimensional space groups are given in Table 9.8.3.5 .

9.8.3.4. Guide to the use of the tables

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In the tables, Bravais classes, point groups, and space groups are given for three-dimensional incommensurate modulated crystals with a modulation of dimension one and for two-dimensional crystals (e.g. surfaces) with one- and two-dimensional modulation (Janssen, Janner & de Wolff, 1980 ). In the following, we discuss briefly the information given. Examples of their use can be found in Subsection 9.8.3.5 .

To determine the symmetry of the modulated phase, one first determines its average structure, which is obtained from the main reflections. Since this structure has three-dimensional space-group symmetry, this analysis is performed in the usual way.

The diffraction pattern of the three-dimensional modulated phase can be indexed by 3 + 1 integers. The Bravais class is determined by the symmetry of the vector module spanned by the 3 + 1 basis vectors. The crystallographic system of the pattern is equal to or lower than that implied by the main reflections. One chooses a conventional basis (q^r = 0) for the vector module, and finds the Bravais class from the general reflection conditions using Table 9.8.3.6 . The relation between indices hklm with respect to the basis a*, b*, c*, and q and HKLm with respect to the conventional basis $[{\bf a}^*_c]$ , $[{\bf b}^*_c]$ , $[{\bf c}^*_c]$ , and $[{\bf q}^i]$ is also given there.

Table 9.8.3.2(a) gives the number labelling the (3 + 1)-dimensional Bravais class, its symbol, its external and internal point group, and the modulation wavevector. Moreover, the superspace conventional basis (for which the rational part q^r vanishes) and the corresponding (3 + 1)-dimensional centring are given. Because the four-dimensional lattices belong to Euclidean Bravais classes, the corresponding class is also given in the notation of Janssen (1969 ) and Brown et al. (1978 ).

The point group of the modulated structure is a subgroup of the holohedry of its lattice Λ. In Table 9.8.3.3 , for each system the (3 + 1)-dimensional point groups are given. Each system contains one or more Bravais classes. Each geometric crystal class contains one or more arithmetic crystal classes. The (3 + 1)-dimensional arithmetic classes belonging to a given geometric crystal class are also listed in Table 9.8.3.3 .

Starting from the space group of the average structure, one can determine the (3 + 1)-dimensional superspace group. In Table 9.8.3.5 , the full list of these (3 + 1)-dimensional superspace groups is given for the incommensurate case and are ordered according to their basic space group. They have a number n.m where n is the number of the basic space group one finds in International Tables for Crystallography, Volume A . The various (3 + 1)-dimensional superspace groups for each basic group are distinguished by the number m. Furthermore, the symbol of the basic space group, the point group, and the symbol for the corresponding superspace group are given . In the last column, the special reflection conditions are listed for typical symmetry elements. These may help in the structure analysis. The (2 + d)-dimensional superspace groups, relevant for modulated surface structures, are given in Table 9.8.3.4 .

Table 9.8.3.4| top | pdf |
(2 + 1)- and (2 + 2)-Dimensional superspace groups

(a) (2 + 1)-Dimensional superspace groups. The number labelling the superspace group is denoted by n.m, where n is the number attached to the two-dimensional basic space group and m numbers the various superspace groups having the same basic space group. The symbol of the basic space group, the symbol for the three-dimensional point group, the number of the three-dimensional Bravais class to which the superspace group belongs (Table 9.8.3.1a ) and the superspace-group symbol are also given.

No.	Basic space group	Point group K_s	Bravais class No.	Group symbol
Oblique
1.1	p1	(1, 1)	1	p1(αβ)
2.1	p2	(2, $[{\bar 1}]$ )	1	p2(αβ)
Rectangular
3.1	pm	(m, 1)	2	pm1(0β)
3.2			2	pm1(0β)s0
3.3			3	pm1( $[{{1}\over{2}}]$ β)
3.4		(m, $[{\bar 1}]$ )	2	p1m(0β)
3.5			3	p1m( $[{{1}\over{2}}]$ β)
4.1	pg	(m, 1)	2	pg1(0β)
4.2			3	pg1( $[{{1}\over{2}}]$ β)
4.3		(m, $[{\bar 1}]$ )	2	p1g(0β)
5.1	cm	(m, 1)	4	cm1(0β)
5.2			4	cm1(0β)s0
5.3		(m, $[{\bar 1}]$ )	4	c1m(0β)
6.1	pmm	(mm, $[1{\bar 1}]$ )	2	pmm(0β)
6.2			2	pmm(0β)s0
6.3			3	pmm( $[{{1}\over{2}}]$ β)
7.1	pmg	(mm, $[1{\bar 1}]$ )	2	pmg(0β)
7.2			2	pgm(0β)
7.3			3	pgm( $[{{1}\over{2}}]$ β)
8.1	pgg	(mm, $[1{\bar 1}]$ )	2	pgg(0β)
9.1	cmm	(mm, $[1{\bar 1}]$ )	4	cmm(0β)
9.2			4	cmm(0β)s0

(b) (2 + 2)-Dimensional superspace groups. The number labelling the superspace group is denoted by n.m, where n is the number attached to the two-dimensional basic space group and m numbers the various superspace groups having the same basic space group. The symbol of the basic space group, the symbol for the four-dimensional point group, the number of the four-dimensional Bravais class to which the superspace group belongs (Table 9.8.3.1b ) and the superspace-group symbol are also given.

No.	Basic space group	Point group K_s	Bravais class No.	Group symbol
Oblique
1.1	p1	(1, 1)	1	p1(αβ, λμ)
2.1	p2	(2, 2)	1	p2(αβ, λμ)
Rectangular
3.1	pm	(m, 1)	2	pm1(0β, 0μ)
3.2			2	pm1(0β, 0μ)s0, 0
3.3			3	pm1( $[{{1}\over{2}}]$ β, 0μ)
3.4			3	pm1( $[{{1}\over{2}}]$ β, 0μ)s0, 0
3.5		(m, 2)	2	p1m(0β, 0μ)
3.6			3	p1m( $[{{1}\over{2}}]$ β, 0μ)
3.7		(m, m)	4	pm1(α0, 0μ)
3.8			4	pm1(α0, 0μ)0s, 0
3.9			5	pm1(α $[{{1}\over{2}}]$ , 0μ)
3.10			5	pm1(α $[{{1}\over{2}}]$ , 0μ)0s, 0
3.11			5	p1m(α $[{{1}\over{2}}]$ , 0μ)
3.12			5	p1m(α $[{{1}\over{2}}]$ , 0μ)0, s0
3.13			6	pm1(α $[{{1}\over{2}}]$ , $[{{1}\over{2}}]$ μ)
3.14			7	pm1(αβ)
4.1	pg	(m, 1)	2	pg1(0β, 0μ)
4.2			3	pg1( $[{{1}\over{2}}]$ β, 0μ)
4.3		(m, 2)	2	p1g(0β, 0μ)
4.4		(m, m)	4	pg1(α0, 0μ)
4.5			7	pg1(α $[{{1}\over{2}}]$ , $[{{1}\over{2}}]$ μ)
5.1	cm	(m, 1)	8	cm1(0β, 0μ)
5.2			8	cm1(0β, 0μ)s0, 0
5.3		(m, 2)	8	c1m(0β, 0μ)
5.4		(m, m)	9	cm1(α0, 0μ)
5.5			9	cm1(α0, 0μ)0s, 0
5.6			10	cm(αβ)
6.1	pmm	(mm, 12)	2	pmm(0β, 0μ)
6.2			2	pmm(0β, 0μ)s0, 0
6.3			3	pmm( $[{{1}\over{2}}]$ β, 0μ)
6.4			3	pmm( $[{{1}\over{2}}]$ β, 0μ)s0, 0
6.5		(mm, mm)	4	pmm(α0, 0μ)
6.6			4	pmm(α0, 0μ)0s, 0
6.7			4	pmm(α0, 0μ)0s, s0
6.8			5	pmm(α $[{{1}\over{2}}]$ , 0μ)
6.9			5	pmm(α $[{{1}\over{2}}]$ , 0μ)0s, 0
6.10			6	pmm(α $[{{1}\over{2}}]$ , $[{{1}\over{2}}]$ μ)
6.11			7	pmm(αβ)
7.1	pmg	(mm, 12)	2	pmg(0β, 0μ)
7.2			2	pmg(0β, 0μ)0s, 0
7.3			2	pgm(0β, 0μ)
7.4			3	pgm( $[{{1}\over{2}}]$ β, 0μ)
7.5		(mm, mm)	4	pgm(α0, 0μ)
7.6			4	pgm(α0, 0μ)0, s0
7.7			5	pmg(α $[{{1}\over{2}}]$ , 0μ)
7.8			5	pmg(α $[{{1}\over{2}}]$ , 0μ)0s, 0
7.9			7	pgm(αβ)
8.1	pgg	(mm, 12)	2	pgg(0β, 0μ)
8.2		(mm, mm)	4	pgg(α0, 0μ)
8.3			7	pgg(αβ)
9.1	cmm	(mm, 12)	8	cmm(0β, 0μ)
9.2			8	cmm(0β, 0μ)0s, 0
9.3		(mm, mm)	9	cmm(α0, 0μ)
9.4			9	cmm(α0, 0μ)0s, 0
9.5			9	cmm(α0, 0μ)0s, s0
9.6			10	cmm(αβ)
Tetragonal
10.1	p4	(4, 4)	11	p4(αβ)
11.1	p4m	(4m, 4m)	12	p4m(α0)
11.2			12	p4m(α0)0, 0s
11.3			13	p4m(α $[{{1}\over{2}}]$ )
11.4		( $[4\dot m]$ , $[4\ddot m]$ )	14	p4m(αα)
11.5			14	p4m(αα)0, 0s
12.1	p4g	(4m, 4m)	12	p4g(α0)
12.2			12	p4g(α0), 0s
12.3		( $[4\dot m]$ , $[4\ddot m]$ )	14	p4g(αα)
12.4			14	p4g(αα)0, 0s
Hexagonal
13.1	p3	(3, 3)	15	p3(αβ)
14.1	p3m1	(3m, 3m)	16	p3m1(α0)
14.2		( $[3\dot m]$ , $[3\ddot m]$ )	17	p3m1(αα)
15.1	p31m	(3m, 3m)	16	p31m(α0)
16.1	p6	(6, 6)	15	p6(αβ)
17.1	p6m	(6m, 6m)	16	p6m(α0)
17.2		( $[6\dot m]$ , $[6\ddot m]$ )	17	p6m(αα)

9.8.3.5. Examples

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(A) Na₂CO₃

Na₂CO₃ has a phase transition at about 753 K from the hexagonal to the monoclinic phase. At about 633 K, one vibration mode becomes unstable and below the transition temperature T_i = 633 K there is a modulated γ-phase (de Wolff & Tuinstra, 1986 ). At low temperature (128 K), a transition to a commensurate phase has been reported.

The main reflections in the modulated phase belong to a monoclinic lattice, and the satellites to a modulation with wavevector q = αa* + γc*, b axis unique. The dimension of the modulation is one. The main reflections satisfy the condition $[hkl0, h+k={\rm even}.]$ Therefore, the lattice of the average structure is C-centred monoclinic. For the satellites, the same general condition holds (hklm, h + k = even). From Table 9.8.3.6 , one sees after a change of axes that the Bravais class of the modulated structure is $[\hbox{No. 4: }2/mC(\alpha0\gamma).]$ Table 9.8.3.2 (a) shows that the point group of the vector module is $[2/m(\bar11)]$ . The point group of the modulated structure is equal to or a subgroup of this one.

The space group of the average structure determined from the main reflections is C2/m (No. 12 in International Tables for Crystallography, Volume A ). The superspace group may then be determined from the special reflection condition $[h0lm, m={\rm even}]$ using Table 9.8.3.5 . There are five superspace groups with basic group No. 12. Among them there are two in Bravais class 4. The reflection condition mentioned leads to the group $[\hbox{No. }12.2=C2/m(\alpha0\gamma)0s = P{^{C2/m}_{\kern5pt{\bar1} \kern5pt s}}]$ In principle, the superspace group could be a subgroup of this, but, since the transition normal–incommensurate is of second order, Landau theory predicts that the basic space group is the symmetry group of the unmodulated monoclinic phase, which is [C2/m] .

Table 9.8.3.5| top | pdf | superspace group finder
(3 + 1)-Dimensional superspace groups

The number labelling the superspace group is denoted by n.m, where n is the number attached to the three-dimensional basic space group and m numbers the various superspace groups having the same basic space group. The symbol of the basic space group, the symbol for the four-dimensional point group K_s, the number of the four-dimensional Bravais class to which the superspace group belongs (Table 9.8.3.2 a), and the superspace-group symbol are also given. The superspace-group symbol is indicated in the short notation, i.e. for the basic group one uses the short symbol from International Tables for Crystallography, Volume A , and then the values of τ are given for each of the generators in this symbol, unless all these values are zero. Then, instead of writing a number of zeros, one omits them all. Finally, the special reflection conditions due to non-primitive translations are given, for hklm if q^r = 0 and for HKLm otherwise. Recall the HKLm are the indices with respect to a conventional basis $[{\bf a}_c^*, {\bf b}_c^*, {\bf c}_c^*, {\bf q}^i]$ as in Table 9.8.3.2 (a). The reflection conditions due to centring translations are given in Table 9.8.3.6 .

No.	Basic space group	Point group K_s	Bravais class No.	Group symbol	Special reflection conditions
1.1	P1	(1, 1)	1	P1(αβγ)
2.1	$[P{\bar 1}]$	( $[{\bar 1}]$ , $[{\bar 1}]$ )	1	P $[{\bar 1}]$ (αβγ)
3.1	P2	(2, $[{\bar 1}]$ )	2	P2(αβ0)
3.2		(2, $[{\bar 1}]$ )	3	P2(αβ $[{{1}\over{2}}]$ )
3.3		(2, 1)	5	P2(00γ)
3.4		(2, 1)	5	P2(00γ)s	00lm: m = 2n
3.5		(2, 1)	6	P2( $[{{1}\over{2}}]$ 0γ)
4.1	P2₁	(2, $[{\bar 1}]$ )	2	P2₁(αβ0)	00l0: l = 2n
4.2		(2, 1)	5	P2₁(00γ)	00lm: l = 2n
4.3		(2, 1)	6	P2₁( $[{{1}\over{2}}]$ 0γ)	00Lm: L = 2n
5.1	B2	(2, $[{\bar 1}]$ )	4	B2(αβ0)
5.2		(2, 1)	7	B2(00γ)
5.3		(2, 1)	7	B2(00γ)s	00lm: m = 2n
5.4		(2, 1)	8	B2(0 $[{{1}\over{2}}]$ γ)
6.1	Pm	(m, 1)	2	Pm(αβ0)
6.2		(m, 1)	2	Pm(αβ0)s	hk0m: m = 2n
6.3		(m, 1)	3	Pm(αβ $[{{1}\over{2}}]$ )
6.4		(m, $[{\bar 1}]$ )	5	Pm(00γ)
6.5		(m, $[{\bar 1}]$ )	6	Pm( $[{{1}\over{2}}]$ 0γ)
7.1	Pb	(m, 1)	2	Pb(αβ0)	hk0m: k = 2n
7.2		(m, 1)	3	Pb(αβ $[{{1}\over{2}}]$ )	HK0m: K = 2n
7.3		(m, $[{\bar 1}]$ )	5	Pb(00γ)	hk00: k = 2n
7.4		(m, $[{\bar 1}]$ )	6	Pb( $[{{1}\over{2}}]$ 0γ)	HK00: K = 2n
8.1	Bm	(m, 1)	4	Bm(αβ0)
8.2		(m, 1)	4	Bm(αβ0)s	hk0m: m = 2n
8.3		(m, $[{\bar 1}]$ )	7	Bm(00γ)
8.4		(m, $[{\bar 1}]$ )	8	Bm(0 $[{{1}\over{2}}]$ γ)
9.1	Bb	(m, 1)	4	Bb(αβ0)	hk0m: k = 2n
9.2		(m, $[{\bar 1}]$ )	7	Bb(00γ)	hk00: k = 2n
10.1	P2/m	(2/m, $[{\bar 1}1]$ )	2	P2/m(αβ0)
10.2		(2/m, $[{\bar 1}1]$ )	2	P2/m(αβ0)0s	hk0m: m = 2n
10.3		(2/m, $[{\bar 1}1]$ )	3	P2/m(αβ $[{{1}\over{2}}]$ )
10.4		(2/m, $[1{\bar 1}]$ )	5	P2/m(00γ)
10.5		(2/m, $[1{\bar 1}]$ )	5	P2/m(00γ)s0	00lm: m = 2n
10.6		(2/m, $[1{\bar 1}]$ )	6	P2/m( $[{{1}\over{2}}]$ 0γ)
11.1	P2₁/m	(2/m, $[{\bar 1}1]$ )	2	P2₁/m(αβ0)	00l0: l = 2n
11.2		(2/m, $[{\bar 1}1]$ )	2	P2₁/m(αβ0)0s	00l0: l = 2n; hk0m: m = 2n
11.3		(2/m, $[1{\bar 1}]$ )	5	P2₁/m(00γ)	00lm: l = 2n
11.4		(2/m, $[1{\bar 1}]$ )	6	P2₁/m( $[{{1}\over{2}}]$ 0γ)	00Lm: L = 2n
12.1	B2/m	(2/m, $[{\bar 1}1]$ )	4	B2/m(αβ0)
12.2		(2/m, $[{\bar 1}1]$ )	4	B2/m(αβ0)0s	hk0m: m = 2n
12.3		(2/m, $[1{\bar 1}]$ )	7	B2/m(00γ)
12.4		(2/m, $[1{\bar 1}]$ )	7	B2/m(00γ)s0	00lm: m = 2n
12.5		(2/m, $[1{\bar 1}]$ )	8	B2/m( $[{{1}\over{2}}]$ 0γ)
13.1	P2/b	(2/m, $[{\bar 1}1]$ )	2	P2/b(αβ0)	hk0m: k = 2n
13.2		(2/m, $[{\bar 1}1]$ )	3	P2/b(αβ $[{{1}\over{2}}]$ )	HK0m: m = 2n
13.3		(2/m, $[1{\bar 1}]$ )	5	P2/b(00γ)	hk00: k = 2n
13.4		(2/m, $[1{\bar 1}]$ )	5	P2/b(00γ)s0	00lm: m = 2n; hk00: k = 2n
13.5		(2/m, $[1{\bar 1}]$ )	6	P2/b( $[{{1}\over{2}}]$ 0γ)	HK00: K = 2n
14.1	P2₁/b	(2/m, $[{\bar 1}1]$ )	2	P2₁/b(αβ0)	00l0: l = 2n; hk0m: k = 2n
14.2		(2/m, $[1{\bar 1}]$ )	5	P2₁/b(00γ)	00lm: l = 2n; hk00: k = 2n
14.3		(2/m, $[1{\bar 1}]$ )	6	P2₁/b( $[{{1}\over{2}}]$ 0γ)	00Lm: L = 2n; HK00: K = 2n
15.1	B2/b	(2/m, $[{\bar 1}1]$ )	4	B2/b(αβ0)	hk0m: k = 2n
15.2		(2/m, $[1{\bar 1}]$ )	7	B2/b(00γ)	hk00: k = 2n
15.3		(2/m, $[1{\bar 1}]$ )	7	B2/b(00γ)s0	00lm: m = 2n; hk00: k = 2n
16.1	P222	(222, $[{\bar 1}{\bar 1}1]$ )	9	P222(00γ)
16.2			9	P222(00γ)00s	00lm: m = 2n
16.3			10	P222(0 $[{{1}\over{2}}]$ γ)
16.4			11	P222( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
17.1	P222₁	(222, $[{\bar 1}{\bar 1}1]$ )	9	P222₁(00γ)	00lm: l = 2n
17.2			10	P222₁(0 $[{{1}\over{2}}]$ γ)	00Lm: L = 2n
17.3			11	P222₁( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 2n
17.4			9	P2₁22(00γ)	h000: h = 2n
17.5			9	P2₁22(00γ)00s	h000: h = 2n; 00lm: m = 2n
17.6			10	P2₁22(0 $[{{1}\over{2}}]$ γ)	H000: H = 2n
18.1	P2₁2₁2	(222, $[{\bar 1}{\bar 1}1]$ )	9	P2₁2₁2(00γ)	h000: h = 2n; 0k00: k = 2n
18.2			9	P2₁2₁2(00γ)00s	h000: h = 2n; 0k00: k = 2n; 00lm: m = 2n
18.3			9	P2₁22₁(00γ)	h000: h = 2n; 00lm: l = 2n
18.4			10	P2₁22₁(0 $[{{1}\over{2}}]$ γ)	H000: H = 2n; 00Lm: L = 2n
19.1	P2₁2₁2₁	(222, $[{\bar 1}{\bar 1}1]$ )	9	P2₁2₁2₁(00γ)	h000: h = 2n; 0k00: k = 2n; 00lm: l = 2n
20.1	C222₁	(222, $[{\bar 1}{\bar 1}1]$ )	13	C222₁(00γ)	00lm: l = 2n
20.2			14	C222₁(10γ)	00Lm: L = 2n
20.3			15	A2₁22(00γ)	h000: h = 2n
20.4			15	A2₁22(00γ)00s	h000: h = 2n; 00lm: m = 2n
21.1	C222	(222, $[{\bar 1}{\bar 1}1]$ )	13	C222(00γ)
21.2			13	C222(00γ)00s	00lm: m = 2n
21.3			14	C222(10γ)
21.4			14	C222(10γ)00s	00Lm: m = 2n
21.5			15	A222(00γ)
21.6			15	A222(00γ)00s	00lm: m = 2n
21.7			16	A222( $[{{1}\over{2}}]$ 0γ)
22.1	F222	(222, $[{\bar 1}{\bar 1}1]$ )	17	F222(00γ)
22.2			17	F222(00γ)00s	00lm: m = 2n
22.3			18	F222(10γ)
23.1	I222	(222, $[{\bar 1}{\bar 1}1]$ )	12	I222(00γ)
23.2			12	I222(00γ)00s	00lm: m = 2n
24.1	I2₁2₁2₁	(222, $[{\bar 1}{\bar 1}1]$ )	12	I2₁2₁2₁(00γ)	h000: h = 2n; 0k00: k = 2n; 00lm: l = 2n
24.2			12	I2₁2₁2₁(00γ)00s	h000: h = 2n; 0k00: k = 2n; 00lm: l + m = 2n
25.1	Pmm2	(mm2, 111)	9	Pmm2(00γ)
25.2			9	Pmm2(00γ)s0s	0klm: m = 2n
25.3			9	Pmm2(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
25.4			10	Pmm2(0 $[{{1}\over{2}}]$ γ)
25.5			10	Pmm2(0 $[{{1}\over{2}}]$ γ)s0s	0KLm: m = 2n
25.6			11	Pmm2( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
25.7		(m2m, $[1{\bar 1}{\bar 1}]$ )	10	Pm2m(0 $[{{1}\over{2}}]$ γ)
25.8			10	Pm2m(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n
25.9		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2mm(00γ)
25.10			9	P2mm(00γ)0s0	h0lm: m = 2n
25.11			10	P2mm(0 $[{{1}\over{2}}]$ γ)
25.12			11	P2mm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
26.1	Pmc2₁	(mm2, 111)	9	Pmc2₁(00γ)	h0lm: l = 2n
26.2			9	Pmc2₁(00γ)s0s	0klm: m = 2n; h0lm: l = 2n
26.3			10	Pmc2₁(0 $[{{1}\over{2}}]$ γ)	H0Lm: L = 2n
26.4			10	Pmc2₁(0 $[{{1}\over{2}}]$ γ)s0s	0KLm: m = 2n; H0Lm: L = 2n
26.5			10	Pcm2₁(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n
26.6			11	Pmc2₁( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	H0Lm: L = 2n
26.7		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2₁am(00γ)	h0lm: h = 2n
26.8			9	P2₁am(00γ)0s0	h0lm: h + m = 2n
26.9			9	P2₁ma(00γ)	hk00: h = 2n
26.10			9	P2₁ma(00γ)0s0	h0lm: m = 2n; hk00: h = 2n
26.11			10	P2₁am(0 $[{{1}\over{2}}]$ γ)	H0Lm: H = 2n
26.12			10	P2₁ma(0 $[{{1}\over{2}}]$ γ)	HK00: H = 2n
27.1	Pcc2	(mm2, 111)	9	Pcc2(00γ)	0klm: l = 2n; h0lm: l = 2n
27.2			9	Pcc2(00γ)s0s	0klm: l + m = 2n; h0lm: l = 2n
27.3			10	Pcc2(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: L = 2n
27.4			11	Pcc2( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: L = 2n
27.5		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2aa(00γ)	h0lm: h = 2n; hk00: h = 2n
27.6			9	P2aa(00γ)0s0	h0lm: h + m = 2n; hk00: h = 2n
27.7			10	P2aa(0 $[{{1}\over{2}}]$ γ)	H0Lm: H = 2n; HK00: H = 2n
28.1	Pma2	(mm2, 111)	9	Pma2(00γ)	h0lm: h = 2n
28.2			9	Pma2(00γ)s0s	0klm: m = 2n; h0lm: h = 2n
28.3			9	Pma2(00γ)ss0	0klm: m = 2n; h0lm: h + m = 2n
28.4			9	Pma2(00γ)0ss	h0lm: h + m = 2n
28.5			10	Pma2(0 $[{{1}\over{2}}]$ γ)	H0Lm: H = 2n
28.6			10	Pma2(0 $[{{1}\over{2}}]$ γ)s0s	0KLm: m = 2n; H0Lm: H = 2n
28.7		(m2m, $[1{\bar 1}{\bar 1}]$ )	10	Pm2a(0 $[{{1}\over{2}}]$ γ)	HK00: H = 2n
28.8			10	Pm2a(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n; HK00: H = 2n
28.9			10	Pc2m(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n
28.10		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2cm(00γ)	h0lm: l = 2n
28.11			9	P2mb(00γ)	hk00: k = 2n
28.12			9	P2mb(00γ)0s0	h0lm: m = 2n; hk00: k = 2n
28.13			10	P2cm(0 $[{{1}\over{2}}]$ γ)	H0Lm: L = 2n
28.14			11	P2cm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	H0Lm: L = 2n
29.1	Pca2₁	(mm2, 111)	9	Pca2₁(00γ)	0klm: l = 2n; h0lm: h = 2n
29.2			9	Pca2₁(00γ)0ss	0klm: l = 2n; h0lm: h + m = 2n
29.3			10	Pca2₁(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: H = 2n
29.4		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2₁ca(00γ)	hk00: h = 2n; h0lm: l = 2n
29.5			9	P2₁ab(00γ)	h0lm: h = 2n; hk00: k = 2n
29.6			9	P2₁ab(00γ)0s0	h0lm: h + m = 2n; hk00: k = 2n
29.7			10	P2₁ca(0 $[{{1}\over{2}}]$ γ)	H0Lm: L = 2n; HK00: H = 2n
30.1	Pcn2	(mm2, 111)	9	Pcn2(00γ)	0klm: l = 2n; h0lm: h + l = 2n
30.2			9	Pcn2(00γ)s0s	0klm: l + m = 2n; h0lm: h + l = 2n
30.3			10	Pcn2(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: H + L = 2n
30.4		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2na(00γ)	h0lm: h + 1 = 2n; hk00: h = 2n
30.5			9	P2an(00γ)	h0lm: h = 2n; hk00: h + k = 2n
30.6			9	P2an(00γ)0s0	h0lm: h + m = 2n; hk00: h + k = 2n
30.7			10	P2na(0 $[{{1}\over{2}}]$ γ)	H0Lm: H + L = 2n; HK00: H = 2n
30.8			11	P2an( $[{{1}\over{2}}{{1}\over{2}}]$ γ)0q0	H0Lm: 2H + m = 4n; HK00: H + K = 2n
31.1	Pmn2₁	(mm2, 111)	9	Pmn2₁(00γ)	h0lm: h + l = 2n
31.2			9	Pmn2₁(00γ)s0s	0klm: m = 2n; h0lm: h + l = 2n
31.3			10	Pmn2₁(0 $[{{1}\over{2}}]$ γ)	H0Lm: H + L = 2n
31.4			10	Pmn2₁(0 $[{{1}\over{2}}]$ γ)s0s	0KLm: m = 2n; H0Lm: H + L = 2n
31.5		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2₁nm(00γ)	h0lm: h + l = 2n
31.6			9	P2₁mn(00γ)	hk00: h + k = 2n
31.7			9	P2₁mn(00γ)0s0	hk00: h + k = 2n; h0lm: m = 2n
31.8			10	P2₁nm(0 $[{{1}\over{2}}]$ γ)	H0Lm: H + L = 2n
32.1	Pba2	(mm2, 111)	9	Pba2(00γ)	0klm: k = 2n; h0lm: h = 2n
32.2			9	Pba2(00γ)s0s	0klm: k + m = 2n; h0lm: h = 2n
32.3			9	Pba2(00γ)ss0	0klm: k + m = 2n; h0lm: h + m = 2n
32.4			11	Pba2( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	0KLm: 2K + m = 4n; H0Lm: 2H + m = 4n
32.5		(m2m, $[1{\bar 1}{\bar 1}]$ )	10	Pc2a(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; HK00: H = 2n
32.6		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2cb(00γ)	h0lm: l = 2n; hk00: k = 2n
33.1	Pbn2₁	(mm2, 111)	9	Pbn2₁(00γ)	0klm: k = 2n; h0lm: h + l = 2n
33.2			9	Pbn2₁(00γ)s0s	0klm: k + m = 2n; h0lm: h + l = 2n
33.3			11	Pbn2₁( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	0KLm: 2K + m = 4n; H0Lm: 2H + 2L + m = 4n
33.4		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2₁nb(00γ)	h0lm: h + l = 2n; hk00: k = 2n
33.5			9	P2₁cn(00γ)	h0lm: l = 2n; hk00: h + k = 2n
34.1	Pnn2	(mm2, 111)	9	Pnn2(00γ)	0klm: k + l = 2n; h0lm: h + l = 2n
34.2			9	Pnn2(00γ)s0s	0klm: k + l + m = 2n; h0lm: h + l = 2n
34.3			11	Pnn2( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	0KLm: 2K + 2L + m = 4n; H0Lm: 2H + 2L + m = 4n
34.4		(2mm, $[{\bar 1}1{\bar 1}]$ )	9	P2nn(00γ)	h0lm: h + l = 2n; hk00: h + k = 2n
34.5			11	P2nn( $[{{1}\over{2}}{{1}\over{2}}]$ γ)0q0	H0Lm: 2H + 2L + m = 4n; HK00: H + K = 2n
35.1	Cmm2	(mm2, 111)	13	Cmm2(00γ)
35.2			13	Cmm2(00γ)s0s	0klm: m = 2n
35.3			13	Cmm2(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
35.4			14	Cmm2(10γ)
35.5			14	Cmm2(10γ)s0s	0KLm: m = 2n
35.6			14	Cmm2(10γ)ss0	0KLm: m = 2n; H0Lm: m = 2n
35.7		(2mm, $[{\bar 1}1{\bar 1}]$ )	15	A2mm(00γ)
35.8			15	A2mm(00γ)0s0	h0lm: m = 2n
35.9			16	A2mm( $[{{1}\over{2}}]$ 0γ)
35.10			16	A2mm( $[{{1}\over{2}}]$ 0γ)0s0	H0Lm: m = 2n
36.1	Cmc2₁	(mm2, 111)	13	Cmc2₁(00γ)
36.2			13	Cmc2₁(00γ)s0s	0klm: m = 2n; h0lm: l = 2n
36.3			14	Cmc2₁(10γ)	H0Lm: L = 2n
36.4			14	Cmc2₁(10γ)s0s	0KLm: m = 2n; H0Lm: L = 2n
36.5		(2mm, $[{\bar 1}1{\bar 1}]$ )	15	A2₁am(00γ)	h0lm: h = 2n
36.6			15	A2₁am(00γ)0s0	h0lm: h + m = 2n
36.7			15	A2₁ma(00γ)	hk00: h = 2n
36.8			15	A2₁ma(00γ)0s0	h0lm: m = 2n; hk00: h = 2n
37.1	Ccc2	(mm2, 111)	13	Ccc2(00γ)	0klm: l = 2n; h0lm: l = 2n
37.2			13	Ccc2(00γ)s0s	0klm: l + m = 2n; h0lm: l = 2n
37.3			14	Ccc2(10γ)	0KLm: L = 2n; H0Lm: L = 2n
37.4			14	Ccc2(10γ)s0s	0KLm: L + m = 2n; H0Lm: L = 2n
37.5		(2mm, $[{\bar 1}1{\bar 1}]$ )	15	A2aa(00γ)	h0lm: h = 2n; hk00: h = 2n
37.6			15	A2aa(00γ)0s0	h0lm: h + m = 2n; hk00: h = 2n
38.1	C2mm	(2mm, $[{\bar 1}1{\bar 1}]$ )	13	C2mm(00γ)
38.2			13	C2mm(00γ)0s0	h0lm: m = 2n
38.3			14	C2mm(10γ)
38.4			14	C2mm(10γ)0s0	H0Lm: m = 2n
38.5		(mm2, 111)	15	Amm2(00γ)
38.6			15	Amm2(00γ)s0s	0klm: m = 2n
38.7			15	Amm2(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
38.8			15	Amm2(00γ)0ss	h0lm: m = 2n
38.9			16	Amm2( $[{{1}\over{2}}]$ 0γ)
38.10			16	Amm2( $[{{1}\over{2}}]$ 0γ)0ss	H0Lm: m = 2n
38.11		(m2m, $[1{\bar 1}{\bar 1}]$ )	15	Am2m(00γ)
38.12			15	Am2m(00γ)s00	0klm: m = 2n
38.13			16	Am2m( $[{{1}\over{2}}]$ 0γ)
39.1	C2mb	(2mm, $[{\bar 1}1{\bar 1}]$ )	13	C2mb(00γ)	hk00: k = 2n
39.2			13	C2mb(00γ)0s0	h0lm: m = 2n; hk00: k = 2n
39.3			14	C2mb(10γ)	HK00: K = 2n
39.4			14	C2mb(10γ)0s0	H0Lm: m = 2n; HK00: K = 2n
39.5		(mm2, 111)	15	Abm2(00γ)	0klm: k = 2n
39.6			15	Abm2(00γ)s0s	0klm: k + m = 2n
39.7			15	Abm2(00γ)ss0	0klm: k + m = 2n; h0lm: m = 2n
39.8			15	Abm2(00γ)0ss	0klm: k = 2n; h0lm: m = 2n
39.9			16	Abm2( $[{{1}\over{2}}]$ 0γ)	0KLm: K = 2n
39.10			16	Abm2( $[{{1}\over{2}}]$ 0γ)0ss	0KLm: K + m = 2n
39.11		(m2m, $[1{\bar 1}{\bar 1}]$ )	15	Ac2m(00γ)	0klm: l = 2n
39.12			15	Ac2m(00γ)s00	0klm: l + m = 2n
39.13			16	Ac2m( $[{{1}\over{2}}]$ 0γ)	0KLm: L = 2n
40.1	C2cm	(2mm, $[{\bar 1}1{\bar 1}]$ )	13	C2cm(00γ)	h0lm: l = 2n
40.2			14	C2cm(10γ)	H0Lm: L = 2n
40.3		(mm2, 111)	15	Ama2(00γ)	h0lm: h = 2n
40.4			15	Ama2(00γ)s0s	0klm: m = 2n; h0lm: h = 2n
40.5			15	Ama2(00γ)ss0	0klm: m = 2n; h0lm: h + m = 2n
40.6			15	Ama2(00γ)0ss	h0lm: h + m = 2n
40.7		(m2m, $[1{\bar 1}{\bar 1}]$ )	15	Am2a(00γ)	hk00: h = 2n
40.8			15	Am2a(00γ)s00	0klm: m = 2n; hk00: h = 2n
41.1	C2cb	(2mm, $[{\bar 1}1{\bar 1}]$ )	13	C2cb(00γ)	h0lm: l = 2n; hk00: k = 2n
41.2			14	C2cb(10γ)	H0Lm: L = 2n; HK00: K = 2n
41.3		(mm2, 111)	15	Aba2(00γ)	0klm: k = 2n; h0lm: h = 2n
41.4			15	Aba2(00γ)s0s	0klm: k + m = 2n; h0lm: h = 2n
41.5			15	Aba2(00γ)ss0	0klm: k + m = 2n; h0lm: h + m = 2n
41.6			15	Aba2(00γ)0ss	0klm: k = 2n; h0lm: h + m = 2n
41.7		(m2m, $[1{\bar 1}{\bar 1}]$ )	15	Ac2a(00γ)	0klm: l = 2n; hk00: h = 2n
41.8			15	Ac2a(00γ)s00	0klm: l + m = 2n; hk00: h = 2n
42.1	Fmm2	(mm2, 111)	17	Fmm2(00γ)
42.2			17	Fmm2(00γ)s0s	0klm: m = 2n
42.3			17	Fmm2(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
42.4			18	Fmm2(10γ)
42.5			18	Fmm2(10γ)s0s	0KLm: m = 2n
42.6			18	Fmm2(10γ)ss0	0KLm: m = 2n; H0Lm: m = 2n
42.7		(2mm, $[{\bar 1}1{\bar 1}]$ )	17	F2mm(00γ)
42.8			17	F2mm(00γ)0s0	h0lm: m = 2n
42.9			18	F2mm(10γ)
42.10			18	F2mm(10γ)0s0	H0Lm: m = 2n
43.1	Fdd2	(mm2, 111)	17	Fdd2(00γ)	0klm: k + l = 4n
43.2			17	Fdd2(00γ)s0s	0klm: k + l + 2m = 4n; h0lm: h + l = 4n
43.3		(2mm, $[{\bar 1}1{\bar 1}]$ )	17	F2dd(00γ)	h0lm: h + l = 4n
44.1	Imm2	(mm2, 111)	12	Imm2(00γ)
44.2			12	Imm2(00γ)s0s	0klm: m = 2n
44.3			12	Imm2(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
44.4		(2mm, $[{\bar 1}1{\bar 1}]$ )	12	I2mm(00γ)
44.5			12	I2mm(00γ)0s0	h0lm: m = 2n
45.1	Iba2	(mm2, 111)	12	Iba2(00γ)	0klm: k = 2n; h0lm: h = 2n
45.2			12	Iba2(00γ)s0s	0klm: k + m = 2n; h0lm: h = 2n
45.3			12	Iba2(00γ)ss0	0klm: k + m = 2n; h0lm: h + m = 2n
45.4		(2mm, $[{\bar 1}1{\bar 1}]$ )	12	I2cb(00γ)	h0lm: l = 2n; hk00: k = 2n
45.5			12	I2cb(00γ)0s0	h0lm: l + m = 2n; hk00: k = 2n
46.1	Ima2	(mm2, 111)	12	Ima2(00γ)	h0lm: h = 2n
46.2			12	Ima2(00γ)s0s	0klm: m = 2n; h0lm: h = 2n
46.3			12	Ima2(00γ)ss0	0klm: m = 2n; h0lm: h + m = 2n
46.4			12	Ima2(00γ)0ss	h0lm: h + m = 2n
46.5		(2mm, $[{\bar 1}1{\bar 1}]$ )	12	I2mb(00γ)	hk00: k = 2n
46.6			12	I2mb(00γ)0s0	h0lm: m = 2n; hk00: k = 2n
46.7			12	I2cm(00γ)	h0lm: l = 2n
46.8			12	I2cm(00γ)0s0	h0lm: l + m = 2n
47.1	Pmmm	(mmm, $[11{\bar 1}]$ )	9	Pmmm(00γ)
47.2			9	Pmmm(00γ)s00	0klm: m = 2n
47.3			9	Pmmm(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
47.4			10	Pmmm(0 $[{{1}\over{2}}]$ γ)
47.5			10	Pmmm(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n
47.6			11	Pmmm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
48.1	Pnnn	(mmm, $[11{\bar 1}]$ )	9	Pnnn(00γ)	0klm: k + l = 2n; h0lm: h + l = 2n; hk00: h + k = 2n
48.2			9	Pnnn(00γ)s00	0klm: k + l + m = 2n; h0lm: h + l = 2n; hk00: h + k = 2n
48.3			11	Pnnn( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	0KLm: 2K + 2L + m = 4n; H0Lm: 2H + 2L + m = 2n; HK00: H + K = 2n
49.1	Pccm	(mmm, $[11{\bar 1}]$ )	9	Pccm(00γ)	0klm: l = 2n; h0lm: l = 2n
49.2			9	Pccm(00γ)s00	0klm: l + m = 2n; h0lm: l = 2n
49.3			9	Pmaa(00γ)	h0lm: h = 2n; hk00: h = 2n
49.4			9	Pmaa(00γ)s00	0klm: m = 2n; h0lm: h = 2n; hk00: h = 2n
49.5			9	Pmaa(00γ)ss0	0klm: m = 2n; h0lm: h + m = 2n; hk00: h = 2n
49.6			9	Pmaa(00γ)0s0	h0lm: h + m = 2n; hk00: h = 2n
49.7			10	Pccm(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: L = 2n
49.8			10	Pmaa(0 $[{{1}\over{2}}]$ γ)	H0Lm: H = 2n; HK00: H = 2n
49.9			10	Pmaa(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n; H0Lm: H = 2n; HK00: H = 2n
49.10			11	Pccm ( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: L = 2n
50.1	Pban	(mmm, $[11{\bar 1}]$ )	9	Pban(00γ)	0klm: k = 2n; h0lm: h = 2n
50.2			9	Pban(00γ)s00	0klm: k + m = 2n; h0lm: h = 2n
50.3			9	Pban(00γ)ss0	0klm: k + m = 2n; h0lm: h + m = 2n
50.4			9	Pcna(00γ)	0klm: l = 2n; h0lm: h + l = 2n; hk00: h = 2n
50.5			9	Pcna(00γ)s00	0klm: l + m = 2n; h0lm: h + l = 2n; hk00: h = 2n
50.6			10	Pcna(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: H + L = 2n; HK00: H = 2n
50.7			11	Pban( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	0KLm: 2K + m = 4n; H0Lm: 2H + m = 4n; HK00: H + K = 2n
51.1	Pmma	(mmm, $[11{\bar 1}]$ )	9	Pmma(00γ)	hk00: h = 2n
51.2			9	Pmma(00γ)s00	0klm: m = 2n; hk00: h = 2n
51.3			9	Pmma(00γ)ss0	0klm: m = 2n; h0lm: m = 2n; hk00: h = 2n
51.4			9	Pmma(00γ)0s0	h0lm: m = 2n; hk00: h = 2n
51.5			9	Pmam(00γ)	h0lm: h = 2n
51.6			9	Pmam(00γ)s00	0klm: m = 2n; h0lm: h = 2n
51.7			9	Pmam(00γ)ss0	0klm: m = 2n; h0lm: h + m = 2n
51.8			9	Pmam(00γ)0s0	h0lm: h + m = 2n
51.9			9	Pmcm(00γ)	h0lm: l = 2n
51.10			9	Pmcm(00γ)s00	0klm: m = 2n; h0lm: l = 2n
51.11			10	Pmma(0 $[{{1}\over{2}}]$ γ)	HK00: H = 2n
51.12			10	Pmma(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n; HK00: H = 2n
51.13			10	Pmam(0 $[{{1}\over{2}}]$ γ)	H0Lm: H = 2n
51.14			10	Pmam(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n; H0Lm: H = 2n
51.15			10	Pmcm(0 $[{{1}\over{2}}]$ γ)	H0Lm: L = 2n
51.16			10	Pmcm(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n; H0Lm: L = 2n
51.17			10	Pcmm(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n
51.18			11	Pcmm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	0KLm: L = 2n
52.1	Pnna	(mmm, $[11{\bar 1}]$ )	9	Pnna(00γ)	0klm: k + l = 2n; h0lm: h + l = 2n; hk00: h = 2n
52.2			9	Pnna(00γ)s00	0klm: k + l + m = 2n; h0lm: h + l = 2n; hk00: h = 2n
52.3			9	Pbnn(00γ)	0klm: k = 2n; h0lm: h + l = 2n; hk00: h + k = 2n
52.4			9	Pbnn(00γ)s00	0klm: k + m = 2n; h0lm: h + l = 2n; hk00: h + k = 2n
52.5			9	Pcnn(00γ)	0klm: l = 2n; h0lm: h + l = 2n; hk00: h + k = 2n
52.6			9	Pcnn(00γ)s00	0klm: l + m = 2n; h0lm: h + l = 2n; hk00: h + k = 2n
52.7			11	Pbnn( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	0KLm: 2K + m = 4n; H0Lm: 2H + 2L + m = 4n; HK00: H + K = 2n
53.1	Pmna	(mmm, $[11{\bar 1}]$ )	9	Pmna(00γ)	h0lm: h + l = 2n; hk00: h = 2n
53.2			9	Pmna(00γ)s00	0klm: m = 2n; h0lm: h + l = 2n; hk00: h = 2n
53.3			9	Pcnm(00γ)	0klm: l = 2n; h0lm: h + l = 2n
53.4			9	Pcnm(00γ)s00	0klm: l + m = 2n; h0lm: h + l = 2n
53.5			9	Pbmn(00γ)	0klm: k = 2n; hk00: h + k = 2n
53.6			9	Pbmn(00γ)s00	0klm: k + m = 2n; hk00: h + k = 2n
53.7			9	Pbmn(00γ)ss0	0klm: k + m = 2n; h0lm: m = 2n; hk00: h + k = 2n
53.8			9	Pbmn(00γ)0s0	0klm: k = 2m; h0lm: m = 2n; hk00: h + k = 2n
53.9			10	Pmna(0 $[{{1}\over{2}}]$ γ)	H0Lm: H + L = 2n; HK00: H = 2n
53.10			10	Pmna(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n; H0Lm: H + L = 2n; HK00: H = 2n
53.11			10	Pcnm(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: H + L = 2n
54.1	Pcca	(mmm, $[11{\bar 1}]$ )	9	Pcca(00γ)	0klm: l = 2n; h0lm: l = 2n; hk00: h = 2n
54.2			9	Pcca(00γ)s00	0klm: l + m = 2n; h0lm: l = 2n; hk00: h = 2n
54.3			9	Pcaa(00γ)	0klm: l = 2n; h0lm: h = 2n; hk00: h = 2n
54.4			9	Pcaa(00γ)0s0	0klm: l = 2n; h0lm: h + m = 2n; hk00: h = 2n
54.5			9	Pbab(00γ)	0klm: k = 2n; h0lm: h = 2n; hk00: k = 2n
54.6			9	Pbab(00γ)s00	0klm: k + m = 2n; h0lm: h = 2n; hk00: k = 2n
54.7			9	Pbab(00γ)ss0	0klm: k + m = 2n; h0lm: h + m = 2n; hk00: k = 2n
54.8			9	Pbab(00γ)0s0	0klm: k = 2n; h0lm: h + m = 2n; hk00: k = 2n
54.9			10	Pcca(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: L = 2n; HK00: H = 2n
54.10			10	Pcaa(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: H = 2n; HK00: H = 2n
55.1	Pbam	(mmm, $[11{\bar 1}]$ )	9	Pbam(00γ)	0klm: k = 2n; h0lm: h = 2n
55.2			9	Pbam(00γ)s00	0klm: k + m = 2n; h0lm: h = 2n
55.3			9	Pbam(00γ)ss0	0klm: k + m = 2n; h0lm: h + m = 2n
55.4			9	Pcma(00γ)	0klm: l = 2n; hk00: h = 2n
55.5			9	Pcma(00γ)0s0	0klm: l = 2n; h0lm: m = 2n; hk00: h = 2n
55.6			10	Pcma(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; HK00: H = 2n
56.1	Pccn	(mmm, $[11{\bar 1}]$ )	9	Pccn(00γ)	0klm: l = 2n; h0lm: l = 2n; hk00: h + k = 2n
56.2			9	Pccn(00γ)s00	0klm: l + m = 2n; h0lm: l = 2n; hk00: h + k = 2n
56.3			9	Pbnb(00γ)	0klm: k = 2n; h0lm: h + l = 2n; hk00: k = 2n
56.4			9	Pbnb(00γ)s00	0klm: k + m = 2n; h0lm: h + l = 2n; hk00: k = 2n
57.1	Pcam	(mmm, $[11{\bar 1}]$ )	9	Pcam(00γ)	0klm: l = 2n; h0lm: h = 2n
57.2			9	Pcam(00γ)0s0	0klm: l = 2n; h0lm: h + m = 2n
57.3			9	Pmca(00γ)	h0lm: l = 2n; hk00: h = 2n
57.4			9	Pmca(00γ)s00	0klm: m = 2n; h0lm: l = 2n; hk00: h = 2n
57.5			9	Pbma(00γ)	0klm: k = 2n; hk00: h = 2n
57.6			9	Pbma(00γ)s00	0klm: k + m = 2n; hk00: h = 2n
57.7			9	Pbma(00γ)ss0	0klm: k + m = 2n; h0lm: m = 2n; hk00: h = 2n
57.8			9	Pbma(00γ)0s0	0klm: k = 2n; h0lm: m = 2n; hk00: h = 2n
57.9			10	Pcam(0 $[{{1}\over{2}}]$ γ)	0KLm: L = 2n; H0Lm: H = 2n
57.10			10	Pmca(0 $[{{1}\over{2}}]$ γ)	H0Lm: L = 2n; HK00: H = 2n
57.11			10	Pmca(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n; H0Lm: L = 2n; HK00: H = 2n
58.1	Pnnm	(mmm, $[11{\bar 1}]$ )	9	Pnnm(00γ)	0klm: k + l = 2n; h0lm: h + l = 2n
58.2			9	Pnnm(00γ)s00	0klm: k + l + m = 2n; h0lm: h + l = 2n
58.3			9	Pmnn(00γ)	h0lm: h + l = 2n; hk00: h + k = 2n
58.4			9	Pmnn(00γ)s00	0klm: m = 2n; h0lm: h + l = 2n; hk00: h + k = 2n
59.1	Pmmn	(mmm, $[11{\bar 1}]$ )	9	Pmmn(00γ)	hk00: h + k = 2n
59.2			9	Pmmn(00γ)s00	0klm: m = 2n; hk00: h + k = 2n
59.3			9	Pmmn(00γ)ss0	0klm: m = 2n; h0lm: m = 2n; hk00: h + k = 2n
59.4			9	Pmnm(00γ)	h0lm: h + l = 2n
59.5			9	Pmnm(00γ)s00	0klm: m = 2n; h0lm: h + l = 2n
59.6			10	Pmnm(0 $[{{1}\over{2}}]$ γ)	H0Lm: H + L = 2n
59.7			10	Pmnm(0 $[{{1}\over{2}}]$ γ)s00	0KLm: m = 2n; H0Lm: H + L = 2n
60.1	Pbcn	(mmm, $[11{\bar 1}]$ )	9	Pbcn(00γ)	0klm: k = 2n; h0lm: l = 2n; hk00: h + k = 2n
60.2			9	Pbcn(00γ)s00	0klm: k + m = 2n; h0lm: l = 2n; hk00: h + k = 2n
60.3			9	Pnca(00γ)	0klm: k + l = 2n; h0lm: l = 2n; hk00: h = 2n
60.4			9	Pnca(00γ)s00	0klm: k + l + m = 2n; h0lm: l = 2n; hk00: h = 2n
60.5			9	Pbna(00γ)	0klm: k = 2n; h0lm: h + l = 2n; hk00: h = 2n
60.6			9	Pbna(00γ)s00	0klm: k + m = 2n; h0lm: h + l = 2n; hk00: h = 2n
61.1	Pbca	(mmm, $[11{\bar 1}]$ )	9	Pbca(00γ)	0klm: k = 2n; h0lm: l = 2n; hk00: h = 2n
61.2			9	Pbca(00γ)s00	0klm: k + m = 2n; h0lm: l = 2n; hk00: h = 2n
62.1	Pnma	(mmm, $[11{\bar 1}]$ )	9	Pnma(00γ)	0klm: k + l = 2n; hk00: h = 2n
62.2			9	Pnma(00γ)0s0	0klm: k + l = 2n; h0lm: m = 2n; hk00: h = 2n
62.3			9	Pbnm(00γ)	0klm: k = 2n; h0lm: h + l = 2n
62.4			9	Pbnm(00γ)s00	0klm: k + m = 2n; h0lm: h + l = 2n
62.5			9	Pmcn(00γ)	h0lm: l = 2n; hk00: h + k = 2n
62.6			9	Pmcn(00γ)s00	0klm: m = 2n; h0lm: l = 2n; hk00: h + k = 2n
63.1	Cmcm	(mmm, $[11{\bar 1}]$ )	13	Cmcm(00γ)	h0lm: l = 2n
63.2			13	Cmcm(00γ)s00	0klm: m = 2n; h0lm: l = 2n
63.3			14	Cmcm(10γ)	H0Lm: L = 2n
63.4			14	Cmcm(10γ)s00	0KLm: m = 2n; H0Lm: L = 2n
63.5			15	Amam(00γ)	h0lm: h = 2n
63.6			15	Amam(00γ)s00	0klm: m = 2n; h0lm: h = 2n
63.7			15	Amam(00γ)ss0	0klm: m = 2n; h0lm: h + m = 2n
63.8			15	Amam(00γ)0s0	h0lm: h + m = 2n
63.9			15	Amma(00γ)	hk00: h = 2n
63.10			15	Amma(00γ)s00	0klm: m = 2n; hk00: h = 2n
63.11			15	Amma(00γ)ss0	0klm: m = 2n; h0lm: m = 2n; hk00: h = 2n
63.12			15	Amma(00γ)0s0	h0lm: m = 2n; hk00: h = 2n
64.1	Cmca	(mmm, $[11{\bar 1}]$ )	13	Cmca(00γ)	h0lm: l = 2n; hk00: h = 2n
64.2			13	Cmca(00γ)s00	0klm: m = 2n; h0lm: l = 2n; hk00: h = 2n
64.3			14	Cmca(10γ)	H0Lm: L = 2n; HK00: H = 2n
64.4			14	Cmca(10γ)s00	0KLm: m = 2n; H0Lm: L = 2n; HK00: H = 2n
64.5			15	Abma(00γ)	0klm: k = 2n; hk00: h = 2n
64.6			15	Abma(00γ)s00	0klm: k + m = 2n; hk00: h = 2n
64.7			15	Abma(00γ)ss0	0klm: k + m = 2n; h0lm: m = 2n; hk00: h = 2n
64.8			15	Abma(00γ)0s0	0klm: k = 2n; h0lm: m = 2n; hk00: h = 2n
64.9			15	Acam(00γ)	0klm: l = 2n; h0lm: h = 2n
64.10			15	Acam(00γ)s00	0klm: l + m = 2n; h0lm: h = 2n
64.11			15	Acam(00γ)ss0	0klm: l + m = 2n; h0lm: h + m = 2n
64.12			15	Acam(00γ)0s0	0klm: l = 2n; h0lm: h + m = 2n
65.1	Cmmm	(mmm, $[11{\bar 1}]$ )	13	Cmmm(00γ)
65.2			13	Cmmm(00γ)s00	0klm: m = 2n
65.3			13	Cmmm(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
65.4			14	Cmmm(10γ)
65.5			14	Cmmm(10γ)s00	0KLm: m = 2n
65.6			14	Cmmm(10γ)ss0	0KLm: m = 2n; H0Lm: m = 2n
65.7			15	Ammm(00γ)
65.8			15	Ammm(00γ)s00	0klm: m = 2n
65.9			15	Ammm(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
65.10			15	Ammm(00γ)0s0	h0lm: m = 2n
65.11			16	Ammm( $[{{1}\over{2}}]$ 0γ)
65.12			16	Ammm( $[{{1}\over{2}}]$ 0γ)0s0	H0Lm: m = 2n
66.1	Cccm	(mmm, $[11{\bar 1}]$ )	13	Cccm(00γ)	0klm: l = 2n; h0lm: l = 2n
66.2			13	Cccm(00γ)s00	0klm: l + m = 2n; h0lm: l = 2n
66.3			14	Cccm(10γ)	0KLm: L = 2n; H0Lm: L = 2n
66.4			14	Cccm(10γ)s00	0KLm: L + m = 2n; H0Lm: L = 2n
66.5			15	Amaa(00γ)	h0lm: h = 2n; hk00: h = 2n
66.6			15	Amaa(00γ)s00	0klm: m = 2n; h0lm: h = 2n; hk00: h = 2n
66.7			15	Amaa(00γ)ss0	0klm: m = 2n; h0lm: h + m = 2n; hk00: h = 2n
66.8			15	Amaa(00γ)0s0	h0lm: h + m = 2n; hk00: h = 2n
67.1	Cmma	(mmm, $[11{\bar 1}]$ )	13	Cmma(00γ)	hk00: h = 2n
67.2			13	Cmma(00γ)s00	0klm: m = 2n; hk00: h = 2n
67.3			13	Cmma(00γ)ss0	0klm: m = 2n; h0lm: m = 2n; hk00: h = 2n
67.4			14	Cmma(10γ)	HK00: H = 2n
67.5			14	Cmma(10γ)s00	0KLm: m = 2n; HK00: H = 2n
67.6			14	Cmma(10γ)ss0	0KLm: m = 2n; H0Lm: m = 2n; HK00: H = 2n
67.7			15	Acmm(00γ)	0klm: l = 2n
67.8			15	Acmm(00γ)s00	0klm: l + m = 2n
67.9			15	Acmm(00γ)ss0	0klm: l + m = 2n; h0lm: m = 2n
67.10			15	Acmm(00γ)0s0	0klm: l = 2n; h0lm: m = 2n
67.11			16	Acmm( $[{{1}\over{2}}]$ 0γ)	0KLm: L = 2n
67.12			16	Acmm( $[{{1}\over{2}}]$ 0γ)0s0	0KLm: L = 2n; H0Lm: m = 2n
68.1	Ccca	(mmm, $[11{\bar 1}]$ )	13	Ccca(00γ)	0klm: l = 2n; h0lm: l = 2n; hk00: h = 2n
68.2			13	Ccca(00γ)s00	0klm: l + m = 2n; h0lm: l = 2n; hk00: h = 2n
68.3			14	Ccca(10γ)	0KLm: L = 2n; H0Lm: L = 2n; HK00: H = 2n
68.4			14	Ccca(10γ)s00	0KLm: L + m = 2n; H0Lm: L = 2n; HK00: H = 2n
68.5			15	Acaa(00γ)	0klm: l = 2n; h0lm: h = 2n; hk00: h = 2n
68.6			15	Acaa(00γ)s00	0klm: l + m = 2n; h0lm: h = 2n; hk00: h = 2n
68.7			15	Acaa(00γ)ss0	0klm: l + m = 2n; h0lm: h + m = 2n; hk00: h = 2n
68.8			15	Acaa(00γ)0s0	0klm: l = 2n; h0lm: h + m = 2n; hk00: h = 2n
69.1	Fmmm	(mmm, $[11{\bar 1}]$ )	17	Fmmm(00γ)
69.2			17	Fmmm(00γ)s00	0klm: m = 2n
69.3			17	Fmmm(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
69.4			18	Fmmm(10γ)
69.5			18	Fmmm(10γ)s00	0KLm: m = 2n
69.6			18	Fmmm(10γ)ss0	0KLm: m = 2n; H0Lm: m = 2n
70.1	Fddd	(mmm, $[11{\bar 1}]$ )	17	Fddd(00γ)	0klm: k + l = 4n; h0lm: h + l = 4n; hk00: h + k = 4n
70.2			17	Fddd(00γ)s00	0klm: k + l + 2m = 4n; h0lm: h + l = 4n; hk00: h + k = 4n
71.1	Immm	(mmm, $[11{\bar 1}]$ )	12	Immm(00γ)
71.2			12	Immm(00γ)s00	0klm: m = 2n
71.3			12	Immm(00γ)ss0	0klm: m = 2n; h0lm: m = 2n
72.1	Ibam	(mmm, $[11{\bar 1}]$ )	12	Ibam(00γ)	0klm: k = 2n; h0lm: h = 2n
72.2			12	Ibam(00γ)s00	0klm: k + m = 2n; h0lm: h = 2n
72.3			12	Ibam(00γ)ss0	0klm: k + m = 2n; h0lm: h + m = 2n
72.4			12	Imcb(00γ)	h0lm: l = 2n; hk00: k = 2n
72.5			12	Imcb(00γ)s00	0klm: m = 2n; h0lm: l = 2n; hk00: k = 2n
72.6			12	Imcb(00γ)ss0	0klm: m = 2n; h0lm: l + m = 2n; hk00: k = 2n
72.7			12	Imcb(00γ)0s0	h0lm: l + m = 2n; hk00: k = 2n
73.1	Ibca	(mmm, $[11{\bar 1}]$ )	12	Ibca(00γ)	0klm: k = 2n; h0lm: l = 2n; hk00: h = 2n
73.2			12	Ibca(00γ)s00	0klm: k + m = 2n; h0lm: l = 2n; hk00: h = 2n
73.3			12	Ibca(00γ)ss0	0klm: k + m = 2n; h0lm: l + m = 2n; hk00: h = 2n
74.1	Imma	(mmm, $[11{\bar 1}]$ )	12	Imma(00γ)	hk00: h = 2n
74.2			12	Imma(00γ)s00	0klm: m = 2n; hk00: h = 2n
74.3			12	Imma(00γ)ss0	0klm: m = 2n; h0lm: m = 2n; hk00: h = 2n
74.4			12	Icmm(00γ)	0klm: l = 2n
74.5			12	Icmm(00γ)s00	0klm: l + m = 2n
74.6			12	Icmm(00γ)ss0	0klm: l + m = 2n; h0lm: m = 2n
74.7			12	Icmm(00γ)0s0	0klm: l = 2n; h0lm: m = 2n
75.1	P4	(4, 1)	19	P4(00γ)
75.2			19	P4(00γ)q	00lm: m = 4n
75.3			19	P4(00γ)s	00lm: m = 2n
75.4			20	P4( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
75.5			20	P4( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q	00Lm: m = 4n
76.1	P4₁	(4, 1)	19	P4₁(00γ)	00lm: l = 4n
76.2			20	P4₁( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 4n
77.1	P4₂	(4, 1)	19	P4₂(00γ)	00lm: l = 2n
77.2			19	P4₂(00γ)q	00lm: 2l + m = 4n
77.3			20	P4₂( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 2n
77.4			20	P4₂( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q	00Lm: 2L + m = 4n
78.1	P4₃	(4, 1)	19	P4₃(00γ)	00lm: l = 4n
78.2			20	P4₃( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 4n
79.1	I4	(4, 1)	21	I4(00γ)
79.2			21	I4(00γ)q	00lm: m = 4n
79.3			21	I4(00γ)s	00lm: m = 2n
80.1	I4₁	(4, 1)	21	I4₁(00γ)	00lm: l = 4n
80.2			21	I4₁(00γ)q	00lm: l + m = 4n
81.1	$[P{\bar 4}]$	( $[{\bar 4}]$ , $[{\bar 1}]$ )	19	P $[{\bar 4}]$ (00γ)
81.2			20	P $[{\bar 4}]$ ( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
82.1	$[I{\bar 4}]$	( $[{\bar 4}]$ , $[{\bar 1}]$ )	21	I $[{\bar 4}]$ (00γ)
83.1	P4/m	(4/m, $[1{\bar 1}]$ )	19	P4/m(00γ)
83.2			19	P4/m(00γ)s0	00lm: m = 2n
83.3			20	P4/m( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
84.1	P4₂/m	(4/m, $[1{\bar 1}]$ )	19	P4₂/m(00γ)	00lm: l = 2n
84.2			20	P4₂/m( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 2n
85.1	P4/n	(4/m, $[1{\bar 1}]$ )	19	P4/n(00γ)	hk00: h + k = 2n
85.2			19	P4/n(00γ)s0	00lm: m = 2n; hk00: h + k = 2n
85.3			20	P4/n( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q0	00Lm: m = 4n; HK00: H = 2n, K = 2n
86.1	P4₂/n	(4/m, $[1{\bar 1}]$ )	19	P4₂/n(00γ)	00lm: l = 2n; hk00: h + k = 2n
86.2			20	P4₂/n( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q0	00Lm: 2L + m = 4n; HK00: H = 2n, K = 2n
87.1	I4/m	(4/m, $[1{\bar 1}]$ )	21	I4/m(00γ)
87.2			21	I4/m(00γ)s0	00lm: m = 2n
88.1	I4₁/a	(4/m, $[1{\bar 1}]$ )	21	I4₁/a(00γ)	00lm: l = 4n; hk00: h = 2n
89.1	P422	(422, $[1{\bar 1}{\bar 1}]$ )	19	P422(00γ)
89.2			19	P422(00γ)q00	00lm: m = 4n
89.3			19	P422(00γ)s00	00lm: m = 2n
89.4			20	P422( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
89.5			20	P422( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q00	00Lm: m = 4n
90.1	P42₁2	(422, $[1{\bar 1}{\bar 1}]$ )	19	P42₁2(00γ)	h000: h = 2n
90.2			19	P42₁2(00γ)q00	00lm: m = 4n; h000: h = 2n
90.3			19	P42₁2(00γ)s00	00lm: m = 2n; h000: h = 2n
91.1	P4₁22	(422, $[1{\bar 1}{\bar 1}]$ )	19	P4₁22(00γ)	00lm: l = 4n
91.2			20	P4₁22( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 4n
92.1	P4₁2₁2	(422, $[1{\bar 1}{\bar 1}]$ )	19	P4₁2₁2(00γ)	00lm: l = 4n; h000: h = 2n
93.1	P4₂22	(422, $[1{\bar 1}{\bar 1}]$ )	19	P4₂22(00γ)	00lm: l = 2n
93.2			19	P4₂22(00γ)q00	00lm: 2l + m = 4n
93.3			20	P4₂22( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 2n
93.4			20	P4₂22( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q00	00Lm: 2L + m = 4n
94.1	P4₂2₁2	(422, $[1{\bar 1}{\bar 1}]$ )	19	P4₂2₁2(00γ)	00lm: l = 2n; h000: h = 2n
94.2			19	P4₂2₁2(00γ)q00	00lm: 2l + m = 4n; h000: h = 2n
95.1	P4₃22	(422, $[1{\bar 1}{\bar 1}]$ )	19	P4₃22(00γ)	00lm: l = 4n
95.2			20	P4₃22( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 4n
96.1	P4₃2₁2	(422, $[1{\bar 1}{\bar 1}]$ )	19	P4₃2₁2(00γ)	00lm: l = 4n; h000: h = 2n
97.1	I422	(422, $[1{\bar 1}{\bar 1}]$ )	21	I422(00γ)
97.2			21	I422(00γ)q00	00lm: m = 4n
97.3			21	I422(00γ)s00	00lm: m = 2n
98.1	I4₁22	(422, $[1{\bar 1}{\bar 1}]$ )	21	I4₁22(00γ)	00lm: l = 4n
98.2			21	I4₁22(00γ)q00	00lm: l + m = 4n
99.1	P4mm	(4mm, 111)	19	P4mm(00γ)
99.2			19	P4mm(00γ)ss0	00lm: m = 2n; 0klm: m = 2n
99.3			19	P4mm(00γ)0ss	0klm: m = 2n; hhlm: m = 2n
99.4			19	P4mm(00γ)s0s	00lm: m = 2n; hhlm: m = 2n
99.5			20	P4mm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
99.6			20	P4mm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)0ss	0KLm: m = 2n; HHLm: m = 2n
100.1	P4bm	(4mm, 111)	19	P4bm(00γ)	0klm: k = 2n
100.2			19	P4bm(00γ)ss0	00lm: m = 2n; 0klm: m = 2n
100.3			19	P4bm(00γ)0ss	0klm: k + m = 2n; hhlm: m = 2n
100.4			19	P4bm(00γ)s0s	00lm: m = 2n; 0klm: k = 2n; hhlm: m = 2n
100.5			20	P4bm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	00Lm: m = 4n; KKLm: 2K + m = 4n
100.6			20	P4bm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qqs	00Lm: m = 4n; KKLm: 2K + m = 4n; H0Lm: m = 2n
101.1	P4₂cm	(4mm, 111)	19	P4₂cm(00γ)	00lm: l = 2n; 0klm: l = 2n
101.2			19	P4₂cm(00γ)0ss	00lm: l = 2n; 0klm: l + m = 2n; hhlm: m = 2n
101.3			20	P4₂cm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 2n; HHLm: L = 2n
101.4			20	P4₂cm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)0ss	00Lm: L = 2n; HHLm: L + m = 2n; H0Lm: m = 2n
102.1	P4₂nm	(4mm, 111)	19	P4₂nm(00γ)	00lm: l = 2n; 0klm: k + l = 2n
102.2			19	P4₂nm(00γ)0ss	00lm: l = 2n; 0klm: k + l + m = 2n; hhlm: m = 2n
102.3			20	P4₂nm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	00Lm: 2L + m = 4n; HHLm: 2H + 2L + m = 4n
102.4			20	P4₂nm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qqs	00Lm: 2L + m = 4n; HHLm: 2H + 2L + m = 4n; H0Lm: m = 2n
103.1	P4cc	(4mm, 111)	19	P4cc(00γ)	0klm: l = 2n; hhlm: l = 2n
103.2			19	P4cc(00γ)ss0	00lm: m = 2n; 0klm: l + m = 2n; hhlm: l = 2n
103.3			20	P4cc( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	HHLm: L = 2n; H0Lm: L = 2n
104.1	P4nc	(4mm, 111)	19	P4nc(00γ)	0klm: k + l = 2n; hhlm: l = 2n
104.2			19	P4nc(00γ)ss0	00lm: m = 2n; 0klm: k + l + m = 2n; hhlm: l = 2n
104.3			20	P4nc( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	00Lm: m = 4n; HHLm: 2H + 2L + m = 4n; H0Lm: L = 2n
105.1	P4₂mc	(4mm, 111)	19	P4₂mc(00γ)	00lm: l = 2n; hhlm: l = 2n
105.2			19	P4₂mc(00γ)ss0	00lm: l + m = 2n; 0klm: m = 2n; hhlm: l = 2n
105.3			20	P4₂mc( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 2n; H0Lm: L = 2n
106.1	P4₂bc	(4mm, 111)	19	P4₂bc(00γ)	00lm: l = 2n; 0klm: k = 2n; hhlm: l = 2n
106.2			19	P4₂bc(00γ)ss0	00lm: l + m = 2n; 0klm: k + m = 2n; hhlm: l = 2n
106.3			20	P4₂bc( $[{{1}\over{2}}{{1}\over{2}}]$ γ)qq0	00Lm: 2L + m = 4n; HHLm: 2H + m = 4n; H0Lm: L = 2n
107.1	I4mm	(4mm, 111)	21	I4mm(00γ)
107.2			21	I4mm(00γ)ss0	00lm: m = 2n; 0klm: m = 2n
107.3			21	I4mm(00γ)0ss	0klm: m = 2n; hhlm: m = 2n
107.4			21	I4mm(00γ)s0s	00lm: m = 2n; hhlm: m = 2n
108.1	I4cm	(4mm, 111)	21	I4cm(00γ)	0klm: l = 2n
108.2			21	I4cm(00γ)ss0	00lm: m = 2n; 0klm: l + m = 2n
108.3			21	I4cm(00γ)0ss	0klm: l + m = 2n; hhlm: m = 2n
108.4			21	I4cm(00γ)s0s	00lm: m = 2n; 0klm: l = 2n; hhlm: m = 2n
109.1	I4₁md	(4mm, 111)	21	I4₁md(00γ)	00lm: l = 4n; hhlm: 2h + l = 4n
109.2			21	I4₁md(00γ)ss0	00lm: l + 2m = 4n; 0klm: m = 2n; hhlm: 2h + l = 4n
110.1	I4₁cd	(4mm, 111)	21	I4₁cd(00γ)	00lm: l = 4n; 0klm: l = 2n; hhlm: 2h + l = 4n
110.2			21	I4₁cd(00γ)ss0	00lm: l + 2m = 4n; 0klm: l + m = 2n; hhlm: 2h + l = 4n
111.1	$[P{\bar 4}2m]$	( $[{\bar 4}2m]$ , $[{\bar 1}{\bar 1}1]$ )	19	P $[{\bar 4}]$ 2m(00γ)
111.2			19	P $[{\bar 4}]$ 2m(00γ)00s	hhlm: m = 2n
111.3			20	P $[{\bar 4}]$ 2m( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
111.4			20	P $[{\bar 4}]$ 2m( $[{{1}\over{2}}{{1}\over{2}}]$ γ)00s	H0Lm: m = 2n
112.1	$[P{\bar 4}2c]$	( $[{\bar 4}2m]$ , $[{\bar 1}{\bar 1}1]$ )	19	P $[{\bar 4}]$ 2c(00γ)	hhlm: l = 2n
112.2			20	P $[{\bar 4}]$ 2c( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	H0Lm: L = 2n
113.1	$[P{\bar 4}2_1m]$	( $[{\bar 4}2m]$ , $[{\bar 1}{\bar 1}1]$ )	19	P $[{\bar 4}]$ 2₁m(00γ)	h000: h = 2n
113.2			19	P $[{\bar 4}]$ 2₁m(00γ)00s	h000: h = 2n; hhlm: m = 2n
114.1	$[P{\bar 4}2_1c]$	( $[{\bar 4}2m]$ , $[{\bar 1}{\bar 1}1]$ )	19	P $[{\bar 4}]$ 2₁c(00γ)	h000: h = 2n; hhlm: l = 2n
115.1	$[P{\bar 4}m2]$	( $[{\bar 4}m2]$ , $[{\bar 1}1{\bar 1}]$ )	19	P $[{\bar 4}]$ m2(00γ)
115.2			19	P $[{\bar 4}]$ m2(00γ)0s0	0klm: m = 2n
115.3			20	P $[{\bar 4}]$ m2( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
116.1	$[P{\bar 4}c2]$	( $[{\bar 4}m2]$ , $[{\bar 1}1{\bar 1}]$ )	19	P $[{\bar 4}]$ c2(00γ)	0klm: l = 2n
116.2			20	P $[{\bar 4}]$ c2( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	HHLm: L = 2n
117.1	$[P{\bar 4}b2]$	( $[{\bar 4}m2]$ , $[{\bar 1}1{\bar 1}]$ )	19	P $[{\bar 4}]$ b2(00γ)	0klm: k = 2n
117.2			19	P $[{\bar 4}]$ b2(00γ)0s0	0klm: k + m = 2n
117.3			20	P $[{\bar 4}]$ b2( $[{{1}\over{2}}{{1}\over{2}}]$ γ)0q0	HHLm: 2H + m = 4n
118.1	$[P{\bar 4}n2]$	( $[{\bar 4}m2]$ , $[{\bar 1}1{\bar 1}]$ )	19	P $[{\bar 4}]$ n2(00γ)	0klm: k + l = 2n
118.2			20	P $[{\bar 4}]$ n2( $[{{1}\over{2}}{{1}\over{2}}]$ γ)0q0	HHLm: 2H + 2L + m = 4n
119.1	$[I{\bar 4}m2]$	( $[{\bar 4}m2]$ , $[{\bar 1}1{\bar 1}]$ )	21	I $[{\bar 4}]$ m2(00γ)
119.2			21	I $[{\bar 4}]$ m2(00γ)0s0	0klm: m = 2n
120.1	$[I{\bar 4}c2]$	( $[{\bar 4}m2]$ , $[{\bar 1}1{\bar 1}]$ )	21	I $[{\bar 4}]$ c2(00γ)	0klm: l = 2n
120.2			21	I $[{\bar 4}]$ c2(00γ)0s0	0klm: l + m = 2n
121.1	$[I{\bar 4}2m]$	( $[{\bar 4}2m]$ , $[{\bar 1}{\bar 1}1]$ )	21	I $[{\bar 4}]$ 2m(00γ)
121.2			21	I $[{\bar 4}]$ 2m(00γ)00s	hhlm: m = 2n
122.1	$[I{\bar 4}2d]$	( $[{\bar 4}2m]$ , $[{\bar 1}{\bar 1}1]$ )	21	I $[{\bar 4}]$ 2d(00γ)	hhlm: 2h + l = 4n
123.1	P4/mmm	(4/mmm, $[1{\bar 1}11]$ )	19	P4/mmm(00γ)
123.2			19	P4/mmm(00γ)s0s0	00lm: m = 2n; 0klm: m = 2n
123.3			19	P4/mmm(00γ)00ss	0klm: m = 2n; hhlm: m = 2n
123.4			19	P4/mmm(00γ)s00s	00lm: m = 2n; hhlm: m = 2n
123.5			20	P4/mmm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
123.6			20	P4/mmm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)00ss	HHLm: m = 2n; H0Lm: m = 2n
124.1	P4/mcc	(4/mmm, $[1{\bar 1}11]$ )	19	P4/mcc(00γ)	0klm: l = 2n; hhlm: l = 2n
124.2			19	P4/mcc(00γ)s0s0	00lm: m = 2n; 0klm: l + m = 2n; hhlm: l = 2n
124.3			20	P4/mcc( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	HHLm: L = 2n; H0Lm: L = 2n
125.1	P4/nbm	(4/mmm, $[1{\bar 1}11]$ )	19	P4/nbm(00γ)	hk00: h + k = 2n; 0klm: k = 2n
125.2			19	P4/nbm(00γ)s0s0	00lm: m = 2n; hk00: h + k = 2n; 0klm: k + m = 2n
125.3			19	P4/nbm(00γ)00ss	hk00: h + k = 2n; 0klm: k + m = 2n; hhlm: m = 2n
125.4			19	P4/nbm(00γ)s00s	00lm: m = 2n; hk00: h + k = 2n; 0klm: k = 2n; hhlm: m = 2n
125.5			20	P4/nbm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q0q0	00Lm: m = 4n; HK00: H = 2n, K = 2n; HHLm: 2H + m = 4n
125.6			20	P4/nbm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q0qs	00Lm: m = 4n; HK00: H = 2n, K = 2n; HHLm: 2H + m = 4n; H0Lm: m = 2n
126.1	P4/nnc	(4/mmm, $[1{\bar 1}11]$ )	19	P4/nnc(00γ)	hk00: h + k = 2n; h0lm: h + l = 2n; hhlm: l = 2n
126.2			19	P4/nnc(00γ)s0s0	00lm: m = 2n; hk00: h + k = 2n; h0lm: h + l + m = 2n; hhlm: l = 2n
126.3			20	P4/nnc( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q0q0	00Lm: m = 4n; HK00: H = 2n, K = 2n; HHLm: 2H + 2L + m = 4n; H0Lm: L = 2n
127.1	P4/mbm	(4/mmm, $[1{\bar 1}11]$ )	19	P4/mbm(00γ)	0klm: k = 2n
127.2			19	P4/mbm(00γ)s0s0	00lm: m = 2n; 0klm: k + m = 2n
127.3			19	P4/mbm(00γ)00ss	0klm: k + m = 2n; hhlm: m = 2n
127.4			19	P4/mbm(00γ)s00s	00lm: m = 2n; 0klm: k = 2n; hhlm: m = 2n
128.1	P4/mnc	(4/mmm, $[1{\bar 1}11]$ )	19	P4/mnc(00γ)	0klm: k + l = 2n; hhlm: l = 2n
128.2			19	P4/mnc(00γ)s0s0	00lm: m = 2n; 0klm: k + l + m = 2n; hhlm: l = 2n
129.1	P4/nmm	(4/mmm, $[1{\bar 1}11]$ )	19	P4/nmm(00γ)	hk00: h + k = 2n
129.2			19	P4/nmm(00γ)s0s0	00lm: m = 2n; hk00: h + k = 2n; 0klm: m = 2n
129.3			19	P4/nmm(00γ)00ss	hk00: h + k = 2n; 0klm: m = 2n; hhlm: m = 2n
129.4			19	P4/nmm(00γ)s00s	00lm: m = 2n; hk00: h + k = 2n; hhlm: m = 2n
130.1	P4/ncc	(4/mmm, $[1{\bar 1}11]$ )	19	P4/ncc(00γ)	hk00: h + k = 2n; 0klm: l = 2n; hhlm: l = 2n
130.2			19	P4/ncc(00γ)s0s0	00lm: m = 2n; hk00: h + k = 2n; 0klm: l + m = 2n; hhlm: l = 2n
131.1	P4₂/mmc	(4/mmm, $[1{\bar 1}11]$ )	19	P4₂/mmc(00γ)	00lm: l = 2n; hhlm: l = 2n
131.2			19	P4₂/mmc(00γ)s0s0	00lm: l + m = 2n; 0klm: m = 2n; hhlm: l = 2n
131.3			20	P4₂/mmc( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 2n; H0Lm: L = 2n
132.1	P4₂/mcm	(4/mmm, $[1{\bar 1}11]$ )	19	P4₂/mcm(00γ)	00lm: l = 2n; 0klm: l = 2n
132.2			19	P4₂/mcm(00γ)00ss	00lm: l = 2n; 0klm: l + m = 2n; hhlm: m = 2n
132.3			20	P4₂/mcm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)	00Lm: L = 2n; HHLm: L = 2n
132.4			20	P4₂/mcm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)00ss	00Lm: L = 2n; HHLm: L + m = 2n; H0Lm: m = 2n
133.1	P4₂/nbc	(4/mmm, $[1{\bar 1}11]$ )	19	P4₂/nbc(00γ)	00lm: l = 2n; hk00: h + k = 2n; 0klm: k = 2n; hhlm: l = 2n
133.2			19	P4₂/nbc(00γ)s0s0	00lm: l + m = 2n; hk00: h + k = 2n; 0klm: k + m = 2n; hhlm: l = 2n
133.3			20	P4₂/nbc( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q0q0	00Lm: 2L + m = 4n; HK00: H = 2n, K = 2n; HHLm: 2H + m = 4n; H0Lm: L = 2n
134.1	P4₂/nnm	(4/mmm, $[1{\bar 1}11]$ )	19	P4₂/nnm(00γ)	00lm: l = 2n; hk00: h + k = 2n; 0klm: k + l = 2n
134.2			19	P4₂/nnm(00γ)00ss	00lm: l = 2n; hk00: h + k = 2n; 0klm: k + l + m = 2n; hhlm: m = 2n
134.3			20	P4₂/nnm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q0q0	00Lm: 2L + m = 4n; HK00: H = 2n, K = 2n; HHLm: 2H + 2L + m = 4n
134.4			20	P4₂/nnm( $[{{1}\over{2}}{{1}\over{2}}]$ γ)q0qs	00Lm: 2L + m = 4n; HK00: H + K = 2n; HHLm: 2H + 2L + m = 4n; H0Lm: m = 2n
135.1	P4₂/mbc	(4/mmm, $[1{\bar 1}11]$ )	19	P4₂/mbc(00γ)	00lm: l = 2n; 0klm: k = 2n; hhlm: l = 2n
135.2			19	P4₂/mbc(00γ)s0s0	00lm: l + m = 2n; 0klm: k + m = 2n; hhlm: l = 2n
136.1	P4₂/mnm	(4/mmm, $[1{\bar 1}11]$ )	19	P4₂/mnm(00γ)	00lm: l = 2n; 0klm: k + l = 2n
136.2			19	P4₂/mnm(00γ)00ss	00lm: l = 2n; 0klm: k + l + m = 2n; hhlm: m = 2n
137.1	P4₂/nmc	(4/mmm, $[1{\bar 1}11]$ )	19	P4₂/nmc(00γ)	00lm: l = 2n; hk00: h + k = 2n; hhlm: l = 2n
137.2			19	P4₂/nmc(00γ)s0s0	00lm: l + m = 2n; hk00: h + k = 2n; 0klm: m = 2n; hhlm: l = 2n
138.1	P4₂/ncm	(4/mmm, $[1{\bar 1}11]$ )	19	P4₂/ncm(00γ)	00lm: l = 2n; hk00: h + k = 2n; 0klm: l = 2n
138.2			19	P4₂/ncm(00γ)00ss	00lm: l = 2n; hk00: h + k = 2n; 0klm: l + m = 2n; hhlm: m = 2n
139.1	I4/mmm	(4/mmm, $[1{\bar 1}11]$ )	21	I4/mmm(00γ)
139.2			21	I4/mmm(00γ)s0s0	00lm: m = 2n; 0klm: m = 2n
139.3			21	I4/mmm(00γ)00ss	0klm: m = 2n; hhlm: m = 2n
139.4			21	I4/mmm(00γ)s00s	00lm: m = 2n; hhlm: m = 2n
140.1	I4/mcm	(4/mmm, $[1{\bar 1}11]$ )	21	I4/mcm(00γ)	0klm: l = 2n
140.2			21	I4/mcm(00γ)s0s0	00lm: m = 2n; 0klm: l + m = 2n
140.3			21	I4/mcm(00γ)00ss	0klm: l + m = 2n; hhlm: m = 2n
140.4			21	I4/mcm(00γ)s00s	00lm: m = 2n; 0klm: l = 2n; hhlm: m = 2n
141.1	I4₁/amd	(4/mmm, $[1{\bar 1}11]$ )	21	I4₁/amd(00γ)	00lm: l = 4n; hk00: h = 2n; hhlm: 2h + l = 4n
141.2			21	I4₁/amd(00γ)s0s0	00lm: l + 2m = 4n; hk00: h = 2n; 0klm: m = 2n; hhlm: 2h + l = 4n
142.1	I4₁/acd	(4/mmm, $[1{\bar 1}11]$ )	21	I4₁/acd(00γ)	00lm: l = 4n; hk00: h = 2n; 0klm: l = 2n; hhlm: 2h + l = 4n
142.2			21	I4₁/acd(00γ)s0s0	00lm: l + 2m = 4n; hk00: h = 2n; 0klm: l + m = 2n; hhlm: 2h + l = 4n
143.1	P3	(3, 1)	23	P3( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
143.2			24	P3(00γ)
143.3			24	P3(00γ)t	00lm: m = 3n
144.1	P3₁	(3, 1)	23	P3₁( $[{{1}\over{3}}{{1}\over{3}}]$ γ)	00Lm: L = 3n
144.2			24	P3₁(00γ)	00lm: l = 3n
145.1	P3₂	(3, 1)	23	P3₂( $[{{1}\over{3}}{{1}\over{3}}]$ γ)	00Lm: L = 3n
145.2			24	P3₂(00γ)	00lm: l = 3n
146.1	R3	(3, 1)	22	R3(00γ)
146.2			22	R3(00γ)t	00lm: m = 3n
147.1	$[P{\bar 3}]$	( $[{\bar 3}]$ , $[{\bar 1}]$ )	23	P $[{\bar 3}]$ ( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
147.2			24	P $[{\bar 3}]$ (00γ)
148.1	$[R{\bar 3}]$	( $[{\bar 3}]$ , $[{\bar 1}]$ )	22	R $[{\bar 3}]$ (00γ)
149.1	P312	(312, $[11{\bar 1}]$ )	23	P312( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
149.2			24	P312(00γ)
149.3			24	P312(00γ)t00	00lm: m = 3n
150.1	P321	(321, $[1{\bar 1}1]$ )	24	P321(00γ)
150.2			24	P321(00γ)t00	00lm: m = 3n
151.1	P3₁12	(312, $[11{\bar 1}]$ )	23	P3₁12( $[{{1}\over{3}}{{1}\over{3}}]$ γ)	00Lm: L = 3n
151.2			24	P3₁12(00γ)	00lm: l = 3n
152.1	P3₁21	(321, $[1{\bar 1}1]$ )	24	P3₁21(00γ)	00lm: l = 3n
153.1	P3₂12	(312, $[11{\bar 1}]$ )	23	P3₂12( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
153.2			24	P3₂12(00γ)	00lm: l = 3n
154.1	P3₂21	(321, $[1{\bar 1}1]$ )	24	P3₂21(00γ)	00lm: l = 3n
155.1	R32	(32, $[1{\bar 1}]$ )	22	R32(00γ)
155.2			22	R32(00γ)t0	00lm: m = 3n
156.1	P3m1	(3m1, 111)	24	P3m1(00γ)
156.2			24	P3m1(00γ)0s0	0klm: m = 2n
157.1	P31m	(31m, 111)	23	P31m( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
157.2			23	P31m( $[{{1}\over{3}}{{1}\over{3}}]$ γ)00s	$[H{\bar H}Lm]$ : m = 2n
157.3			24	P31m(00γ)
157.4			24	P31m(00γ)00s	hhlm: m = 2n
158.1	P3c1	(3m1, 111)	24	P3c1(00γ)	0klm: l = 2n
159.1	P31c	(31m, 111)	23	P31c( $[{{1}\over{3}}{{1}\over{3}}]$ γ)	$[H{\bar H}Lm]$ : L = 2n
159.2			24	P31c(00γ)	hhlm: l = 2n
160.1	R3m	(3m, 11)	22	R3m(00γ)
160.2			22	R3m(00γ)0s	hhlm: m = 2n
161.1	R3c	(3m, 11)	22	R3c(00γ)	hhlm: l = 2n
162.1	$[P{\bar 3}1m]$	( $[{\bar 3}1m]$ , $[{\bar 1}11]$ )	23	P $[{\bar 3}]$ 1m( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
162.2			23	P $[{\bar 3}]$ 1m( $[{{1}\over{3}}{{1}\over{3}}]$ γ)00s	$[H{\bar H}Lm]$ : m = 2n
162.3			24	P $[{\bar 3}]$ 1m(00γ)
162.4			24	P $[{\bar 3}]$ 1m(00γ)00s	hhlm: m = 2n
163.1	$[P{\bar 3}1c]$	( $[{\bar 3}1m]$ , $[{\bar 1}11]$ )	23	P $[{\bar 3}]$ 1c( $[{{1}\over{3}}{{1}\over{3}}]$ γ)	$[H{\bar H}Lm]$ : L = 2n
163.2			24	P $[{\bar 3}]$ 1c(00γ)	hhlm: l = 2n
164.1	$[P{\bar 3}m1]$	( $[{\bar 3}m1]$ , $[{\bar 1}11]$ )	24	P $[{\bar 3}]$ m1(00γ)
164.2			24	P $[{\bar 3}]$ m1(00γ)0s0	0klm: m = 2n
165.1	$[P{\bar 3}c1]$	( $[{\bar 3}m1]$ , $[{\bar 1}11]$ )	24	P $[{\bar 3}]$ c1(00γ)	0klm: l = 2n
166.1	$[R{\bar 3}m]$	( $[{\bar 3}m]$ , $[{\bar 1}1]$ )	22	R $[{\bar 3}]$ m(00γ)
166.2			22	R $[{\bar 3}]$ m(00γ)0s	hhlm: m = 2n
167.1	$[R{\bar 3}c]$	( $[{\bar 3}m]$ , $[{\bar 1}1]$ )	22	R $[{\bar 3}]$ c(00γ)	hhlm: l = 2n
168.1	P6	(6, 1)	24	P6(00γ)
168.2			24	P6(00γ)h	00lm: m = 6n
168.3			24	P6(00γ)t	00lm: m = 3n
168.4			24	P6(00γ)s	00lm: m = 2n
169.1	P6₁	(6, 1)	24	P6₁(00γ)	00lm: l = 6n
170.1	P6₅	(6, 1)	24	P6₅(00γ)	00lm: l = 6n
171.1	P6₂	(6, 1)	24	P6₂(00γ)	00lm: l = 3n
171.2			24	P6₂(00γ)h	00lm: 2l + m = 6n
172.1	P6₄	(6, 1)	24	P6₄(00γ)	00lm: l = 3n
172.2			24	P6₄(00γ)h	00lm: 2l + m = 6n
173.1	P6₃	(6, 1)	24	P6₃(00γ)	00lm: l = 2n
173.2			24	P6₃(00γ)h	00lm: 3l + m = 6n
174.1	$[P{\bar 6}]$	( $[{\bar 6}]$ , $[{\bar 1}]$ )	24	P $[{\bar 6}]$ (00γ)
175.1	P6/m	(6/m, $[1{\bar 1}]$ )	24	P6/m(00γ)
175.2			24	P6/m(00γ)s0	00lm: m = 2n
176.1	P6₃/m	(6/m, $[1{\bar 1}]$ )	24	P6₃/m(00γ)	00lm: l = 2n
177.1	P622	(622, $[1{\bar 1}{\bar 1}]$ )	24	P622(00γ)
177.2			24	P622(00γ)h00	00lm: m = 6n
177.3			24	P622(00γ)t00	00lm: m = 3n
177.4			24	P622(00γ)s00	00lm: m = 2n
178.1	P6₁22	(622, $[1{\bar 1}{\bar 1}]$ )	24	P6₁22(00γ)	00lm: l = 6n
179.1	P6₅22	(622, $[1{\bar 1}{\bar 1}]$ )	24	P6₅22(00γ)	00lm: l = 6n
180.1	P6₂22	(622, $[1{\bar 1}{\bar 1}]$ )	24	P6₂22(00γ)	00lm: l = 3n
180.2			24	P6₂22(00γ)h00	00lm: 2l + m = 6n
181.1	P6₄22	(622, $[1{\bar 1}{\bar 1}]$ )	24	P6₄22(00γ)	00lm: l = 3n
181.2			24	P6₄22(00γ)h00	00lm: 2l + m = 6n
182.1	P6₃22	(622, $[1{\bar 1}{\bar 1}]$ )	24	P6₃22(00γ)
182.2			24	P6₃22(00γ)h00	00lm: 3l + m = 6n
183.1	P6mm	(6mm, 111)	24	P6mm(00γ)
183.2			24	P6mm(00γ)ss0	00lm: m = 2n; 0klm: m = 2n
183.3			24	P6mm(00γ)0ss	0klm: m = 2n; hhlm: m = 2n
183.4			24	P6mm(00γ)s0s	00lm: m = 2n; hhlm: m = 2n
184.1	P6cc	(6mm, 111)	24	P6cc(00γ)	0klm: l = 2n; hhlm: l = 2n
184.2			24	P6cc(00γ)s0s	00lm: m = 2n; 0klm: l = 2n; hhlm: l + m = 2n
185.1	P6₃cm	(6mm, 111)	24	P6₃cm(00γ)	00lm: l = 2n; 0klm: l = 2n
185.2			24	P6₃cm(00γ)0ss	00lm: l = 2n; 0klm: l + m = 2n; hhlm: m = 2n
186.1	P6₃mc	(6mm, 111)	24	P6₃mc(00γ)	00lm: l = 2n; hhlm: l = 2n
186.2			24	P6₃mc(00γ)0ss	00lm: l = 2n; 0klm: m = 2n; hhlm: l + m = 2n
187.1	$[P{\bar 6}m2]$	( $[{\bar 6}m2]$ , $[{\bar 1}1{\bar 1}]$ )	24	P $[{\bar 6}]$ m2(00γ)
187.2			24	P $[{\bar 6}]$ m2(00γ)0s0	0klm: m = 2n
188.1	$[P{\bar 6}c2]$	( $[{\bar 6}m2]$ , $[{\bar 1}1{\bar 1}]$ )	24	P $[{\bar 6}]$ c2(00γ)	0klm: l = 2n
189.1	$[P{\bar 6}2m]$	( $[{\bar 6}2m]$ , $[{\bar 1}{\bar 1}1]$ )	24	P $[{\bar 6}]$ 2m(00γ)
189.2			24	P $[{\bar 6}]$ 2m(00γ)00s	hhlm: m = 2n
190.1	$[P{\bar 6}2c]$	( $[{\bar 6}2m]$ , $[{\bar 1}{\bar 1}1]$ )	24	P $[{\bar 6}]$ 2c(00γ)	hhlm: l = 2n
191.1	P6/mmm	(6/mmm, $[1{\bar 1}11]$ )	24	P6/mmm(00γ)
191.2			24	P6/mmm(00γ)s0s0	00lm: m = 2n; 0klm: m = 2n
191.3			24	P6/mmm(00γ)00ss	0klm: m = 2n; hhlm: m = 2n
191.4			24	P6/mmm(00γ)s00s	00lm: m = 2n; hhlm: m = 2n
192.1	P6/mcc	(6/mmm, $[1{\bar 1}11]$ )	24	P6/mcc(00γ)	0klm: l = 2n; hhlm: l = 2n
192.2			24	P6/mcc(00γ)s00s	00lm: m = 2n; 0klm: l = 2n; hhlm: l + m = 2n
193.1	P6₃/mcm	(6/mmm, $[1{\bar 1}11]$ )	24	P6₃/mcm(00γ)	00lm: l = 2n; 0klm: l = 2n
193.2			24	P6₃/mcm(00γ)00ss	00lm: l = 2n; 0klm: l + m = 2n; hhlm: m = 2n
194.1	P6₃/mmc	(6/mmm, $[1{\bar 1}11]$ )	24	P6₃/mmc(00γ)	00lm: l = 2n; hhlm: l = 2n
194.2			24	P6₃/mmc(00γ)00ss	00lm: l = 2n; 0klm: m = 2n; hhlm: l + m = 2n

Table 9.8.3.6| top | pdf |
Centring reflection conditions for (3 + 1)-dimensional Bravais classes

The centring reflection conditions are given for the 24 Bravais classes, belonging to six systems (with number and symbol according to Table 9.8.3.2 a). If qⁱ = q these are the usual conditions for hklm, the indices of the reflections expressed with respect to a*, b*, c*, q. Otherwise the conditions are for indices HKLm with respect to a conventional basis $[{\bf a}_c^*, {\bf b}_c^*, {\bf c}_c^*, {\bf q}^i]$ of the vector module M*. The relation between indices HKLm and hklm is given in the fourth column. Planar monoclinic and axial monoclinic mean a monoclinic lattice of main reflections and with the (irrational part of the) modulation wavevector in the mirror plane, or along the unique axis, respectively.

System	qⁱ vector	Reflection conditions	Relation of indices	Bravais class
System	qⁱ vector	Reflection conditions	Relation of indices	No.	Symbol
Triclinic	(αβγ)			1	$[{\bar 1}]$ P(αβγ)
Planar monoclinic	(αβ0)			2	2/mP(αβ0)
				3	2/mP(αβ $[{{1}\over{2}}]$ )
				4	2/mB(αβ0)
Axial monoclinic	(00γ)			5	2/mP(00γ)
				6	2/mP( $[{{1}\over{2}}]$ 0γ)
				7	2/mB(00γ)
				8	2/mB(0 $[{{1}\over{2}}]$ γ)
Orthorhombic	(00γ)			9	mmmP(00γ)
				10	mmmP(0 $[{{1}\over{2}}]$ γ)
				11	mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
				12	mmmI(00γ)
				13	mmmC(00γ)
				14	mmmC(10γ)
				15	mmmA(00γ)
				16	mmmA( $[{{1}\over{2}}]$ 0γ)
				17	mmmF(00γ)
				18	mmmF(10γ)
Tetragonal	(00γ)			19	4/mmmP(00γ)
				20	4/mmmP( $[{{1}\over{2}}{{1}\over{2}}]$ γ)
				21	4/mmmI(00γ)
Hexagonal/Trigonal	(00γ)			22	$[{\bar 3}]$ mR(00γ)
				23	$[{\bar 3}]$ 1mP( $[{{1}\over{3}}{{1}\over{3}}]$ γ)
				24	6/mmmP(00γ)

(B) ThBr₄

Thorium tetrabromide has an incommensurately modulated phase below [T_i] = 95 K (Currat, Bernard & Delamoye, 1986 ). Above that temperature, the structure has space group [I4_1/amd] (No. 141 in International Tables for Crystallography, Volume A ). At [T_i] , a mode becomes unstable and a modulated β-phase sets in with modulation wavevector γc*. The dimension of the modulation is one, consequently.

The main reflections belong to a tetragonal lattice. The general reflection condition is $[hklm, h+k+l \hbox{ even}.]$ Looking at Table 9.8.3.6 , one finds the Bravais class to be No. 21 = I4/mmm(00γ). Table 9.8.3.2 (a) gives $[4/mmm(1\bar111)]$ for the point group of the vector module.

For the determination of the symmetry group of the modulated structure, one has the special reflection conditions $[hk00, h\hbox{ even}; \quad hhl0, 2h+l=4n; \quad (00l0, l=4n) \atop 0klm\hbox{ (and $h0lm$) absent for {\it m}} =1.]$ Higher-order satellites have not been observed. The main reflections lead to the basic group [I4_1/amd] . If one generalizes the reflection condition observed for 0klm to 0klm, m = even, the superspace group is found from Table 9.8.3.5 under the groups 141.x as $[\hbox{No. }141.2=I4_1/amd(00\gamma)s0s0 = P^{I4_1/amd}_{\kern5pt s\kern5pt \bar1 \,s\,1}.]$

Bis(n-propylammonium) tetrachloromanganate (PAMC) has several phase transitions. Above about 395 K, it is orthorhombic with space group Abma. At [T_i] , this β-phase goes over into the incommensurately modulated γ-phase (Depmeier, 1986 ; Kind & Muralt, 1986 ). The wavevector of the modulation is $[\alpha{\bf a}^*+{\bf c}^*]$ . Therefore, the dimension of the modulation is one. Interchanging the a and c axes, one sees from Table 9.8.3.2 (a) that the Bravais class is No. 14 = mmmC(10γ). In this new setting, the conventional basis of the vector module is a*, b*, c*, and γc* and the general reflection condition becomes $[HKLm, H+K+m={\rm even}.]$ Therefore, if one considers the vector module as the projection of a four-dimensional lattice, the reflection condition corresponds to a $[({1\over2}{1\over2}0{1\over2})]$ centring in four dimensions.

The point group of the vector module is $[mmm(11\bar1)]$ . The basic space group being Abma (or Ccmb in the new setting), the superspace group follows from Table 9.8.3.5 as $[\hbox{No. }64.3=Ccmb(10\gamma)=L^{Ccmb}_{\kern 5pt 1\kern0.5pt1\kern0.5pt\bar1}]$ or, in the original setting $[\hbox{No. }64.3=Abma(\alpha01)=N^{Abma}_{\kern 5pt\bar1\kern.5pt1\kern.5pt1} \, .]$ No. 64.4 can be excluded because the reflections do not show the special reflection condition 0KLm, m = even.

9.8.3.6. Ambiguities in the notation

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The invariant part [v^o_s] of the translation part [v_s] of a (3 + 1)-dimensional superspace-group element is uniquely determined by (9.8.3.5). This does not imply that for each element of the point group there is a translation for which the invariant part is unique up to lattice vectors. The reason is that, for a given element R of the point group and given origin, [v_s] may be changed when R is combined with a three-dimensional lattice translation $[w_s=({\bf w},0)]$ . This situation is well known in ordinary three-dimensional crystallography. For example, the twofold rotation $[(x,y,z)\rightarrow(-x,z,y)]$ in the space group [P4_132] has according to Volume A of International Tables for Crystallography a translation part $[(\,{1\over4},{3\over4},{1\over4}\,)]$ . Its invariant part is $[(0,{1\over2},{1\over2}\,)]$ . However, when the translation part is equivalently taken as $[(\,{1\over4},{3\over4},-{3\over4}\,)]$ , the invariant part vanishes. Therefore, in the symbol for that space group, the corresponding generator is given as the rotation `2' and not as the screw axis ` [2_1] '.

The same situation may occur in 3 + 1 dimensions. This can be seen very clearly from the definition of τ [equation (9.8.3.8)]. Since v is only determined modulo a lattice vector, one may add to it a lattice vector that has a non-vanishing product with q^r. This results in a change for τ. For example, the (3 + 1)-dimensional space group $[Pmmm(\,{1\over2}0\gamma)000=A^{Pm m\,m}_{\hskip 4.5pt 1\ 1\ \bar1}]$ has a mirror perpendicular to the a axis with associated value τ = 0. The parallel mirror at a distance a/2 has v = a and consequently $[\tau={1\over2}]$ . Hence, the symbols $[Pmmm(\,{1\over2}0\gamma)000]$ and $[Pmmm(\,{1\over2}0\gamma)s00]$ indicate the same group. This non-uniqueness in the symbol, however, does not have serious practical consequences.

Another source of ambiguity is the fact that the assignment of a satellite to a main reflection is not unique. For example, the reflection conditions for the group $[I2cb(00\gamma)0s0=P\kern.5pt^{I2cb}_{\kern3pt\bar1s\bar1}]$ are h + k + l = even because of the centring and l + m = even and h + m = even for h0lm because of the two glide planes perpendicular to the b axis. When one takes for the modulation vector q = γ′c* = (1 − γ)c*, the new indices are h, k, l′, and m′ with l′ = l + m and m′ = −m. Then the reflection conditions become l′ = even and h + m = even for [h0l'm'] . The first of these conditions implies the symbol $[I2cb(00\gamma)000=P\kern.5pt^{I2 c b}_{\kern 3pt\bar11\bar1}]$ for the group considered. This, however, is the symbol for the nonequivalent group with condition h = even for h0lm. This difficulty may be avoided by sometimes using a non-standard setting of the three-dimensional space group (see Yamamoto et al., 1985 ). In this case, the setting I2ab instead of I2cb avoids the problem.

9.8.4. Theoretical foundation

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9.8.4.1. Lattices and metric

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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 $[{\bf a}_1,{\bf a}_2,{\bf a}_3]$ : $[{\bf n}=\textstyle\sum\limits^3_{i=1}n_i{\bf a}_i, \quad n_i\hbox{ integers}. \eqno (9.8.4.1)]$ These translations are linearly independent and span a lattice Λ. The dimension of Λ is the dimension of the space spanned by $[{\bf a}_1,{\bf a}_2,{\bf a}_3]$ 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, $[\{\Lambda\}=V\quad\hbox{ and }\quad \Lambda \approx Z^3. \eqno (9.8.4.2)]$ The elements of [Z^3] 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): $[{\bf H}=\textstyle\sum\limits^3_{i=1}h_i{\bf a}^*_i, \quad h_i\hbox{ integers} \eqno (9.8.4.3)]$ where $[\{{\bf a}^*_i\}]$ is the reciprocal basis $[{\bf a}_i\cdot {\bf a}^*_k=\delta_{ik}.]$ The two corresponding metric tensors g and [g^*] , $[g_{ik}={\bf a}_i\cdot{\bf a}_k\quad\hbox{ and }\quad g^*_{ik}={\bf a}^*_i\cdot{\bf a}^*_k, \eqno (9.8.4.4)]$ are positive definite and dual: $[\textstyle\sum\limits^3_{k=1}g_{ik} g^*_{kj}=\delta_{ij}.]$ 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 $[{\bf a}^*_1]$ , $[\ldots]$ , $[{\bf a}^*_n]$ : $[{\bf H}=\textstyle\sum\limits^n_{i=1}h_i {\bf a}^*_i, \quad h_i\hbox{ integers}. \eqno (9.8.4.5)]$ The components $[(h_1,\ldots,h_n)]$ are the indices labelling the corresponding Bragg reflection peaks.

A crystal is incommensurate when d > 0 and the vectors $[{\bf a}^*_i]$ 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 $[{\bf a}^*_i]$ 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 $[{\bf a}^*_1, \ldots,{\bf a}^*_n]$ . 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* $[\{M^*\}=V \quad\hbox{ and }\quad M^*\approx Z^n. \eqno (9.8.4.6)]$ The elements of [Z^n] 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 $[\Sigma^*]$ in an n-dimensional Euclidean space [V_s] . $[M^*\rightarrow\Sigma^*, \quad\{\Sigma^*\}=V_s\quad \hbox{ and }\quad\Sigma^*\approx Z^n, \eqno (9.8.4.7)]$ with the orthogonal projection $[\pi_E]$ of [V_s] onto V defined by $[M^*=\pi_E\Sigma^*. \eqno (9.8.4.8)]$ This corresponds to attaching to the diffraction peak with indices $[(h_1,\ldots,h_n)]$ the point of an n-dimensional reciprocal lattice having the same set of coordinates. The orthocomplement of V in [V_s] is called internal space and denoted by [V_I] . The embedding is uniquely defined by the relations $[a^*_{si}=({\bf a}^*_i,{\bf a}^*_{Ii}), \quad i=1,\ldots, n, \eqno (9.8.4.9)]$ where $[\{a^*_{si}\}]$ is a basis of $[\Sigma^*]$ and $[\{{\bf a}^*_i\}]$ a basis of M*. The vectors $[{\bf a}^*_{Ii}]$ span [V_I] .

The crystal density ρ in V can also be embedded as $[\rho_s]$ in [V_s] by identifying the Fourier coefficients $[\hat\rho]$ at points of M* and of $[\Sigma^*]$ having correspondingly the same components. $[\hat\rho_s(h_1,\ldots,h_n)\equiv\hat\rho(h_1,\ldots,h_n). \eqno (9.8.4.10)]$ Then $[\rho_s]$ is invariant with respect to translations of the lattice $[\Sigma]$ with basis $[a_{si}=({\bf a}_i, {\bf a}_{Ii}) \eqno (9.8.4.11)]$ 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 [R_s] in [V_s] , leaving invariant the Euclidean metric tensors: $[g_{sik}=a_{si}\cdot a_{sk}\quad \hbox{ and }\quad g^*_{sik}=a^*_{si}\cdot a^*_{sk}. \eqno (9.8.4.12)]$ 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 $[\eqalign{ a^*_{si} &=({\bf a}^*_i,0) \cr a^*_{s(3+j)}&=({\bf a}^*_{3+j},{\bf d}^*_j)}\ \quad \eqalign{ i&=1,2,3, \cr j&=1,2,\ldots, d=n-3,} \eqno (9.8.4.13)]$ and, correspondingly, $[\eqalign{ a_{si}&=({\bf a}_i,{\bf a}_{Ii}) \cr a_{s(3+j)} &=(0,{\bf d}_j)}\ \quad\eqalign{ i&=1,2,3, \cr j&=1,2,\ldots, d,} \eqno (9.8.4.14)]$ with $[{\bf d}^*_i\cdot{\bf d}_k=\delta_{ik}]$ and $[{\bf a}^*_i\cdot{\bf a}_k=\delta_{ik}]$ . These are called standard lattice bases.

9.8.4.2. Point groups

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9.8.4.2.1. Laue class

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Definition 1. The Laue point group [P_L] 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 $[R{\bf a}^*_i=\textstyle\sum\limits^{3+d}_{j=1}\Gamma^*(R)_{ji}{\bf a}^*_j, \quad i=1, \ldots,3+d. \eqno (9.8.4.15)]$ The (3 + d) × (3 + d) matrices $[\Gamma^*(R)]$ form a finite group of integral matrices $[\Gamma^*(K)]$ for K equal to [P_L] 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 $[\Gamma^*(R)]$ on that basis are orthogonal and represent (3 + d)-dimensional orthogonal transformations [R_s] . The corresponding group is a (3 + d)-dimensional crystallographic group denoted by [K_s] . Because R is already an orthogonal transformation on V, [R_s] is reducible and can be expressed as a pair [(R,R_I)] of orthogonal transformations, in 3 and d dimensions, respectively. The basis on which acts according to $[\Gamma^*(R)]$ is denoted by $[\{({\bf a}^*_i,{\bf a}^*_{Ii})\}]$ . It spans a lattice $[\Sigma^*]$ that is the reciprocal of the lattice $[\Sigma]$ with basis elements $[({\bf a}_i,{\bf a}_{Ii})]$ . The pairs [(R, R_I)] , sometimes also noted [(R_E,R_I)] , leave $[\Sigma]$ invariant: $[(R, R_I)({\bf a}_i, {\bf a}_{Ii}) \equiv (R{\bf a}_i,R_I{\bf a}_{Ii})=\textstyle\sum\limits^{3+d}_{j=1}\Gamma(R)_{ji}({\bf a}_j,{\bf a}_{Ij}), \eqno (9.8.4.16)]$ where Γ(R) is the transpose of $[\Gamma^*(R^{-1})]$ .

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 $[\Gamma(R)=\left(\matrix{ \Gamma_E(R)&0 \cr\Gamma_M(R)&\Gamma_I(R)}\right). \eqno (9.8.4.17)]$ The transformation of main reflections and satellites is then given by $[\Gamma^*(R)]$ as in (9.8.4.15), the relation with Γ(R) being (as already said) $[\Gamma^*(R)= \tilde\Gamma(R^{-1}),]$ where the tilde indicates transposition. Accordingly, on a standard basis one has $[\Gamma^*(R)=\left(\matrix{ \Gamma^*_E(R)&\Gamma^*_M(R) \cr 0&\Gamma^*_I(R)}\right). \eqno (9.8.4.18)]$ The set of matrices $[\Gamma_E(R)]$ for R elements of K forms a crystallographic point group in three dimensions, denoted [K_E] , having elements R of O(3), and the corresponding set of matrices $[\Gamma_I(R)]$ forms one in d dimensions denoted by [K_I] with elements [R_I] of O(d).

For a modulated crystal, one can choose the $[{\bf a}^*_i]$ (i = 1, 2, 3) of a standard basis. These span the (reciprocal) lattice of the basic structure. One can then express the additional vectors $[{\bf a}^*_{3+j}]$ (which are modulation wavevectors) in terms of the basis of the lattice of main reflections: $[{\bf a}^*_{3+j}=\textstyle\sum\limits^3_{i=1}\sigma_{ji}{\bf a}^*_i, \quad j=1,2, \ldots, d. \eqno (9.8.4.19)]$ The three components of the jth row of the (d × 3)-dimensional matrix σ are just the three components of the jth modulation wavevector $[{\bf q}_j={\bf a}^*_{3+j}]$ with respect to the basis $[{\bf a}^*_1, {\bf a}^*_2, {\bf a}^*_3]$ . It is easy to show that the internal components $[{\bf a}_{Ii}]$ (i = 1, 2, 3) of the corresponding dual standard basis can be expressed as $[{\bf a}_{Ii}=-\textstyle\sum\limits^d_{j=1}\sigma_{ji}{\bf d}_j, \quad i=1,2,3. \eqno (9.8.4.20)]$ 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 $[\Gamma_E(R)]$ , $[\Gamma_I(R)]$ , and $[\Gamma_M(R)]$ of the matrix Γ(R): $[-\Gamma_I(R)\sigma + \sigma\Gamma_E(R)=\Gamma_M(R). \eqno (9.8.4.21)]$ Notice that the elements of $[\Gamma_M(R)]$ 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 $[\sigma^i]$ and $[\sigma^r]$ as follows. $[\sigma =\sigma^i+\sigma^r,\quad\hbox{ with }\quad\sigma^i\equiv{1\over N}\sum_R\Gamma_I(R)\sigma\Gamma_E(R)^{-1}, \eqno (9.8.4.22)]$ where the sum is over all elements of the Laue group of order N. It follows from this definition that $[\Gamma_I(R)\sigma^i\Gamma_E(R)^{-1}=\sigma^i.\eqno (9.8.4.23)]$ This implies $[\Gamma_M(R)=-\Gamma_I(R)\sigma^r+\sigma^r\Gamma_E(R). \eqno (9.8.4.24)]$ The matrix $[\sigma^r]$ has rational entries and is called the rational part of σ. The part $[\sigma^i]$ is called the irrational (or invariant) part.

The above equations simplify for the case d = 1. The elements $[\sigma_{1i}=\sigma_i]$ are the three components of the wavevector q, the row matrix $[\sigma\Gamma_E(R)]$ has the components of $[R^{-1}{\bf q}]$ and $[\Gamma_I(R)]$ = ɛ = ±1 since, for d = 1, q can only be transformed into ±q. One has the corresponding relations $[{\bf q} = {\bf q}^i+{\bf q}^r,\quad\hbox{ with }\quad{\bf q}^i\equiv{1\over N}\sum_R\varepsilon R{\bf q}, \eqno (9.8.4.25)]$ and $[R{\bf q}\equiv\varepsilon{\bf q}\hbox{ (modulo reciprocal lattice} \Lambda^*)\semi \quad R{\bf q}^i=\varepsilon{\bf q}^i. \eqno (9.8.4.26)]$ The reciprocal-lattice vector that gives the difference between $[R{\bf q}]$ and $[\varepsilon{\bf q}]$ has as components the elements of the row matrix $[\Gamma_M(R)]$ .

9.8.4.2.2. Geometric and arithmetic crystal classes

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According to the previous section, in the case of modulated structures a standard basis can be chosen (for M* and correspondingly for $[\Sigma^*]$ ). 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 [R_E] in the external space (the physical one, so that [R_E=R] ) and a point-group transformation [R_I] in the internal space, such that the pair [(R,R_I)] is a (3 + d)-dimensional orthogonal transformation [R_s] leaving a (3 + d)-dimensional lattice $[\Sigma]$ 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 $[{\bf a}^*_{3+j}]$ basis vectors]: there is exactly one [R_I] for each R. This is so because in the incommensurate case the correspondence between M* and $[\Sigma^*]$ 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 [K_E] of elements [R_E] to the group [K_I] of elements [R_I] that transforms products into products, i.e. is a group homomorphism. A point group [K_s] of the (3 + d)-dimensional lattice constructed for an incommensurate crystal, therefore, consists of a three-dimensional crystallographic point group [K_E] , a d-dimensional crystallographic point group [K_I] , and a homomorphism from [K_E] to [K_I] .

Definition 2. Two (3 + d)-dimensional point groups [K_s] and [K'_s] are geometrically equivalent if they are connected by a pair of orthogonal transformations [(T_E,T_I)] in [V_E] and [V_I] , respectively, such that for every [R_s] from the first group there is an element [R'_s] of the second group such that [R_ET_E=T_ER'_E] and [R_IT_I=T_IR'_I] .

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.

9.8.4.3. Systems and Bravais classes

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9.8.4.3.1. Holohedry

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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 $[\Sigma]$ * 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 $[\Sigma]$ * is the subgroup of the direct product O(3) × O(d), i.e. the group of all pairs of orthogonal transformations [R_s=(R,R_I)] that leave the lattice invariant.

This choice is possible because the point groups are reducible , i.e. leave the subspaces V and [V_I] of the direct sum space [V_s] invariant. In the case of an incommensurate crystal, the projection of $[\Sigma]$ * on M* is one-to-one as one can see as follows. The vector $[H_s=\textstyle\sum\limits^3_{i=1}\,h_i({\bf a}^*_i,0)+\textstyle\sum\limits^d_{j=1}\,m_j({\bf q}_j,{\bf d}^*_j) \eqno (9.8.4.27)]$ of $[\Sigma]$ * is projected on $[{\bf H}=\sum_i\,h_i{\bf a}^*_i+\sum_j\,m_j{\bf q}_j]$ . The vectors projected on the null vector satisfy, therefore, the relation $[\sum_i\,h_i{\bf a}^*_i+\sum_j\,m_j{\bf q}_j=0]$ . For an incommensurate phase, the basis vectors are rationally independent, which means that [h_i=0] and [m_j=0] for any i and j. Consequently, precisely one vector of $[\Sigma]$ * is projected on each given vector of M*.

Suppose now R = 1. This transformation leaves the component of every vector belonging to $[\Sigma^*]$ in V invariant. If [R_I] is the corresponding orthogonal transformation in [V_I] of an element [R_s] of the point group, a vector with component $[{\bf H}_I]$ is transformed into a vector with component [H'_I] . Since a given H is the component of only one vector of $[\Sigma]$ *, this implies $[{\bf H}_I={\bf H}'_I]$ . Consequently, [R_I] is also the identity transformation. Therefore, for incommensurate modulated phases, there are no point-group elements with [R=R_E=1] and $[R_I\neq1]$ . 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).

9.8.4.3.2. Crystallographic systems

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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 [V_I] , respectively, such that the holohedral point groups of the two lattices are represented by the same set of matrices.

9.8.4.3.3. Bravais classes

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Definition 6. Two lattices belong to the same Bravais class if their holohedral point groups are arithmetically equivalent.

This means that each of them admits a lattice basis of standard form such that their holohedral point group is represented by the same set of integral matrices.

9.8.4.4. Superspace groups

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9.8.4.4.1. Symmetry elements

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The elements of a (3 + d)-dimensional superspace group are pairs of Euclidean transformations in 3 and d dimensions, respectively: $[g_s=(\{R|{\bf v}\}, \{R_I|{\bf v}_I\})\in E(3)\times E(d), \eqno (9.8.4.28)]$ i.e. are elements of the direct product of the corresponding Euclidean groups. The elements $[\{R|{\bf v}\}]$ form a three-dimensional space group, but the same does not hold for the elements $[\{R_I|{\bf v}_I\}]$ of [E(d)] . This is because the internal translations $[{\bf v}_I]$ 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 $[\Sigma]$ does not simply split into one basis for V and one for [V_I] .

As for elements of a three-dimensional space group, the translational component $[\upsilon_s=({\bf v},{\bf v}_I)]$ of the element [g_s] can be decomposed into an intrinsic part $[\upsilon^o_s]$ and an origin-dependent part $[\upsilon^a_s]$ : $[( \boldupsilon, \boldupsilon_I)=({\bf v}^o,{\bf v}^o_I)+({\bf v}^a,{\bf v}^a_I),]$ with $[({\bf v}^o,{\bf v}^o_I)={1\over n}\sum^n_{m=1}\,(R^m{\bf v}, R^m_I{\bf v}_I), \eqno (9.8.4.29)]$ where n denotes the order of the element R. In particular, for d = 1 the intrinsic part $[{\bf v}^o_I]$ of $[{\bf v}_I]$ is equal to $[{\bf v}_I]$ if [R_I] = ɛ = +1 and vanishes if ɛ = −1. The latter means that for d = 1 there is always an origin in the internal space such that the internal shift $[{\bf v}_I]$ can be chosen to be zero for an element with ɛ = −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 $[({\bf a}_i,{\bf a}_{Ii}), (0,{\bf d}_j),\quad i=1,2,3,\quad j=1, \ldots, d. \eqno (9.8.4.30)]$ The internal part of the first three basis vectors is $[{\bf a}_{Ii}=-\Delta{\bf a}_i=-\textstyle\sum\limits^d_{j=1}\sigma_{ji}{\bf d}_j \eqno (9.8.4.31)]$ according to equation (9.8.4.20). The three-dimensional translation $[{\bf v}=\sum_i\upsilon_i{\bf a}_i]$ then entails a d-dimensional translation −Δv in [V_I] given by $[\Delta{\bf v}=\Delta\left(\textstyle\sum\limits^3_{i=1}\upsilon_i{\bf a}_i\right)=\textstyle\sum\limits^3_{i=1}\upsilon_i\Delta{\bf a}_i. \eqno (9.8.4.32)]$ These are the so-called compensating translations . Hence, the internal translation $[{\bf v}_I]$ can be decomposed as $[{\bf v}_I=-\Delta{\bf v}+ {\bolddelta}, \eqno (9.8.4.33)]$ where $[{\bolddelta}=\sum^d_{j=1}\upsilon_{3+j}{\bf d}_j]$ .

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 −Δv = $[-{\bf q}\cdot{\bf v}]$ to $[-{\bf q}\cdot {\bf v}-{\bf q}\cdot(1-R){\bf s}]$ . This implies that for the case that ɛ = 1 the part δ changes to $[\delta+{\bf q}\cdot(1-R){\bf s}=\delta+{\bf q}^r\cdot(1-R){\bf s}]$ , because $[{\bf q}^i]$ is invariant under R. Therefore, δ changes, in general. The internal translation $[\tau=\delta-{\bf q}^r\cdot{\bf v}, \eqno (9.8.4.34)]$ 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 $[g_s=\{R_s|v_s\}]$ if referred to a basis of the space can be represented isomorphically by an (n + 1)-dimensional matrix, of the form $[A(g_s)=\left(\matrix{ R_s&v_s\cr 0&1}\right) \eqno (9.8.4.35)]$ with [R_s] an n × n matrix and [v_s] an n-dimensional column matrix, all with real entries.

9.8.4.4.2. Equivalent positions and modulation relations

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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 $[{\bf d}_1]$ and $[{\bf d}^*_1]$ and give only the corresponding components.

An element of the (3 + 1)-dimensional superspace group is a pair $[g_s=(\{R|{\bf v}\}, \{\varepsilon|\upsilon_I\}) \eqno (9.8.4.36)]$ of Euclidean transformations in V and [V_I] , respectively. This element maps a point located at $[r_s=({\bf r}, t)]$ to one at $[(R{\bf r} + {\bf v}, \varepsilon t+\upsilon_I)]$ . Suppose the probability for the position $[{\bf n}+{\bf r}_j]$ to be occupied by an atom of species A is given by $[P_A({\bf n},j,t)=p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)+t], \eqno (9.8.4.37)]$ where [p_j(x)=p_j(x+1)] . By [g_s] , the position $[{\bf n}+{\bf r}_j]$ is transformed to the equivalent position $[{\bf n}'+{\bf r}_{j\,'}]$ = Rn + Rr_j + v. As the crystal is left invariant by the superspace group, the occupation probability on equivalent points has to be the same: $[P_A({\bf n}',j\,',t)=P_A[{\bf n},j,\varepsilon(t-\upsilon_I)]. \eqno (9.8.4.38)]$ This implies that for the structure in the three-dimensional space one has the relation $[P_A({\bf n}',j\,',0)=P_A({\bf n},j,-\varepsilon\upsilon_I). \eqno (9.8.4.39)]$ In terms of the modulation function [p_j] this means $[p_{j\,'}[{\bf q}\cdot({\bf n}'+{\bf r}_{j\,'})] = p_j[{\bf q}\cdot({\bf n}+{\bf r}_j)-\varepsilon\upsilon_I]. \eqno (9.8.4.40)]$ In the same way, one derives the following property of the modulation function: $[p_{j\,'}(x)=p_j[\varepsilon(x-\delta)+{\bf K}\cdot({\bf r}_{j\,'}-{\bf v})], \quad\hbox{where }R{\bf q}=\varepsilon {\bf q} + {\bf K}. \eqno (9.8.4.41)]$ Analogously, for a displacive modulation, the position $[{\bf n}+{\bf r}_j]$ with displacement $[{\bf u}_j(t_o)]$ , where $[t_o={\bf q}\cdot({\bf n}+{\bf r}_j)]$ , is transformed to $[{\bf n}'+ {\bf r}_{j\,'}]$ with displacement $[{\bf u}_{j\,'}(t'_o)=R{\bf u}_j(t_o-\varepsilon\upsilon_I). \eqno (9.8.4.42)]$ To be invariant, the displacement function has to satisfy the relation $[{\bf u}_{j\,'}(x)=R{\bf u}_j[\varepsilon x-\varepsilon\delta+{\bf K}\cdot({\bf r}_{j\,'}-{\bf v})], \quad\hbox{where }R{\bf q}=\varepsilon {\bf q}+{\bf K}. \eqno (9.8.4.43)]$ The expressions for $[d \, \gt \, 1]$ are straightforward generalizations of these.

9.8.4.4.3. Structure factor

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The scattering from a set of atoms at positions $[{\bf r}_n]$ is described in the kinematic approximation by the structure factor: $[S_{\bf H}=\textstyle\sum\limits_n\,f_n({\bf H})\exp(2\pi i{\bf H}\cdot{\bf r}_n), \eqno (9.8.4.44)]$ where $[f_n({\bf H})]$ is the atomic scattering factor. For an incommensurate crystal phase, this structure factor $[S_{\bf H}]$ is equal to the structure factor $[S_{H_S}]$ of the crystal structure embedded in 3 + d dimensions, where H is the projection of [H_s] on [V_E] . This structure factor is expressed by a sum of the products of atomic scattering factors [f_n] and phase factors $[\exp(2\pi iH_s\cdot r_{sn})]$ 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) $[\eqalignno{ S_{\bf H}&=\textstyle\sum\limits_A\textstyle\sum\limits_j\textstyle\int\limits_\Omega{\rm d}{\bf t}\ f_A({\bf H})P_{Aj}({\bf t}) \cr &\quad\times\exp\{2\pi i({\bf H,H}_I)\cdot [{\bf r}_j+{\bf u}_j({\bf t}), {\bf t}]\}, & (9.8.4.45)}]$ 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 $[{\bf d}_1,\ldots,{\bf d}_d]$ in [V_I] ; [f_A] is the atomic scattering factor of species A, $[P_{Aj}({\bf t})]$ 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 $[P_j(t)=1\semi \quad {\bf u}_j(t)={\bf U}_j\sin[2\pi({\bf q}\cdot {\bf r}_j+t+\varphi_j)]. \eqno (9.8.4.46)]$ According to (9.8.4.45), the structure factor is $[\eqalignno{ S_{\bf H} &=\textstyle\sum\limits_j\textstyle\int\limits^1_0{\rm d} t\ f_j({\bf H})\exp(2\pi i{\bf H}\cdot{\bf r}_j)\exp(2\pi imt) \cr &\quad\times\exp[2\pi i{\bf H}\cdot{\bf U}_j\sin2\pi({\bf q}\cdot{\bf r}_j+t+\varphi_j)]. &(9.8.4.47)}]$ For a diffraction vector H = K + mq, this reduces to $[\eqalignno{ S_{\bf H} &=\textstyle\sum\limits_j\,f_j({\bf H})\exp(2\pi i{\bf K\cdot r}_j)J_{-m}(2\pi{\bf H}\cdot{\bf U}_j) \cr &\quad\times\exp(-2\pi im\varphi_j). & (9.8.4.48)}]$ For a general one-dimensional modulation with occupation modulation function [p_j(t)] and displacement function $[{\bf u}_j(t)]$ , the structure factor becomes $[\eqalignno{ S_{\bf H}&=\textstyle\sum\limits_j\textstyle\int\limits^1_0{\rm d} t\ f_j({\bf H})p_j({\bf q}\cdot {\bf r}_j+t+\psi_j)\exp[2\pi i({\bf H}\cdot{\bf r}_j+mt)] \cr&\quad\times\exp[2\pi i{\bf H}\cdot {\bf u}_j({\bf q}\cdot{\bf r}_j+t+\varphi_j)]. & (9.8.4.49)}]$ Because of the periodicity of [p_j(t)] and $[{\bf u}_j(t)]$ , one can expand the Fourier series: $[\eqalignno{ &p_j({\bf q}\cdot{\bf r}_j+t+\psi_j)\exp[2\pi i{\bf H}\cdot{\bf u}_j({\bf q}\cdot{\bf r}_j+t+\varphi_j)] \cr&\quad=\textstyle\sum\limits_k\,C_{j,k}({\bf H})\exp[2\pi ik({\bf q}\cdot{\bf r}_j+t)], &(9.8.4.50)}]$ and consequently the structure factor becomes $[S_{\bf H}=\textstyle\sum\limits_j\,f_j({\bf H})\exp(2\pi i{\bf K}\cdot{\bf r}_j)C_{j,-m}({\bf H}), \quad\hbox{where }{\bf H}={\bf K}+m{\bf q}. \eqno (9.8.4.51)]$ The diffraction from incommensurate crystal structures has been treated by de Wolff (1974 ), Yamamoto (1982a ,b ), Paciorek & Kucharczyk (1985 ), Petricek, Coppens & Becker (1985 ), Petříček & Coppens (1988 ), Perez-Mato et al. (1986 , 1987 ), and Steurer (1987 ).

9.8.5. Generalizations

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9.8.5.1. Incommensurate composite crystal structures

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The basic structure of a modulated crystal does not always have space-group symmetry. Consider, for example, composite crystals (also called intergrowth crystals ). Disregarding modulations, one can describe these crystals as composed of a finite number of subsystems, each with its own space-group symmetry. The lattices of these subsystems can be mutually incommensurate. In that case, the overall symmetry is not a space group, the composite crystal is incommensurate and so also is its basic structure. The superspace approach can also be applied to such crystals. Let the subsystems be labelled by an index ν. For the subsystem ν, we denote the lattice by $[\Lambda_\nu]$ with basis vectors $[{\bf a}_{\nu i}]$ (i = 1, 2, 3), its reciprocal lattice by $[\Lambda^*_\nu]$ with basis vectors $[{\bf a}^*_{\nu i}]$ (i = 1, 2, 3), and the space group by $[G_\nu]$ . The atomic positions of the basic structure are given by $[{\bf n}_\nu+{\bf r}_{\nu j}, \eqno (9.8.5.1)]$ where $[{\bf n}_\nu]$ is a lattice vector belonging to $[\Lambda_\nu]$ . In the special case that the subsystems are mutually commensurate, there are three basis vectors a*, b*, c* such that all vectors $[{\bf a}^*_{\nu i}]$ are integral linear combinations of them. In general, however, more than three basis vectors are needed, but never more than three times the number of subsystems. Suppose that the vectors $[{\bf a}^*_i]$ $[(i=1, \ldots, n)]$ form a basis set such that every $[{\bf a}^*_{\nu i}]$ can be expressed as an integral linear combination of them: $[{\bf a}^*_{\nu i}=\textstyle\sum\limits^n_{k=1}\,Z^\nu_{ik}{\bf a}^*_k,\quad Z^\nu_{ik}\hbox{ integers}, \eqno (9.8.5.2)]$ with [n=3+d_o] and $[d_o\, \gt \, 0]$ . Then the vectors of the diffraction pattern of the unmodulated system are again of the form (9.8.4.5) and generate a vector module [M^*_o] of dimension three and rank [(3+d_o)] , which can be considered as projection of a -dimensional lattice $[\Sigma^*_o]$ .

We now assume that one can choose $[{\bf a}^*_{Ii}=0]$ for i = 1, 2, 3 and we denote $[{\bf a}^*_{I3+j}]$ by $[{\bf d}^*_j]$ . This corresponds to assuming the existence of a subset of Bragg reflections at the positions of a three-dimensional reciprocal lattice $[\Lambda]$ *. Then there is a standard basis for the lattice $[\Sigma_o]$ , which is the reciprocal of $[\Sigma^*_o]$ , given by $[({\bf a}_i,{\bf a}_{Ii}), \quad(0, {\bf d}_j), \quad i=1,2,3,\quad j=1,\ldots, d_o. \eqno (9.8.5.3)]$ In order to find the [(3+d_o)] -dimensional periodic structure for which this composite crystal is the three-dimensional intersection, one associates with a translation t in the internal space [V_I] three-dimensional independent shifts, one for each subsystem. These shifts of the subsystems replace the phase shifts adopted for the modulated structures: [V_I] is now the space of the variable relative positions of the subsystems. Again, a translation in the superspace can give rise to a non-Euclidean transformation in the three-dimensional space of the crystal, because of the variation in the relative positions among subsystems. Each subsystem, however, is rigidly translated. For the basis vectors $[{\bf d}_j]$ , the shift of the subsystem ν is defined in terms of projection operators $[\pi_\nu]$ : $[\pi_\nu{\bf d}_j=\textstyle\sum\limits^3_{i=1}\, Z^\nu_{i3+j}{\bf a}_{\nu i}, \quad j=1,\ldots, d_o. \eqno (9.8.5.4)]$ Then an arbitrary translation $[{\bf t}=\sum_j t_j{\bf d}_j]$ in [V_I] displaces the subsystem ν over a vector $[\sum_j\,t_j(\pi_\nu{\bf d}_j)]$ . A translation $[({\bf a},{\bf a}_I+{\bf d})]$ belonging to the [(3+d_o)] -dimensional lattice $[\Sigma_o]$ induces for the subsystem ν in ordinary space a relative translation over vector $[{\bf a}+\pi_\nu({\bf a}_I+{\bf d})]$ . The statement is that this translation is a vector of the lattice $[\Lambda_\nu]$ and leaves therefore the subsystem ν invariant. So the lattice translations belonging to $[\Sigma_o]$ form a group of symmetry operations for the composite crystal as a whole.

The proof is as follows. If k belongs to $[\Lambda^*_\nu]$ , the vector $[({\bf k},{\bf k}_I)]$ belongs to $[\Sigma^*_o]$ . In particular, for $[{\bf k}={\bf a}^*_{\nu i}]$ , one has, because of (9.8.5.2) and (9.8.5.4), $[{\bf a}^*_{\nu i}\cdot\pi_\nu{\bf d}_j=Z^\nu_{i3+j}, \quad j=1, \ldots, d_o, \eqno (9.8.5.5)]$ and $[{\bf k}_I=\textstyle\sum\limits^{d_o}_{j=1}\,Z^\nu_{i3+j}{\bf d}^*_j\quad\hbox{and therefore }\quad {\bf k}_I\cdot{\bf d}_j=Z^\nu_{i 3+j}.]$

Note that one has $[{\bf k}_I\cdot {\bf t}={\bf k}\cdot\pi_\nu{\bf t}]$ , for any t from [V_I] as $[\pi_\nu]$ is a linear operator. Because of the linearity, this holds for every k from $[\Lambda^*_\nu]$ as well. Since $[({\bf k}, {\bf k}_I)]$ belongs to $[\Sigma^*_o]$ and $[({\bf a,a}_I+{\bf d})]$ to $[\Sigma_o]$ , one has for their inner product: $[{\bf k\cdot a+k}_I\cdot{\bf a}_I+{\bf k}_I\cdot {\bf d}={\bf k}\cdot({\bf a}+\pi_\nu{\bf a}_I+\pi_\nu{\bf d})\equiv 0\eqno \hbox{(modulo 1),}]$ which implies that $[{\bf a}+\pi_\nu{\bf a}_I+\pi_\nu{\bf d}]$ is an element of $[\Lambda_\nu]$ .

In conclusion, one may state that the composite structure is the intersection with the ordinary space (t = 0) of a pattern having atomic position vectors given by $[({\bf n}_\nu+{\bf r}_{\nu j}-\pi_\nu{\bf t}, {\bf t})\quad\hbox{for any {\bf t} of }V_I. \eqno (9.8.5.6)]$ Such a pattern is invariant under the [(3+d_o)] -dimensional lattice $[\Sigma_o]$ . Again, orthogonal transformations R of O(3) leaving the vector module [M^*_o] invariant can be extended to orthogonal transformation [R_s] of $[O(3)\times O(d_o)]$ allowing a Euclidean structure to be given to the superspace. One can then consider the superspace-group symmetry of the basic structure defined by atomic positions as in (9.8.5.6). A superspace-group element [g_s] as in (9.8.4.28) induces (in three-dimensional space) for the subsystem ν the transformation $[g_s:{\bf n}_\nu+{\bf r}_{\nu j}\rightarrow R{\bf n}_\nu+R{\bf r}_{\nu j} +{\bf v}+R\pi_{\nu}R^{-1}_I{\bf v}_I, \eqno (9.8.5.7)]$ changing the position $[{\bf n}_\nu+{\bf r}_{\nu j}]$ into an equivalent one of the composite structure, not necessarily, however, within the same subsystem ν.

Finally, the composite structure can also be modulated. For the case of a one-dimensional modulation of each subsystem ν, the positions are $[{\bf n}_\nu+{\bf r}_{\nu j}+{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j})]. \eqno (9.8.5.8)]$ Possibly the modulation vectors can also be expressed as integral linear combinations of the $[{\bf a}^*_i]$ $[(i=1,\ldots,3+d_o)]$ . Then, the dimension of [V_I] is again [d_o] . In general, however, one has to consider [(d-d_o)] additional vectors, in order to ensure the validity of (9.8.4.5), now for n = 3 + d. We can then write $[{\bf q}_\nu=\textstyle\sum\limits^{3+d}_{j=1}\, Q^\nu_j{\bf a}^*_j,\quad Q^\nu_j\hbox{ integers}. \eqno (9.8.5.9)]$ The peaks of the diffraction pattern are at positions defined by a vector module M*, which can be considered as the projection of a (3 + d)-dimensional lattice $[\Sigma]$ *, the reciprocal of which leaves invariant the pattern of the modulated atomic positions in the superspace given by $[\eqalignno{ \{{\bf n}_\nu+r_{\nu j}-\pi_\nu{\bf t}+{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j}-\pi_\nu{\bf t})+{\bf q}_{I\nu}&\cdot{\bf t}], {\bf t}\}, \cr \hbox{ for any {\bf t} of }V_I& &(9.8.5.10)}]$ with $[\pi_\nu{\bf d}_j=0]$ for $[j \, > \, d_o]$ , where $[{\bf q}_{I\nu}]$ is the internal part of the (3 + d)-dimensional vector that projects on $[{\bf q}_{\nu}]$ . Their symmetry is a (3 + d)-dimensional superspace group [G_s] . The transformation induced in the modulated composite crystal by an element under [g_s] of [G_s] is now readily written down. For the case [d=d_o=1] and [g_s] = ({R|v}, {ɛ|Δ}) , the position $[{\bf n}_\nu+{\bf r}_{\nu j}]$ is transformed into $[R({\bf n}_\nu+ {\bf r}_{\nu j})+{\bf v}+\varepsilon R\pi_\nu \Delta{\bf d}_1, \eqno (9.8.5.11)]$ and the modulation $[{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j})]]$ into $[R{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j}+\varepsilon\pi_\nu\Delta{\bf d}_1)-\varepsilon{\bf q}_{I\nu}\cdot\Delta{\bf d}_1].]$

This shows how the superspace-group approach can be applied to a composite (modulated) structure. Note that composite systems do not necessarily have an invariant set of (main) reflections at lattice positions.

9.8.5.2. The incommensurate versus the commensurate case

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As said earlier, it sometimes makes sense also to use the description as developed for incommensurate crystal phases for a (commensurate) superstructure. In fact, very often the modulation wavevector also shows, in addition to continuously varying (incommensurate) values, several rational values at various phase transitions of a given crystal or in different compounds of a given structural family. In these cases, there is three-dimensional space-group symmetry. Generally, the space groups of the various phases are different. The description as used for incommensurate phases then gives the possibility of a more unified characterization for the symmetry of related modulated crystal phases. In fact, the theory of higher-dimensional space groups for modulated structures is largely independent of the assumption of irrationality. Only some of the statements need to be adapted. The most important change is that there is no longer a one-to-one correspondence between the points of the reciprocal lattice $[\Sigma]$ * and its projection on V defining the positions of the Bragg peaks. Furthermore, the projection of the lattice $[\Sigma]$ on the space [V_I] forms a discrete set. The latter means the following. For an incommensurate modulation, the incommensurate structure, which is the intersection of a periodic structure with the hyperplane $[{\bf r}_I=0]$ , is also the intersection of the same periodic structure with a hyperplane $[{\bf r}_I]$ = constant , where this constant is of the form $[\textstyle\sum\limits^3_{i=1}h_i{\bf a}_{Ii}+\textstyle\sum\limits^d_{j=1}m_j{\bf a}_{I3+j}. \eqno (9.8.5.12)]$ Because for an incommensurate structure these vectors form a dense set in [V_I] , the phase of the modulation function with respect to the basic structure is not determined. For a commensurate modulation, however, the points (9.8.5.12) form a discrete set, even belong to a lattice, and the phase (or the phases) of the modulation are determined within vectors of this lattice. Notice that the grid of this lattice becomes finer as the denominators in the rational components become larger.

When [G_s] is a (3 + d)-dimensional superspace group, its elements, in general, do not leave the ordinary space V invariant. The subgroup of all elements that do leave V invariant, when restricted to V, is a group of distance-preserving transformations in three dimensions and thus a subgroup of E(3), the three-dimensional Euclidean group. In general, this subgroup is not a three-dimensional space group. It is so when the modulation wavevectors all have rational components only, i.e. when σ is a matrix with rational entries. Because the phase of the modulation function is now determined (within a given rational number smaller than 1), the space group depends in general on this phase.

As an example, consider a one-dimensional modulation of a basic structure with orthorhombic space group Pcmn. Suppose that the modulation wavevector is γc*. Then the mirror [R=m_z] perpendicular to the c axis is combined with [R_I] = ɛ = −1. Suppose, furthermore, that the glide reflection perpendicular to the a axis and the b mirror are both combined with a phase shift $[1\over2]$ . In terms of the coordinates x, y, z with respect to the a, b and c axes, and internal coordinate t, the generators of the (3 + 1)-dimensional superspace group Pcmn(00γ)ss0 act as $[\eqalignno{&(x,y,z,t)\rightarrow(x+k, y+l, z+m, t-\gamma m+n),&\cr &\qquad\qquad\qquad\qquad k,l,m,n\hbox{ integers},&(9.8.5.13a)}]$ $[\eqalignno{& (x,y,z,t)&\cr &\quad \rightarrow(-x+k+\textstyle{1\over2}, y+l, z+{1\over2}+m, t-\gamma/2-\gamma m+\textstyle{1\over2}+n),&\cr &&(9.8.5.13b)}]$ $[(x,y,z,t)\rightarrow(x+k, -y+l+\textstyle{1\over2}, z+m, t-\gamma m+{1\over2}+n), \eqno (9.8.5.13c)]$ $[\eqalignno{& (x,y,z,t)&\cr&\quad \rightarrow(x+\textstyle{1\over2}+k, y+{1\over2}+l,-z+{1\over2}+m, -t-\gamma/2-\gamma m+n).&\cr &&(9.8.5.13d)}]$ Note that these positions are referred to a split basis (i.e. of basis vectors lying either in V or in [V_I] ) and not to a basis of the lattice $[\Sigma]$ . When the superstructure is the intersection of a periodic structure with the plane at [t=t_o] , its three-dimensional space group follows from equation (9.8.5.13) by the requirement [t'=t_o] . When γ has the rational value r/s with r and s relatively prime, the conditions for each of the generators to give an element of the three-dimensional space group are, respectively: $[\eqalignno{ -rm+sn&=0&(9.8.5.14a) \cr-2rm+2sn&=r-s&(9.8.5.14b) \cr -2rm+2sn&=-s&(9.8.5.14c) \cr -2rm+2sn&=4st, &(9.8.5.14d)}%fd9.8.5.14b9.8.5.14c9.8.5.14d]$ for m, n, r, s integers and t real. These conditions are never satisfied simultaneously. It depends on the parity of both r and s which element occurs, and for the elements with ɛ = −1 it also depends on the value of the `phase' t, or more precisely on the product τ = 4st. The translation group is determined by the first condition as in (9.8.5.14a ). Its generators are $[{\bf a}, {\bf b}, \hbox{and }s{\bf c},]$ where the last vector is the external part of the lattice vector s(c, −r/s) + r(0, 1). The other space-group elements can be derived in the same way. The possible space groups are:

γ = r/s	τ even integer	τ odd integer	otherwise
r even, s odd	$[11\displaystyle{2_1\over n}]$	2₁2₁2₁	112₁
r odd, s even	$[ 1\displaystyle{2_1\over c}1]$	2₁cn	1c1
r odd, s odd	$[\displaystyle{2_1\over c}11]$	c 2₁n	c 11

In general, the three-dimensional space groups compatible with a given (3 + d)-dimensional superspace group can be determined using analogous equations.

As one can see from the table above, the orthorhombic (3 + d)-dimensional superspace group leads in several cases to monoclinic three-dimensional space groups. The lattice of main reflections, however, still has orthorhombic point-group symmetry. Description in the conventional way by means of three-dimensional groups then neglects some of the structural features present. Even if the orthorhombic symmetry is slightly broken, the orthorhombic basic structure is a better characterization than a monoclinic one. Note that in that case the superspace-group symmetry is also only an approximation.

When the denominators of the wavevector components become small, additional symmetry operations become possible. Because the one-to-one correspondence between $[\Sigma]$ * and M* is no longer present, there may occur symmetry elements with trivial action in V but with nontrivial transformation in [V_I] . For d = 1, these possibilities have been enumerated. The corresponding Bravais classes are given in Table 9.8.3.2 (b).

Appendix A9.8.1

A9.8.1. Glossary of symbols

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M *	Vector module in m-dimensional reciprocal space (m = 1, 2, 3; normally m = 3), isomorphic to Zⁿ with $[n\,\ge \,m]$ . The dimension of M* is m, its rank n.
$[{\bf a}^*_i]$	$[(i=1,\ldots, n.)]$ Basis of a vector module M* of rank n; if n = 4 and q is modulation wavevector (the n = 4 case is restricted in what follows to modulated crystals), the basis of M* is chosen as a, b, c, q, with a, b, c a basis of the lattice of main reflections.
Λ*	Lattice of main reflections, m-dimensional reciprocal lattice.
a , b, c*	(Conventional) basis of Λ* for m = 3.
Λ	Direct m-dimensional lattice, dual to Λ*.
	Superspace; Euclidean space of dimension n = m + d; $[V_s=V\oplus V_I]$ .
V	Physical (or external) space of dimension m (m = 1,2 or 3), also indicated by .
	Internal (or additional) space of dimension d.
$[\Sigma^*]$	Reciprocal lattice in n-dimensional space, whose orthogonal projection on V is M*.
$[\Sigma]$	Lattice in n-dimensional superspace for which $[\Sigma]$ * is the reciprocal one.
$[a^*_{si}]$	Lattice basis of $[\Sigma]$ * in $[(i=1,\ldots, n)]$ ; if n = 4, this basis can be chosen as {(a, 0), (b, 0), (c, 0), (q, 1)} and is called standard. An equivalent notation is (q, 1) = (q, d); for n = 3 + d, the general form of a standard basis is (a, 0), (b, 0), (c, 0), $[({\bf q}_1,{\bf d}^_1),\ldots,({\bf q}_j,{\bf d}^_j),\ldots,]$ $[({\bf q}_d,{\bf d}^_d)]$ .
$[a_{si}]$	( $[i = 1,\ldots,n.]$ ) Lattice basis of $[\Sigma]$ in dual to $[\{a^*_{si}\}]$ ; if n = 4, the standard basis is $[({\bf a}, -{\bf q} \cdot {\bf a}), ({\bf b},-{\bf q}\cdot {\bf b})]$ , $[({\bf c, -q\cdot c})]$ , (0, 1) = (0, d); for n = 3 + d, a standard basis is dual to the standard one given above.
q _j	Modulation wavevector(s) $[{\bf q}_j=\sum^3_{i=1}\sigma_{ji}{\bf a}^_i]$ ; if n = 4, $[{\bf q}=\sum^3_{i=1}\sigma_i{\bf a}^_i= \alpha{\bf a}^+\beta{\bf b}^+\gamma{\bf c}^*;]$ σ = (α, β, γ); if n = 4, $[{\bf q}={\bf q}^i+{\bf q}^r]$ , with $[{\bf q}^i=(1/N)\sum_{R {\, {\rm in}\,} K}\varepsilon(R)R{\bf q}]$ , where $[\varepsilon(R)=R_I]$ , and N is the order of K.
H	Bragg reflections: $[{\bf H}=\sum^n_{i=1}h_i{\bf a}^_i=(h_1,h_2,\ldots,h_n)]$ ; if n = 4, $[{\bf H}=\sum^4_{i=1}h_i{\bf a}^_i=h{\bf a}^+k{\bf b}^+l{\bf c}^*+m{\bf q} =]$ (h, k, l, m).
	Embedding of H in : for $[{\bf H}=(h_1,\ldots,h_n) =]$ $[\sum^n_{i=1}h_i{\bf a}^_i]$ , one has correspondingly $[({\bf H},{\bf H}_I)=]$ $[\sum^n_{i=1}h_ia^_{si}]$ .
	Laue point group.
O (m)	Orthogonal group in m dimensions.
R	Orthogonal point-group transformation, element of O(m).
K	Point group, crystallographic subgroup of O(m).
	Superspace point-group element: element of O(m) × O(d) with external, and internal part of , respectively; if n = 4, superspace point-group element: [R, ɛ(R)] with ɛ(R) = ±1, also written (R, ɛ).
	Point group, crystallographic subgroup of O(m) × O(d).
	External part of , crystallographic point group, subgroup of O(m) with as elements the external part transformations of .
	Internal part of , crystallographic point group, subgroup of O(d) with as elements the internal part transformations of .
$[{\bf r}_o({\bf n},j)]$	Atomic positions in the basic structure: $[{\bf r}_o({\bf n},j)=]$ $[{\bf n}+{\bf r}_j]$ with $[{\bf n}\in \Lambda]$ .
r (n, j)	Atomic positions in the displacively modulated structure; (d = 1): r(n, j) = $[{\bf r}_o({\bf n}, j)+{\bf u}_j[{\bf q}\cdot{\bf r}({\bf n},j)+\varphi_j]]$ . In general, however, different phases $[\varphi_{j\alpha}]$ may occur for different components $[u_{j\alpha}]$ along the crystallographic axes.
$[{\bf u}_j(x)]$	Modulation function for displacive modulation with $[{\bf u}_j(x+1)={\bf u}_j(x)]$ .
	Modulation function for occupation modulation with .
g	Euclidean transformation in m dimensions; g = {R\|v} element of the space group G with rotational part R and translational part v.
$[{\bf v}^o]$	Intrinsic translation part (origin independent).
	Superspace group transformation (d = 1): $[g_s=\{(R,\varepsilon)\|({\bf v},\Delta)\}=(\{R\|{\bf v}\}, \{\varepsilon\|\Delta\})=\{R_s\|\upsilon_s\}]$ element of the superspace group . In the (3 + d)-dimensional case: $[g_s=\{(R,R_I)\|({\bf v},{\bf v}_I)\}=(\{R\|{\bf v}\}, \{ R_I\|{\bf v}_I\})]$ .
$[\upsilon_I]$	Internal shift (d = 1): $[\upsilon_I=\Delta=\delta-{\bf q}\cdot {\bf v}]$ .
τ	Intrinsic internal shift (d = 1): τ = δ − $[{\bf q}^r\cdot {\bf v}]$ .
$[\Gamma^*(R)]$	Point-group transformation R with respect to a basis of M* and at the same time superspace point-group transformation with respect to a corresponding basis of $[\Sigma^*]$ .
Γ(R)	Superspace point-group transformation with respect to a lattice basis of $[\Sigma]$ dual to that of $[\Sigma^]$ leading to $[\Gamma^(R)]$ . The mutual relation is then $[\Gamma^*(R)=\tilde\Gamma(R^{-1})]$ with the tilde denoting transposition.
$[\Gamma_E(R), \Gamma_I(R), \Gamma_M(R)]$	external, internal, and mixed blocks of Γ(R), respectively.
$[\Gamma^_E(R), \Gamma^_I(R), \Gamma^*_M(R)]$	external, internal, and mixed blocks of $[\Gamma^*(R)]$ , respectively.
$[S_{\bf H}]$	Structure factor: $[S_{\bf H}=\textstyle\sum\limits_{\bf n}\textstyle\sum\limits_j\, f_j({\bf H})\exp[2\pi i{\bf H}\cdot{\bf r}({\bf n}, j)].]$
$[f_j({\bf H})]$	Atomic scattering factor for atom j.

Appendix A9.8.2

A9.8.2. Basic definitions

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In the following, we give a short definition of the most important notions appearing in the theory and of the equivalence relations used in the tables. The latter are especially adapted to the case of modulated crystal phases.

[i] Vector module. A set of all integral linear combinations of a finite number of vectors. The dimension of the vector module is the dimension (m) of the space V (also indicated as and called external) generated by it over the real numbers. Its rank (n) is the minimal number of rationally independent vectors that generate the vector module. If this rank is equal to the dimension, the vector module is also a lattice. In general, a vector module of rank n and dimension m is the orthogonal projection on the m-dimensional space V of an n-dimensional lattice. We shall restrict ourselves mainly to the case m = 3 and n = 4, but the following definitions are valid for modulated phases of arbitrary dimension and rank. The dimension of the modulation (d) is n − m. The modulation phases span a d-dimensional space (called internal or additional).
[ii] Superspace. is an n-dimensional Euclidean space that is the direct sum of an m-dimensional external space V (of the crystal) and a d-dimensional internal space (for the additional degrees of freedom). V is sometimes denoted by .
[iii] Split basis. For the space $[V_s=V\oplus V_I]$ , this is a basis with m basis vectors in V and d = n − m basis vectors in .
[iv] Standard basis. For the (m + d)-dimensional space $[V_s=V\oplus V_I]$ , a standard basis in direct space is one having the last d basis vectors lying in (d = dimension of = dimension of the modulation). A standard basis in reciprocal space (V* identified with V) is one with the first m basis vectors lying in V (m = dimension of V).
[v] Conventional basis. For a lattice Λ in three dimensions, it is a basis such that (i) the lattice generated by it is contained in Λ as a sublattice and (ii) there is the standard relationship between the basis vectors (e.g. for a cubic lattice a conventional basis consists of three mutually perpendicular vectors of equal length).

The lattice Λ is obtained from the lattice spanned by the conventional basis by adding (a small number of) centring vectors. [For example, the b.c.c. lattice is obtained from the conventional cubic lattice by centring the unit cell with $[(\,{1\over2}{1\over2}{1\over2}\,)]$ .] The reciprocal basis for the conventional basis is a conventional basis for the reciprocal lattice Λ*.

In the (m + d)-dimensional superspace, a conventional basis for the lattice $[\Sigma]$ satisfies the same conditions (i) and (ii) as formulated above for the three-dimensional case. In addition, however, one requires that the basis is standard and such that the non-vanishing external components satisfy the relations of an (m = 3) conventional basis and that the corresponding internal components only involve the irrational components of the modulation vector(s) (for d = 1 the basis is such that $[{\bf q}^r=0]$ , thus $[{\bf q}^i={\bf q}]$ ). Again a conventional basis for $[\Sigma]$ * is dual to the same for $[\Sigma]$ .
[vi] Holohedry. The holohedry of a vector module is the group of orthogonal transformations of the same dimension that leaves the vector module invariant. The holohedry of an (m + d)-dimensional lattice is the subgroup of O(m) × O(d) that leaves the lattice invariant.
[vii] Point group. An (m + d)-dimensional crystallographic point group is a subgroup of $[O(m)\times O(d)]$ . With respect to a standard lattice basis its elements are of the form $[\Gamma(R)=\left(\matrix{\Gamma_E(R) &0\cr\Gamma_M(R) &\Gamma_I(R)}\right),]$ where all the entries are integers and R is an element of an m-dimensional point group K, which is actually the same as . For an incommensurate modulated crystal, and K are isomorphic groups. If d = 1, $[\Gamma_I(R)]$ = ɛ = ±1.
[viii] Geometric crystal class. Two point groups and of pairs of orthogonal transformations [ belongs to O(m) and to O(d)] are geometrically equivalent if and only if there are orthogonal transformations and of O(m) and O(d), respectively, such that $[R'_E=T_E\cdot R_E\cdot T^{-1}_E]$ and $[R'_I=T_I\cdot R_I\cdot T^{-1}_{I}]$ for some group isomorphism $[(R_E,R_I)\rightarrow(R'_E,R'_I)]$ . For d = 1, that condition takes a simpler form because = ɛ = ±1.

[ix] Arithmetic crystal class. A group of integral matrices $[\Gamma^*(R)]$ [for $[R\in K]$ of O(m)] is determined on a basis $[\{{\bf a}^*_i; i=1,\ldots,n\}={\bf a}^*, {\bf b}^*,{\bf c}^*, {\bf q}_1,\ldots, {\bf q}_d]$ of a vector module in reciprocal space by an m-dimensional point group K (here m = 3). For modulated crystals, the transformations in direct space are given by matrices Γ(R) = transpose of $[\Gamma^*(R^{-1})]$ which are of the form (9.8.4.17). Two groups Γ′(K′) and Γ(K) are arithmetically equivalent if and only if there is an (m + d)-dimensional matrix S of the form $[S=\left(\matrix{S_E &0\cr S_M &S_I}\right)]$ with integral entries and determinant ±1 such that $[\Gamma'(K')=S\Gamma(K)\cdot S^{-1}]$ . Here [S_E] is m × m and [S_I] is d × d dimensional. An alternative formulation is: the matrix groups Γ(K) and Γ′(K′) determined as in equation (9.8.1.16) or in equation (9.8.1.21) are arithmetically equivalent if

(a) the groups K and K′ are geometrically equivalent m-dimensional point groups [the corresponding (m + d)-dimensional point groups and are then also geometrically equivalent];
(b) there are vector module bases a*, $[\ldots]$ , q_d and a*′, $[\ldots]$ , $[{\bf q}^{\prime}_{d}]$ such that K on the first basis gives the same group of matrices as K′ on the second basis.

[x] Bravais class. Two vector modules are in the same Bravais class if the groups of matrices determined by their holohedries are arithmetically equivalent. Two (m + d)-dimensional lattices are in the same Bravais class if their holohedries are arithmetically equivalent. In both cases, one can find bases for the two structures such that the holohedries take the same matrix form. In the (m + d)-dimensional case, the lattice bases both have to be standard.
[xi] Superspace group. An (m + d)-dimensional superspace group is an n-dimensional space group (n = m + d) such that it has a d-dimensional lattice of internal translations. (This latter property reflects the periodicity of the modulation.) It is determined on a standard lattice basis by the matrices Γ(R) of the point-group transformations and by the components $[\upsilon_i(R)]$ $[(i=1,\ldots, m+d)]$ of the translation parts of its elements. The matrices Γ(R) represent at the same time the elements R of the m-dimensional point group K and the corresponding elements of the (n + d)-dimensional point groups . Two (m + d)-dimensional superspace groups are equivalent if there is an origin and a standard lattice basis for each group such that the collection $[\{\Gamma(K),\upsilon_s(K)\}]$ is the same for both groups. [In previous formulae, $[\upsilon_s(R)]$ is often simply indicated as $[\upsilon_s]$ .]

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International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 907-955
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