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

International Tables for Crystallography (2006). Vol. C, ch. 9.8, pp. 943-944
https://doi.org/10.1107/97809553602060000624

## Appendix A9.8.1. Glossary of symbols

T. Janssen,a A. Janner,a A. Looijenga-Vosb and P. M. de Wolffc

aInstitute 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

 M* Vector module in m-dimensional reciprocal space (m = 1, 2, 3; normally m = 3), isomorphic to Zn with . The dimension of M* is m, its rank 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 Physical (or external) space of dimension m (m = 1,2 or 3), also indicated by . Internal (or additional) space of dimension d. Reciprocal lattice in n-dimensional space, whose orthogonal projection on V is M*. Lattice in n-dimensional superspace for which * is the reciprocal one. Lattice basis of * in ; 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), . () Lattice basis of in dual to ; if n = 4, the standard basis is , , (0, 1) = (0, d); for n = 3 + d, a standard basis is dual to the standard one given above. qj Modulation wavevector(s) ; if n = 4, σ = (α, β, γ); if n = 4, , with , where , and N is the order of K. H Bragg reflections: ; if n = 4, (h, k, l, m). Embedding of H in : for , one has correspondingly . 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, (R)] with (R) = ±1, also written (R, ). 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 . Atomic positions in the basic structure: with . r(n, j) Atomic positions in the displacively modulated structure; (d = 1): r(n, j) = . In general, however, different phases may occur for different components along the crystallographic axes. Modulation function for displacive modulation with . 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. Intrinsic translation part (origin independent). Superspace group transformation (d = 1): element of the superspace group . In the (3 + d)-dimensional case: . Internal shift (d = 1): . τ Intrinsic internal shift (d = 1): τ = δ − . 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 . Γ(R) Superspace point-group transformation with respect to a lattice basis of dual to that of leading to . The mutual relation is then with the tilde denoting transposition. external, internal, and mixed blocks of Γ(R), respectively. external, internal, and mixed blocks of , respectively. Structure factor: Atomic scattering factor for atom j.