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 [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*.
[V_s] 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 [V_E].
[V_I] 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 [V_s] [(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 [V_s] 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.
qj 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).
[H_s] Embedding of H in [V_s]: for [{\bf H}=(h_1,\ldots,h_n) =] [\sum^n_{i=1}h_i{\bf a}^*_i], one has correspondingly [H_s=] [({\bf H},{\bf H}_I)=] [\sum^n_{i=1}h_ia^*_{si}].
[P_L] 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).
[R_s] Superspace point-group element: [R_s=(R_E,R_I)=] [(R,R_I)] element of O(m) × O(d) with [R_E=R] external, and [R_I] internal part of [R_s], respectively; if n = 4, superspace point-group element: [R, [epsilon](R)] with [epsilon](R) = ±1, also written (R, [epsilon]).
[K_s] Point group, crystallographic subgroup of O(m) × O(d).
[K_E] External part of [K_s], crystallographic point group, subgroup of O(m) with as elements the external part transformations of [K_s].
[K_I] Internal part of [K_s], crystallographic point group, subgroup of O(d) with as elements the internal part transformations of [K_s].
[{\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)].
[p_j(x)] Modulation function for occupation modulation with [p_j(x+1)=p_j(x)].
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).
[g_s] 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 [G_s]. 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 [R_s] 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.








































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