M* 
Vector module in mdimensional reciprocal space (m = 1, 2, 3; normally m = 3), isomorphic to Z^{n} 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, mdimensional reciprocal lattice. 
a*, b*, c* 
(Conventional) basis of Λ* for m = 3. 
Λ 
Direct mdimensional 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 ndimensional space, whose orthogonal projection on V is M*. 

Lattice in ndimensional 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. 
q_{j} 
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 pointgroup transformation, element of O(m). 
K 
Point group, crystallographic subgroup of O(m). 

Superspace pointgroup element: element of O(m) × O(d) with external, and internal part of , respectively; if n = 4, superspace pointgroup 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 = {Rv} 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 − . 

Pointgroup transformation R with respect to a basis of M* and at the same time superspace pointgroup transformation with respect to a corresponding basis of . 
Γ(R) 
Superspace pointgroup 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. 