# Superspace groups for one-dimensional modulations A crystal whose atoms are displaced, or whose sites are occupied, by a wave of wave vector $\mathbf{q}$ that is not a reciprocal-lattice vector has no three-dimensional translation symmetry left. It is periodic in a four-dimensional superspace instead, and its symmetry is a (3+1)D superspace group [[Wolff *et al.*, 1981](references.html.md#id140)]. This chapter is the operator algebra `rietx` builds those groups with. It is the layer every later part of a modulated model rests on (the allowed waves at a site, the satellite absences, the structure factor), and none of those later parts is built yet: what is here is the groups, how two of them are compared, their one-line symbols, and the three-dimensional group of a commensurate section. ## The operators The coordinates are three external ones, $\mathbf{x}$, and one internal one, $x_4 = \mathbf{q}\cdot\mathbf{x} + t$, the argument of the modulation function in the section $t$. In this basis the superspace lattice is $\mathbb{Z}^4$ [[Stokes *et al.*, 2011](references.html.md#id142)], and an operation is the augmented matrix $$ A(g) \;=\; \begin{pmatrix} R & 0 & \mathbf{v} \\ \mathbf{M} & \varepsilon & \delta \\ 0 & 0 & 1 \end{pmatrix}, \qquad \mathbf{q}R \;=\; \varepsilon\,\mathbf{q} + \mathbf{M}, $$ *Source:* [`rietx.crystallography.superspace.SuperspaceOperator`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/superspace/operators.py#L170) with $R$ and $\mathbf{v}$ the operation of the basic space group, $\varepsilon = \pm 1$ its action on the internal axis, $\mathbf{M}$ an integer row fixed by $\mathbf{q}$, and $\delta$ the internal translation. The second equation is the compatibility condition: an incommensurate $\mathbf{q}$ is mapped to $\pm\mathbf{q}$ up to a reciprocal-lattice vector, which forces its irrational part onto a subspace on which every point operation acts as a sign. In the section phase the operation reads $t' = \varepsilon t + \delta - \mathbf{q}\cdot\mathbf{v}$. Composition is the matrix product, and its only non-trivial entry is the internal translation of the product, $$ \delta_{ab} \;=\; \mathbf{M}_a\cdot\mathbf{v}_b + \varepsilon_a\,\delta_b + \delta_a \pmod 1, $$ *Source:* [`rietx.crystallography.superspace.SuperspaceOperator`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/superspace/operators.py#L170) which is the rule $\tau_3 = \tau_1 + \varepsilon_1\tau_2$ of [[Wolff *et al.*, 1981](references.html.md#id140)] written in the lattice basis. ## Generating the groups Given a basic space group and the form of $\mathbf{q}$, everything in [(11.1)](#equation-ss-operator) is fixed except $\delta$, and closure under [(11.2)](#equation-ss-composition) is a set of linear congruences modulo 1 in the unknown $\delta$ of each coset representative. Its real solutions are the shifts of the internal origin alone, which move every $\varepsilon = -1$ operation by the same amount and can be fixed by one equation; what is left is a finite set, solved exactly by integer row reduction. Two solutions describe the same group when an affine map of superspace that normalises the basic group, together with $\mathbf{q} \to \pm\mathbf{q} + \mathbf{H}$ for a reciprocal-lattice vector $\mathbf{H}$, takes one onto the other [[Smaalen *et al.*, 2013](references.html.md#id143)]: $$ S \;=\; \begin{pmatrix} P & 0 & \mathbf{p} \\ \mathbf{S}_M & S_\varepsilon & S_\delta \\ 0 & 0 & 1 \end{pmatrix}, \qquad g_2 \;=\; S\,g_1\,S^{-1}. $$ *Source:* [`rietx.crystallography.superspace.equivalent`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/superspace/generation.py#L1054) The generator counts 775 inequivalent groups over the 230 basic groups and 24 Bravais classes of (3+1)D lattices, the numbers of [[Yamamoto *et al.*, 1985](references.html.md#id141)] and [[Stokes *et al.*, 2011](references.html.md#id142)], and the test suite asserts both. Groups are compared as operator lists under [(11.3)](#equation-ss-equivalence), never as symbols: a symbol is unique only together with $\mathbf{q}$ and the reflection conditions, and $\mathbf{q} \to \mathbf{c}^* - \mathbf{q}$ turns the bottom line of one group into the printed bottom line of an inequivalent one [[Yamamoto *et al.*, 1985](references.html.md#id141)]. ## Symbols A one-line symbol is the basic group’s symbol, $\mathbf{q}$, and one letter per generator of that symbol, the generator’s internal intrinsic translation $$ \tau \;=\; \delta - \mathbf{q}_r\cdot\mathbf{v}, $$ *Source:* [`rietx.crystallography.superspace.symbol`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/superspace/symbols.py#L565) written 0, s, t, q or h for 0, 1/2, 1/3, 1/4 or 1/6, with $\mathbf{q}_r$ the rational part of $\mathbf{q}$. An $\varepsilon = -1$ generator always prints 0. When $\mathbf{q}_r \neq 0$, adding a lattice translation to a generator changes $\tau$, so a convention picks one: the printer follows the four steps of [[Stokes *et al.*, 2011](references.html.md#id142)] § 5.2, and returns the ITC-C spelling for the eleven groups where that convention and ITC-C differ. ## Commensurate sections At a rational $\mathbf{q}$ the structure is a superstructure, and which three-dimensional space group it has depends on the section phase $t_0$. An operation survives in the section when, for some lattice vector $(\mathbf{l}, l_4)$, $$ \delta + l_4 - \mathbf{q}\cdot(\mathbf{v} + \mathbf{l}) + (\varepsilon - 1)\,t_0 \;=\; 0, $$ *Source:* [`rietx.crystallography.superspace.SuperspaceGroup.section`](https://github.com/yue-here/rietx/blob/v1.10.0/src/rietx/crystallography/superspace/operators.py#L750) and becomes the operation $\{R \mid \mathbf{v} + \mathbf{l}\}$ of the supercell [[Orlov *et al.*, 2008](references.html.md#id144)]. An $\varepsilon = +1$ operation survives at every $t_0$ or at none; an $\varepsilon = -1$ one only at a discrete set of sections, so at rational $\mathbf{q}$ the phase $t_0$ is a parameter of the model. The test suite reproduces every cell of the two worked tables of [[Orlov *et al.*, 2008](references.html.md#id144)].