11. 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]. 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.

11.1. 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], and an operation is the augmented matrix

(11.1)\[\begin{split}A(g) \;=\; \begin{pmatrix} R & 0 & \mathbf{v} \\ \mathbf{M} & \varepsilon & \delta \\ 0 & 0 & 1 \end{pmatrix}, \qquad \mathbf{q}R \;=\; \varepsilon\,\mathbf{q} + \mathbf{M},\end{split}\]

Source: rietx.crystallography.superspace.SuperspaceOperator

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,

(11.2)\[\delta_{ab} \;=\; \mathbf{M}_a\cdot\mathbf{v}_b + \varepsilon_a\,\delta_b + \delta_a \pmod 1,\]

Source: rietx.crystallography.superspace.SuperspaceOperator

which is the rule \(\tau_3 = \tau_1 + \varepsilon_1\tau_2\) of [Wolff et al., 1981] written in the lattice basis.

11.2. Generating the groups

Given a basic space group and the form of \(\mathbf{q}\), everything in (11.1) is fixed except \(\delta\), and closure under (11.2) 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]:

(11.3)\[\begin{split}S \;=\; \begin{pmatrix} P & 0 & \mathbf{p} \\ \mathbf{S}_M & S_\varepsilon & S_\delta \\ 0 & 0 & 1 \end{pmatrix}, \qquad g_2 \;=\; S\,g_1\,S^{-1}.\end{split}\]

Source: rietx.crystallography.superspace.equivalent

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] and [Stokes et al., 2011], and the test suite asserts both. Groups are compared as operator lists under (11.3), 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].

11.3. 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

(11.4)\[\tau \;=\; \delta - \mathbf{q}_r\cdot\mathbf{v},\]

Source: rietx.crystallography.superspace.symbol

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] § 5.2, and returns the ITC-C spelling for the eleven groups where that convention and ITC-C differ.

11.4. 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 \((\mathbf{l}, l_4)\),

(11.5)\[\delta + l_4 - \mathbf{q}\cdot(\mathbf{v} + \mathbf{l}) + (\varepsilon - 1)\,t_0 \;=\; 0,\]

Source: rietx.crystallography.superspace.SuperspaceGroup.section

and becomes the operation \(\{R \mid \mathbf{v} + \mathbf{l}\}\) of the supercell [Orlov et al., 2008]. 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].