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

International Tables for Crystallography (2006). Vol. C, ch. 9.8, pp. 941-943

Section 9.8.5. Generalizations

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

9.8.5. Generalizations

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The basic structure of a modulated crystal does not always have space-group symmetry. Consider, for example, composite crystals (also called intergrowth crystals). Disregarding modulations, one can describe these crystals as composed of a finite number of subsystems, each with its own space-group symmetry. The lattices of these subsystems can be mutually incommensurate. In that case, the overall symmetry is not a space group, the composite crystal is incommensurate and so also is its basic structure. The superspace approach can also be applied to such crystals. Let the subsystems be labelled by an index ν. For the subsystem ν, we denote the lattice by [\Lambda_\nu] with basis vectors [{\bf a}_{\nu i}] (i = 1, 2, 3), its reciprocal lattice by [\Lambda^*_\nu] with basis vectors [{\bf a}^*_{\nu i}] (i = 1, 2, 3), and the space group by [G_\nu]. The atomic positions of the basic structure are given by [{\bf n}_\nu+{\bf r}_{\nu j}, \eqno (]where [{\bf n}_\nu] is a lattice vector belonging to [\Lambda_\nu]. In the special case that the subsystems are mutually commensurate, there are three basis vectors a*, b*, c* such that all vectors [{\bf a}^*_{\nu i}] are integral linear combinations of them. In general, however, more than three basis vectors are needed, but never more than three times the number of subsystems. Suppose that the vectors [{\bf a}^*_i] [(i=1, \ldots, n)] form a basis set such that every [{\bf a}^*_{\nu i}] can be expressed as an integral linear combination of them: [{\bf a}^*_{\nu i}=\textstyle\sum\limits^n_{k=1}\,Z^\nu_{ik}{\bf a}^*_k,\quad Z^\nu_{ik}\hbox{ integers}, \eqno (]with [n=3+d_o] and [d_o\, \gt \, 0]. Then the vectors of the diffraction pattern of the unmodulated system are again of the form ([link] and generate a vector module [M^*_o] of dimension three and rank [(3+d_o)], which can be considered as projection of a [(3+d_o)]-dimensional lattice [\Sigma^*_o].

We now assume that one can choose [{\bf a}^*_{Ii}=0] for i = 1, 2, 3 and we denote [{\bf a}^*_{I3+j}] by [{\bf d}^*_j]. This corresponds to assuming the existence of a subset of Bragg reflections at the positions of a three-dimensional reciprocal lattice [\Lambda]*. Then there is a standard basis for the lattice [\Sigma_o], which is the reciprocal of [\Sigma^*_o], given by [({\bf a}_i,{\bf a}_{Ii}), \quad(0, {\bf d}_j), \quad i=1,2,3,\quad j=1,\ldots, d_o. \eqno (]In order to find the [(3+d_o)]-dimensional periodic structure for which this composite crystal is the three-dimensional intersection, one associates with a translation t in the internal space [V_I] three-dimensional independent shifts, one for each subsystem. These shifts of the subsystems replace the phase shifts adopted for the modulated structures: [V_I] is now the space of the variable relative positions of the subsystems. Again, a translation in the superspace can give rise to a non-Euclidean transformation in the three-dimensional space of the crystal, because of the variation in the relative positions among subsystems. Each subsystem, however, is rigidly translated. For the basis vectors [{\bf d}_j], the shift of the subsystem ν is defined in terms of projection operators [\pi_\nu]: [\pi_\nu{\bf d}_j=\textstyle\sum\limits^3_{i=1}\, Z^\nu_{i3+j}{\bf a}_{\nu i}, \quad j=1,\ldots, d_o. \eqno (]Then an arbitrary translation [{\bf t}=\sum_j t_j{\bf d}_j] in [V_I] displaces the subsystem ν over a vector [\sum_j\,t_j(\pi_\nu{\bf d}_j)]. A translation [({\bf a},{\bf a}_I+{\bf d})] belonging to the [(3+d_o)]-dimensional lattice [\Sigma_o] induces for the subsystem ν in ordinary space a relative translation over vector [{\bf a}+\pi_\nu({\bf a}_I+{\bf d})]. The statement is that this translation is a vector of the lattice [\Lambda_\nu] and leaves therefore the subsystem ν invariant. So the lattice translations belonging to [\Sigma_o] form a group of symmetry operations for the composite crystal as a whole.

The proof is as follows. If k belongs to [\Lambda^*_\nu], the vector [({\bf k},{\bf k}_I)] belongs to [\Sigma^*_o]. In particular, for [{\bf k}={\bf a}^*_{\nu i}], one has, because of ([link] and ([link], [{\bf a}^*_{\nu i}\cdot\pi_\nu{\bf d}_j=Z^\nu_{i3+j}, \quad j=1, \ldots, d_o, \eqno (]and [{\bf k}_I=\textstyle\sum\limits^{d_o}_{j=1}\,Z^\nu_{i3+j}{\bf d}^*_j\quad\hbox{and therefore }\quad {\bf k}_I\cdot{\bf d}_j=Z^\nu_{i 3+j}.]

Note that one has [{\bf k}_I\cdot {\bf t}={\bf k}\cdot\pi_\nu{\bf t}], for any t from [V_I] as [\pi_\nu] is a linear operator. Because of the linearity, this holds for every k from [\Lambda^*_\nu] as well. Since [({\bf k}, {\bf k}_I)] belongs to [\Sigma^*_o] and [({\bf a,a}_I+{\bf d})] to [\Sigma_o], one has for their inner product: [{\bf k\cdot a+k}_I\cdot{\bf a}_I+{\bf k}_I\cdot {\bf d}={\bf k}\cdot({\bf a}+\pi_\nu{\bf a}_I+\pi_\nu{\bf d})\equiv 0\eqno \hbox{(modulo 1),}]which implies that [{\bf a}+\pi_\nu{\bf a}_I+\pi_\nu{\bf d}] is an element of [\Lambda_\nu].

In conclusion, one may state that the composite structure is the intersection with the ordinary space (t = 0) of a pattern having atomic position vectors given by [({\bf n}_\nu+{\bf r}_{\nu j}-\pi_\nu{\bf t}, {\bf t})\quad\hbox{for any {\bf t} of }V_I. \eqno (]Such a pattern is invariant under the [(3+d_o)]-dimensional lattice [\Sigma_o]. Again, orthogonal transformations R of O(3) leaving the vector module [M^*_o] invariant can be extended to orthogonal transformation [R_s] of [O(3)\times O(d_o)] allowing a Euclidean structure to be given to the superspace. One can then consider the superspace-group symmetry of the basic structure defined by atomic positions as in ([link]. A superspace-group element [g_s] as in ([link] induces (in three-dimensional space) for the subsystem ν the transformation [g_s:{\bf n}_\nu+{\bf r}_{\nu j}\rightarrow R{\bf n}_\nu+R{\bf r}_{\nu j} +{\bf v}+R\pi_{\nu}R^{-1}_I{\bf v}_I, \eqno (]changing the position [{\bf n}_\nu+{\bf r}_{\nu j}] into an equivalent one of the composite structure, not necessarily, however, within the same subsystem ν.

Finally, the composite structure can also be modulated. For the case of a one-dimensional modulation of each subsystem ν, the positions are [{\bf n}_\nu+{\bf r}_{\nu j}+{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j})]. \eqno (]Possibly the modulation vectors can also be expressed as integral linear combinations of the [{\bf a}^*_i] [(i=1,\ldots,3+d_o)]. Then, the dimension of [V_I] is again [d_o]. In general, however, one has to consider [(d-d_o)] additional vectors, in order to ensure the validity of ([link], now for n = 3 + d. We can then write [{\bf q}_\nu=\textstyle\sum\limits^{3+d}_{j=1}\, Q^\nu_j{\bf a}^*_j,\quad Q^\nu_j\hbox{ integers}. \eqno (]The peaks of the diffraction pattern are at positions defined by a vector module M*, which can be considered as the projection of a (3 + d)-dimensional lattice [\Sigma]*, the reciprocal of which leaves invariant the pattern of the modulated atomic positions in the superspace given by [\eqalignno{ \{{\bf n}_\nu+r_{\nu j}-\pi_\nu{\bf t}+{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j}-\pi_\nu{\bf t})+{\bf q}_{I\nu}&\cdot{\bf t}], {\bf t}\}, \cr \hbox{ for any {\bf t} of }V_I& &(}]with [\pi_\nu{\bf d}_j=0] for [j \, > \, d_o], where [{\bf q}_{I\nu}] is the internal part of the (3 + d)-dimensional vector that projects on [{\bf q}_{\nu}]. Their symmetry is a (3 + d)-dimensional superspace group [G_s]. The transformation induced in the modulated composite crystal by an element under [g_s] of [G_s] is now readily written down. For the case [d=d_o=1] and [g_s] = ({R|v}, {[epsilon]|Δ}) , the position [{\bf n}_\nu+{\bf r}_{\nu j}] is transformed into [R({\bf n}_\nu+ {\bf r}_{\nu j})+{\bf v}+\varepsilon R\pi_\nu \Delta{\bf d}_1, \eqno (]and the modulation [{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j})]] into [R{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j}+\varepsilon\pi_\nu\Delta{\bf d}_1)-\varepsilon{\bf q}_{I\nu}\cdot\Delta{\bf d}_1].]

This shows how the superspace-group approach can be applied to a composite (modulated) structure. Note that composite systems do not necessarily have an invariant set of (main) reflections at lattice positions. The incommensurate versus the commensurate case

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As said earlier, it sometimes makes sense also to use the description as developed for incommensurate crystal phases for a (commensurate) superstructure. In fact, very often the modulation wavevector also shows, in addition to continuously varying (incommensurate) values, several rational values at various phase transitions of a given crystal or in different compounds of a given structural family. In these cases, there is three-dimensional space-group symmetry. Generally, the space groups of the various phases are different. The description as used for incommensurate phases then gives the possibility of a more unified characterization for the symmetry of related modulated crystal phases. In fact, the theory of higher-dimensional space groups for modulated structures is largely independent of the assumption of irrationality. Only some of the statements need to be adapted. The most important change is that there is no longer a one-to-one correspondence between the points of the reciprocal lattice [\Sigma]* and its projection on V defining the positions of the Bragg peaks. Furthermore, the projection of the lattice [\Sigma] on the space [V_I] forms a discrete set. The latter means the following. For an incommensurate modulation, the incommensurate structure, which is the intersection of a periodic structure with the hyperplane [{\bf r}_I=0], is also the intersection of the same periodic structure with a hyperplane [{\bf r}_I] = constant , where this constant is of the form [\textstyle\sum\limits^3_{i=1}h_i{\bf a}_{Ii}+\textstyle\sum\limits^d_{j=1}m_j{\bf a}_{I3+j}. \eqno (]Because for an incommensurate structure these vectors form a dense set in [V_I], the phase of the modulation function with respect to the basic structure is not determined. For a commensurate modulation, however, the points ([link] form a discrete set, even belong to a lattice, and the phase (or the phases) of the modulation are determined within vectors of this lattice. Notice that the grid of this lattice becomes finer as the denominators in the rational components become larger.

When [G_s] is a (3 + d)-dimensional superspace group, its elements, in general, do not leave the ordinary space V invariant. The subgroup of all elements that do leave V invariant, when restricted to V, is a group of distance-preserving transformations in three dimensions and thus a subgroup of E(3), the three-dimensional Euclidean group. In general, this subgroup is not a three-dimensional space group. It is so when the modulation wavevectors all have rational components only, i.e. when σ is a matrix with rational entries. Because the phase of the modulation function is now determined (within a given rational number smaller than 1), the space group depends in general on this phase.

As an example, consider a one-dimensional modulation of a basic structure with orthorhombic space group Pcmn. Suppose that the modulation wavevector is γc*. Then the mirror [R=m_z] perpendicular to the c axis is combined with [R_I] = [epsilon] = −1. Suppose, furthermore, that the glide reflection perpendicular to the a axis and the b mirror are both combined with a phase shift [1\over2]. In terms of the coordinates x, y, z with respect to the a, b and c axes, and internal coordinate t, the generators of the (3 + 1)-dimensional superspace group Pcmn(00γ)ss0 act as [\eqalignno{&(x,y,z,t)\rightarrow(x+k, y+l, z+m, t-\gamma m+n),&\cr &\qquad\qquad\qquad\qquad k,l,m,n\hbox{ integers},&(}] [\eqalignno{& (x,y,z,t)&\cr &\quad \rightarrow(-x+k+\textstyle{1\over2}, y+l, z+{1\over2}+m, t-\gamma/2-\gamma m+\textstyle{1\over2}+n),&\cr &&(}] [(x,y,z,t)\rightarrow(x+k, -y+l+\textstyle{1\over2}, z+m, t-\gamma m+{1\over2}+n), \eqno (] [\eqalignno{& (x,y,z,t)&\cr&\quad \rightarrow(x+\textstyle{1\over2}+k, y+{1\over2}+l,-z+{1\over2}+m, -t-\gamma/2-\gamma m+n).&\cr &&(}]Note that these positions are referred to a split basis (i.e. of basis vectors lying either in V or in [V_I]) and not to a basis of the lattice [\Sigma]. When the superstructure is the intersection of a periodic structure with the plane at [t=t_o], its three-dimensional space group follows from equation ([link] by the requirement [t'=t_o]. When γ has the rational value r/s with r and s relatively prime, the conditions for each of the generators to give an element of the three-dimensional space group are, respectively: [\eqalignno{ -rm+sn&=0&( \cr-2rm+2sn&=r-s&( \cr -2rm+2sn&=-s&( \cr -2rm+2sn&=4st, &(}%fd9.8.5.14b9.8.5.14c9.8.5.14d]for m, n, r, s integers and t real. These conditions are never satisfied simultaneously. It depends on the parity of both r and s which element occurs, and for the elements with [epsilon] = −1 it also depends on the value of the `phase' t, or more precisely on the product τ = 4st. The translation group is determined by the first condition as in ([link]). Its generators are [{\bf a}, {\bf b}, \hbox{and }s{\bf c},]where the last vector is the external part of the lattice vector s(c, −r/s) + r(0, 1). The other space-group elements can be derived in the same way. The possible space groups are:

γ = r/sτ even integerτ odd integerotherwise
r even, s odd [11\displaystyle{2_1\over n}] 212121 1121
r odd, s even [ 1\displaystyle{2_1\over c}1] 21cn 1c1
r odd, s odd [\displaystyle{2_1\over c}11] c21n c11

In general, the three-dimensional space groups compatible with a given (3 + d)-dimensional superspace group can be determined using analogous equations.

As one can see from the table above, the orthorhombic (3 + d)-dimensional superspace group leads in several cases to monoclinic three-dimensional space groups. The lattice of main reflections, however, still has orthorhombic point-group symmetry. Description in the conventional way by means of three-dimensional groups then neglects some of the structural features present. Even if the orthorhombic symmetry is slightly broken, the orthorhombic basic structure is a better characterization than a monoclinic one. Note that in that case the superspace-group symmetry is also only an approximation.

When the denominators of the wavevector components become small, additional symmetry operations become possible. Because the one-to-one correspondence between [\Sigma]* and M* is no longer present, there may occur symmetry elements with trivial action in V but with nontrivial transformation in [V_I]. For d = 1, these possibilities have been enumerated. The corresponding Bravais classes are given in Table[link](b).

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