Incommensurate composite crystal structures

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 941-942

Section 9.8.5.1. Incommensurate composite crystal structures

T. Janssen,^a A. Janner,^a A. Looijenga-Vos^b and P. M. de Wolff^c ^‡

^a Institute 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

9.8.5.1. Incommensurate composite crystal structures

| top | pdf |

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 $[{\bf n}_\nu+{\bf r}_{\nu j}, \eqno (9.8.5.1)]$ 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: $[{\bf a}^*_{\nu i}=\textstyle\sum\limits^n_{k=1}\,Z^\nu_{ik}{\bf a}^*_k,\quad Z^\nu_{ik}\hbox{ integers}, \eqno (9.8.5.2)]$ 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 (9.8.4.5) and generate a vector module [M^*_o] of dimension three and rank [(3+d_o)] , which can be considered as projection of a -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 $[({\bf a}_i,{\bf a}_{Ii}), \quad(0, {\bf d}_j), \quad i=1,2,3,\quad j=1,\ldots, d_o. \eqno (9.8.5.3)]$ 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 $[\pi_\nu]$ : $[\pi_\nu{\bf d}_j=\textstyle\sum\limits^3_{i=1}\, Z^\nu_{i3+j}{\bf a}_{\nu i}, \quad j=1,\ldots, d_o. \eqno (9.8.5.4)]$ 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 (9.8.5.2) and (9.8.5.4), $[{\bf a}^*_{\nu i}\cdot\pi_\nu{\bf d}_j=Z^\nu_{i3+j}, \quad j=1, \ldots, d_o, \eqno (9.8.5.5)]$ 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 $[({\bf n}_\nu+{\bf r}_{\nu j}-\pi_\nu{\bf t}, {\bf t})\quad\hbox{for any {\bf t} of }V_I. \eqno (9.8.5.6)]$ 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 (9.8.5.6). A superspace-group element [g_s] as in (9.8.4.28) induces (in three-dimensional space) for the subsystem ν the transformation $[g_s:{\bf n}_\nu+{\bf r}_{\nu j}\rightarrow R{\bf n}_\nu+R{\bf r}_{\nu j} +{\bf v}+R\pi_{\nu}R^{-1}_I{\bf v}_I, \eqno (9.8.5.7)]$ 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 $[{\bf n}_\nu+{\bf r}_{\nu j}+{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j})]. \eqno (9.8.5.8)]$ 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 (9.8.4.5), now for n = 3 + d. We can then write $[{\bf q}_\nu=\textstyle\sum\limits^{3+d}_{j=1}\, Q^\nu_j{\bf a}^*_j,\quad Q^\nu_j\hbox{ integers}. \eqno (9.8.5.9)]$ 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 $[\eqalignno{ \{{\bf n}_\nu+r_{\nu j}-\pi_\nu{\bf t}+{\bf u}_{\nu j}[{\bf q}_\nu\cdot({\bf n}_\nu+{\bf r}_{\nu j}-\pi_\nu{\bf t})+{\bf q}_{I\nu}&\cdot{\bf t}], {\bf t}\}, \cr \hbox{ for any {\bf t} of }V_I& &(9.8.5.10)}]$ 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}, {ɛ|Δ}) , the position $[{\bf n}_\nu+{\bf r}_{\nu j}]$ is transformed into $[R({\bf n}_\nu+ {\bf r}_{\nu j})+{\bf v}+\varepsilon R\pi_\nu \Delta{\bf d}_1, \eqno (9.8.5.11)]$ 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.

References

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 941-942