The basic ideas of higher-dimensional crystallography

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

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International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 908-909

Section 9.8.1.2. The basic ideas of higher-dimensional crystallography

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

9.8.1.2. The basic ideas of higher-dimensional crystallography

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Incommensurate modulated crystals are systems that do not obey the classical requirements for crystals. Nevertheless, their long-range order is as perfect as that of ordinary crystals. In the diffraction pattern they also show sharp, well separated spots, and in the morphology flat faces. Dendritic crystallization with typical point-group symmetry is observed in both commensurate and incommensurate materials. Therefore, we shall consider both as crystalline phases and generalize for that reason the concept of a crystal. The positions of the Bragg diffraction peaks given in (9.8.1.1) are a special case. In general, they are elements of a vector module M* and can be written as $[{\bf H}=\textstyle\sum\limits^n_{i=1}h_i{\bf a}^*_i,\quad\hbox{integers $h_i$.} \eqno (9.8.1.2)]$ This leads to the following definition of crystal.

An ideal crystal is considered to be a matter distribution having Fourier wavevectors expressible as an integral linear combination of a finite number (say n) of them and such that its diffraction pattern is characterized by a discrete set of resolved Bragg peaks, which can be indexed accordingly by a set of n integers $[h_1,h_2,\ldots,h_n]$ .

Implicit in this definition is the possibility of neglecting diffraction intensities below a given threshold, allowing one to identify and to label individual Bragg peaks even when n is larger than the dimension m of the crystal, which is usually three. Actually, for incommensurate crystals, the unresolved Bragg peaks of arbitrarily small intensities form a dense set, because they may come arbitrarily close to each other.

Here some typical examples are indicated. In the normal crystal case, n = m = 3 and $[{\bf a}^*_1]$ , $[{\bf a}^*_2]$ , $[{\bf a}^*_3]$ are conventionally denoted by a*, b*, c* and the indices [h_1, h_2, h_3] by h, k, l.

In the case of a one-dimensionally modulated crystal, which can be described as a periodic plane wave deformation of a normal crystal (defining the basic structure), one has n = 4. Conventionally, $[{\bf a}^*_1]$ , $[{\bf a}^*_2]$ , $[{\bf a}^*_3]$ are chosen to be a*, b*, and c* generating the positions of the main reflections, whereas $[{\bf a}^*_4]$ = q is the wavevector of the modulation. The corresponding indices are usually denoted by h, k, l and m. Also, crystals having two- and three-dimensional modulation are known: in those cases, n = 5 and n = 6, respectively.

In the case of a composite crystal, one can identify two or more subsystems, each with its own space-group symmetry. As an example, consider the case where two subsystems share a* and b* of their reciprocal-lattice basis, whereas they differ in periodicity along the c axis and have, respectively, $[{\bf c}^*_1]$ and $[{\bf c}^*_2]$ as third basis vector. Then again, n = 4 and $[{\bf a}^*_1,\ldots, {\bf a}^*_4]$ can be chosen as $[{\bf a}*]$ , $[{\bf b}*]$ , $[{\bf c}^*_1]$ , and $[{\bf c}^*_2]$ . The indices can be denoted by h, k, l₁, l₂. In general, the subsystems interact, giving rise to modulations and possibly (but not necessarily) to a larger value of n. In addition to the main reflections from the undistorted subsystems, satellite reflections then occur.

In the case of a quasicrystal, the Bragg reflections require more than three indices, but they do not arise from structurally different subsystems having lattice periodicity. So, for the icosahedral phase of AlMn alloy, n = 6, and one can take for $[{\bf a}^*_1]$ , $[{\bf a}^*_2]$ , $[{\bf a}^*_3]$ a cubic basis a*, b*, and c* and for $[{\bf a}^*_4]$ , $[{\bf a}^*_5]$ , $[{\bf a}^*_6]$ then τa*, τb*, and τc*, respectively, with τ the golden number $[[1+\sqrt5]/2]$ .

The Laue point group [P_L] of a crystal is the point symmetry group of its diffraction pattern. This subgroup of the orthogonal group O(3) is of finite order. The finite order of the group follows from the discreteness of the (resolved) Bragg peak positions, implying a finite number only of peaks of the same intensity lying at a given distance from the origin.

Under a symmetry rotation R, the indices of each reflection are transformed into those of the reflection at the rotated position. Therefore, a symmetry rotation is represented by an $[n\times n]$ non-singular matrix Γ(R) with integral entries. Accordingly, a Laue point group admits an n-dimensional faithful integral representation $[\Gamma(P_L)]$ . Because any finite group of matrices is equivalent with a group of orthogonal matrices of the same dimension, it follows that:

(1) is isomorphic to an n-dimensional crystallographic point group;
(2) there exists a lattice basis $[a^*_{s1}, \ldots, a^*_{sn}]$ of a Euclidean n-dimensional (reciprocal) space, which projects on the Fourier wavevectors $[{\bf a}^*_1]$ , $[\ldots, {\bf a}^*_n]$ ;
(3) the three-dimensional Fourier components $[\hat\rho(h_1,\ldots, h_n)]$ of the crystal density function ρ(r) can be attached to corresponding points of an n-dimensional reciprocal lattice and considered as the Fourier components of a density function $[\rho_s(r_s)]$ having lattice periodicity in that higher-dimensional space ( is an n-dimensional position vector).

Such a procedure is called the superspace embedding of the crystal.

Note that this procedure only involves a reinterpretation of the structural data (expressed in terms of Fourier coefficients): the structural information in the n-dimensional space is exactly the same as that in the three-dimensional description.

The symmetry group of a crystal is then defined as the Euclidean symmetry group of the crystal structure embedded in the superspace.

Accordingly, the symmetry of a crystal whose diffraction pattern is labelled by n integral indices is an n-dimensional space group.

The equivalence relation of these symmetry groups follows from the requirement of invariance of the equivalence class with respect to the various possible choices of bases and embeddings. Because of that, the equivalence relation is not simply the one valid for n-dimensional crystallography.

In the case of modulated crystals, such an equivalence relation has been worked out explicitly, giving rise to the concept of the (3 + d)-dimensional superspace group.

The concepts of point group, lattice holohedry, Bravais classes, systems of non-primitive translations, and so on, then follow from the general properties of n-dimensional space groups together with the (appropriate) equivalence relations.

A glossary of symbols is given in Appendix 9.8.1 and a list of definitions in Appendix 9.8.2.

References

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 908-909