Types of quasiperiodic crystals

Janssen, T.

doi:10.1107/97809553602060000637

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.10, pp. 243-244

Section 1.10.1.2. Types of quasiperiodic crystals

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.10.1.2. Types of quasiperiodic crystals

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One may distinguish various families of quasiperiodic systems. [Sometimes these are also called incommensurate systems (Janssen & Janner, 1987).] It is not a strict classification, because one may have intermediate cases belonging to more than one family as well. Here we shall consider a number of pure cases.

An incommensurately modulated structure or incommensurate crystal (IC) phase is a periodically modified structure that without the modification would be lattice periodic. Hence there is a basic structure with space-group symmetry. The periodicity of the modification should be incommensurate with respect to the basic structure. The position of the jth atom in the unit cell with origin at the lattice point $[{\bf n}]$ is $[{\bf n} + {\bf r}_{j}]$ ( $[j = 1, 2, \ldots, s]$ ).

For a displacive modulation, the positions of the atoms are shifted from a lattice periodic basic structure. A simple example is a structure that can be derived from the positions of the basic structure with a simple displacement wave. The positions of the atoms in the IC phase are then $[{\bf n} + {\bf r}_{j} + {\bf f}_{j}({\bf Q}\cdot{\bf n})\quad [{\bf f}_{j}(x) = {\bf f}_{j}(x + 1)]. \eqno(1.10.1.2)]$ Here the modulation wavevector $[{\bf Q}]$ has irrational components with respect to the reciprocal lattice of the basic structure. One has $[{\bf Q} = \alpha{\bf a}^{*} + \beta{\bf b}^{*} + \gamma{\bf c}^{*},\eqno(1.10.1.3)]$ where at least one of α, β or γ is irrational. A simple example is the function $[{\bf f}_{j}(x) = {\bf A}_{j}\cos(2\pi{x} + \varphi_{j})]$ , where $[{\bf A}_{j}]$ is the polarization vector and $[\varphi_{j}]$ is the phase of the modulation. The diffraction pattern of the structure (1.10.1.2) shows spots at positions $[{\bf H} = h_{1}{\bf a}^{*} + h_{2}{\bf b}^{*} + h_{3}{\bf c}^{*} + h_{4}{\bf Q}. \eqno(1.10.1.4)]$ Therefore, the rank is four and $[{\bf a}_{4}^{*} = {\bf Q}]$ . In a more general situation, the components of the atom positions in the IC phase are given by $[{\bf n}^{\alpha} + {\bf r}_{j}^{\alpha} + \textstyle\sum\limits_{m}{\bf A}_{j}^{\alpha} ({\bf Q}_{m})\cos(2\pi {\bf Q}_{m}\cdot{\bf n} + \varphi_{jm\alpha}), \quad\alpha = x, y, z. \eqno(1.10.1.5)]$ Here the vectors $[{\bf Q}_{m}]$ belong to the Fourier module of the structure. Then there are vectors $[{\bf Q}_{j}]$ such that any spot in the diffraction pattern can be written as $[{\bf H} = \textstyle\sum\limits_{i=1}^{3}h_{i}{\bf a}_{i}^{*} + \textstyle\sum\limits_{j=1}^{d} h_{3+j}{\bf Q}_{j}\eqno(1.10.1.6)]$ and the rank is [3+d] . The peaks corresponding to the basic structure [the combinations of the three reciprocal-lattice vectors $[{\bf a}_{i}^*]$ ( [i= 1, 2, 3] )] are called the main reflections, the other peaks are satellites. For the latter, at least one of the $[h_{4},\ldots, h_{n}]$ is different from zero.

A second type of modulation is the occupation or composition modulation. Here the structure can again be described on the basis of a basic structure with space-group symmetry. The basic structure positions are occupied with a certain probability by different atom species, or by molecules in different orientations. In CuAu(II), the two lattice positions in a b.c.c. structure are occupied by either Cu and Au or by Au and Cu with a certain probability. This probability function is periodic in one direction with a period that is not a multiple of the lattice constant. In NaNO₂, the NO₂ molecules are situated at the centre of the orthorhombic unit cell. There are two possible orientations for the V-shaped molecule, and the probability for one of the orientations is a periodic function with periodicity along the a axis. In this case, the modulation wavevector $[\alpha{\bf a}^{*}]$ has a component $[\alpha]$ that strongly depends on temperature in a very narrow temperature range.

If the probability of finding species A in position $[{\bf n} + {\bf r}_{j}]$ or of finding one orientation of a molecule in that point is given by $[P_{j}({\bf Q}\cdot{\bf n})]$ , the probability for species B or the other orientation is of course $[1-P_{j}({\bf Q}\cdot{\bf n})]$ . In the diffraction pattern, the spots belong to the Fourier module with basic vectors $[{\bf a}^{*}]$ , $[{\bf b}^{*}]$ , $[{\bf c}^{*}]$ and $[{\bf Q}]$ . The analogous expression for a more general situation with more modulation wavevectors, or with more species or orientations, is a straightforward generalization.

The first examples of IC phases were found in magnetic systems (see Section 1.5.1.2.3 ). For example, holmium has a spiral spin arrangement with a periodicity of the spiral that does not fit with the underlying lattice. For the $[\alpha]$ component ( $[\alpha = x,y,z]$ ) of the magnetic moment at position $[{\bf n}+{\bf r}_{j}]$ one has in an incommensurate magnetic system a superposition of waves $[S_{\alpha}({\bf n}j) = \textstyle\sum\limits_{m}M_{m\alpha j}\cos (2\pi {\bf Q}_{m}\cdot{\bf n} + \varphi_{m\alpha}).\eqno(1.10.1.7)]$ The most general expression is $[S_{\alpha}({\bf n}j) = \textstyle\sum\limits_{{\bf H}\in M^*}M_{\alpha j}({\bf H})\exp (i{\bf H}\cdot{\bf n}),\eqno(1.10.1.8)]$ where [M^*] is the Fourier module (1.10.1.1).

A following class of quasiperiodic materials is formed by incommensurate composite structures. To this belong misfit structures, intercalates and incommensurate adsorbed layers. An example is Hg_3−xAsF₆. This consists of a subsystem of AsF₆ octahedra forming a (modulated) tetragonal system and two other subsystems consisting of Hg chains, one system of chains in the x direction and one in the y direction. Because the average spacing between the Hg atoms is irrational with respect to the lattice constant of the host AsF₆ system in the same direction, the total structure does not have lattice periodicity in the a or b direction.

In general, there are two or more subsystems, labelled by $[\nu]$ , and the atomic positions are given by $[{\bf n}_{\nu} + {\bf r}_{\nu j} + {\rm modulation},\eqno(1.10.1.9)]$ where $[{\bf n}_{\nu}]$ belongs to the $[\nu]$ th lattice, and where the modulation is a quasiperiodic displacement from the basic structure. The diffraction pattern has wavevectors $[{\bf H} = \textstyle\sum\limits_{\nu}\textstyle\sum\limits_{i_{\nu}=1}^{3}h_{i_{\nu}}{\bf a}_{\nu i_{\nu}}^{*} = \textstyle\sum\limits_{i=1}^{n}h_{i}{\bf a}_{i}. \eqno(1.10.1.10)]$ Each of the reciprocal-lattice vectors $[{\bf a}_{\nu j}^{*}]$ belongs to the Fourier module [M^*] and can be expressed as a linear combination with integer coefficients of the n basis vectors $[{\bf a}_{i}^{*}]$ .

Very often, composite structures consist of a host system in the channels of which another material diffuses with a different, and incommensurate, lattice constant. Examples are layer systems in which foreign atoms intercalate. Another type of structure that belongs to this class is formed by adsorbed monolayers, for example a noble gas on a substrate of graphite. If the natural lattice constant of the adsorbed material is incommensurate with the lattice constant of the substrate, the layer as a whole will be quasiperiodic.

In general, the subsystems can not exist as such. They form idealized lattice periodic structures. Because of the interaction between the subsystems the latter will, generally, become modulated, and even incommensurately modulated because of the mutual incommensurability of the subsystems. The displacive modulation will, generally, contain wavevectors that belong to the Fourier module (1.10.1.10). However, in principle, additional satellites may occur due to other mechanisms, and this increases the rank of the Fourier module.

The last class to be discussed here is that of quasicrystals. In 1984 it was found (Shechtman et al., 1984) that in the diffraction pattern of a rapidly cooled AlMn alloy the spots were relatively sharp and the point-group symmetry was that of an icosahedron, a group with 120 elements and one that can not occur as point group of a three-dimensional space group. Later, ternary alloys were found with the same symmetry of the diffraction pattern, but with spots as sharp as those in ordinary crystals. These structures were called quasicrystals. Others have been found with eight-, ten- or 12-fold rotation symmetry of the diffraction pattern. Such symmetries are also noncrystallographic symmetries in three dimensions. Sometimes this noncrystallographic symmetry is considered as characteristic of quasicrystals.

Mathematical models for quasicrystals are quasiperiodic two- and three-dimensional tilings, plane or space coverings, without voids or overlaps, by copies of a finite number of `tiles'. Examples are the Penrose tiling or the standard octagonal tiling in two dimensions, and a three-dimensional version of Penrose tiling, a quasiperiodic space filling by means of two types of rhombohedra. For Penrose tiling, all spots of the diffraction pattern are linear combinations of the five basis vectors $[{\bf a}_{m}^{*} = \{a\cos[2\pi(m-1)/5], a\sin[2\pi (m-1)/5]\} \quad (m=1,\ldots 5).\eqno(1.10.1.11)]$ Because the sum of these five vectors is zero, the rank of the spanned Fourier module is four. The Fourier module of the standard octagonal tiling is spanned by $[{\bf a}_{m}^{*} = \{a\cos[(m-1)\pi/4], a\sin[(m-1)\pi/4]\} \quad (m=1,\ldots, 4).\eqno(1.10.1.12)]$ The rank of the Fourier module is four. The rank of the Fourier module of the three-dimensional Penrose tiling, consisting of two types of rhombohedra with a ratio of volumes of $[(\sqrt{5}+1)/2]$ , is six and basis vectors point to the faces of a regular dodecahedron.

An atomic model can be obtained by decorating the tiles with atoms, each type of tile in a specific way. Some quasicrystals can really be considered as decorated tilings.

References

Janssen, T. & Janner, A. (1987). Incommensurability in crystals. Adv. Phys. 36, 519–624.Google Scholar

Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.10, pp. 243-244