Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 4.2, pp. 526-528   | 1 | 2 |

Section 4.2.6. Disorder diffuse scattering from aperiodic crystals

F. Frey,a H. Boysena and H. Jagodzinskia

aDepartment für Geo- und Umweltwissenschaften, Sektion Kristallographie, Ludwig-Maximilians Universität, Theresienstrasse 41, 80333 München, Germany

4.2.6. Disorder diffuse scattering from aperiodic crystals

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The preceding sections of this chapter have either been related to disorder phenomena in conventional crystals defined by the presence of 3D translational symmetry, i.e. by a 3D lattice function, at least in the averaged sense, or to solids with crystalline order in only lower (2 or 1) dimensions where the ordering principle along `nonperiodic' directions shows a gradual transition from long- to short-range order, in particular to a liquid-like behaviour and, finally, to an (almost) random ordering behaviour of structural units. So-called aperiodic crystals do not fit this treatment because aperiodicity denotes a different type of order, which is nonperiodic in 3D space but where translational order may be restored in higher-dimensional (n > 3) direct space. Three types of solids are commonly included in the class of aperiodic structures: (i) incommensurately modulated structures (IMSs), where structural parameters of any kind (coordinates, occupancies, orientations of extended molecules or subunits in a structure, atomic displacement parameters) deviate periodically from the average (`basic') structure and where the modulation period is incommensurate compared to that of the basic structure; (ii) composite structures (CSs), which consist of two (or more) intergrown incommensurate substructures with mutual interactions giving rise to mutual (incommensurate) modulations; and (iii) quasicrystals (QCs), which might simply be viewed as made up by n (≥ 2) different tiles – analogous to the elementary cell in crystals – which are arranged according to specific matching rules and which are decorated by atoms or atomic clusters. Most common are icosahedral (i-type) quasicrystals with 3D aperiodic order (3D `tiles') and decagonal quasicrystals (d-phases) with 2D aperiodicity and one unique axis along which the QC is periodically ordered. The basic `crystallography' of aperiodic crystals is given in Chapter 4.6[link] of this volume; see also the review articles by Janssen & Janner (1987[link]) and van Smaalen (2004[link]) (and further references therein). We only point out some of the aspects here to provide the background for a short discussion about disorder diffuse scattering in aperiodic crystals.

As outlined in Section 4.6.1[link] , a d-dimensional (dD) ideal aperiodic crystal can be defined as a dD irrational section of an nD (`hyper'-)crystal with nD crystal symmetry. Corresponding to the section of the nD hypercrystal with the dD (d = 1, 2, 3) direct physical (= `external' or `parallel') space we have a projection of the (weighted) nD reciprocal hyperlattice onto the dD reciprocal physical space. The occurrence of Bragg reflections as a signature of an ordered (in at least an averaged sense) aperiodic crystal form a countable dense pattern of the projected reciprocal (hyper-)lattice vectors. Disorder phenomena, i.e. deviations from the periodicity in higher-dimensional direct (hyper-)space, are thus related to diffuse phenomena in reciprocal (hyper-)space projected down to the reciprocal physical space. Therefore only fluctuations from the aperiodic order infer diffuse scattering in its true sense, which might be hard to discern in a dense pattern of discrete (Bragg) reflections. As a consequence, in the higher-dimensional description of aperiodic crystals the atoms must be replaced by `hyperatoms' or `atomic surfaces', which are extended along (nd) dimensions. If, for example, an IMS exhibits a modulation in only one direction (1D IMS), we have n = 2 and a 1D atomic surface (which is a continuous modulation function extended along the 1D `internal' or `perpendicular' subspace). The physical subspace is spanned by 1 (and +2 `unaffected') dimensions. In a second example, a decagonal QC with aperiodic order in two dimensions is described by n = 4 (plus the remaining coordinate along the unaffected periodic direction) and d = 2, i.e. by 2D atomic surfaces. The atomic surfaces are, as shown in Chapter 4.6[link] , continuous n-dimensional objects for IMSs and CSs, and discrete ones in the case of QCs. In addition to the `hyperlattice' fluctuations within the physical (parallel) subspace we have therefore an additional quality of disorder phenomena related to positions and shape and size fluctuations of the atomic surfaces within the perpendicular subspace. Therefore, one has to consider disordering effects in aperiodic crystals related to fluctuations in the (external) physical subspace as well as in the internal subspace, but one should bear in mind that the two types are often coupled. Disordering due to displacements along directions within the physical subspace are – in the context of aperiodic crystals – commonly termed `phonon-like', in contrast to `phason-like' disorder related to displacements parallel to directions within the internal subspace. The term phason originates from a particular type of (dynamical) fluctuations of an IMS (see below). Incommensurately modulated structures

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Quite generally, small domains with periodic or aperiodic structures give rise to broadening of reflections. Diffuse satellite scattering is due to the limited coherence length of a modulation wave within a crystal or due to short-range order of twin or nanodomains (with an `internal' modulated structure) embedded in a periodic matrix structure. This type of diffuse scattering can be treated by the rules outlined in Section[link]. Structural fluctuations with limited correlation lengths are, for example, responsible for diffuse scattering in 1D organic conductors (Pouget, 2004[link]). In some cases, modulated structures are intermediate phases within a limited temperature range between a high- and low-temperature phase (e.g. quartz) where the structural change is driven by a dynamical instability (a soft mode). In addition to (inelastic) soft-mode diffuse scattering, dynamic fluctuations of phase and amplitude of the modulation wave give rise to diffuse intensity. In particular, the phase fluctuations, phasons, give rise to low-frequency scattering which might be observable as an additional diffuse contribution around condensing reflections. Composite structures

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In the case of CSs, diffuse scattering relates to interactions between the component substructures which are responsible for mutual modulations: each substructure becomes modulated with the period of the other. If one of the substructures is low-dimensional, for example chain-like structural elements embedded in tubes of a host structure, one observes diffuse planes if direct interchain correlations are mostly absent. The diffuse planes are, however, not only due to the included subsystem, but also due to corresponding modulations of the matrix structure. On the other hand, maxima superimposed on the diffuse planes reflect the influence of the matrix structure on the chain system. The thickness of the diffuse planes depends on the degree of short- or long-range order along the unique direction (cf. Section[link]). This might be either a consequence of faults in the surrounding matrix which interrupt the (longitudinal) coherence of the chains (Rosshirt et al., 1991[link]) or, intrinsically, due to the degree of misfit between the periods of the mutually incommensurate structures of the host and guest structures. If this misfit becomes large, e.g. as a consequence of different thermal expansion coefficients of the two substructures, the modulation is only preserved within small domains and one observes a series of diffuse satellite planes (Weber et al., 2000[link]). Equivalent considerations relate to composite systems made up from stacks of planar molecules (e.g. van Smaalen et al., 1998[link]) or an intergrowth of layer-like substructures where the modulation is mainly due to a one-dimensional stacking along the normal to the layers. Correspondingly, quite extended diffuse streaks occur at positions of the satellites in reciprocal space.

A further discussion of the more complicated disorder diffuse scattering of higher- (than one-) dimensionally modulated IMSs and CSs and the corresponding diffuse patterns in reciprocal space is beyond the scope of this chapter. We refer to Section 4.6.3[link] and the references cited therein. For examples, see also Petricek et al. (1991[link]). Quasicrystals

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As in the case of conventional crystals, there is no unique theory of diffuse scattering by quasicrystals. Chemical disorder, phonon- and phason-like displacive disorder, topological glass-like disorder and domain disorder exist. The term domain covers those with an aperiodic structure as well as periodic approximant domains, which are also known as approximant phases (cf. Section 4.6.3[link] ). Approximant phases exhibit local atomic clusters which do not differ significantly from those of related aperiodic QCs. In d-phases, disorder diffuse scattering occurs which is related to the periodic direction. Reviews of disorder diffuse scattering from quasicrystals are given, e.g., by Steurer & Frey (1998[link]) and Estermann & Steurer (1998[link]).

Chemical disorder. Many of the quasicrystals are ternary intermetallic phases consisting of atomic clusters where two components are transition metals (TMs) with only a small difference in Z (the number of electrons). Most of them exhibit a certain amount of substitutional (chemical) disorder, in particular with respect to the distribution of the TMs. This behaviour is equally true in the approximant phases. Chemical short-range order between the TMs, if any, is largely uninvestigated because conventional X-ray diffraction and electron-microscopy methods are not sensitive enough to provide significant contrast between the TMs. If the X-ray form-factor difference Δf is small, X-ray patterns do not show diffuse scattering, whereas an analogous neutron pattern could reveal a diffuse component due to the different scattering contrast Δb of the TMs (where the b's are the neutron scattering lengths). In practice, the chemical disorder phenomena might be more complex as the compositional stability of the QCs is often rather extended and the atomic distribution in a sample is not always structurally homogeneous. Depending on the crystal-growth process, microstructures may occur with locally coexisting small QC domains with fluctuating chemical content and, moreover, coexisting QC and approximant domains. Chemical disorder might also be caused by phasonic disorder.

Phonon-type (static or dynamic) displacements, fluctuations or straining relate to the physical subspace coordinates giving rise to continuous local distortions of the atomic structure. Phason-type disorder describes, as indicated above, discontinuous atomic jumps between different sites in an aperiodic structure. Qualitatively, dominant phonon- or phason-related scattering may be separated by analysing the dependence of the diffracted intensities on the components of the scattering vector in the external and the internal subspace, He (Qe) and Hi (Qi), respectively (cf. above and also Section 4.6.3[link] ). There are different kinds of phason-like disorder, including random phason fluctuations, phason-type modulations and phason straining, and also shape and size fluctuations of the hyperatomic surfaces. If displacing an atomic surface (hyperatom) parallel to an internal space component by any kind of phason fluctuation which is equivalent to an infinitesimal rotation of the external (physical) space, it might happen that the hyperatom no longer intersects the physical space. Then an empty site is created, which is compensated by a (real) atom `appearing' at a different position in physical space. In addition, there might even be a change of atomic species as a hyperatom may be chemically different at different sites of the atomic surface. Randomly distributed phason strains are responsible for Bragg-peak broadening and Huang-type diffuse scattering close to Bragg peaks. There are well developed theories based on the elastic theory of icosahedral (e.g. Jaric & Nelson, 1988[link]) or decagonal (Lei et al., 1999[link]) quasicrystals that also include phonon–phason coupling. An example of the quantitative analysis of diffuse scattering by an i-phase (Al–Pd–Mn) is given by de Boissieu et al. (1995[link]) and by a d-phase (Al–Ni–Fe) by Weidner et al. (2004[link]). Dislocations in quasicrystals have partly phonon- and partly phason-like character; a discussion of the specific dislocation-related diffuse scattering will not be given here (cf. Section 4.6.3[link] ). Arcs and rings of localized diffuse scattering are observed in various i-phases and could be modelled in terms of some short-range `glass-like' ordering of icosahedral clusters (Goldman et al., 1988[link]; Gibbons & Kelton, 1993[link]). Phason-related diffuse scattering phenomena are discussed in more detail in Section 4.6.3[link] . Depending on the exact stoichiometry and the growth conditions, d-phases very often show intergrown domain structures where the internal atomic structure of an individual domain varies between a (periodic) approximant structure, more or less strained aperiodic domains, or transient aperiodic variants (Frey & Weidner, 2002[link]). Apart from finite size effects of domains of any kind which cause peak broadening, there are complex diffraction patterns of satellite reflections, diffuse maxima and diffuse streaking in d-Al–Ni–Co and other d-phases (Weidner et al., 2001[link]). In various d-phases one also observes, in addition to the Bragg layers, prominent diffuse layers perpendicular to the unique `periodic' axis. They correspond to n-fold (n = 2, 4, 8) superperiods along this direction. In different d-phases there is a gradual change from almost completely diffuse planes to layers of superstructure reflections (satellites). The picture of a kind of stacking disorder of aperiodic layers does not match such observations. An explanation is rather due to 1D columns of atomic icosahedral clusters along the unique direction (Steurer & Frey, 1998[link]). The temperature behaviour of these diffuse layers was studied by in situ neutron diffraction (Frey & Weidner, 2003[link]), which shed some light on the complicated order/disorder phase transitions in d-Al–Ni–Co.


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