Displacive modulated structure

Here let us consider the structure of Bai+ Fe2S4 from the viewpoint of displacive modulated structures. As clearly seen from Fig. 2.38, Ba ions in the basic compound BaFe2S4 occupy regularly half of the face-capped tetragonal prisms (FCTP twelve coordinations) formed by S ions. The FCTP sites are at z = 0 and along the c-axis. The other sites for Ba ions are at z = j and I, where the Ba ions are eight coordinated by S ions (square anti-prism, SAP). The former sites are more stable than the latter ones, however, the occupation of all of the former sites seems to be impossible, due to the repulsive force between Ba ions. 

In general, these defect-free modulated structures can, to a first approximation, be divided into two parts. One part is a conventional structure that behaves like a normal crystal, but a second part exists that is modulated5 in one, two, or three dimensions. The fixed part of the structure might be, for example, the metal atoms, while the anions might be modulated in some fashion. The primary modulation might be in the position of the atoms, called a displacive modulation (Fig. 4.35a). Displacive modulations sometimes occur when a crystal structure is transforming from one... 

Figure 4.35 Modulated crystal structures (a) crystal showing a displacive modulation of one set of atoms and (b) a crystal showing a compositional modulation of one set of atoms. (The change in the average chemical nature of the atom is represented by differing circle diameters.)...

Different kinds of modulations can be considered the displacive modulation is related to the shift of the atom position from the average structure the occupational (or substitutional) modulation is related to the changes of the atomic occupation probability depending on the position. Generally, in alloys, displacive modulation is small but not negligible (Yamamoto 1996) whereas substitutional modulation often occurs. 

A special case of long-period structure to be considered is the oI40-AuCu(II) type structure which has ID substitutional and displacive modulations (Fig. 3.41). We must first mention that ordering of the Au-Cu face-centred cubic (cF4-Cu type) solid solution, having a 50-50 atomic composition, re-distributes Cu and Au atoms... 

The small displacements of the atoms produce several sorts of modulated structures. 

In some crystalline materials a phase transition on lowering the temperature may produce a modulated structure. This is characterized by the appearance of satellite or superstructure reflections that are adjacent reflections (called fundamental reflections) already observed for the high temperature phase. The superstructure reflections, usually much weaker than fundamental reflections, can in some cases be indexed by a unit cell that is a multiple of the high temperature cell. In such a case the term commensurate modulated structure is commonly used. However, the most general case arises when the additional reflections appear in incommensurate positions in reciprocal space. This diffraction effect is due to a distortion of the high temperature phase normally due to cooperative displacements of atoms, ordering of mixed occupied sites, or both. Let us consider the case of a displacive distortion. 

The structural parameters characterizing the modulated structure correspond to the average positions of the atoms, x, in the average unit cell as well as the components of the cosine and sine vector terms Cj and Sy . Clearly, the structure factor is similar to that of a conventional crystal structure but with an atomic contribution weighted by the rather cumbersome function g (h) containing the information about the modulation displacements. 

The symmetry treatment of incommensurate structures is beyond the scope of this chapter. From Equation (33) it is readily seen that for indexing, whatever the reflection of the diffraction pattern of an incommensurately modulated structure, we need to specify 3 + d integers (h, k, I, m, m2... m fl. It can be demonstrated that the observed 3D structure can be considered as a projection of a periodic structure m3 + d dimensions over the real 3D space, which is a hyper-plane not cutting the points of the 3 + d lattice except the origin. The superspace approach of de Wolff, Janssen and Janner is now well established and has become the routine way of treating the symmetry of the displacive incommensurate structures. The same approach has been extended to study general quasiperiodic structures (composite structures and quasicrystals). 

The diffraction pattern from a normal crystal is characterised by an array of spots separated by a distance /a = a that arise from the parent structure, together with a set of commensurate superlattice reflections that arise as a consequence of the additional ordering. In this case the spot spacing is 1 /na = a /n, where n is an integer, (Figure 8.21a, b). In modulated structures, the modulation might be in the position of the atoms, called a displacive modulation, (Figure 8.21c). Displacive modulations... 

Figure 8.21 Schematic representations of normal and modulated crystal structures and diffraction patterns (a) a normal superlattice, formed by the repetition of an anion substitution (b) part of the diffraction pattern of (a) (c) a crystal showing a displacive modulation of the anion positions (d) a crystal showing a compositional modulation of the anion conditions, (the change in the average chemical nature of the anion is represented by differing circle diameters) (e) part of the diffraction pattern from (c) or (d) (f) a modulation wave at an angle to the unmodulated component (g) part of the diffraction pattern from (f). Metal atoms are represented by shaded circles and non-metal atoms by open circles...

Fig. 7.11. Cu02 layer of the deformation modulated structure. The displacements of Cu (black dots) are indicated by arrows they are strongly exaggerated. Perovskite and superstructure meshes are indicated by subscripts p and s respectively.

The three intermediate phases II, III, and IV of thiourea are believed to be incommensurate, and the incommensurately modulated structures in phase II and rv have been analyzed [50]. Another commensurate phase which is stable between phase I and phase II in a narrow temperature region of about 2 K has been recognized many years ago [27c,/], and was refined at 170 K by Tanisaki et al. [27b]. This ninefold superstructure of thiourea is characterized by a rotation of the (NH2)2CS molecule along the c axis coupled with a displacement of the center of mass in a plane perpendicular to the axis. Around mirror planes (y = 1/4 and 3/4) the local structure is isostructural with phase I. Therefore the superstructure is constructed of alternately polarized layers which are sandwiched by domain walls (or discom-mensurations) whose local structure is that of the paraelectric room temperature phase (V) around y = 0 and 1/2. 

Figure 1.12 shows a simple example of an incommensurate (or modulated) structure. The square grid represents a perfect crystal lattice. However, the atoms do not occupy the corners of the squares. They are displaced relative to the ideal positions according to a plane sinusoidal modulation wave whose wavelength A is incommensurate with the length of the translation b kjb is an irrational number. 

What are the essential features of the refined modulated structure Most of the effect of the modulation is observed in the H-bond network. The most significant change concerns the alternate intermolecular bonds 029-H-. N13 and 029-H. 027. These bonds are mutually exclusive and are equally distributed in space. Obviously, the displacement of the quinoline illustrated above is directly linked to the H-bond formation with N13. The modulation with the shape of a crenel function is associated to a weak H-bond Cl 1-H-. 023. The two H atoms bonded to C11 of the vinyl group form alternatively a link to 023. which explains the two conformations observed in Fig. 6. 

Modulated structures can be obtained from the structures having translational symmetry by say a displacement of atomic layers by an integral number of lattice translations. This results into a superlattice still retaining translational symmetry. But if the displacement is not an integral multiple of lattice vectors, the resulting lattice wiU lose its commensurability with the basic structure and also the translational symmetry. The diffraction of the modulated wave in addition to the determination of r of the atoms is within the domain of Incommensurate Crystallography. ... 

The parent structure of the anion-deficient fluorite structure phases is the cubic fluorite structure (Fig. 4.7). As in the case of the anion-excess fluorite-related phases, diffraction patterns from typical samples reveals that the defect structure is complex, and the true defect structure is still far from resolved for even the most studied materials. For example, in one of the best known of these, yttria-stabilized zirconia, early studies were interpreted as suggesting that the anions around vacancies were displaced along < 111 > to form local clusters, rather as in the Willis 2 2 2 cluster described in the previous section, Recently, the structure has been described in terms of anion modulation (Section 4.10). In addition, simulations indicate that oxygen vacancies prefer to be located as second nearest neighbors to Y3+ dopant ions, to form triangular clusters (Fig. 4.11). Note that these suggestions are not... 

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