Modulated crystal structures displacive modulations

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.)...

Fig. 21 Crystal structure (a) and temperature dependences of magnetization (b), modulation wavenumber (c), dielectric constant (d), and polarization (e) in TbMnOs crystal [38] (modulations of magnetic moment and lattice displacement are shown at the bottom left)...

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. 

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...

The intensity variation along the rod (i.e. as a function of or /) is solely contained in the structure factor it is thus related to the z-co-ordinates of the atoms within the unit-cell of this quasi-two dimensional crystal. In general, the rod modulation period gives the thickness of the distorted layer and the modulation amplitude is related to the magnitude of the normal atomic displacements. This is the case of a reconstructed surface, for which rods are found for fractional order values of h and k, i.e. outside scattering from the bulk. 

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 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. 

The summation is over all the p atoms in the unit cell vvdth atomic scattering factor The modulation describes the individual atomic displacement relative to its basic position r. The integration variable t is related to the fourth dimension by the expression X4 = q. r -1- r (see Fig. 2). The first line of this expression corresponds to the structure factor of a conventional crystal (/i4 is equal to zero, and thus H = h). The second line of the expression is specific to aperiodic structures and contains the information related to the atomic modulation function in superspace. In addition to a displacement parameter, the modulation can also be affected by a variable occupation parameter p. ... 

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