Incommensurate phase

Dzyaloshinskii (1964) belong to this category. The term modulated structures has been used to describe incommensurate perturbation (i.e. one in which the ratio of the imposed periodicity to that of the unit cell is irrational). A broader definition of modulated structure can be used to describe any periodic or partly periodic perturbation of a cystal structure with a repetition distance appreciably greater than the basic unit cell dimensions such a definition would include a variety of superstructures as well (Cowley et al, 1979). 

In conducting solids, the conduction electron density is spatially modulated, forming charge density waves (CDW) the periodic distortion accompanying the CDW (due to interaction between the conduction electron and the lattice) is responsible for the incommensurate phase (Overhauser, 1962 Di Salvo Rice, 1979 Riste, 1977). The occurrence of CDW and the periodic distortion can be understood in terms of the model proposed by Peierls and Frdhlich for one-dimensional metals. Let us consider a row of uniformly spaced chain of ions (spacing = a) associated with conduction electrons of energy E k) and a wave vector k. At 0 K, all the states are filled up to the Fermi energy, = E(kp). If the electron density is sinusoidally modulated as in Fig. 4.15 such that 

Besides electron-phonon coupling, the shape of the Fermi surface is important in deciding CDW formation. Fermi surfaces of systems containing linear or planar arrays of atoms reflect their one- or two-dimensional character. Since the Fermi surfaces connect many states with the same wave vector, Q, a periodic distortion having the wave vector Q will produce gaps at those portions connected by Q. The energy gained 

K2[Pt(CN)4]BrQ jxHjO (KCP) is an example of a one-dimensional compound (see Chapter 6 for a discussion of low-dimensional solids) consisting of parallel stacks of 

Besides its anisotropic electrical and optical properties, this solid exhibits superconductivity at low temperatures. 

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Considering the changes of E upon fi for different values of the parameters, Villain has concluded that imperfections in the surface structure may have a very strong influence on the behavior of incommensurate phases and on the C-IC transition. The usual lowering of symmetry during the C-IC transition does not occur here and the nature of the C-IC transition may be quite different from that on the surface free of defects. 

Of course, the above discussion apphes only to systems exhibiting domain wall structure, i.e., to weakly inhomogeneous phases formed on surfaces with low corrugation of the gas-solid potential and characterized by the presence of more then one type of equivalent sublattices. When this is not the case, i.e., when the dense incommensurate phase can be considered to be... 

From LEED measurements of H monolayers adsorbed on Fe(110) Imbihl et al. proposed a phase diagram as shown in Fig. IS. In addition to lattice gas and lattice fluid phases, two commensurate ordered phases were identifled, denoted as (2 x 1) and (3 x 1) in the figure (cf. Fig. 16). The shaded regions are interpreted as incommensurate phases or as phases composed of antiphase domains their signature is that the LEED spot does not occur at the Bragg position but rather the peak is splitted and satellites appear (Fig. 17). 

A particular complex problem has been the modelling of Si/W(l 10) Amar et have included pairwise interactions up to the sixth nearest neighbor shell, as estimated experimentally from field-ion microscopic studies The predicted phase diagram (Fig. 30) exhibits (5 x 1), (6 x 1) and p(2 x 1) commensurate phases, as well as a broad regime of an incommensurate phase. In contrast to the ANNNI model the present model does seem to have a finite-temperature Lifshitz point, where the incommensurate, commensurate... 

Fig. 28. Soliton lattice solution of the FvdM model with regularly spaced, distance 1, domain walls. The dashed line corresponds to the incommensurate phase with negligible potential modulation (K = 0). The plateaus represent the commensurate domains.

Fig. 29). Using Landau theory, Bak et al. (BMVW) have shown that it is the wall crossing energy A which determines the symmetry of the weakly incommensurate phase and the nature of the phase transition ... 

The FvdM as well as the BMVW model neglects thermal fluctuation effects both are T = 0 K theories. Pokrovsky and Talapov (PT) have studied the C-SI transition including thermal effects. They found that, for T 0 K the domain walls can meander and collide, giving rise to an entropy-mediated repulsive force of the form F where I is the distance between nearest neighbor walls. Because of this inverse square behavior, the inverse wall separation, i.e. the misfit m, in the weakly incommensurate phase should follow a power law of the form... 

The basic modem data describing the atomic stmcture of matter have been obtained by the using of diffraction methods - X-ray, neutron and electron diffraction. All three radiations are used not only for the stmcture analysis of various natural and synthetic crystals - inorganic, metallic, organic, biological crystals but also for the analysis of other condensed states of matter - quasicrystals, incommensurate phases, and partly disordered system, namely, for high-molecular polymers, liquid crystals, amorphous substances and liquids, and isolated molecules in vapours or gases. This tremendous... 

Figure 4.21 Portion of the (010) plane in the reciprocal lattice of K2Se04. On the left is the lattice for the high-temperature phase and on the right that for the ferroelastic phase. In the high-temperature phase, a softening occurs at a point displaced by 5 from q = (j, 0, 0), shown by a cross. In the incommensurate phase a satellite reflection develops at X.

Misfit layered compounds of the type, (RX) (TX2) , where R is a rare earth, Pb or Bi, T = Ti, V, Nb or Ta and X = S or Se, are typical incommensurate phases that have been investigated in detail (Rouxel Meerschaut, 1994). The incommensuration here arises from a structural misfit between the rocksalt-like RX and layered TXj dichalcogenide units which are stacked alternately. The interplay of structure and electron-transfer from the RX to TXj units gives rise to novel electronic properties. For... 

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