Commensurate phase

In the next paper [160], Villain discussed the model in which the local impurities are to some extent treated in the same fashion as in the random field Ising model, and concluded, in agreement with earlier predictions for RFIM [165], that the commensurate, ordered phase is always unstable, so that the C-IC transition is destroyed by impurities as well. The argument of Villain, though presented only for the special case of 7 = 0, suggests that at finite temperatures the effects of impurities should be even stronger, due to the presence of strong statistical fluctuations in two-dimensional systems which further destabilize the commensurate phase. 

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... 

In a recent He diffraction study we have shown that the adsorption system Xe/Pt(l 11) is dominated by the existence of a (y/3 x. 3) R30° commensurate phase, shown schematically in Fig. 30. The C-phase has been found to be stable in an extended temperature (62 K-99 K) and coverage range (0xe 1 /3)-... 

Figure 4.19 Model structures of (a) incommensurate and (b) commensurate phases of K2Pb[Cu(N02)6]. The displacement pattern of Jahn-Teller active phonons is shown by arrows. In (a) the phonon mode has wave-vector k — (0.425, 0.425, 0) and in (b), wave-vector of the phonon mode is k = (, j, ). (After Yamada, 1977.)...

Keywords Copper metaborate, magnetic state, magnetic field, commensurate phase,... 

Figure 11. Temperature-magnetic field (parallel to the basal plane) phase diagram of single crystalline copper metaborate 1 - incommensurate phase of helix magnet 2 - commensurate phase of weak ferromagnet.

The presence of the higher order harmonics of the magnetic helix in the incommensurate phase is characteristic for the temperature interval where the Lifshits invariant is comparable with an anisotropy invariant [11], For magnetic systems with a one-parametric thermodynamic potential the propagation vector q is not equal to zero already at the temperature where the system orders, T1 =TP. As an anisotropy invariant is proportional to rj for a crystal with tetragonal symmetry, then it becomes comparable with Lifshits invariant proportional to q t] 2 much below Ti near the transition into a low-temperature commensurate phase. However, in copper metaborate q grows sharply from approximately zero at temperature 7) < Tp (Fig. 7) [5],... 

Previously, sudden changes of amplitudes of magnetic satellites observable below 2 K [7] were interpreted in terms of lock-in transitions into phases with commensurate propagation vectors. Within the framework of the thermodynamic potential (5) it is impossible to describe a low-temperature transition from an incommensurate phase into a commensurate phase at zero external magnetic field. 

Figure 1. Crystal structure of CUB2O4. Positions of Cu(A), Cu(B), B(I) and B(II) sites are indicated. Arrows in the figure denote the magnetic structure in the commensurate phase reported in ref. [7],...

In the commensurate phase (T < T < Tn) the spectral line splits into several lines, and in the incommensurate phase below T the spectral... 

Figure 6. Angular dependence of the 11B-NMR spin echo spectrum in the commensurate phase. T = 16 K. B T c-axis. The angle is measured from the crystal a-axis in the tetragonal crystal.

The NMR experiment was performed on nB in C11B2O4. The spectrum in the commensurate phase is discussed. The magnetic moment at the Cu(A) site is estimated to be 0.45 /tg, which is almost 50% of the moment derived from the neutron diffraction experiment. The magnetic moment at the Cu(B) sites is absent in the commensurate phase. The asymmetric nature of the spectral pattern is not understood so far. A phenomenological understanding of the commensurate to incommensurate transition is discussed on the basis of the molecular-field approximation. More precise discussion of the commensurate phase and also of the incommensurate phase will be presented in the near future. 

If the system is nearly commensurate, discommensurations will naturally form and separate ordered commensurate phases. These discommensurations have a gapless excitation spectrum [60]. This subject has been reviewed in Ref. 61. 

Sub-steps, similar to those in Figure 9.7, have been observed with both methane and ethane (Bienfait, 1980, 1985). It has been possible to construct 2-D phase diagrams for several of these systems (Gay et al., 1986 Suzanne and Gay, 19%). LEED and neutron diffraction have provided information on the 2-D structures. For example, seven different 2-D phases have been reported for ethane on graphite over the temperature range 64-140 K. Thus, three solid commensurate phases were identified at temperatures <85 K, the S3 phase apparently having a close-packed hexagonal structure, with ofCjHj) = 0.157 nm2. 

Higher order phase transitions in systems with two differently ordered commensurate phases or with a commensurate and an incommensurate phase have been suggested in 2D Meads UPD overlayers [3.93, 3.94, 3.110-3.114, 3.223]. However, there has not been clear experimental evidence supporting this assumption. 

Merwin LH, Sebald A, Seifert F (1989) The incommensurate-commensurate phase transition in akermanite, Ca2MgSi207, observed by in-situ Si MAS NMR spectroscopy. Phys Chem Min 16 752-756... 

The systems exhibiting the 1x2 structure are then also expected to show commensurate and incommensurate phases, quite similar to those observed for the ANNNI model [75]. In the lattice gas systems the presence of incommensurate phases is restricted to the situations in which the substrate lattice can be divided into a certain number of equivalent interpenetrating sublattices and the ordered state corresponds to the preferential occupation of one of those sublattices. Incommensurability is manifested by the presence of regions with different occupied sublattices and the formation of walls between the domains of commensurate phase. In the case of the discussed here systems exhibiting 1x2 ordered phase we have two sublattices, since particles occupy alternate rows. Figure 6 shows examples of equilibrium configurations demonstrating the formation of incommensurate structure when the ordered 1x2 phase is heated up. 

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