Commensurate-incommensurate phase

Figure 9. Model for understanding the commensurate-incommensurate phase transition. The Cu(A) system is ordered. The Cu(B) system is disordered because of the frustration caused by the geometrically competing interaction. The molecular field from the Cu(B) system disturbs the order in the Cu(A) system.

Commensurability. Incommensurate lattice distortions and commensurate-incommensurate phase transitions are often observed in these materials. The incommensurability comes either from an incommensurate Fermi wave vector (2A F, 4kF scattering in charge-transfer salts) or from the counterion stacks (e.g., triiodide-containing materials). 

Ar and evidently much larger for Ne. Therefore, we would expect Kr and Xe to undergo the commensurate-incommensurate phase changes more easily than Ar or Ne. 

Seifert F, Czank M, Simons B, Schmahl W (1987) A commensurate-incommensurate phase transition in iron-bearing Akermanites. Phys Chem Min 14 26-35... 

A serious drawback of lattice gas models is their inadequacy to describe properly the commensurate - incommensurate phase transitions, often observed in real systems [144 - 150]. The possibility of the formation of incommensurate phases results directly from the finitness of potential berriers between adjacent potential minima and from the off-lattice motion of adsorbed particles. Although attempts have been made to extend the lattice-gas models and include the possibility of the formation of incommensurate solid phases [151,152], but it is commonly accepted (and intuitively obvious) that the continuous-space theories are much better suited to describe behaviour of adsorbed films exhibiting incommensurate phases. Theoretical calculations of the gas - solid potential for a variety of systems [88] have shown that, in most cases, the lateral corrugation is rather low. Nevertheless, it appears to have a very big influence on the behaviour of adsorbed layers. 

ADA/BIL] Adam, A., Billerey, D., Terrier, C., Bartholin, H., Regnault, L. P., Rossat-Mignod, J., Hydrostatic pressure effect on the commensurate-incommensurate phase transition of NiBr2, Phys. Lett., 84A, (1981), 24-27. Cited on pages 132, 386. 

Unfortunately, not much is known for N2 and CO on graphite or boron nitride with respect to the nature of their commensurate-incommensurate transitions, and the order and mechanism of their melting transition, although various speculations and claims can be found in the literature. Therefore, and in view of the excellent reviews already available, we did not include detailed discussions of the theoretical implications of these topics. Many aspects related to the vast subject of commensurate-incommensurate phase transitions can be found in Refs. 10, 205, 260, and 364 from a theoretical perspective, whereas experimental results for Kr on graphite are described in Refs. 40, 323, and 329, for H2, HD, and D2 on graphite in Refs. 83, 84, 120-123, 243, 377, and 379, and for Kr and Xe on Pt(lll) in Refs. 167 and 168. The possible peculiarities of the melting transition, when restricted to two dimensions, are covered, for example, in Refs. 129, 173, 205, 252, and 332. 

The introduction of Xe in a layer of Kr stabilizes the V3xV3 commensurate structure [84]. The analysis of the corresponding binary phase diagram shows a first order transition between the commensurate and incommensurate sohds [85,86]. Mixtures of Ar and Xe exhibit a large tendency to form commensurate alloys [87]. A first order commensurate-incommensurate phase transition is also observed in that case. 

The commensurate-incommensurate phase transition was also observed for Kr on six different graphites at 77.3 K by both volumetric and calorimetric methods [46], Moreover, LEED experiments [139] indicated confirmation of the second-order nature of that transition... 

For a long time, there were major imcertainties about the commensurate-incommensurate phase transition, mainly because the experimental studies indicated a different order of that transition at high and low temperatures [113]. These uncertainties have been resolved by the proven existence of a reentrant fluid between the two solid regions [15,143]. The final proposed form for the phase diagram is shown in Fig. 5 [144], where the points are experimental data obtained from thermodynamic methods. 

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). 

It is not clear whether in the centered rectangular lattice gas of section 3.2 such a Kosterlitz-Thouless transition occurs, or whether the disordered phase extends, though being incommensurate, down to the commensurate (3x1) phase (then this transition is believed to belong to a new chiral universality class ), or whether there is another disorder line for (3 x 1) correlations. However, Kosterlitz-Thouless type transitions have been found for various two-dimensional models the XY ferromagnet , the Coulomb gas . ... 

Fig. 20. Phase diagram of the triangular lattice gas model with nearest-neighbor repulsion and next-nearest neighbor attraction, for JmJJ = — 1, in the coverage-temperature plane. For fl=0.5 a Kost tz-Thouless transition occurs at Ti and a commensurate-incommensurate transition at Tj. Two commensurate. 3 x 3 phases (with ideal coverages of 1/3 and 2/3, respectively) occur, as well as several two-phase regions, as indicated in the figure. Here l.g. stands for lattice gas and LI for lattice liquid . (From Landau. )...

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.

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