Commensurate-incommensurate

Kosterlitz-Thouless and commensurate-incommensurate transitions in the triangular lattice 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. )...

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.

There is, however, another type of transition possible in two dimensions, a transition between states without LRO. This is the Kosterlitz-Thouless transition [8] mentioned in Sections II and V.B.l. It is relevant to superconductivity, commensurate-incommensurate transitions [61], planar magnetism, the electron gas system, and to many other systems in two dimensions. It involves vortices (thus the requirement of a two-component order parameter) characterized by a winding number q = (1/2-rr) dr V0, in which 0 is the phase of the order parameter (see also Ref. 4), the amplitude being fixed. These free vortices have an energy [see Eq. (28)] given by... 

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. 

Nho K., and Manousakis, E. (2003). Commensurate—incommensurate transitions in quantum films submonolayer molecular hydrogen on graphite, Phys. Rev. B, 67, 195411-1-7. 

Boninsegni, M., Lee S., and Crespi V.H. (2001). HeUum in one-dimensional nanopores free dispersion, localization, and commensurate/incommensurate transitions with nonrigid orbitals. Phys. Rev. Lett., 86, 3360—3. 

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. 

K. Kern, R. David, P. Zeppenfeld, R. Palmer, and G. Comsa. Symmetry Breaking Commensurate-Incommensurate Transition of Monolayer Xe Physisorbed on Pt(l 11). Solid State Commun. 62 391 (1987). 

H. Freimuth, H. Wiechert, and H.J. Lauter. The Commensurate-Incommensurate Transition of Hydrogen Monolayers Physisorbed on Graphite. Surf. Sci. 189/190 548 (1987). 

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. 

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