Commensurate-incommensurate transition

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. )...

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... 

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. 

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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