Long range ordering problems models

Percolation. Application of local interactions to the problem of generating long-range order, including phase transitions and morphogenesis, to large-scale discrete models. 

The theoretical calculation of the thermodynamic properties of liquids presents an even more difficult problem than that of solids, for in a liquid the molecules are in a state of disorder intermediate between the random motion of a gas, and the long range order of a solid. We must therefore be content with an even cruder approximation than for solids. ) A model which leads to a useful semi-quantitative correlation is based upon the idea of the free volume. 

RMC is a variation of the standard Metropolis Monte Carlo (MMC) method (Metropolis et al., 1953 see also Chapters l and 5). The principle is that we wish to generate an ensemble of atoms, i.e. a structural model, which corresponds to a total structure factor (set of experimental data) within its errors. These are assumed to be purely statistical and to have a normal distribution. Usually the level and distribution of statistical errors in the data is not a problem, but systematic errors can be. We shall initially consider materials that are macro-scopically isotropic and that have no long range order, i.e. glasses, liquids and gases. The basic algorithm, as applied to a monatomic system with a single set of experimental data, is as follows ... 

Because of the small size of the configuration (even if it contains many thousands of atoms) the statistical accuracy of the three-dimensional distribution g(r) is too poor to enable direct transformation to the single crystal structure factor T(Q). In addition there are truncation problems due to long range order, as mentioned earlier for powders. We therefore use essentially the same method as in Section 2.5.4, e.g. equation (24). It is, if possible, sensible to choose the Q points at which the measurement is to be made to be appropriate to the model, otherwise it is necessary to interpolate the data onto these points. For example, if the model is cubic and contains 10 x 10 x 10 unit cells then Q = 0.1 (/iA, B, /C) where A, B and C are the reciprocal lattice vectors and h, k, l are integers. 

The crystalline solids possess both short- and long-range order whereby, through the construction of the symmetry-adapted Bloch waves of Eq. (64), the calculations become feasible. This is not the case for liquids that may possess short-range but not long-range order. In order to circumvent this problem one has to consider a simplified model system, whereby one most often considers a... 

These results remain valid for the geneanl case when 6, 5 and p are simultaneously different from zero. The present refined model and the crude approximation of Ch. IX are almost equivalent. Therefore it does not seem worthwhile to use these more complicated relations when the results of Ch. IX can be applied just as well. This is however only the case when lattice deformations may be neglected (molecules of the same size, long range order...). The interest of the approach we have considered is precisely that it permits the discussion of local lattice deformations. This problem could not be discussed by the method of Ch. IX in which a single set of average potential constants is used. 

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