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We studied the phase transitions and magnetic properties of the Ising model on a square lattice by the replica Monte Carlo method and by the method of transfer-matrix, the maximum eigenvalue of which was found by Lanczos method. The competing exchange interactions between nearest neighbors J 1 , second J 2 , third neighbors J 3 and an external magnetic field were taken into account. We found the frustration points and expressions for the frustration fields, at crossing of which cardinal changes of magnetic structures (translational invariance changes discontinuously) take place. A comparative analysis with 1D Ising model was performed and it was shown that the behavior of magnetic properties of the 1D model and the 2D model with J 1 and J 3 interactions reveals detailed similarity only distinguishing in scales of magnetic field and temperature.

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In part I of this series the problem of immanent chaotization of crystal structures was described in simple models and the mathematical approach to the calculation of diffuse scattering (DS) was described. It was formulated that the loose packing of most real crystals appears to be the universal microscopical cause of DS. In the present paper the microscopical nature of chaotization is analysed and a quantitative description of the structural loose packing is given for cubic perovskites; these are particularly convenient examples because the most vivid and thorough experimental data on DS were obtained for such crystals: KNbO3, BaTiO~, NaNbO3, KMnF3.

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O n the basis of numerous experiments using the mono-Laue method on the diffuse scattering of X-rays and electrons we have developed the Comes-Lambert-Guinier concept of the existence of co-operative thermal vibrations in perfect crystals in which one-dimensional or two-dimensional atomic objects, i.e. chains or planes of atoms, are moving in a double-well potential with conservation of coherence. For an approximate description of such a motion we have utilized the pseudospin king model approach which Is widely used in the theory of structural phase transitions. These ideas proved to be successful for the description of diffuse scattering patterns, and their temperature evolution in close connection with structural phase transitions.The developed theory gives the possibility through the observation of diffuse scattering at high temperatures and its analysis to predict the structural phase transition (and even the subsequent cascades of transitions) and to give the correct explanation for physical anomalies in the crystals, either with distortive (weak) or reconstructive (strong) transitions. A wide variety of crystallochemical situations and transitions in crystals of the perovskite family and in metals with a bcc lattice are analysed in detail. A number of new experimental diffuse scattering patterns from crystals of various types are obtained.KEY WORDS: mono-Laue method, crystal structure loose-packing, movable coherent extended objects, relrods and relplanes. I INTRODUCTIONIn this paper we shall consider only the spontaneous (occurring with temperature variation without outer interference) structural phase transitions of the displacement type. Two different types of such transitions are assumed to exist: distortive and reconstructive. Distortive transitions are weak structural phase transitions at which the structure is changed only slightly, and the atom displacements u are far less than the interatomic distances or lattice parameters, a so that u < a. Usually u/a % 10p2-10-3. At such a transition some symmetry elements of the initial phase vanish, but in such a way that the final and initial phases possess the subgroup relation: G, c Go. This is the fundamental relation for Landau theory and that is why the theory of such transitions is rather well developed. Among them one can find either second-order transitions or the so-called first-order transitions close to the secondorder ones. 89 Downloaded by [University of Cambridge] at 02:40 05 November 2014 90 F. A. KASSAN-OGLY, V. E. NAISH A N D I. V. SAGARADZEReconstructive transitions are strong first-order transitions at which a substantial crystal structure reconstruction with alteration of the actual structure type takes place, for example, bcc-fcc, bcc-hcp and so on. The displacements of atoms at these transitions are much greater (u i a) and the subgroup relation is absent completely; hence Landau theory in the usual way is not applicable to them. For this reason alone the theory of reconstructive transitions is far less developed.In the present paper, an...

A diagram technique for calculation of correlation functions of ferromagnet being described by the Ising model is proposed here and its zero approximation corresponds to the selfconsistent field. The essential dependence of a numerical factor of a diagram upon its structure is the important peculiarity of this technique in comparison with the usual diagram technique for fermion or boson operators. It leads to the special character of analytical series corresponding to the certain diagram series and to the special ways of their summarizing. The exact forma1 expression for the average spontaneous moment in terms of irreducible many-particle correlation functions had been obtained. A certain series of graphs for spontaneous moment which corresponds to the Gaussian fluctuations of the selfconsistent field being taken into account had been extracted. A relation between the proposed diagram technique and the expansion of functional integrals for the correlation functions of the Ising model had been obtained as well. In conclusion the referencies to authors papers where a diagram technique for Heisenberg and s-d models are developed are given there