%0 Book Section %1 statphys23_0127 %A Gadjiev, B.R. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K connectivity critical distribution indexes phase statphys23 topic-2 transition %T Effect of defects distributions on phase transitions in crystals %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=127 %X We consider close-packed structure with defects, which undergoes structural phase transition, and we investigate dependence of critical exponents on distribution of defects. We do not consider single-domain crystals with randomly distributed defects. We discuss the growth condition which of lead to inhomogeneous distribution of defects over a sample 1. We consider close-packed structure with defects as result of growth process. As a result we obtain some structure which represents a network. Symmetry of such structure is a symmetry corresponding graph. Irreducible representation (non identity) groups of graph define transformation properties of the order parameter. Integer basis of considered irreducible representation allows to write down free energy and to define corresponding critical indexes. We in details analyze a case when presence of defects induces structure with incommensurate or quasicrystal ordering 2. The structure such space graph can be obtained as a projection highly dimensional regular graph 3. The point group symmetry of such structure is isomorphic to permutation group. Hence, for phase transitions without change of an elementary cell volume, a transformation property of the order parameter is defined non identity irreducible representation of the permutation group. Using concrete algorithms for the growth process, it is possible to introduce corresponding equation the solution of which is distribution functions of defects in structure 4. For the analysis of defects distribution dependence critical exponents, we introduce a free energy functional, which depends on an order parameter and on connectivity distribution of defects of structure. The symmetry of an order parameter is defined by an irreducible representation of a space group of structure. After the procedure an average of a free energy functional on connectivity of defects (practically it implies calculation a moments of the distribution) we obtain, that the critical behavior strongly depends on the form of the distribution function. We consider a case when the solution the equation for growth process leads to fractal distribution of defects connectivity, namely, to Tsallis distribution 5. We show that the critical behavior strongly depended on the form distribution of connections and differs strongly from mean - field behavior. References\\ 1) A.P. Levanyuk, A.S. Sigov, Defects and structural phase transition (Taylor and Francis, 1988).\\ 2) T. Janssen, A. Janner, Advanced in physics, 36, n.5, 519 (1987).\\ 3) F. Harary, Graph theory (Addison-Wesley, Reading, MA, 1969).\\ 4) R. Albert, A.-L. Barabasi, Rev. Mod. Phys. 74, 47 (2002).\\ 5) C. Tsallis, Braz. J. Phys., 29:1 (1999).

@incollection{statphys23_0127, abstract = {We consider close-packed structure with defects, which undergoes structural phase transition, and we investigate dependence of critical exponents on distribution of defects. We do not consider single-domain crystals with randomly distributed defects. We discuss the growth condition which of lead to inhomogeneous distribution of defects over a sample [1]. We consider close-packed structure with defects as result of growth process. As a result we obtain some structure which represents a network. Symmetry of such structure is a symmetry corresponding graph. Irreducible representation (non identity) groups of graph define transformation properties of the order parameter. Integer basis of considered irreducible representation allows to write down free energy and to define corresponding critical indexes. We in details analyze a case when presence of defects induces structure with incommensurate or quasicrystal ordering [2]. The structure such space graph can be obtained as a projection highly dimensional regular graph [3]. The point group symmetry of such structure is isomorphic to permutation group. Hence, for phase transitions without change of an elementary cell volume, a transformation property of the order parameter is defined non identity irreducible representation of the permutation group. Using concrete algorithms for the growth process, it is possible to introduce corresponding equation the solution of which is distribution functions of defects in structure [4]. For the analysis of defects distribution dependence critical exponents, we introduce a free energy functional, which depends on an order parameter and on connectivity distribution of defects of structure. The symmetry of an order parameter is defined by an irreducible representation of a space group of structure. After the procedure an average of a free energy functional on connectivity of defects (practically it implies calculation a moments of the distribution) we obtain, that the critical behavior strongly depends on the form of the distribution function. We consider a case when the solution the equation for growth process leads to fractal distribution of defects connectivity, namely, to Tsallis distribution [5]. We show that the critical behavior strongly depended on the form distribution of connections and differs strongly from mean - field behavior. References\\ 1) A.P. Levanyuk, A.S. Sigov, Defects and structural phase transition (Taylor and Francis, 1988).\\ 2) T. Janssen, A. Janner, Advanced in physics, 36, n.5, 519 (1987).\\ 3) F. Harary, Graph theory (Addison-Wesley, Reading, MA, 1969).\\ 4) R. Albert, A.-L. Barabasi, Rev. Mod. Phys. 74, 47 (2002).\\ 5) C. Tsallis, Braz. J. Phys., 29:1 (1999).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Gadjiev, B.R.}, biburl = {https://www.bibsonomy.org/bibtex/26a32fe855f9c76aac5093a9c65074dfc/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {49b4b6b87f0ce81f493fb5376f04ec1c}, intrahash = {6a32fe855f9c76aac5093a9c65074dfc}, keywords = {connectivity critical distribution indexes phase statphys23 topic-2 transition}, month = {9-13 July}, timestamp = {2007-06-20T10:16:12.000+0200}, title = {Effect of defects distributions on phase transitions in crystals}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=127}, year = 2007 }