ISSN 1063-7788, Physics of Atomic Nuclei, 2012, Vol. 75, No. 10, pp. 1182–1194. c  Pleiades Publishing, Ltd., 2012. ELEMENTARY PARTICLES AND FIELDS Theory Quantum 3D Spin-Glass System on the Scales of Space–Time Periods of External Electromagnetic Fields* A. S. Gevorkyan ** Institute for Informatics and Automation Problems, NAS of Armenia, Yerevan; Joint Institute for Nuclear Research, Dubna, Russia Received March 6, 2012 Abstract—A dielectric medium consisting of rigidly polarized molecules has been treated as a quantum 3D disordered spin system. It is shown that using Birkhoff’s ergodic hypothesis the initial 3D disordered spin problem on scales of space–time periods of external field is reduced to two conditionally separable 1D problems. The first problem describes a 1D disordered N -particle quantum system with relaxation in random environment while the second one describes statistical properties of ensemble of disordered 1D steric spin chains of certain length. Basing on constructions which are developed in both problems, the coefficient of polarizability related to collective orientational effects under the influence of external field was calculated. On the basis of these investigations the equation of Clausius–Mossotti (CM) has been generalized as well as the equation for permittivity. It is shown that under the influence of weak standing electromagnetic fields in the equation of CM arising of catastrophe is possible, that can substantially change behavior of permittivity in the X-ray region on the macroscopic scale of space. DOI: 10.1134/S1063778812100043 1. INTRODUCTION Formation and control of periodic nanostructures in various media (medium with periodically modu- lated refractive index) are now among the most im- portant problems in applied physics and material sci- ence. First of all, it is related to the possibility of creating compact UV or X-ray Free-Electron Lasers (FEL) based on the emission of transition radiation (TR) (see, for example, [1]). Currently the following two problems are discussed intensively: 1. A gas–plasma medium with periodically varied ionization density [2–10]. 2. Special periodical structures in solid states (solid-state superlattice-like (SSL) structure), which are composed of layers with different refraction in- dexes [11–19]. Let us remind that TR is generated as a result of crossing of an electronic beam interface with various frequency-dependent dielectric constants (permittivities) and the radiation intensity in this case is proportional to the value [χ R 1 (w) − χ R 2 (w)] 2 , where χ 1 (w) and χ 2 (w) are correspondingly permit- tivities of neighboring layers, in addition χ R 1;2 (w)= Re[χ 1;2 (w)] [20]. Therefore, the investigation of ∗ The text was submitted by the author in English. ** E-mail: g_ashot@sci.am possibilities of creation and controlling parameters of permittivity’s superlattice by external field is highly important scientific problem and can have many applications in modern technics and technologies. In other words our aim is creation of controlled superlat- tice, where intensity of TR would be proportional to the value [χ R 1 (w, g) − χ R 2 (w, g)] 2 , where g describes the controlling parameters of external field, χ 1 (w, g) and χ 2 (w, g) are the permittivities of neighboring layers. According to the theoretical and experimental studies, the periodical structures of permittivity may be formed in condensed matters by means of external electromagnetic or acoustic fields [21–24]. This idea was recently implemented in TR gener- ation experiments [25]. In particular, experimentally it was shown that the 20-MeV beam of electrons at the passage through the amorphous silicon dioxide SiO 2 in the presence of the standing electromagnetic wave with the frequency of 10 GHz, generates an anomalously intensive short-wave radiation. Pre- liminary studies explain this high intensity of radia- tion as a result of multiple passage of electron beam through interfaces between regions with appreciably different permittivities. Theoretically the formation of superlattice in disordered medium is explained by medium polarization due to the orientational relax- ation of elastic molecular dipoles in the direction of external electromagnetic field propagation [26]. 1182