New version of the gauge theory of linear defects in crystals with a polyatomic lattice
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RY OF CRYSTAL STRUCTURES
New Version of the Gauge Theory of Linear Defects in Crystals with a Polyatomic Lattice M. Kh. Kharrasov, I. R. Kyzyrgulov, and A. T. Khusainov Bashkir State University, Ufa, 450074 Russia email: [email protected] Received June 23, 2009
Abstract—The gauge theory of dislocations and disclinations has been generalized to the case of crystals with a polyatomic lattice. An infinite anisotropic continuous medium with dislocations and disclinations is used as a real crystal model based on the gauge group SO(3N)λT(3N). DOI: 10.1134/S1063774510050020
INTRODUCTION The methods of gauge field theory are considered the most beautiful and efficient ways for describing phenomena in the mechanics of continuous media as well as in the physics of elementary particles [1]. They have been recently successfully applied to describe the elastic [2], magnetoelastic [3], and ferroelastic [4] media with continuously distributed dislocations and disclinations. Attempts to describe these systems in terms of the classical methods were not successful. Analogs for all electrodynamic effects, including relativistic ones, were found in the dislocation theory. However, the material Lagrangian in Men’shenin’s theory [3] describes a massless magnetoelastic field in a defectfree continuum, whereas in electrodynamics the material Lagrangian describes a massive electron– positron field, disregarding the electromagnetic inter action. Men’shenin, Kadic, and Edelen [2, 3] consid ered the interaction of defects with elastic fields, replacing the partial derivatives in the magnetoelastic field Lagrangian with covariant derivatives. As a result, new constant parameters (coupling constants) appeared in the theory. Kadic and Edelen [2] could not determine how these constants are related to the known characteristics of solids: density, elasticity moduli, etc. An analysis of the abovementioned anal ogies with electrodynamics shows that the material Lagrangian of the dislocation theory must be the Lagrangian describing particlelike objects (dislo cations) rather than the elastic field Lagrangian in a defectfree medium. A theory of linear defects for crystals with a polyatomic lattice was proposed in [5, 6]. This study is an attempt to generalize the Kadic– Edelen model to a crystal with a polyatomic lattice. Note that the theory presented below does not make it possible to describe edge dislocations, as was shown in [7] for the Kadic–Edelen model. It is demonstrated how one can describe dislocations and disclinations in
crystals with a polyatomic lattice within the calcula tion scheme proposed in [2–4]. A polyatomic lattice consists of N different interacting subcontinua; specif ically, of N incorporated sublattices. In this study, using the Born model, we consider a crystal with a complex lattice containing N atoms per unit cell to be a set of N interpenetrating (inserted into each other) subcontinua. Equations can be written separately for each subcontinuum to obtain N equations in total. Instead of this, one can use 3Nd
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