Kinetic properties of adatoms in Sine-Gordon model
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Kinetic properties of adatoms in Sine-Gordon model I.V.Baryakhtar Institute for Low Temperature Physics and Engineering, Kharkov 61164, Ukraine ABSTRACT The model for describing kinetic properties of adatoms on an anisotropic substrate is proposed. The kinetic equation for 1D adatom localized structures is constructed in the frame of the Sine-Gordon model. The self-diffusion coefficient for adatoms is calculated and the process of their homogenization is discussed. INTRODUCTION The dynamics of adatoms attracts a steady interest as an example of a dynamical behavior of low dimensional crystals. Typically the 2D crystals are formatted from a gas or liquid phase [1]. The one dimensional line type of adatoms ordering is of especial interest because it assumes the detailed theoretical analysis and it has been investigated experimentally well enough. 1D adatom structures can be formatted on the anisotropic substrate; La atoms on the W substrate and atoms of an inert gas on a graphite substrate are well known examples of the line type ordering (see [1]). When the amplitude of the periodic potential of the substrate is much less then the averaging energy of the adatoms interaction, the Hamiltonian of the adatom displacement can be reduced to the well known Sine-Gordon model [1], which is an exactly integrable model (see [2]). The one kink solution and the periodic solution of this model were successfully applied for the interpretation of some physical properties of low dimensional systems, for example, the electron and neutron scattering on structures in 2D crystals [1]. Except for the kink type solution, the Sine-Gordon equation has the two-soliton solution (breather), which can be interpreted as the bound state of two adatom structures. The common property of solitons in the exactly integrable systems is their interaction without changing of the velocity and the only result is a shift of their coordinates. The specific type of soliton interaction in the frame of the Sine-Gordon model, as well as in other exactly integrable models, leads to completely different kinetics as compared with usual particles or quasiparticles [3,4]. In particular, the main mechanism of a relaxation deal with soliton - soliton collisions with shifts of their coordinates. This property takes place in the systems, which are close to the integrable ones also. Here the kinetic equation for a system of 1D adatom structures has been constructed and the self-diffusion coefficients has been calculated. The process of adatoms’ homogenization is discussed. MODEL Let us consider the model of a 2D rhombic crystal with inhomogeneity on the y coordinate only. This model is reasonable in the case of weak interaction of adatoms with the r surface of a substrate. The energy of the displacement u ( y, z ) of the system of adatoms can be written in the form [1]
P3.2.1
H=
[
]
r r 1 ρ u 2t + λ 1 u 2yy + 2λ 2 u 2zz + 2λ 6 u 2yz + λ 12 u yy u zz + 2V0 cos( 2π κ u ) dy dz ∫ 2
(1)
Here the u ik are the components of the strain tensor, i, k = y, z; V0 is the const
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