Dr. Alexander Filonov

Abstract:

A dynamic theory of ordering of the DFK hole dislocations is put forward.

The results of the developed FK and DFK models are incorporated into the AIM theory to answer the question: how does the process of ordering dislocations of the DFK model proceed into a commensurate crystal?

Consider the DFK model, a system of two N atomic elastically periodic chains (A, B).

Let the elasticity of the springs of chain (B) be equal to infinity, and opposite forces are applied to the ends of chain (A) by jerk. As a result of a strong jerk atoms of chain A leave the region of interaction with chain B, and a commensurate crystal with a doubled period is formed. Each element of this crystal is a Frenkel-Kontorova hole dislocation (FK) [1]. 

An FK is a formation that has a number of properties that coincide with the characteristics of a point particle, namely: – mass M; incompressible size equal to 2; kinetic and potential energy, etc. 

A commensurate crystal after a jerk is not formed immediately, but as atoms fly out of the region of interaction of chains, during the movement of dislocations to the center of the system.

On a chain (length N), dislocations appear at its edges and are arranged in pairs symmetrically relative to the center.

It is of interest to write down the spatiotemporal equation of the FK ordering process, starting from the departure of the first two edge atoms of the stretched chain to the departure of its last L atoms.

Let us associate the emitted J-atom of the DFK chain with a dislocation with number J at its end, J≤J₀, J₀ = L/2. The remaining FK dislocations numbered i, 1 ≤ i < J, are located between the center of the chain and its edge. FK move towards the center of the chain. 

We will describe the dynamics of dislocation ordering depending on the number of the ejected J-atom using the AIM model.

The theory states:

1. DFK dislocation with number i is an analogue of the i-th moment of time, located at the moment of time J at a distance from the center of the system. 

2. R(J) - discrete Lagrangian of the AIM model.

R(J) has the form:

                                    (1)

3. determined from the equation:

                                                                                                                                 (2)

4. Our goal is to find all values of ; 1≤ i 0, , with the final solution:

                                                                                        (3)

where L is the main parameter of the model.

5. Parameters M_J, V(J) are found from the system of inequalities:

                                                                                                            (4)

where M_J is the mass of atoms in a chain of 2J dislocations, atoms are called DFK dislocations; - period of an elastic-periodic chain, with an elasticity coefficient equal to 1; 

V(J) is the periodic potential in which the dislocation chain is located.