Dr. Alexander Filonov

Abstract:

The DFK model is formed by replacing the external periodic potential of the FK model with a second elastically periodic chain of atoms. The properties of the FK and DFK models are basically the same, but the role of the boundary atoms of the DFK model in the structural phase transitions of the "commensurate- incommensurate " phase (IC) has increased. The IC transitions have become asymmetrical in parameter of incommensurate.

In 1938, the authors: Yakov Ilyich Frenkel and Tatyana Abramovna Kontorova (LFTI) put forward the Frenkel-Kontorova model (FK model) [1], which served as the basis for the creation of many theories of highly nonlinear processes [2].

It is of interest to develop the FK model in order to expand the scope of its natural science applications [3-4].

In order to develop the FK model, the DFK model (Developed Frenkel-Kontorova model) is put forward: - two one-dimensional sequences of N and L point atoms, masses m and M; with coordinates {x_i} and {y_j}, connected by elastic springs with the laws of elastic dispersion Φ₁(x) and Φ₂(y). Chains CH1 and CH2 interact with each other by potential V_i,j.

The Hamiltonian of the DFK model has the form: 

                                    (1)

From the analysis of the ground state of the DFK model (N = L) [3], the following conclusion: - when one of the Hooke’s chains is stretched by force F, an abrupt transition to the incommensurate phase occurs (F>F_c), in which part of the atoms of the stretched chain CH1 leaves the interaction space with CH2. The number of atoms falling out of the V_i,j interaction space , where V₀ = max V_i,j.

With strong interaction (V₀ ~ 1) and strong stretching (F>F_c ~1), the size of the dislocation is 2, and the number of precipitated atoms is N/2. In this case, the incommensurate phase will be a periodic chain of hole dislocations, i.e., commensurate crystal with doubled period.