Pairs of inversely occupied electronic states in the energy range best suited to wave generation

Abstract

For a model electronic spectrum of a bcc crystal, characteristic s-surfaces are found that separate, in the quasimomentum space, pairs of electronic states potentially active in the generation of elastic waves at the stage of growth of a martensite crystal. Numerical estimation of the fraction of active electronic states (R eff /R) is performed for the energy range ∆ = 0.2 eV near the Fermi level. The ratio R eff /R is calculated as a function of the parameter ratio for the interaction between the nearest and the second neighbors for both bcc and fcc lattices.In the wave model of the growth of a martensite crystal [1] of the fcc-bcc (γ-α) type, the central part is played by the mechanism by which the waves of displacement of atoms by nonequilibrium 3d-electrons are generated. The key parameter is the number of pairs of inversely occupied electronic states which are able to effectively participate in the generation of phonons, R eff . The nonequilibrium character of the electronic subsystem in the interphase region at the stage of crystal growth under reconstructive martensitic transformations is set mainly by the chemical potential gradient ∇µ (we designate its direction by e). The pairs of electronic states potentially active in the generation of displacement waves possess the opposite signs of the nonequilibrium components ∆f of the Fermi electron distribution function. In the quasi-momentum space, these pairs of states are separated by s-surfaces on which the quantity ∆f proportional to the scalar product (v, ∇µ), where v is the group velocity of the electrons, vanishes.If the sections of the s-surfaces whose energies of states, ε, are in an acceptable (of the order of 0.1 eV) range of values of the deviations ∆ from the Fermi level µ are considerable in area, the number of pairs of inversely occupied electronic states able to effectively participate in the generation of phonons, R eff , will be great, and this will ensure the fulfillment of the condition for wave generationwhere σ 0 ~ (v, ∇µ) is the initial inverted population of states and σ th is their threshold inverted population.Clearly, for a known analytic form of the law of electron dispersion, ε(k), it is possible to determine the field of velocities v(k) = ∇ε k and then to find the s-surfaces that separate pairs of inversely occupied electronic states in the k-space. The goal of this work is to construct the s-surfaces typical of the tight-binding approximation for bcc crystals and to calculate the areas ∆S of sections of the s-surfaces (note that R eff ~ ∆S) for various values of µ and parameters of the interaction between the nearest and the second neighbors for both bcc and fcc lattices for a fixed electron energy range, ∆ ≈ 0.2 eV, near the Fermi level.