Formation of a large polaron crystal from a homogeneous, dilute polaron gas

We introduce a very simple method to study the phase transition from the homogeneous dilute large polaron gas toward the formation of a cubic crystal within the framework of the Vlasov [Many-Particle Theory and its Applications to Plasma (Gordon and Breach, New York, 1962)] nonlinear kinetic equation and the equations for the Bogolyubov [Problems of Dynamic Theory in Statistical Physics. Selected Works on Statistical Physics (Moscow State University, Moscow, 1979)] distribution functions. Two different critical temperatures T₁^cr and T₂^cr are introduced, defining the range of stability of the crystal phase (T₂^cr<T<T₁^cr). The existence and properties of the crystal phase are discussed as a function of the ionicity parameters of the medium, the electron-phonon Fröhlich coupling constant and the density of the dilute polaron gas. In particular, the lattice parameter is found to increase when the ionicity of the medium decreases whereas it is independent of the Fröhlich constant α and the polaron density. Furthermore, it increases upon decreasing the temperature from T₁^cr toward T₂^cr, showing a behavior different from that observed for common solids. We find also that a drift velocity of the polaron gas tends to reduce the stability of the crystal. To illustrate the simplicity of the calculation scheme, we apply the same technique to study the formation of the argon (Ar) crystal starting from the atomic gas interacting through a reliable Lennard-Jones potential. We calculate the crystallization temperature and the Ar atom distribution. The lattice parameter is an increasing function of the temperature and the second critical temperature is not defined. We try to relate the different dependence on the temperature of the lattice parameter to the features of the polaron-polaron and atom-atom interactions.