Phase Diagram and Phase Separation of a Trapped Interacting Bose--Fermi Gas Mixture

In six different regimes for a spatial phase diagram of a trapped interacting Bose--Fermi gas mixture at low temperatures, we present the conditions for the spatial demixing and separation of bosons and fermions. Starting from a semiclassically thermodynamic model for the local density functional of thermal bosons and fermions, the explicit analytical expressions for the fugacities of bosons and fermions are derived in different regimes by means of a first-order perturbation method in a local-density approximation. The critical values of the fermion--boson interaction strength as a function of the fractional composition of fermions have a general feature: increase, extreme and decrease with increasing the fermionic composition slightly above Bose--Einstein critical temperature.