  Experimental realizations of a variety of atomic binary Bose-Fermi mixtures
have brought opportunities for studying composite quantum systems with
different spin-statistics. The binary atomic mixtures can exhibit a structural
transition from a mixture into phase separation as the boson-fermion
interaction increases. By using a path-integral formalism to evaluate the grand
partition function and thermodynamic grand potential, we obtain the effective
potential of binary Bose-Fermi mixtures. Thermodynamic quantities in a broad
range of temperatures and interactions are also derived. The structural
transition can be identified as a loop of the effective potential curve, and
the volume fraction of phase separation can be determined by the lever rule.
For $^6$Li-$^7$Li and $^6$Li-$^{41}$K mixtures, we present the phase diagrams
of the mixtures in a box potential at zero and finite temperatures. Due to the
flexible densities of atomic gases, the construction of phase separation is
more complicated when compared to conventional liquid or solid mixtures where
the individual densities are fixed. For harmonically trapped mixtures, we use
the local density approximation to map out the finite-temperature density
profiles and present typical trap structures, including the mixture, partially
separated phases, and fully separated phases.
