A Boson-Fermion theory that goes beyond the BCS approximations for superconductors
A detailed analysis is given of the effects of common and recurring approximations used in conventional superconductivity theories on the condensation energy values, whose magnitudes are notoriously smaller than those of other energies as the superconducting energy gap and the chemical potential. These approximations come from using the density of states $N(ε)$ and the chemical potential $μ(T)$ either constant or temperature-dependent, respectively. We use these approximations, a total of three, to calculate the critical temperature $T_{c}$, the superconductor energy gap $Δ(T)$, the chemical potential $μ(T)$ and the thermodynamic potential $Ω(T)$ which are needed to obtain the condensation energy, and compare them with the exact case, i.e., where no approximations are used. To do this, we use a ternary Boson-Fermion theory of superconductivity composed of unbound electrons (or holes) as fermions plus two-electron and two-hole Cooper pairs, both as bosons. Although all these approximations lead to reasonable values of $T_{c}$ and $Δ(T)$, the resulting thermodynamic and chemical potentials are quite different, so that the condensation energy value could be incorrect. However, when $N(ε)$ and $μ(T)$ variables are used, together with a correct physical interpretation of the condensation energy as the sum of the thermodynamic and chemical potential differences, it leads to a better agreement with reported experimental data, compared to the one obtained when taking them as constants, particularly so for low temperatures.