Applying the generalized Bose–Einstein condensation (GBEC) formalism, we extend the BCS-Bose crossover theory by explicitly including hole Cooper pairs (2hCPs). From this follows a phase diagram with two pure phases, one with 2hCPs and the other with electron Cooper pairs (2eCPs), plus a mixed phase with arbitrary proportions of 2eCPs and 2hCPs. One has a special-case phase when there is perfect symmetry (i.e., with ideal 50–50 proportions between 2eCPs and 2hCPs). Explicitly including 2hCPs leads to an extended BCS-Bose crossover which predicts [Formula: see text] values for some well-known conventional superconductors (SCs) (i.e., assuming electron–phonon dynamics). These compare reasonably well with experimental data. We compare with experimental [Formula: see text] values for some conventional SCs associated with the new dimensionless number density [Formula: see text] with theoretical curves associated with the extended crossover for the special case of perfect symmetry. They all obey the Bogoliubov et al. upper limit, thus vindicating it.
The generalized Bose-Einstein condensation (GBEC) theory of superconductivity (SC) is briefly surveyed. It hinges on three distinct new ingredients: (i) Treatment of Cooper pairs (CPs) as actual bosons since they obey Bose statistics, in contrast to BCS pairs which do not obey Bose commutation relations; (ii) inclusion of two-hole Cooper pairs (2hCPs) on an equal footing with two-electron Cooper pairs (2eCPs), thus making this a complete boson-fermion (BF) model; and (iii) inclusion in the resulting ternary ideal BF gas with particular BF vertex interactions that drive boson formation/disintegration processes. GBEC subsumes as special cases both BCS (having its 50-50 symmetry of both kinds of CPs) and ordinary BEC theories (having no 2hCPs), as well as the now familiar BCS-Bose crossover theory. We extended the crossover theory with the explicit inclusion of 2hCPs and construct a phase diagram of Tc/T F versus n/n f , where Tc and T F are the critical and Fermi temperatures, n is the total number density and n f that of unbound electrons at T = 0. Also, with this extended crossover one can construct the energy gap ∆(T )/∆(0) versus T /Tc for some elemental SCs by solving at least two equations numerically: a gap-like and a number equation. In 50-50 symmetry, the energy gap curve agrees quite well with experimental data. But ignoring 2hCPs altogether leads to the gap curve falling substantially below that with 50-50 symmetry which already fits the data quite well, showing that 2hCPs are indispensable to describe SCs.
The new generalized Bose–Einstein condensation (GBEC) quantum-statistical theory starts from a noninteracting ternary boson-fermion (BF) gas of two-hole Cooper pairs (2hCPs) along with the usual two-electron Cooper pairs (2eCPs) plus unpaired electrons. Here we obtain the entropy and heat capacity and confirm once again that GBEC contains as a special case the Bardeen–Cooper–Schrieffer (BCS) theory. The energy gap is first calculated and compared with that of BCS theory for different values of a new dimensionless coupling parameter n/n[Formula: see text] where n is the total electron number density and n[Formula: see text] that of unpaired electrons at zero absolute temperature. Then, from the entropy, the heat capacity is calculated. Results compare well with elemental-superconductor data suggesting that 2hCPs are indispensable to describe superconductors (SCs).
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