On the Bose-Einstein condensation of nonzero-momentum cooper pairs. A second-order phase transition

Based on the BCS Hamiltonian, the normal-to-super phase transition is investigated, approaching the critical temperatureT _c from the high-temperature side. Nonzero-momentum Cooper pairs, that is, pairs of electrons (holes) with antiparallel spins and nearly opposite momenta aboveT _c in the bulk limit, are shown to move like independent bosons with the energy vs. momentump relation=1/2v_F , where _F represents the Fermi velocity (1/2m* _F ² _FFermi energy). The system of free Cooper pairs undergoes a phase transition of the second order with the critical temperatureT _c given byk _B T _c=1/2(²³ _F ³ n/1.20257)^1/3 wheren is the number density of Cooper pairs. The ratio of the jump of the heat capacity, C, to the maximum heat capacity,C _s, is a universal constant: C/C _s=0.60874; this number is close to the universal constant 0.588 obtained by the finite-temperature BCS theory. The physical significance of these results is discussed, referring to the well-known BCS theory, which treats the many-Cooper-pair ground state exactly and the thermodynamic state belowT _c approximately. An explanation is proposed on the question why sodium should remain normal down to 0 K, based on the band structures with the hypothesis that the supercondensate composed of zero-momentum electron and hole Cooper pairs is electrically neutral.