BCS Superconductivity with Fixed Number Parity
  We investigate superconductivity in a grand canonical ensemble with {\it fixed number parity} (even or odd). In the low temperature limit we find small corrections to the BCS gap equation and energy spectrum $E(k)$. The even-odd free energy difference in the same limit decreases linearly with temperature, in accordance with the behavior observed experimentally and previously arrived at from a quasiparticle model. The theory yields deviations from the BCS predictions for the specific heat, ultrasound attenuation, NSR relaxation rate, and electromagnetic absorption. 
Boldizsar Janko, Anders Smith, and Vinay Ambegaokar