Thermodynamically Consistent BCS Model

We now start with the exponential form of the heat capacity equation (Equation 26.12) and work backward to find the free energy of the superconducting state and eventually obtain the critical magnetic field as a function of temperature. 

The entropy of the superconducting state must equal the electronic entropy of the normal state at T= Tc, which requires 

The integral has the form of an exponential integral (Ei) and the constant thus becomes 

The heat capacity may then be related to the critical temperature by 

Taking the free energy of the normal state to be zero at T = 0, the free energy of the superconducting state at T = 0 is obtained by integrating the entropy from Tc to 0 and adding this to the Gcs(Tc) which is the same as Gn(Tc). 

---