Thermodynamics of the superconducting transition

In the following we provide first a simple thermodynamic treatment of the superconducting transition, which sheds some light onto the behavior of the physical system, and then we give an elementary account of the BCS theory of superconductivity. We conclude this section with a short discussion of the high-temperature superconductors and their differences from the conventional ones. 

In order to discuss the thermodynamics of the superconducting transition as a function of the external magnetic field H and temperature T we consider a simplified description in terms of two components, the normal (denoted by the superscript ( )) and superconducting (denoted by the superscript (5)) parts. The system attains equilibrium under given H and T by transferring some amount from one to the other part. We wifi assume that the normal part occupies a volume and the superconducting part occupies a volume Qj, which add up to the fixed total volume of the specimen -f 2. For simplicity we consider a type I superconductor in 

Differentiating the last expression with respect to T we obtain 

From the definition of /q we can view the two partial derivatives on the left-hand side of the above equation as derivatives of Helmholtz free energies at constant magnetization M (in this case zero), whereupon, using standard thermodynamic relations (see Appendix C), we can equate these derivatives to the entropies per unit volume of the normal and superconducting parts, s  

Differentiating both sides of Eq. (8.3) with respect to temperature, and using the standard definition of the specific heat C = dQjdT = 7 (d5/d7 ), we find for the difference in specific heats per unit volume  

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