1.4　Superconducting thermodynamics15

The phase transition from the normal state to the superconducting state can be dealt with the general principles of thermodynamics. Considering the magnetization effect for a magnetic medium, the Gibbs free energy G can be expressed as

Here, U is the internal energy, T is the temperature, S is the entropy, p is pressure, V is the volume，μ0 is the permeability of free space, and M is the magnetization.At constant temperature and constant pressure conditions，

The Gibbs free energy can be obtained by the integral

A magnetic field which is applied to the superconductor can produce negative magnetization, i. e.the magnetization M=−H, the magnetic flux of the negative magnetization exactly offsets that caused by the external magnetic field, This is the Meisner-Ochsenfeld effect describes the superconductor, i.e.B=0.Therefore, the Gibbs free energy, g per unit volume

where the subscript s indicates the superconducting state, and the magnetic energy density，μ0 H2/2，is independent of temperature.Thus, the superconducting state free energy in an external magnetic field will be increased due to the negative magnetization of superconductors.

In the normal state, M=χH, magnetic susceptibilityχis very small（about 10−5），it can be described as

where the subscript n indicates the normal state. The Gibbs free energy of the superconductor in the normal state is unchanged before and after applying the magnetic field.According to the balance conditions of phase transitions between superconducting state and normal state in the critical magnetic field Hc ，

then Eq.（1. 4.4）becomes

The Gibbs free energy in the superconducting state depends on the value of the critical field at that temperature. This confirms that there is a close relationship between superconductivity and magnetism.It shows that the free energy density of the superconducting state is lower than that of the normal state.Usually, this energy density is called as the condensation energy of the superconducting state.The thermodynamic critical field Hc of the type I superconductor can be derived from the Eq.（1.4.7）.

The above discussion deals with the phase transition between the normal state and the superconducting state at T＜Tc in a magnetic field. It is a first-order transition and which requires a latent heat for the phase change.At T=Tc the superconducting phase transition has no latent heat, but it has a specific heat jump.This shows that the superconducting phase transition at T=Tc is a second order phase transition.