High Temperature Superconducting Magnetic Levitation

6 Ë 1 Fundamentals of superconductivity

The Gibbs free energy can be obtained by the integral

G(T,p,H)−G(T,p, 0 )= −휇 0

H

X

0

MdH. (1.4)

A magnetic field which is applied to the superconductor can produce negative ma-
gnetization, i.e. the magnetizationM= −H. The magnetic flux of the negative ma-
gnetization exactly offsets that caused by the external magnetic field – this is the
Meissner-Ochsenfeld effect that describes the superconductor, i.e.B=0. Therefore,
the Gibbs free energy,gper unit volume

gs(T,p,H)=gs(T,p, 0 )+^1

2

휇 0 H^2 , (1.5)

where the subscripts indicates the superconducting state, and the magnetic energy
density,휇 0 H^2 /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=휒H, magnetic susceptibility휒is very small (about 10−^5 ),
it can be described as

gn(T,p,H)≈gn(T,p, 0 ), (1.6)

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 fieldHc,

gn(T,p,Hc)=gs(T,p,Hc), (1.7)

then Eq. (1.5) becomes

gn(T,p, 0 )−gs(T,p, 0 )=^1

2

휇 0 H^2 c. (1.8)

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 fieldHcof type I superconductor can be derived from Eq. (1.8).