Page 396 - Electrical Properties of Materials

378 Superconductivity

d W =–VHμ 0 (d H +d M)
=–μ 0 VH d H – μ 0 VH d M. (14.10)
The first term on the right-hand side of eqn (14.10) gives the increase of
energy in the vacuum, and the second term is due to the presence of the
magnetic material. Thus, the work done on the material is

d W =–μ 0 VH d M. (14.11)
Hence, for a paramagnetic material the work is negative, but for a diamagnetic
material (where M is opposing H) the work is positive, which means that the
system needs to do some work in order to reduce the magnetic field inside the
material.
Now to describe the phase transition in a superconductor, we have to define
a ‘magnetic Gibbs function’. Remembering that P dV gives positive work (an
expanding gas does work) and H d M gives negative work, we have to replace
PV in eqn (14.3) by –μ 0 VHM. Our new Gibbs function takes the form,
G = E – μ 0 VHM – TS, (14.12)

and
dG =dE – μ 0 VH d M – μ 0 VM d H – T dS – S dT. (14.13)

Taking account of the first law for magnetic materials (again replacing
pressure and volume by the appropriate magnetic quantities),

dE = T dS + μ 0 VH d M. (14.14)
Equation (14.13) reduces to

dG =–S dT + μ 0 VM d H. (14.15)
This is exactly what we wanted. It follows immediately from the above equa-
tion that for a constant temperature and constant, magnetic field process,
G remains constant while the
superconducting phase transition
dG = 0. (14.16)
takes place.
For a perfect diamagnet,

M =–H, (14.17)
which substituted into eqn (14.15) gives

dG =–S dT + μ 0 VH d H. (14.18)
G s (0) is the Gibbs free energy at Integrating at constant temperature, we get
zero magnetic field, and the sub- 1 2
script s refers to the superconduct- G s (H)= G s (0) + μ 0 H V. (14.19)
2
ing phase.
Since superconductors are practically non-magnetic above their critical
temperatures, we can write for the normal phase,
G n (H)= G n (0) = G n . (14.20)