Critical field for a type I superconductor

[quasar_4]

Homework Statement

Consider a type I superconducting material with a parabolic coexistence curve separating the uniform superconducting and normal phases. H is the external magnetic field and T is temperature. Ignore the tiny magnetization of the normal phase. The critical field is
Hc = H0 +a*T + b*T^2.

a) Why must the coefficient a =0?
b) Find the latent heat per unit volume as a function of T along the coexistence curve
c) Calculate the discontinuity in the specific heat per unit volume at constant H along the coexistence curve

Homework Equations

Hc = H0 +a*T + b*T^2

The Attempt at a Solution

I'm mostly at a loss on this one. I know that with a type I superconductor, the Meissner effect vanishes instantly if the external magnetic field exceeds this critical field, Hc, but I have no idea why that means the coefficient a has to vanish. Is it just that we want the vertex of the parabola to lie on the field axis?

I'm guessing that for part b I want to use the Clausius-Clapeyron relation, but not sure... and as for c I'm afraid I can't say much until I do a and b. Any hints? Any good references that actually discuss this? This is supposed to be part of "thermodynamics review" from undergrad days, but my undergrad text says almost nothing about superconductors, so it's not helping too much.

 

[mark726]

a) The coefficient a must be zero because at the critical field, the superconducting and normal phases have the same free energy. This means that the slope of the coexistence curve at the critical field must be zero, and therefore the coefficient a must be zero.

b) To find the latent heat per unit volume, we can use the Clausius-Clapeyron relation, which relates the latent heat to the slope of the coexistence curve:

L = T*(dHc/dT)

Since a=0, the critical field is only dependent on T through the term b*T^2. Therefore, we can rewrite the above equation as:

L = T*(2bT)

L = 2b*T^2

c) The discontinuity in specific heat per unit volume at constant H is given by the difference in specific heat between the two phases at the coexistence curve, divided by the temperature range:

ΔC = (Csc - Cn)/ΔT

Where Csc and Cn are the specific heats per unit volume of the superconducting and normal phases, respectively. At the coexistence curve, the two phases have the same free energy, so:

Csc = Cn

Therefore, the discontinuity in specific heat is zero along the coexistence curve.