Simple BCS theory: question about the internal energy

[FranzDiCoccio]

Hi all,

I'm am reading chapter 3 of Tinkham's "Introduction to superconductivity".
At some point several thermodynamic quantities are considered (sec. 3.6.3).

In particular fig. 3 compares the internal energies of the superconducting and normal states,
showing that they are different below the critical temperature whereas they overlap above it.
Of course this makes definitely sense, but |'m puzzled by the calculations.

The internal energies are evaluated as integrals of the relevant specific heats. Since
the specific heats are different below T_c and overlap above T_c I would expect that for T > T_c

$$U_{es}(T) = U_{en}(T) + \int_0^{T_c} [C_{es}(T)-C_{en}(T)] dT$$

where "es" and "en" refer to the superconductive and normal states, respectively.
Hence the integral should vanish for the two quantities to coincide, right?

Now it seems to me that this hardly applies for the curves in fig. 3b, but this could
very well a be graphical problem, like a poor choice of the curves.

The above condition is somehow enforced in the calculations just before eq. (3.60), where,
if I get it right, the author sets

$$U_{es}(T_c) = U_{en}(T_c)$$

"since the specific heat remains finite there". I'm not sure I get this argument. The finiteness of the "es" specific heat ensures that the "es" internal energy is continuous, but not that it has a particular value (the same as the "en" internal energy).
So this feature of the specific heat does not seem to me a sufficient reason for equalling
the two internal energies at the critical temperature.
Am I missing something?

Thanks a lot for your help

FPS the equation I'm seeing in my preview of this post are altogether different from what
I've typed. I hope that the submitted version of this post is ok. If this is not the case, placing the mouse pointer on the equations seems to correctly give the equation I've typed, although in latex format.

PPS unfortunately the equations are not displayed correctly... I do not understand what the problem is... I've always used this syntax... I've tried to modify my post, but I can't manage to have the equations display properly.

 

[ZapperZ]

LaTex function isn't working properly yet since the recent server upgrade. This problem is being worked on.

Zz.

 

[FranzDiCoccio]

Oh, I see...

well, luckily my equations are not very complex. I hope they can be understood from
their latex syntax, which appears correctly when the mouse pointer is placed on the
equation image.

Using a "mixed style" they are:

U_es(T) = U_en(T) + S₀^Tc (C_es-C_en) dT

(where the big S stands for the integral symbol) and

U_es(T_c) = U_en(T_c)

Franz

 

[FranzDiCoccio]

It's me again...

Actually the first equation in my previous message(s) is not correct. It should be:


U_es(T) = U_en(T) + S₀^Tc (C_es-C_en) dT - U_en(0) + U_es(0)

(T>T_c) which means that the equality of the two energies does not yield the vanishing of the integral.
So much for my first observation.

I still do not completely understand why U_es(T_c) = U_en(T_c).