Problem 7.7 and its solution from A Quantum Approach to Condensed Matter

[MathematicalPhysicist]

Homework Statement

Problem 7.7 asks for:
How does the electronic specific heat of a superconductor vary with temperature $T$ as $T\to 0$?

Relevant Equations

See the attachment below.

Well, I don't understand the integral part of $1/(VD) = \int_0^{\hbar \omega_D}\frac{\tanh(\beta E/2}{E}dE$ and $\tanh(\beta E/2) \approx 1-2\exp(-\beta E)$, then he writes the following (which I don't understand how did he get it):
$$\frac{1}{VD} = \sinh^{-1} (\hbar \omega/\Delta(0)) = \sinh^{-1}(\hbar \omega/ \Delta(T)) - 2\int_0^{\hbar \omega_D}\exp(-\beta E)/E dE$$

If I plug the approximation of $\tanh$ I get the following:
$$1/(VD)=\log(\hbar \omega_D)-\log 0 -2\int \exp(-\beta E)/E dE$$

Doesn't seem to converge.
I don't understand this solution...
Any help?

Thanks!

 

[mfb]

The given approximation is good for large $\beta E$ (which is generally what you want), but the integral needs the tanh at small $\beta E$.
$\tanh(\beta E/2) \approx 1-2\exp(-\beta E)$ makes the integral diverge as this approximation doesn't go to zero for $E\to 0$.

I don't have the book, could there be a mistake with the integral borders?

 

[vela]

I noticed in the integral, it's with respect to $d\hat \varepsilon$ rather than $dE$. How is $\hat \varepsilon$ defined?

 

[MathematicalPhysicist]

I noticed in the integral, it's with respect to $d\hat \varepsilon$ rather than $dE$. How is $\hat \varepsilon$ defined?

Yes, I think you are correct.
On page 249 in Eq. (7.5.7) we have the following identity:
$$E_{\vec{k}}=[\hat{\epsilon}_{\vec{k}}^2+\Delta^2_k(T)]^{1/2}$$

I think I can see how the calculation is done, and I believe it should be $\frac{1}{VD} = \sinh^{-1} (\hbar \omega/\Delta(0)) - \sinh^{-1}(\hbar \omega/ \Delta(T)) - 2\int_0^{\hbar \omega_D}\exp(-\beta E)/E d\hat{\epsilon}$, I'll do the calculation and I'll let you know if I need more help.

Thanks!