- #1
MathematicalPhysicist
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- 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!
$$\frac{1}{VD} = \sinh^{-1} (\hbar \omega/\Delta(0)) = \sinh^{-1}(\hbar \omega/ \Delta(T)) - 2\int_0^{\hbar \omega_D}\exp(-\beta E)/E dE$$
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!