- #1
neophysicist
- 3
- 0
Homework Statement
Hello everyone!
I'm using the text:
"Elements of Solid State Physics - JP Srivastava (2006)"
I have followed the argument leading up to the derivation of the Debye formula for specific heat capacity, so we now have;
[tex]
C_V = \frac{9N}{\omega_D^3} \frac{\partial}{\partial T} \int_0^{\omega_D}\frac{\hbar \omega^3}{exp(\frac{\hbar \omega}{k_BT})-1}d\omega
[/tex]
The next equation presented is the final form which I am having difficulty deriving.
[tex]
C_V = 9Nk_B(\frac{T}{\theta_D})^3 \int_0^{\frac{\theta_D}{T}}\frac{x^4e^x}{(e^x-1)^2}dx
[/tex]
Homework Equations
We are supposed to use the substitutions;
[tex]
\theta_D = \frac{\hbar\omega_D}{k_B}
[/tex]
and
[tex]
x = \frac{\hbar\omega}{k_BT}
[/tex]
The Attempt at a Solution
[tex]
d\omega = \frac{k_BTx}{\hbar}dx
[/tex]
[tex]
\therefore \
C_V = \frac{9Nk_B}{\theta_D^3} \int_0^{\frac{\theta_D}{T}}
\frac{\partial}{\partial T}(\frac{T^4x^3}{e^x-1})dx
[/tex]
Now is where I run into difficulty. Applying the quotient rule I get;
[tex]
C_V = 9Nk_B(\frac{T}{\theta_D})^3 \int_0^{\frac{\theta_D}{T}}
\frac{x^3(e^x-1)+x^4e^x}{(e^x-1)^2})dx
[/tex]
So I can see that I am tantalizingly close but clearly I must be making a dumb mistake somewhere.
I would be grateful if anybody could help me out as this is really bugging me and it's chewed up enough of my revision time already!