When Does the Specific Heat of Free Electrons Surpass the Lattice Specific Heat?

[steph_mil]

Homework Statement

At what temperature T does the specific heat of the free electrons become larger than the specific heat of the lattice? Express T in terms of the Debye temperature and the electron concentration. Calculate T for copper (Debye = 343 K).

Homework Equations

Unfortunately, I have a ton of equations but have no clue which one to use. There are different equations for low and high temperatures, so I'm not sure where to start with this problem.

The Attempt at a Solution

Once I have the first part, I can solve for T for copper. I think I'm supposed to use some equations involving heat capacity and temperature, but no clue which one? Any hints may help! My book (Kittel's intro to solid state) is not helping...

Thanks!

 

[Mute]

I'm assuming this is being done in the low temperature limit, where the total (constant volume) heat capacity of the metal goes as c_v = \alpha T + \gamma T^3, where the linear term corresponds to the electrons and the cubic term corresponds to the lattice (phonons). (In the high temperature limit both capacities, per particle, should be the same, making for a much less interesting problem!) I'm assuming you have what \alpha and \gamma are in your list of equations - I can't remember what they are, but the debye temperature is buried in \gamma, and presumably the electron concentration is buried in \alpha.

Anyways, from there it's a simple matter: determine when \alpha T/(\gamma T^3) = \alpha/(\gamma T^2) > 1.