Debye's T^3 Law: Specific heat, Latice and Electronic terms

[12x4]

Homework Statement

QUESTION ADDED AS ATTACHMENT AS NEED TO SEE GRAPH.

Homework Equations

C = (12Nk_Bπ⁴/5)(T/θ_D)³ for T<<θ_D
C = 3Nk_B for T>>θ_D

The Attempt at a Solution

a.)[/B] So I assume the expression for the specific heat as a function of temperature that the question must want:
C = (12Nk_Bπ⁴/5)(T/θ_D)³
otherwise I thought that the specific heat wasn't dependent on temperature?

The second part of this question really stumped me. I have looked though all of the books in my university library that aren't currently on loan, of which few relevant ones are left for some reason, and can't find anything that helps me extract lattice and electronic terms in the specific heat. Does anyone have any Ideas?

EDIT: Just found something that shows the electronic contribution:
C_e = (π²/3)g(E_Fk_b²T = γT

Therefore the electronic contribution = γ = (π²/3)(3N_A/2E_F)k_B²

Still not sure on the lattice part.
\EDITb.) I think I can do.

c.) The low temperature heat capacity for KCl plotted as to demonstrate the T³ law at low temperatures. The fact that the graph of C/T vs T² goes through the origin indicates the absence of a term linear in T. I.e there is no contribution to the energy via conduction electrons.

Am I along the right lines here?

 

[Nic_1995]

how did you do part b?

 

[12x4]

γ is taken as the y intercept on the graph. Then rearrange the electronic contribution formula for E_F.

 

[Nic_1995]

also the lattice term is just C for low temperature, so the term you have written for C