PX284 - M4 - heat capacity

the only accessible empty energy states are those close to $E_{F}$
only the electrons very close to $E_{F}$ can change their momenta and carry any electrical/thermal energy
at $T > 0$ , $f_{F D}$ is not a step function, but 'smeared out' by $\sim k_{B} T$ , where $k_{B} T ≪ E_{F}$

roughly, the fraction of electrons that gain energy at $T \sim \frac{k_{B} T}{E_{F}}$
so, the energy gained $\sim k_{B} T$
the internal energy due to $T > 0 :$

U = N \frac{k_{B} T}{E_{F}} k_{B} T ≃ \frac{N k_{B}^{2} T^{2}}{E_{F}}

∴ C_{electron} ≃ 2 \frac{N k_{B}^{2} T}{E_{F}}

the full solution is: $c_{electron} = \frac{π^{2}}{2} \frac{N k_{B}^{2} T}{E_{F}}$
therefore, for a metal at low temperature:

C_{t o t a l} = C_{e l e c t r o n} + C_{l a t t i c e} = α (\frac{T}{T_{F}}) + β {(\frac{T}{θ_{D}})}^{3}

where, $C_{l a t t i c e}$ comes from the debye model

so, $C_{l a t t i c e}$ dominates, except at very low temperatures