The Electronic Heat Capacity of Metals

$\displaystyle \shadowbox{ $ \displaystyle \bar E \approx E_0 + \gamma T^2 $\ }$

(34)

where

$\displaystyle E_0$	$\displaystyle = \frac{4 \pi V}{5} \, (2s+1) \, \left( \frac{m}{2 \pi^2 \hbar^2}\right)^{3/2} \, {\epsilon_F}^{5/2}$ and	(35)
$\displaystyle \gamma$	$\displaystyle = \frac{\pi^3 V (2s+1)}{3} \, \left( \frac{m}{2 \pi^2 \hbar^2}\right)^{3/2} \, {\epsilon_F}^{1/2} \, {k_B}^2.$	(36)

This means that the heat capacity for a collection of fermions has the form

$\displaystyle \shadowbox{ $ \displaystyle C_v = \left. \frac{\partial\bar E}{\partial T}\right\vert _V = 2 \gamma T. $\ }$

(37)

If the electrons in metals behave like non-interacting fermions in a box, then this result also should describe the heat capacity of metals. In fact it does. But only at very very low temperatures. You'll be able to see this yourself if you do the heat capacity lab. Most metals' heat capacity is more accurately described by

$\displaystyle C_V = 2 \gamma T + \alpha T^3,$ (38)

at least for low temperatures. The term comes from quantized lattice vibrations, known as phonons, rather than electrons' kinetic energy. It results from another mechanism and will be discussed elsewhere. For now, we're done talking about free electrons in a box.

$\displaystyle \bar E$	$\displaystyle \approx$	$\displaystyle 2 \pi V \, (2s+1) \, \left( \frac{m}{2 \pi^2 \hbar^2}\right)^{3/2... ...\epsilon^{3/2} \, d\epsilon + \frac{\pi^2}{4} \, \sqrt{\mu} \, (k_BT)^2 \right]$	(31)
	$\displaystyle =$	$\displaystyle 2 \pi V \, (2s+1) \, \left( \frac{m}{2 \pi^2 \hbar^2}\right)^{3/2... ...ft[ \frac{2}{5} \, \mu^{5/2} + \frac{\pi^2}{4} \, \mu^{1/2} \, (k_BT)^2 \right]$	(32)
	$\displaystyle \approx$	$\displaystyle 2 \pi V \, (2s+1) \, \left( \frac{m}{2 \pi^2 \hbar^2}\right)^{3/2... ... \right)^2 \right) + \frac{\pi^2}{4} \, {\epsilon_F}^{1/2} \, (k_BT)^2 \right],$	(33)