# Applying the Sommerfeld model

1. Oct 20, 2014

### CAF123

1. The problem statement, all variables and given/known data
a) From the graph attached (see also Philips book) deduce the Fermi energy for copper using the Sommerfeld model
b) Estimate $v_s$ in copper, where $v_s$ is the speed of the low temperature phonon vibration.

2. Relevant equations
Sommerfeld model, $c_v = \pi^2/2 (k_B T/\epsilon_f) k_B$ per electron in sample.

3. The attempt at a solution
So I rewrite the equation above like $$c_v/T = \pi^2/2 (k_B /\epsilon_f)k_B.$$ From the sketch $C/T = A + BT^2$ so at $T=0, C/T = A$ in the given units. Sub this in gives $$\epsilon_f = \frac{\pi^2}{2} \left(\frac{(k_B)^2 (m^2 kg s^{-2}K^{-1})^2}{A \, mJ \, mol^{-1} deg^{-2})}\right) = \frac{\pi^2}{2} \left(\frac{(k_B)^2 J \text{mol}}{A}\right)$$ by cancelling out the units. Then there are avagadro number of atoms in a mole and copper has one conduction electron per atom, so the Fermi energy I get is $\epsilon_f \times 6.02 \times 10^23 \approx 3.75 eV$. The value I found from another source is about 7eV so the numbers are comparable but I was wondering if my method was right?

Thanks

#### Attached Files:

• ###### Graph1.jpg
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2. Oct 25, 2014