## Friday, December 9, 2011

### Fermi Energy

The Fermi temperature of Cu is about 80,000 K. Which of the following is most nearly equal to the average speed of a conduction electron in Cu?
A. $2\times{10^{-2}}$ m/s
B. $2$ m/s
C. $2\times{10^{2}}$ m/s
D. $2\times{10^{4}}$ m/s
E. $2\times{10^{6}}$ m/s
Solution:
Fermi Energy: $E_F=kT_F=\frac{1}{2}mv^2$
where
$k=$ Boltzmann's constant
$T_F=$ Fermi temperature
$m=$ mass of the particle
$v=$ speed of the particle
Since
$T_F =8\times{10^4}$ K
$k = 1.38\times{10^{-23}}$ Joule/K
$m_e = 9.11\times{10^{-31}}$ kg
($k$ and $m_e$ are provided by ETS in the problem sheet)
Therefore,
$v=\sqrt{\frac{2kT_F}{m_e}}$
$v=\sqrt{\frac{2\cdot1.38\cdot8}{9.11}\cdot\frac{10^{-23}\cdot10^4}{10^{-31}}}\approx{\sqrt{4\cdot{10^{12}}}}=2\cdot{10^{6}}$