Question

1. At absolute zero temperature, all states within the Fermi sphere are occupied by electrons with spin state of half. Take density  cm-3. Calculate

i. Fermi wave vector.

(3)

ii. Fermi energy.

(3)

iii. Fermi momentum.

Answer 1

Solution.

$n\approx10^{22}cm^{-3}=10^{28}m^{-3};$

$n=\dfrac{k_F^3}{3\pi^2}\implies k_F=(3\pi^2n)^{1/3};$

$k_F=(3\sdot3.14^2\sdot10^{28})^{1/3}=6.53\sdot10^9m^{-1};$

$E_F=\dfrac{h^2k_F^2}{2m};$

$E_F=\dfrac{(6.626\sdot10^{-34})^2\sdot(6.53\sdot10^9)^2}{2\sdot9.1\sdot10^{-31}}=15.75\sdot10^{-19}J;$

$p_F=hk_F;$

$p_F=6.626\sdot10^{-34}\sdot6.53\sdot10^9=43.27\sdot 10^{-25}kgm/s;$

Answer: I.$k_F=(3\sdot3.14^2\sdot10^{28})^{1/3}=6.53\sdot10^9m^{-1};$

II.$E_F=15.75\sdot10^{-19}J;$

III.$p_F==43.27\sdot 10^{-25}kgm/s.$