 Solution

ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/section-a/a5/solution

A spin-half gas has two spin states. Filling the Fermi sphere at zero temperature gives

n = k_{F}^{3} / (3 π^{2})

, so

k_{F} = (3 π^{2} ρ / m_{3})^{1/3}, E_{F} = \frac{ℏ ^{2} k _{F}^{2}}{2 m _{3}}, v_{F} = \frac{ℏ k _{F}}{m _{3}} .

With

m_{3} ≃ 5.008 \times 1 0^{- 27} kg

1.62 \times 1 0^{28} m^{- 3}

and

k_{F} ≃ 7.82 \times 1 0^{9} m^{- 1}

. Thus

E_{F} ≃ 6.80 \times 1 0^{- 23} J ≃ 0.424 meV, v_{F} ≃ 165 m s^{- 1} .

The associated Fermi temperature is

T_{F} = E_{F} / k_{B} ≃ 4.92 K .

Fermionic degeneracy matters on cooling to a few kelvin or below in this model, and is pronounced for

T ≪ T_{F}

 New to topics? Read the docs here!