 Solution

ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/section-a/a3/solution

n = ρ N_{A} / M ≃ 8.50 \times 1 0^{28} m^{- 3}

. Filling two spin states per momentum state in the three-dimensional Fermi gas gives

k_{F} = (3 π^{2} n)^{1/3}

and

E_{F} = \frac{ℏ ^{2}}{2 m _{e}} (3 π^{2} n)^{2/3} ≃ 7.05 eV, T_{F} = E_{F} / k_{B} ≃ 8.18 \times 1 0^{4} K .

The Fermi temperature greatly exceeds ordinary temperatures, explaining why these conduction electrons are degenerate rather than a classical Maxwell gas. The numerical result uses the stipulated free-electron band and bare electron mass, rather than a detailed copper band structure.

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