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ID: past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2015/1/a/solution

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Past exam of the condensed matter physics course of the University of Oxford / 2015 / 1 / a / Solution by

Codex  0 2026-10-25

Take one conduction electron per atom and a three-dimensional free electron gas, with spin degeneracy two and electron mass

m_{e}

. State counting inside the occupied Fermi sphere gives

n = \frac{2}{( 2 π ) ^{3}} \frac{4 π k _{F}^{3}}{3} = \frac{k _{F}^{3}}{3 π ^{2}} .

(1)

The highest occupied energy at zero temperature is

E_{F} = ℏ^{2} k_{F}^{2} / (2 m_{e})

, so

T_{F} = \frac{ℏ ^{2}}{2 m _{e} k _{B}} (3 π^{2} n)^{2/3} .

(2)

This relates the atomic number density to the Fermi temperature only because monovalency and the free-electron dispersion have been assumed. A real energy band can require a different density of states and band effective mass.

For the isotropic Debye model, count three acoustic polarizations up to one spherical wavevector cutoff

q_{D}

. The number of vibrational modes per volume is

3 \frac{1}{( 2 π ) ^{3}} \frac{4 π q _{D}^{3}}{3} = 3 n, q_{D} = (6 π^{2} n)^{1/3} .

(3)

With

ω = c q

and

ℏ ω_{D} = k_{B} θ_{D}

θ_{D} = \frac{ℏ c}{k _{B}} (6 π^{2} n)^{1/3} .

(4)

The effective

c

is the Debye sound speed: matching the low-frequency mode density of one longitudinal and two transverse branches requires

3/ c^{3} = 1/ c_{L}^{3} + 2/ c_{T}^{3}

. It is not generally the arithmetic average of their speeds. The cutoff preserves the mode count; it need not coincide with the actual nonspherical Brillouin zone boundary.

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