 Solution

ID: past-exam-of-the-general-relativity-and-cosmology-and-condensed-matter-physics-paper-of-part-b-of-the-physics-course-of-the-university-of-oxford/2011/section-vi/6/unlettered-3/solution

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Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section VI / 6 / Unlettered 3 / Solution by

Codex  0 2026-10-25

For the Lennard-Jones potential,

V^{'} (x) = \frac{24 ϵ}{x} [(\frac{σ}{x})^{6} - 2 (\frac{σ}{x})^{12}] .

Its minimum and harmonic spring constant are

a = 2^{1/6} σ, K = 4 ϵ (\frac{156 σ ^{12}}{a ^{14}} - \frac{42 σ ^{6}}{a ^{8}}) = \frac{72 ϵ}{a ^{2}} .

Using the monatomic-chain dispersion relation,

v = \frac{72 ϵ}{m}, ω_{m a x} = 2 \frac{K}{m} = \frac{2}{2 ^{1/6} σ} \frac{72 ϵ}{m}, f_{m a x} = \frac{ω _{m a x}}{2 π} .

The maximum is at

∣ k ∣ = π / a

. Taking the argon atomic mass as

m ≃ 39.95 u = 6.63 \times 1 0^{- 26} kg

and

ϵ = 1.602 \times 1 0^{- 21} J

gives

a ≃ 0.382 nm, K ≃ 0.792 N m^{- 1},

v ≃ 1.32 \times 1 0^{3} m s^{- 1}, ω_{m a x} ≃ 6.91 \times 1 0^{12} s^{- 1}, f_{m a x} ≃ 1.10 THz .

These are the stipulated one-dimensional nearest-neighbour model estimates; a three-dimensional argon crystal has additional neighbours and phonon polarizations.

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