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ID: past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2012/paper-1/1/a/solution

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Past exam of the thermal and statistical physics course of the University of Cambridge / 2012 / Paper 1 / 1 / a / Solution by

Codex  0 2026-10-20

Use thermal de Broglie wavelength

λ_{T} = h / 2 πm k_{B} T

and spacing

a = n^{- 1/3}

. Degeneracy onset from thermal wavelength occurs at

T_{λ} = \frac{h ^{2} n ^{2/3}}{2 πm k _{B}} .

(1)

The numerical factor depends on the convention for a typical momentum, but the scale

ℏ^{2} n^{2/3} / (m k_{B})

does not. For fermions the Fermi temperature

T_{F} = ℏ^{2} (3 π^{2} n)^{2/3} / (2 m k_{B})

gives a comparable scale for spin degeneracy two.

Representative estimates, with the assumed densities stated explicitly, are:

System	Assumed number density	Assumed particle mass	$T_{λ}$
Conduction electrons	$1 0^{29} m^{- 3}$	$m_{e}$	$1.2 \times 1 0^{5} K$
Dilute heavy atomic cloud	$1 0^{19} m^{- 3}$	$87 u$	$1.6 \times 1 0^{- 7} K$
Neutrons at nuclear-scale density	$1 0^{44} m^{- 3}$	$m_{n}$	$6.5 \times 1 0^{11} K$

Here

u

is the atomic mass constant. Thus the characteristic orders are $1 0^{5} K$ for electrons, sub-microkelvin for a dilute heavy atomic gas, and $1 0^{12} K$ for neutrons. There is no density-independent quantum-degeneracy temperature for atomic gases: for example, taking

n = 2.5 \times 1 0^{25} m^{- 3}

and mass

4 u

instead gives about

0.065 K

. These are illustrative input estimates, not values supplied by the paper. Denser neutron-star cores require relativistic and interaction corrections; the estimate identifies the degeneracy scale rather than a precise stellar temperature.

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