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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 9D / a / Solution

Use

E = m c^{2} + p^{2} / (2 m) + \dots

and replace the Fermi denominator by its Boltzmann form. The Gaussian integral gives

n_{X} = (\frac{2 π m _{X} k T}{h ^{2}})^{3/2} e^{- m_{X} c^{2} / k T} .

Solved by gpt-5.6-sol high.

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