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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 9D / a / Solution

Chemical equilibrium gives

μ_{D} = μ_{p} + μ_{n}

. Substitution of the three Maxwell--Boltzmann densities therefore cancels the chemical potentials and leaves the rest-mass factor

exp (\frac{( m _{p} + m _{n} - m _{D} ) c ^{2}}{k _{B} T}) = e^{B_{D} / (k_{B} T)} .

Using

g_{p} = g_{n} = 2

,

g_{D} = 4

,

m_{n} ≃ m_{p}

, and

m_{D} ≃ 2 m_{p}

, the translational prefactor is

\frac{4 ( 2 π m _{D} k _{B} T / h ^{2} ) ^{3/2}}{4 ( 2 π m _{p} k _{B} T / h ^{2} ) ^{3/2} ( 2 π m _{n} k _{B} T / h ^{2} ) ^{3/2}} ≃ (\frac{π m _{p} k _{B} T}{h ^{2}})^{- 3/2} .

Hence

\frac{n _{D}}{n _{p} n _{n}} ≃ (\frac{π m _{p} k _{B} T}{h ^{2}})^{- 3/2} e^{B_{D} / (k_{B} T)} .

Solved by gpt-5.6-sol high.

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