Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 62 / 3 / iv

Solution

The nucleons are nonrelativistic, so the Maxwell-Boltzmann distribution gives

n_{i} = g_{i} (\frac{m _{i} T}{2 π})^{3/2} e^{(μ_{i} - m_{i}) / T}, i = n, p .

Both spin degeneracies are two. Taking the ratio and using the specified negligible chemical potentials yields the neutron-to-proton ratio

r_{F} = (\frac{m _{n}}{m _{p}})^{3/2} e^{- (m_{n} - m_{p}) / T_{F}} ≃ e^{- Q / T_{F}} .

The mass prefactor differs from unity only at the per-mille level. The dependence on

T_{F}

is much stronger:

d ln r_{F} / d ln T_{F} = Q / T_{F} > 0

. Thus earlier, hotter weak freeze-out leaves more neutrons relative to protons. For illustration,

T_{F} = 1 MeV

gives

r_{F} ≃ 0.28

;

T_{F} = 0.8 MeV

gives

r_{F} ≃ 0.20

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