The Big Bang — Question 4

Question 4

Neutron decay changes both sides of the neutron-to-proton ratio

At freeze-out assume $n/p=e^{-1.293/.800}$. During the next 180 s, neutrons decay with lifetime 880 s into protons; total baryon number stays fixed. Ignore further interconversion.

1. Find the initial neutron fraction Xn=n/(n+p).

2. Find its final value and the final n/p ratio, retaining the newly created protons.

Question 4 - Solution

Neutron decay changes both sides of the neutron-to-proton ratio

(a) Method and calculation

$$\boxed{r_f=e^{-1.293/.800}=.198642}.$$$$\boxed{X_{n,f}=r_f/(1+r_f)=.165723}.$$

(b) Interpretation and further calculation

$$\boxed{X_n=X_{n,f}e^{-180/880}=.135067}.$$$$\boxed{n/p=X_n/(1-X_n)=.156159}.$$

Physical check

Multiplying n/p directly by the decay survival factor neglects the increase in proton number. Applying exponential decay to the neutron fraction conserves total baryons.