Alpha Decay — Question 5

Question 5

The Coulomb barrier extends beyond the nuclear surface

Outside a daughter nucleus of charge $Z_D=82$, an alpha particle sees $V(r)=K/r$ with $K=2Z_D(1.44)\,\mathrm{MeV\,fm}$. Take nuclear matching radius $R=9.00\,\mathrm{fm}$ and relative alpha-daughter energy Q=5.00 MeV.

1. Find the Coulomb energy at R and the outer classical turning radius b.

2. Find the classically forbidden interval width and explain why an escaping alpha is classically forbidden there.

Question 5 - Solution

The Coulomb barrier extends beyond the nuclear surface

(a) Method and calculation

$$\boxed{K=236.16\,\mathrm{MeV\,fm},\quad V(R)=K/R=26.24\,\mathrm{MeV}}.$$$$V(b)=Q\Rightarrow\boxed{b=K/Q=47.232\,\mathrm{fm}}.$$

(b) Interpretation and further calculation

$$\boxed{b-R=38.232\,\mathrm{fm}}.$$$$R<r<b:\quad\boxed{p_r^2/(2\mu)=Q-V(r)<0}\quad\text{classically forbidden}.$$

Physical check

Quantum tunneling permits a finite transmission probability through this region. The relative-motion energy Q and reduced mass belong in the two-body barrier model.