Alpha Decay — Question 9

Question 9

Angular-momentum conservation can add a centrifugal barrier

An alpha particle has spin-parity $0^+$. Consider alpha decay of a $0^+$ parent to a $2^+$ daughter. Relative orbital angular momentum is ell. Use $V_\ell(r)=\hbar^2\ell(\ell+1)/(2\mu r^2)$ with $\mu c^2=3600\,\mathrm{MeV}$ and $\hbar c=197.3\,\mathrm{MeV\,fm}$.

1. Find the required ell and verify parity conservation.

2. Evaluate the centrifugal contribution at r=9.00 fm and compare it with ell=0 decay to a $0^+$ daughter.

Question 9 - Solution

Angular-momentum conservation can add a centrifugal barrier

(a) Method and calculation

$$\mathbf J_P=\mathbf J_D+\mathbf\ell,\quad J_P=0,\ J_D=2\Rightarrow\boxed{\ell=2}.$$$$\pi_P=\pi_D\pi_\alpha(-1)^\ell=(+)(+)(+)=+\Rightarrow\boxed{\text{parity allowed}}.$$

(b) Interpretation and further calculation

$$\boxed{V_2(9)=\frac{197.3^2(2)(3)}{2(3600)(9^2)}=.4005\,\mathrm{MeV}}.$$$$\boxed{V_0=0,\quad V_2>V_0}\Rightarrow\text{additional tunneling hindrance at equal Q}.$$

Physical check

The centrifugal term is part of the relative-motion barrier. Branch rates cannot be compared from angular momentum alone when Q values or preformation amplitudes differ.