Alpha Decay — Question 6

Question 6

A rectangular-barrier model quantifies tunneling sensitivity

For a simplified barrier, use $P=e^{-2\kappa L}$, where $\kappa=\sqrt{2\mu c^2(V_0-Q)}/(\hbar c)$. Let $\mu c^2=3600\,\mathrm{MeV}$, $V_0-Q=10.0\,\mathrm{MeV}$, $L=10.0\,\mathrm{fm}$, and $\hbar c=197.3\,\mathrm{MeV\,fm}$.

1. Evaluate kappa and the penetration probability.

2. Find the ratio of probabilities if the barrier width increases by 1.00 fm while all other quantities remain fixed.

Question 6 - Solution

A rectangular-barrier model quantifies tunneling sensitivity

(a) Method and calculation

$$\boxed{\kappa=\sqrt{2(3600)(10.0)}/197.3=1.360\,\mathrm{fm^{-1}}}.$$$$\boxed{P=e^{-2\kappa L}=1.54\times10^{-12}}.$$

(b) Interpretation and further calculation

$$P(L+\Delta L)/P(L)=e^{-2\kappa\Delta L}.$$$$\boxed{P(11.0)/P(10.0)=e^{-2(1.360)(1.00)}\simeq.0659}.$$

Physical check

The rectangular barrier is a supplied approximation, not the true Coulomb shape. It demonstrates how a small geometric change can strongly alter the decay rate.