# Thermally Assisted Alpha Particle Decay

I am attempting to solve the following exercise from Condensed Matter Field Theory by Altland and Simons:

Consider a heavy nucleus having a finite rate of $$\alpha$$-decay. The nuclear forces are short range so that the rate of $$\alpha$$-emission is controlled by the tunneling of $$\alpha$$-particles under a Coulomb barrier. Taking the effective potential to be spherically symmetric, with a deep well of radius $$r_0$$ beyond which it decays as

$$U(r) = \frac{2(Z - 1)e^2}{r},$$

where $$Z$$ is the nuclear charge, find the temperature $$T$$ of the nuclei above which $$\alpha$$-decay is thermally assisted if the energy of the emitted particles is $$E_0$$. Estimate the mean energy of the $$\alpha$$ particles as a function of $$T$$.

I have the following questions about the problem:

1. What is $$E_0$$? Is it the energy of a metastable state, modeled as a resonant scattering state?
2. What does thermally assisted mean, i.e., what is the mathematical criterion for a process to be thermally assisted at a temperature $$T$$?

My attempt at a solution is the following: Assume $$V(r) = V_0 < U(r_0)$$ for $$r < r_0$$. Model the particles inside the well as being three-dimensional free particles with velocities distributed according to the canonical ensemble, so they have average energies $$\frac{3}{2} T$$ (in units where $$k_b = 1$$). We say that the process is thermally assisted if $$U(r_0) - E_0 \leq \langle E \rangle$$, where $$\langle E \rangle$$ is the average energy of the particles in the canonical ensemble, so we have the criterion $$T \geq \frac{2}{3}(U(r_0) - E_0)$$. Then, assuming the energy is modeled according to the canonical ensemble, I guess we would have $$\langle E\rangle = E_0 + \frac{3}{2} T$$ (but, if this is right, I'm not really sure how to justify it).

• 2d edition? I can't find the exercise you are referring to.
– LPZ
Commented Aug 5 at 12:31
• @LPZ It's in the third chapter of the third edition right after they do some example tunneling calculations with instantons. Commented Aug 5 at 18:56