Thermonuclear Fusion: The Basic Process — Question 4

Question 4

A Coulomb estimate explains why classical thermal contact is rare

Two singly charged nuclei must approach within 5.00 fm in a crude contact model. Use $e^2/(4\pi\varepsilon_0)=1.44$ MeV fm and $k_B=8.617\times10^{-5}$ eV/K. A plasma has T=$10^8$ K.

1. Estimate the contact Coulomb energy and compare it with kT.

2. Calculate the Boltzmann exponential factor $e^{-U_C/(kT)}$ and state why fusion can occur below the classical contact energy.

Question 4 - Solution

A Coulomb estimate explains why classical thermal contact is rare

(a) Method and calculation

$$\boxed{U_C=1.44/5.00=.288\,\mathrm{MeV}=288\,\mathrm{keV}}.$$$$\boxed{kT=8.617\,\mathrm{keV},\quad U_C/(kT)=33.4223}.$$

(b) Interpretation and further calculation

$$\boxed{e^{-U_C/(kT)}=3.05\times10^{-15}}.$$$$P_{\mathrm{tunnel}}\sim e^{-2G}>0\quad\text{even when}\quad E<U_C.$$

Physical check

The exponential is a suppression factor, not the full Maxwell energy-tail probability. Quantum tunneling and the high-energy tail both matter.