Classical description of fusion

[Aidan Davis]

Fusion is, in most cases (stars, etc.), considered probabilistic. The Gamow-Sommerfeld factor is used to calculate the probability that two colliding nuclei will undergo fusion, considering the fact that the particles have a chance of fusing by quantum tunneling. However, one can calculate an energy (and corresponding speed) that a particle must have to overcome the coloumb barrier and approach within a set distance of the target nuclei, using only classical mechanics. This energy is 1.44 MeV•Z(1)•Z(2)/D, where D is the set distance in fermis, aka 10^-15 meters. If a particle is incoming onto a target nucleus enough energy to come close enough to fuse, even by classical mechanics, is fusion guaranteed or is there still a probabilistic nature to it?

 

[mfb]

You cannot guarantee fusion. In addition, often this energy is sufficient to get various other reactions.

 

[Aidan Davis]

You cannot guarantee fusion. In addition, often this energy is sufficient to get various other reactions.

Ah, pair production of electrons and positrons occurs at these high energies. So fusion would be optimized best by using the Gamow factor https://en.m.wikipedia.org/wiki/Gamow_factor (Ok, wiki isn't the best source but it gives the equations straightforwardly. Here's a derivation of sorts: http://www.astro.princeton.edu/~gk/A403/fusion.pdf)
to calculate the probability of fusion f fusion at a certain energy and then treating that as an optimization problem to get the lowest energy per fusion. I worked it this way, and came to the conclusion that if optimized in this way, D-D fusion, for example, peaks at 246.6 KeV, and all combinations peak at a probability of 13.535% (e^-2). The peak energy is proportional to the product of the charges of the two nuclei and the square root of their reduced mass.