# I Classical description of fusion

1. Jul 8, 2017

### 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?

2. Jul 8, 2017

### Staff: Mentor

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

3. Jul 8, 2017

### Aidan Davis

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.