Nuclear Collsions fusion cross section

[MadRocketSci2]

Hello,

I am curious about fusion reaction cross sections, and fusion power in general, and have been trying to self-teach myself in this area. I saw some emprical data for cross section curves in a paper that I read recently, and thought I understood enough of what made the curve (cross section vs rel particle energy) look that way to model it with a classical point charge model.

I derived an expression for the radius of an approach envelope for particle A to get within some radius ri of particle B (presumably where the strong nuclear force takes over and reactions start happening). The curve only has one unknown fudge factor (the radius ri), and so only one degree of freedom. Other than that, on first inspection, the curve I derived seemed correct - the behavior as vA -> infinity was for ra->ri, and there was some vmin where ra->0 (where there isn't enough energy to overcome coulomb repulsion even with a dead-on trajectory).

ra = sqrt(ri^2 - 2*ri*C*q1*q2/((ma*mb)/(ma+mb)*vab^2)))))

When I plugged the numbers into the equation though, I couldn't get the curve to converge to emprical data on the D-D fusion cross section. When I chose ri such that the cutoff was similar, the cross section was 4 orders of magnitude too high (and it was a very large ri from the standpoint of atomic diameters). When I chose ri small enough such that the cross sections were in the right order of magnitude, the relative particle energy required was orders of magnitude too high.

The problem could be either in my derivation or in the model. If it's in my derivation, a numerical model I'm turning the crank on should let me know (good old forward Euler method - what would I do without it). If it is my model though, I wonder if you guys could help point out what is wrong.

I realize that at low enough particle energies, some sort of quantum mechanical effects take over, spreading the cross section out due to particle-position diffraction. That's why neutron cross sections look like they do - the spatial extent of the neutron spreads faster as the energy approaches 0. I was banking that that wouldn't be important enough in the model to cause such gross inaccuracies, which takes place with particles massed 1+ proton mass in the 10-100keV range, but I may be wrong.

Other things that may be wrong with my model - the strong nuclear force extends some distance from the nucleus and causes some attenuation of the coulomb repulsion at distances greater than the reaction distance.
Even when the particle hits the reaction distance, there is only a 1/10000 chance that it will react for some reason.

Do any of you have experience with this? What do you think?

 

[fatra2]

Hi there,

Just as simple answer, it is pratically impossible to summarise nuclear cross-section into an equation, for every matter out there. This is one of the big issues with nuclear cross section.

For the rest, your comment seems (at first glance) correct.

Cheers

 

[hagopbul]

maybe start with neutron cross section [fusion]

 

[hamster143]

The easiest thing you could do is to make a simple correction. Your model of a target nucleus is a simple ball (so that the projectile either hits it when it gets close than r_min, or it misses).

Replace that with an exponentially decaying cloud. So that, say, when your projectile passes at the distance of r > r_min, the probability of interaction is (r_min/r) exp(1-r/r_min).

r_min is the inverse of pion mass, it should be around 1.5 femtometers (1.5 * 10^-15 m).

A more complicated calculation could go like this. Do a Born approximation of scattering on the Yukawa potential of the nucleus

http://electron6.phys.utk.edu/QM2/modules/m7/born.htm

But instead of modelling the incoming particle as a simple plane wave ~exp(i*k*x), throw in some factor that reflects the difficulty of getting to close to the center due to Coulomb interaction. That way you account for the spread of the strong interaction outside r_min, and you account for the quantum tunneling of the projectile through the Coulomb barrier.

 

[rdm]

what is the fusion cross section formula for DT fusion (or for fusion of lighter nuclei) as a function of centre of mass energy?