1) This is a difficult question to answer. I can give you factors, but you probably won't be able to do much with them as this is QCD which is VERY difficult to do. You would need to know:
i) the incoming and outgoing 4-momenta of all particles
ii) the coupling constant (probably just -i g(sub s) for the first level tree diagram, and then more for the higher order diagrams
iii) some other mathematics involving the propagator (gluons), balancing energy and momentum, and then some more math
You'll end up with an expression involving the matrix element of the reaction, or the amplitude for this process to occur. Once you have that, you have to square it, multiply it by a whole ton of other stuff, and then you can get the cross section. It's not fun.
You can also get the matrix element directly by drawing the feynmann diagram for the process, but as it involves a number of quarks in each neutron and a whole ton in the nucleus, I would really not recommend doing this unless you really want to do a few hundred (thousand?) integrals.
2) There isn't any energy barrier AFAIK. That's why neutrons are so useful for imaging nuclei. (search for work done by Bertram Neville Brockhouse... nobel prize for neutron work).
3) I would say so yes. As the strong force acts on short distance scales the expectation value of r for the neutron would matter here.
(I guess the simple answer to 1 would be you would need to know the momentum/energy of the nucleus and neutron, their trajectories, and the "strength" of the strong force... that's as basic as it'll get I guess. If you want to actually calculate numbers, you'll need some QCD)
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