Can somebody explain to me how a spectrum is found or determined from a particle capture?

You can't get a spectrum from one particle. You have to collect thousands and graph their energy distribution.

Physicists use different kind of detectors to measure charge, momentum and energy of particles, either from particle accelerator collision like at the LHC, or from cosmic rays. The exact analysis depends on the experiment which depends on what they are looking for - they generally don't spend millions of dollars just "to see what happens".

Offhand, I don't recall the term "spectrum" used to refer to particles. It's usually used for waves e.g. electromagnetic radiation.