 Neutron detection by proton recoil

In neutron detection by proton recoil, a neutron strikes a hydrogen atomic nucleus, namely a proton, and the moving proton produces detectable ionization. Paraffin wax provides many hydrogen targets. There are two distinct questions: how much kinetic energy can a collision transfer, and how often does it occur? Elastic collision kinematics answers the first. It does not determine a collision cross-section or a detector's efficiency.

For the following derivations the target is initially at rest in the laboratory but is free to recoil. Its mass is finite. In particular a proton must not be treated as an immovable target. All massive particles are nonrelativistic particles, their translational kinetic energies are $m{v}^2/2$, and the collision changes no internal state. Target binding and thermal motion are neglected relative to the incident kinetic energy. These assumptions suit an introductory description of fast-neutron elastic collisions; they do not cover every neutron capture or other nuclear reaction.