The answer by Lewis Miller is adequate qualitatively.
Please note though the very important fact that there exist no free neutrons to be set up as targets. Free neutrons have a lifetime of minutes and can be used as projectiles , but not as targets. Thus the neutron neutron scattering cross section, the scientific term for what you are asking , has to be inferred from neutron scattering on some target that has both protons and neutron and the contribution of neutrons to the scattering can be ascertained by subtracting the known proton crossections.
In this link you will find a lot of neutron scattering crossections on various targets . A search on google will give a number of books where the subject is treated. There is little interest in pure neutron neutron scattering, although the extraction has been made using deuterons as a target, since the neutron proton cross section is known form experiments on hydrogen. It is not simple:
The theory of scattering and production reactions on deuterium is developed with a view to the extraction of neutron target amplitudes from analysis of deuterium experiments. Corrections to the spectator model due to the following effects are examined: target particle binding corrections to the Impulse Approximation; (ii) the Pauli principle in charge exchange reactions; (iii) internal or „Fermi” motion of the target particles; (iv) multiple scattering and final state interactions. The closure approximation and the general analysis of single-arm spectrometer experiments are discussed. The connection at high energies and small momentum transfers between the general Watson Theory and the Glauber model is established, and the results of calculations of deuterium corrections for pion and rho photoproduction at high energies are reviewed. “Fresnel” and recoil corrections to Glauber theory are examined. An approximation scheme suitable to the analysis of high energy wide angle coincidence or bubble chamber experiments is developed. A special examination is made of the breakdown of the impulse approximation in the 3-3 resonance region, and of the applicability of lowest order binding and in multiple scattering corrections in this sensitive kinematical region.