A common way is to use a nuclear reactor. The chain reaction produces a lot of neutrons. By construction a neutron transparent “window” into the reactor, some of the neutrons will escape through the window, in a crude beam. You can then place the experiment in the beam.
Some experiments, can simply be done in the reactor – and a small test tube containing the experiment can just be installed into the reactor core.
While fission reactors are low tech and can produce very large numbers of neutrons, they are limited in the energy of neutrons that they can produce. Many designs of nuclear reactor use a “moderator” to reduce the neutron energy for the chain reaction, and this will reduce the energy of the neutrons available for experiments. Even “fast” reactors without a moderator are limited to the maximum energy of fission neutrons.
Where faster neutrons are required, a fusion reactor can be used. However, while fusion reactors can be constructed extremely cheaply and compactly, this type of design has been limited to very low power and therefore very low neutron production rate. A new generation of advanced “compact neutron source” fusion reactors optimised for very high neutron production rates (https://www.phoenixwi.com/product/high-yield-neutron-generator).
Some of the latest technology versions are so powerful and efficient, that they are often easier, cheaper and safer to use than fission reactors. Due to the risk of fission reactors being subverted for production of nuclear weapons, many governments are pushing for research institutions to replace their fission reactors with fusion sources. You can always add a moderator to reduce the neutron energy if the fusion neutrons are too fast for your experiment.
For experiments requiring neutrons which are faster than the maximum fusion neutron energy, then you need to use a particle accelerator and a “spallation target”. A proton accelerator is produce a beam of high energy protons – which are directed at a target made from a heavy metal. The protons hit the nuclei in the target and smash off neutrons. These can have energies as high as the proton beam energy.