In a nuclear reactor the neutrons that produced are very fast and has lot of energies. These neutrons are not useful for condensed matter research. So we need to slow down the neutrons or decrease its energy by using moderators. The fast neutrons enter the moderator gets scattered and rescattered by its atoms of the moderator each time losing its energy. Finally the neutrons loose a lot energy and their energies become equal to that comparable to the thermal energy of the atoms in a solid at room temperature. This process is called thermalization. You must note that the neutrons release a lot of energy to the moderator which gets very hot. So one must take out the heat by running water.  The moderator must be kept at a particular emperature. If it is kept around room temperature then we get thermal neutrons which are very useful for studing condensed matter. In some applications we also need colder neutrons. Then we must use a cold source of liquid H or D. In some other applications we need hotter neutrons. Then we must use a hot source which is kept at a higher temperatures. The details are for the neuclear engineers who construct and maintain nuclear reactors. If you are a condensed matter scientists then you just use these neutrons.

how we produce the first neutron to fissile the first nuclei to continue the chain of reaction?

It depends on the reactor design, after removal of the safety/control rods the spontaneous fission of 238U may be enough to start the chain reaction, otherwise a spontaneous fission source is used (e.g. 252Cf).

Startup neutron source is a neutron source used for stable and reliable initiation of nuclear chain reaction in nuclear reactors, when they are loaded with fresh nuclear fuel, whose neutron flux from spontaneous fission is insufficient for a reliable startup, or after prolonged shutdown periods. Neutron sources ensure a constant minimal population of neutrons in the reactor core, sufficient for a smooth startup. Without them, the reactor could suffer fast power excursions during startup from state with too few self-generated neutrons (new core or after extended shutdown).

Few startup sources;

·         Californium-252 (spontaneous fission)

·         Plutonium-238-beryllium, plutonium-239-beryllium or americium-beryllium (α,n) nuclear reactions.

·         Sb-Be photoneutron source; antimony becomes radioactive in the reactor and its strong gamma emissions (1.7 MeV for 124Sb) interact with beryllium-9 by an (γ,n) reaction and provide photoneutrons. In a PWR reactor one neutron source rod contains 160 grams of antimony, and stay in the reactor for 5–7 years. The sources are often constructed as an antimony rod surrounded by beryllium layer and clad in stainless steel.

As we see from the above two answers from supoosedly nuclear engineer colleagues the details of a neuclear reactor operation is highly complex. Over the years right from the days of Fermi in Chicago, operations of nuclear reactors have been optimized and settled down and adapted by a team of specialists along with automatic safety systems. We condensed matter physicists must leave these operations to these excellent engineers who overs years gave us opportunities to use neutron in condensed matter research. Our business would be to use the neutron beam as best as we can by constructing diffractometers and spectrometers and using them to find out microscopic origin of the various fundamental properties of the condensed states of matter.

Thank you all.

Given on net. Cf 252 spontaneous fission, Cosmic rays, reactions.