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Typical nuclear reactors use neutrons in the $10^{-2}$ eV range (thermal neutrons) which corresponds to a fission cross section of a few hundred barns. What stops us from using a super-cold moderator, resulting in $10^{-5}-10^{-3} eV$ neutrons and dramatically increasing the cross section to a few thousand barns (as this would generate more power)? We have experience with handling extremely cold materials (liquid H2 in rockets), so is the intuition wrong or is there simply no reason to do this?
First problem is you want to use a reactor to create heat (and then use the heat to generate electricity). Using a cool moderator is directly detrimental to this aim. Also burning the fuel as fast as anybody wants is easy with current moderators, the trick is you don't want to burn it too fast, otherwise the plant and the neighboring city might get damaged.
In a reactor, the neutrons are slowed down because they get a thermal equilibrium with the coolant. This is why are they called thermal.
There is no way to create neutrons colder than the coolant on this way, it is probably even impossible without serious changes in the current reactor constructions.
