wouldn't it be possible for an individual in the united states to construct a mini-nuclear reactor?
Short answer - no. For reasons why see long answer below.
Based on my experience in the Navy as a nuclear reactor operator I can say:
Assuming you could get permission and unless one is very, very rich building a home reactor wouldn't be feasible IMO.
You need enriched U238 for a reactor which is very hard to get and would be very expensive even if you could. You could use Plutonium also. There are a few other isotopes of uranium and some non uranium isotopes that are fissionable that would work too. The primary requirement though is the the material be fissionable with thermal neutrons. Materials that fission with fast neutrons (like U235) will work but are difficult to control and are generally restricted to use in breeder reactors.
You don't need a neutron source. Enough neutrons are given off by the normal decay process to start the fission reaction going
IF there is enough fissile material and it is in the correct geometry. With out a way to absorb and moderate the neutrons the fissile material will go supercritical (this is a bad thing - a really, really bad thing). You use an absorber to control the number of neutrons available in order to control the rate of fission and a moderator to thermalize the neutrons since U238 splits when a thermalized (low energy neutron) hits it. It is necessary to thermalize the neutrons because a fast neutron just bounces of the uranium atom kind of like a ping pong ball bouncing off of a pool ball - it will not be absorbed and in order for the fission reaction to occur the neutron must be absorbed. NOTE: there are isotopes that will fission when a fast neutron hits them but reactors that use fast neutrons are difficult to control so they are not common.
In order to keep the reaction from going supercritical you need a neutron absorber like hafnium (expensive materials) which is what the reactor control rods are made of. The most common moderator is water which thermalizes neutrons and that in and of itself is what allows a controlled fission reaction to take place. Usually the water is also the medium by which the heat of the core is transferred to what ever medium is used to generate power. In US Naval reactors the hot water goes thru a heat exchanger called a steam generator and that steam drives turbines hooked up to electric generators or to the reduction gears and the shaft. The actual core size for a pressurized water reactor (most common) is about the size of a garbage can though core sizes for can be larger and much, much larger in a boiling water reactor (most common type used for commercial production of electricity).
Add to all of the above the electronics you would need to monitor and control the reactor, the expense of shielding it, the expense of the high quality steel that goes into the piping, all the auxiliary control systems, pumps, containment vessel, the containment building, then the expense of disposing of the waste after 20 years or so and building a home reactor becomes an exercise in wasting money.
On the other hand one might be able to make what is called a snap reactor which uses radioactive materials to somehow make electricity. The Russians used to use those to power their satellites but that is a technology I am not familiar with.
