Question

Neutrons are born out of fission at a high energy and must be slowed down to the low energy region, called the “thermal energy” region, to cause another fission. A material called a “moderator” is used to slow the neutrons down, or “moderate” them, by elastic scattering collisions.The average neutron energy coming out of fission of thermal neutrons is 1.98 MeV in kinetic energy. What is the number of collisions required to bring its average kinetic energy to 0.1 eV if it elastically scatters off of the following moderators:

a. Hydrogen-1

b. Hydrogen-2

c. Carbon-12

d. Uranium-238

Answer 1

Consider the moderator material to be . We now assume that the moderator is initially not moving, but after a collision, gets a very small momentum (although that's not necessary, but otherwise we have some trivial solutions, which we would like to avoid). Then, by conservation of momentum and energy (the dashed terms represent quantities after collision), we have:

Now, using the fact that , these can be easily solved, yielding:

This gives an easy multiplication recursion. Thus, after collisions, we have:

Now, this is exactly solvable for , as we know the mass and initial energy of a neutron, and even the final energy of the neutron. Also, for a given moderator, we know it's mass. The solution for looks like (with a modulus to withhold the direction of final motion):

(a) The mass of Hydrogen-1 is basically the mass of a proton, which is and the mass of neutron is . Putting in the values, we have:

(b) Similarly, mass of Hydrogen 2 is . Hence:

(c) Mass of Carbon 12 is . Thus:

(d) Mass of Uranium-238 is . Thus: