A neutron with initial kinetic energy K makes a head-on elastic collision with a resting atom of mass m. (a) We have to show that the fractional energy loss of the neutron is given by

,

In which is the neutron mass. (b) We have to find if the resting atom is hydrogen, deuterium, carbon, or lead. (c) If initially, we have to find how many such collisions would it take to reduce the neutron energy to thermal values (0.025 eV) if the material is deuterium, a commonly used moderator. (Note: In actual moderators, most collisions are not “head-on.”)

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(a)

Let the speed of the incident neutron be v. We assume that the neutron has a head-on collision with a nuclide of mass m and it rebounds with speed and that the atom of mass m moves forward with speed V after the head-on collision. By applying the principle of conservation of momentum, we write the equation

As the collision is elastic, conservation of kinetic energy gives us the second algebraic equation,

We eliminate V from equations (1) and (2), and solve for . We assume that , the solution that gives a recoil speed to neutron on head-on collision with an atom of mass m is easily found to be

Therefore, the change in the kinetic energy of the neutron after its head-on collision with an atom of mass m will be

(b)

In the next part we will calculate if the resting atom is hydrogen, deuterium, carbon, or lead.

,

and

,

and

.

,

and

,

and

(c)

We have to estimate the number of head-on collisions with deuterium atoms that would reduce the energy of 1.00 MeV neutron to o.025 eV.

With each collision the kinetic energy of the neutron gets changed from to , or equivalently by the factor

Therefore, after n collisions the energy of the neutron will change to

.

We have to solve for n given that and

For deuterium atom, , and therefore

That is after about 8 head-on collisions with stationary deuterium atoms, a 1.00 MeV neutron will have its energy reduced to 0.025 eV.