The Nuclear Reactor — Question 3

Question 3

An elastic collision shows why light nuclei can moderate neutrons

A neutron of mass m and speed v collides head-on elastically with a stationary nucleus of mass Am. Use one-dimensional momentum and kinetic-energy conservation.

1. Derive the fraction of neutron kinetic energy remaining after the collision.

2. Evaluate this limiting fraction for A=1 and A=12 and explain why it is not an average over all scattering angles.

Question 3 - Solution

An elastic collision shows why light nuclei can moderate neutrons

(a) Method and calculation

$$mv=mv\prime+AmV,\quad v^2=v\prime{}^2+AV^2\Rightarrow v\prime=\frac{1-A}{1+A}v.$$$$\boxed{E\prime/E=\left(\frac{A-1}{A+1}\right)^2}.$$

(b) Interpretation and further calculation

$$\boxed{A=1:\ E\prime/E=0;\qquad A=12:\ E\prime/E=121/169=.71598}.$$$$\boxed{\Delta E/E=1-E\prime/E}\quad\text{is largest for the head-on transfer considered}.$$

Physical check

For A greater than one the neutron reverses direction in this head-on case. General scattering leaves a distribution of outgoing energies.