Elementary processes

In a nuclear reactor neutrons are produced, slowed down and captured. Furthermore, energy is produced by the fission process, and, to a lesser extent, by radioactive decay. The most important nuclear quantities of a nucleus present in a reactor are, therefore:

• The fission cross section $\sigma_{F}$.

• The capture cross section $\sigma_{c}$

• The number of neutrons $\eta$ emitted following the capture of a neutron by a fissile nucleus. This quantity is basic for the possibility of establishing a chain reaction. It can be decomposed in two factors:
  • The probability that a capture leads to fission: ${\frac{\sigma_{F} }{\sigma_{F}+\ \sigma_{c}}}\ =\ {\frac{1}{1\ +\ \alpha}}$ where $\alpha ={\frac{\sigma_{c}}{\sigma_{F}}}.$

  • The mean number of neutrons $\nu$ emitted per fission.

  Thus, $\eta\ =\ {\frac{\nu}{1\ +\ \alpha}}.$

• The scattering cross sections, either elastic $\sigma_{s}$ or inelastic $\sigma_{in}$which control the propagation of neutrons, in the medium.

• The atomic mass of the nucleus A which controls the amount of slowing down of the neutron following an elastic scattering. After scattering at an angle $\theta$ in the center of mass, the final laboratory energy of the neutron of initial energy E₀, is given by $E_{f}=\frac{E_{0}}{2}\left( 1+\left( \frac{A-1}{A+1}\right) ^{2}+\left( 1-\left( \frac{A-1} {A+1}\right) ^{2}\right) \cos(\theta)\right)$. If the scattering in the center of mass is isotropic it follows that all final energies between E₀ and $E_{0}\left( \left( \frac{A-1}{A+1}\right) ^{2}\right)$ are equiprobable. Since, in this latter case, the neutron energy loss is proportional to its initial energy, it is convenient to measure energies in terms of lethargy $u=\ln\frac{E_{0}}{E}$ where E₀ is some arbitrary initial energy (usually the average energy of fission neutrons), and E the actual neutron energy. We define
  

  $$\varpi=\left( \frac{A-1}{A+1}\right) ^{2}$$ (4.1)

  
  
  It is convenient to define the average lethargy gain per collision
  

  $$\xi=\int_{E_{0}}^{\varpi E_{0}}\ln\frac{E_{0}}{E}\frac{dE}{E_{0}(1-\varpi )}=1+\frac{\varpi}{1-\varpi}\ln\varpi$$ (4.2)

  
  
  which, expressed as function of the mass A yields
  

  $$\xi=1+\left( \frac{A-1}{1}\right) ^{2}\frac{1}{2A}\ln\frac{A-1}{A+1}$$ (4.3)

  
  
  For large A, $\xi\simeq\frac{2}{A+\frac{2}{3}}$.