Energy spectra

We give a very simple treatment of neutron slowing down under the simplifing assumptions:

• No absorption

• No direct contribution of the neutron source at energy E

• Isotropic elastic scattering

• No inelastic scattering

We consider an infinite medium with a source of N₀ neutrons/sec. The number of neutrons which are scattered each second from an energy larger than E to an energy lower than E is evidently equal to N₀. A neutron of energy $E^{\prime}$ can be scattered equiprobably at energies between $E^{\prime}$ and $\varpi E^{\prime}$, $\varpi$ being defined in equation 3.1. It follows that the number of collisions past the energy E is

$$N_{0}=\int_{E}^{\frac{E}{\varpi}}\Sigma_{s}(E^{\prime})\frac{... ...\prime}}{E^{\prime}-\varpi E^{\prime}}n(E^{\prime})dE^{\prime}$$ (4.38)

where $n(E)dE=dE\iiint\varphi(\overrightarrow{r} ,E)d^{3}r$ is the integral of the neutron flux over the whole system with energy E within dE. Assuming that $\Sigma_{s}(E)$ is independent of E, we see that

$$\frac{d}{dE}\int_{E}^{\frac{E}{\varpi}}\left( \frac{E}{E^{\pr... ...\frac{\varpi}{1-\varpi}\right) n(E^{\prime})dE^{\prime}\equiv0$$ (4.39)

which reads

$$\int_{E}^{\frac{E}{\varpi}}\frac{1}{E^{\prime}(1-\varpi)}n(E^{\prime })dE^{\prime}-n(E)\equiv0$$ (4.40)

which, after differentiation, gives:

$$\frac{d}{dE}n(E)=\frac{n(\frac{E}{\varpi})}{E(1-\varpi)}-\fra... ...}\left( \frac{1}{E}n(\frac{E}{\varpi} )-\frac{1}{E}n(E)\right)$$

which is realised if

$$n(E)=\frac{C}{E}$$

To determine C we use

$$N_{0}=\Sigma_{s}\int_{E}^{\frac{E}{\varpi}}\frac{E-\varpi E^{\prime} }{E^{\prime}-\varpi E^{\prime}}n(E^{\prime})dE^{\prime}$$

and get

$$N_{0}=\frac{\Sigma_{s}C}{(1-\varpi)}\left( 1-\varpi+\varpi\ln\varpi\right)$$

thus:

$$n(E)=\frac{N_{0}(1-\varpi)}{\Sigma_{s}\left( 1-\varpi+\varpi\ln\varpi\right) E}=\frac{N_{0}}{\xi\Sigma_{s}E}$$

This expression is true for heavy scatterers, and, also, for hydrogen scatterer. One observes the so called $\frac{1}{E}$ slowing down spectrum, and that the neutron flux is approximately proportionnal to the mass of the scatterer. This allows to consider the use of heavy medium for transmuting fission products in the resonance region, as tested in the TARC experiment[36].

Similarly, the case of an absorbing medium can be treated for these two extremes. We treat the case of the heavy scatterer which will present itself later. We give a simple derivation of the evolution of the neutron distributions profiles as a function of energy and distance to the source, based on the random walk process.