Inclusion of absorption in the model

Everytime a collision takes place the neutron may be absorbed and survives with probability $\gamma(E)={\frac{\Sigma_{s}(E)}{\Sigma_{a}(E)+\Sigma_{s} (E)}}={\frac{1}{1+{\frac{\Sigma_{a}(E)}{\Sigma_{s}(E)}}}}.$ It follows that the probability distribution after n collisions becomes:

$$Q(r:n)=P(r:n)\prod_{1}^{n}\gamma\left( E_{i}\right)$$ (4.50)

In the limit when the macroscopic absorption cross-section remains always small as compared to a constant scattering cross-section one obtains:

$$Q(r:n)=P(r:n)e^{-\ {\sum_{1}^{n}\frac{\Sigma_{a}(E_{i})}{\Sigma_{s}(E)}}}$$ (4.51)

Since the average energy interval is $\xi E_{i}$ the sommation may be approximated by an integral:

$$\sum_{1}^{n}\Sigma_{a}(E_{i})=\int_{E_{0}}^{E_{n}}{\frac{\Sigma_{a}(E)}{\xi E\Sigma_{s}(E)}}dE$$ (4.52)

Thus we can obtain the age distribution, including small absorption:

$$q(r,\tau)=\frac{e^{-\frac{r^{2}}{4\tau}}e^{-\int_{E_{0}}^{E(\... ...a}(E)}{\xi E\Sigma_{s}(E)}}dE}}{\left( 4\pi\tau\right) ^{3/2}}$$ (4.53)

A very similar expression is obtained for an hydrogen scatterer.