Absorption by a strong resonance

The possible interest of transmutation in the resonance region[36] comes from two effects:

1.

The presence of very strong resonances in the nucleus to be transmuted

2.

The fact that the change of energy in very heavy scatterers like lead is very progressive, allowing several attempts by the slowing down neutron to interact with a nucleus to be transmuted.

We give a very schematic treatment of the effect of a very strong resonance absorption characterized by a width $\Gamma$ and energy E_R and a cross-section $\sigma_{abs}$. We want to compute the number of capture in the absorbing nucleus per incident neutron, as a function of the concentration of the absorber. The Breit and Wigner formula reads:

$$\sigma(E)=\frac{\sigma_{0}}{\left( 1+\frac{\left( E-E_{R}\rig... ...ight) ^{2}}\right) }=\frac{\sigma_{0}}{\left( 1+x^{2}\right) }$$ (4.54)

with $x=\frac{E-E_{R}}{\frac{\Gamma}{2}}$. A neutron at energy E enters the medium with n_aabsorbing nuclei and a macroscopic scattering cross-section of $\Sigma_{s}$. We define $\Sigma_{a}^{(0)}=n_{a}\sigma_{0}$ and $\Sigma(E)=n_{a}\sigma(E)$. The probability that the neutron survives an interaction, i.e. that it is not captured is $P_{surv}=\frac{\Sigma_{s} (E)}{\Sigma_{s}(E)+\Sigma(E)}$. After n interactions the survival probability is:

$$P_{surv}=\underset{i=1,n}{\prod}\frac{\Sigma_{s}(E_{i})}{\Sig... ...,n}{\prod}\frac{1}{1+\frac{\Sigma(E_{i} )}{\Sigma_{s}(E_{i})}}$$

thus
$$\begin{align*}\ln P_{surv} & =-\sum\ln\left( 1+\frac{\Sigma(E_{i})}{\Sigma_{s}(E... ...um\ln\left( 1+x_{i}^{2} +\frac{\Sigma_{a}^{(0)}}{\Sigma_{s}}\right) \end{align*}$$
the interval between two successive values of E_i is $\xi E_{R}$ and that between two x_i is, accordingly, $\frac{2\xi E_{R}}{\Gamma}$. Using the integral approximation of the sum, and integrating from 0 to $\infty$ we get
$$\begin{align}\ln P_{surv} & =\frac{\Gamma}{2\xi E_{R}}\int\ln\left( 1+x^{2}\righ... ...ft( \sqrt{1+\frac{\Sigma_{a}^{(0)}} {\Sigma_{s}}}-1\right) \nonumber \end{align}$$
and

$$P_{surv}=e^{-\frac{\pi\Gamma}{\xi E_{R}}\left( \sqrt{1+\frac{\Sigma_{0} }{\Sigma_{s}}}-1\right) }$$ (4.55)