@@ -833,7 +833,7 @@ In Eq. (II B1 a.1)\ref{eq:sigma-in-terms-of-X} the summations are over those cha
The total cross section (for non-Coulomb initial states) is the sum of Eq. (II B1 a.1)\ref{eq:sigma-in-terms-of-X} over all possible final-state particle-pairs $\alpha'$, assuming the scattering matrix is unitary (i.e., assuming that the sum over $c'$ of $|U_{cc'}^2=1$). Written in terms of the $X$ matrix, the total cross section has the form
Here $c$ is restricted to those channels of the $J^\pi$ spin group from which the particle pair is $\alpha$, and $c'$ to those channels for which the particle-pair is $\alpha'$.
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@@ -878,6 +878,7 @@ The capture cross section for the eliminated radiation channels can be calculate
\noindent
or may be found by subtracting the sum of all reaction cross sections from the absorption cross section. In Eq. (II B1 a.6)\ref{eq:sig-cap-in-terms-of-X}, the sum over $c$ includes all incident particle channels for the $J^\pi$ spin group, and the sum over $c'$ includes \textbf{all} particle channels, both incident and exit, for that spin group.
\newpage
\noindent
\textbf{One-level two-channel case}
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@@ -909,7 +910,141 @@ For this simple case, the X matrix of Eq. (II B1.4)\ref{eq:X-matrix} takes the f
in which the subscript on the penetrabilities denotes the channel number (not the angular momentum), the symbol $D$ has been used for $E_\lambda-E-i\overline{\gamma}^2_{\lambda\gamma}$, and the subscript $\lambda$ has been omitted from the reduced-width amplitudes for simplicity's sake. This equation can be rewritten as
i P_{1} P_{2}\gamma_{1}^{3}\gamma_{2}+P_{2} D \gamma_{1}\gamma_{2}-i P_{1} P_{2}\gamma_{1}^{3}\gamma_{2}& i P_{1} P_{2}\gamma_{1}^{2}\gamma_{2}^{2}+P_{2}\gamma_{2}^{2} D-i P_{1} P_{2}\gamma_{1}^{2}\gamma_{2}^{2}
\end{array}\right] \\
&\qquad\times\left[\begin{array}{cc}
\sqrt{P_{1}}& 0 \\
0 &\sqrt{P_{2}}
\end{array}\right] \\
& = \frac{1}{\left(D^{2}-i P_{1}\gamma_{1}^{2} D-i P_{2}\gamma_{2}^{2} D\right)}\left[\begin{array}{cc}\frac{1}{\sqrt{P_{1}}} & 0 \\ 0 & \frac{1}{\sqrt{P_{2}}}\end{array}\right]\left[\begin{array}{cc}P_{1} \gamma_{1}^{2} D & P_{1} D \gamma_{1} \gamma_{2} \\ P_{2} D \gamma_{1} \gamma_{2} & P_{2} \gamma_{2}^{2} D\end{array}\right]\\
in which $\Gamma$ is the sum of the partial widths $\Gamma_1+\Gamma_2+\Gamma_\gamma$.
In this form, $X$ can be substituted into the equations for the various cross sections. Assuming the second channel is a reaction channel, Eq. (II B1 a.2)\ref{eq:tot-in-terms-of-X} for the total cross section becomes
in which $d$ has been used to represent $|E_\lambda-E-i\Gamma/2|^2=(E-E_\lambda)^2+(\Gamma/2)^2$. Similarly, the elastic cross section, Eq. (II B1 a.3)\ref{eq:elastic-in-terms-of-X}, can be expressed as
Angular distributions (elastic, inelastic, or other reaction) cross sections for incident neutrons can be calculated from Reich-Moore resonance parameters. Following Blatt and Biedenharn [JB52]\cite{blatt-biedenharn-1952} with some notational changes, the angular distribution cross section in the center-of-mass system may be written