finish single-case examples for reich-moore

eb7486d5 · Jesse Brown · 7abef3de · eb7486d5 · eb7486d5 · eb7486d5
Commit eb7486d5 authored 2 years ago by Jesse Brown
--- a/docs/tex/sammy-main.pdf
+++ b/docs/tex/sammy-main.pdf
--- a/docs/tex/sammyrefs.bib
+++ b/docs/tex/sammyrefs.bib
@@ -1249,5 +1249,16 @@ abstract = "The response of the RPI 16-segment NaI(Tl) multiplicity detector sys
    year = {{Invited paper, October 12-16, 1992, ed. Charles L. Dunford, pp. 285-294, 1993}},
 }

+@article{blatt-biedenharn-1952,
+  title={The angular distribution of scattering and reaction cross sections},
+  author={Blatt, John M and Biedenharn, LC},
+  journal={Reviews of Modern Physics},
+  volume={24},
+  number={4},
+  pages={258},
+  year={1952},
+  publisher={APS}
+}
+

 @Comment{jabref-meta: databaseType:bibtex;}
--- a/docs/tex/scattering-theory.tex
+++ b/docs/tex/scattering-theory.tex
@@ -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

-\begin{equation}
+\begin{equation}\label{eq:tot-in-terms-of-X}
    \sigma_{total}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \left[ \left(\sin^2\phi_c + X^i_{cc}\cos(2\phi_c)- X_{cc}^r\sin(2\phi_c)\right)\right],
 \end{equation}

@@ -842,7 +842,7 @@ where again the sum over $c$ includes only those channels of the $J^\pi$ spin gr

 The angle integrated elastic cross section is given by

-\begin{equation}
+\begin{equation}\label{eq:elastic-in-terms-of-X}
    \begin{aligned}
        \sigma_{elastic}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \Biggl[ \sin^2\phi_c(1-2X^i_{cc}) &-X^r_{cc}\sin(2\phi_c) \\
        & +\sum_{c'}\left({X^i_{cc'}}^2+{X^r_{cc'}}^2\right) \Biggr].
@@ -854,8 +854,8 @@ In this case, both $c$ and $c'$ are limited to those channels of the $J^\pi$ spi

 Similarly, the reaction cross section from particle pair $\alpha$ to particle pair $\alpha'$ (where $\alpha'$ is not equal to $\alpha$) is

-\begin{equation}
-    \sigma_{reaction}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c\sum_{c'}\left[{X^i_{cc'}}^2+{X^r_{cc'}}^2\right]
+\begin{equation}\label{eq:reax-in-terms-of-X}
+    \sigma_{reaction}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c\sum_{c'}\left[{X^i_{cc'}}^2+{X^r_{cc'}}^2\right].
 \end{equation}

 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 $\alpha'$.
@@ -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}

@@ -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

+\begin{equation}
+    \begin{array}{r}
+    X=\frac{i P_{1} P_{2} D}{i D}\left[\begin{array}{cc}
+    \frac{1}{\sqrt{P_{1}}} & 0 \\
+    0 & \frac{1}{\sqrt{P_{2}}}
+    \end{array}\right]\left[\begin{array}{cc}
+    P_{2}\left(D-i P_{1} \gamma_{1}^{2}\right) & -i P_{1} P_{2} \gamma_{1} \gamma_{2} \\
+    -i P_{1} P_{2} \gamma_{1} \gamma_{2} & P_{1}\left(D-i P_{2} \gamma_{2}^{2}\right)
+    \end{array}\right]^{-1}\left[\begin{array}{cc}
+    \gamma_{1}^{2} & \gamma_{1} \gamma_{2} \\
+    \gamma_{1} \gamma_{2} & \gamma_{2}^{2}
+    \end{array}\right] \\ \\
+
+    \times\left[\begin{array}{cc}
+    \sqrt{P_{1}} & 0 \\
+    0 & \sqrt{P_{2}}
+    \end{array}\right] \\ \\
+
+    =\frac{P_{1} P_{2}}{P_{1} P_{2}\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}\left(D-i P_{2} \gamma_{2}^{2}\right) & i P_{1} P_{2} \gamma_{1} \gamma_{2} \\
+    i P_{1} P_{2} \gamma_{1} \gamma_{2} & P_{2}\left(D-i P_{1} \gamma_{1}^{2}\right)
+    \end{array}\right] \\ \\
+
+    \times\left[\begin{array}{cc}
+    \gamma_{1}^{2} & \gamma_{1} \gamma_{2} \\
+    \gamma_{1} \gamma_{2} & \gamma_{2}^{2}
+    \end{array}\right]\left[\begin{array}{cc}
+    \sqrt{P_{1}} & 0 \\
+    0 & \sqrt{P_{2}}
+    \end{array}\right],
+    \end{array}
+\end{equation}
+
+\noindent
+or,
+
+\begin{equation}
+    \begin{aligned}
+        X &=\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] \\
+        & \qquad \times\left[\begin{array}{cc}
+        P_{1} \gamma_{1}^{2} D-i P_{1} P_{2} \gamma_{1}^{2} \gamma_{2}^{2}+i P_{1} P_{2} \gamma_{1}^{2} \gamma_{2}^{2} & P_{1} D \gamma_{1} \gamma_{2}-i P_{1} P_{2} \gamma_{1} \gamma_{2}^{3}+i P_{1} P_{2} \gamma_{1} \gamma_{2}^{3} \\
+        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] \\ 
+        & \qquad \times\left[\begin{array}{cc}\sqrt{P_{1}} & 0 \\ 0 & \sqrt{P_{2}}\end{array}\right] \\
+        & =\frac{1}{\left(D-i P_{1} \gamma_{1}^{2}-i P_{2} \gamma_{2}^{2}\right)}\left[\begin{array}{cc}
+        P_{1} \gamma_{1}^{2} & \sqrt{P_{1} P_{2}} \gamma_{1} \gamma_{2} \\
+        \sqrt{P_{1} P_{2}} \gamma_{1} \gamma_{2} & P_{2} \gamma_{2}^{2}
+        \end{array}\right] ,
+    \end{aligned}
+\end{equation}
+
+\noindent
+or finally, 
+
+\begin{equation}
+    \begin{aligned}
+        X &=\frac{1}{\left(E_{\lambda}-E-i \bar{\gamma}_{\gamma}^{2}-i P_{1} \gamma_{1}^{2}-i P_{2} \gamma_{2}^{2}\right)}\left[\begin{array}{cc}
+        P_{1} \gamma_{1}^{2} & \sqrt{P_{1} P_{2}} \gamma_{1} \gamma_{2} \\
+        \sqrt{P_{1} P_{2}} \gamma_{1} \gamma_{2} & P_{2} \gamma_{2}^{2}
+        \end{array}\right] \\
+        &=\frac{1}{\left(E_{\lambda}-E-i \Gamma / 2\right)}\left[\begin{array}{cc}
+        \Gamma_{1} / 2 & \sqrt{\Gamma_{1} \Gamma_{2}} / 2 \\
+        \sqrt{\Gamma_{1} \Gamma_{2}} / 2 & \Gamma_{2} / 2
+        \end{array}\right],
+    \end{aligned}
+\end{equation}
+
+\noindent
+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
+
+\begin{equation}
+    \begin{aligned}
+        \sigma_{\text {total }}(E) &=\frac{4 \pi}{k_{\alpha}^{2}} g_{J}\left[\sin ^{2} \phi_{c}+\frac{\Gamma \Gamma_{1}}{4 d} \cos \left(2 \phi_{c}\right)-\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{2 d} \sin \left(2 \phi_{c}\right)\right] \\
+        &=\frac{2 \pi}{k_{\alpha}^{2}} g_{J}\left[1-\left(1-\frac{\Gamma \Gamma_{1}}{2 d}\right) \cos \left(2 \phi_{c}\right)-\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{d} \sin \left(2 \phi_{c}\right)\right],
+    \end{aligned}
+\end{equation}
+
+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
+
+\begin{equation}
+    \begin{aligned}
+        \sigma_{\text {elastic }}(E)=& \frac{4 \pi}{k_{\alpha}^{2}} \sum_{J} g_{J \alpha} \sum_{c}\left[\sin ^{2} \phi_{c}\left(1-2 \frac{\Gamma \Gamma_{1}}{4 d}\right)\right.\\
+        &\left.-\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{2 d} \sin \left(2 \phi_{c}\right)+\left(\frac{\Gamma \Gamma_{1}}{4 d}\right)^{2}+\left(\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{2 d}\right)^{2}\right]
+    \end{aligned}
+\end{equation}
+
+\noindent
+which reduces to
+
+\begin{equation}
+    \begin{aligned}
+        \sigma_{\text {elastic }}(E)=\frac{2 \pi}{k_{\alpha}^{2}} \sum_{J} g_{J \alpha} \sum_{c} & {\left[1-\cos 2 \phi_{c}\left(1-\frac{\Gamma \Gamma_{1}}{2 d}\right)\right.} \\
+        &\left.-\sin 2 \phi_{c} \frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{d}-\frac{\Gamma_{1}\left(\Gamma_{\gamma}+\Gamma_{2}\right)}{2 d}\right].
+    \end{aligned}
+\end{equation}
+
+\noindent
+The reaction cross section, Eq. (II B1 a.4)\ref{eq:reax-in-terms-of-X}, becomes
+
+\begin{equation}
+    \begin{aligned}
+        \sigma_{\text {reaction }}(E) &=\frac{4 \pi}{k_{\alpha}^{2}} g\left[\left(\frac{\Gamma \sqrt{\Gamma_{1} \Gamma_{2}}}{4 d}\right)^{2}+\left(\frac{\left(E-E_{\lambda}\right) \sqrt{\Gamma_{1} \Gamma_{2}}}{2 d}\right)^{2}\right] \\
+        &=\frac{\pi g}{k_{\alpha}^{2}}\left[\frac{\Gamma_{1} \Gamma_{2}}{d}\right],
+    \end{aligned}
+\end{equation}
+
+\noindent
+and, finally, the capture cross section, Eq. (II B1 a.6)\ref{eq:sig-cap-in-terms-of-X}, is
+
+\begin{equation}
+    \begin{aligned}
+        \sigma_{\text {capture }}(E) &=\frac{4 \pi g}{k_{\alpha}^{2}}\left[\frac{\Gamma \Gamma_{1}}{4 d}-\left\{\left(\frac{\Gamma \Gamma_{1}}{4 d}\right)^{2}+\left(\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{2 d}\right)^{2}\right.\right.\\
+        &\qquad\left.\left.+\left(\frac{\Gamma \sqrt{\Gamma_{1} \Gamma_{2}}}{4 d}\right)^{2}+\left(\frac{\left(E-E_{\lambda}\right) \sqrt{\Gamma_{1} \Gamma_{2}}}{2 d}\right)^{2}\right\}\right] \\
+        &= \frac{4 \pi g}{k_{\alpha}^{2}}\left[\frac{\Gamma \Gamma_{1}}{4 d}-\left\{\frac{\Gamma_{1}^{2}}{4 d}+\frac{\Gamma_{1} \Gamma_{2}}{4 d}\right\}\right]=\frac{\pi g}{k_{\alpha}^{2}}\left[\frac{\Gamma_{1} \bar{\Gamma}_{\gamma}}{d}\right] .
+    \end{aligned}
+\end{equation}
+
+\newpage
+\noindent
+\textbf{Angular distributions}

+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

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Simulation of Full R-Matrix} \label{subsec:r-matrix-full}