fix inconsistencies with old manual

docs/tex/scattering-theory.tex

@@ -95,11 +95,11 @@ The scattering matrix $U$ can be written in terms of matrix $W$ as
 where $\Omega$ is given by
 
 \begin{equation}\label{eq:omega}
-    \Omega_c = e^{i(w_c-\phi_c)} \:. 
+    \Omega_c = e^{i(w_c-\varphi_c)} \:. 
 \end{equation}
 
 \noindent
-Here again, $w_c$ is zero for non-Coulomb channels, and the potential scattering phase shifts for non-Coulomb interactions $\phi_c$ are defined in many references (e.g., \cite{lane_thomas_1958}) and shown in Table \ref{tab:penetrabilities}. The matrix $W$ in Eq. \ref{eq:scat-matrix} is related to the R-matrix (in matrix notation with indices suppressed) via
+Here again, $w_c$ is zero for non-Coulomb channels, and the potential scattering phase shifts for non-Coulomb interactions $\varphi_c$ are defined in many references (e.g., \cite{lane_thomas_1958}) and shown in Table \ref{tab:penetrabilities}. The matrix $W$ in Eq. \ref{eq:scat-matrix} is related to the R-matrix (in matrix notation with indices suppressed) via
 
 \begin{equation}\label{eq:W-matrix}
     W = P^{1/2}\left(I-RL\right)^{-1}\left(I-RL^*\right)P^{-1/2} \:. 
@@ -131,28 +131,28 @@ where $\rho$ is related to the center-of-mass momentum which in turn is related
 \noindent
 as shown in Section II.C.2 \ref{subsec:kinematic-conventions}. Here $\Xi_\alpha$ is the energy threshold for the particle pair $\alpha$, $m_\alpha$ and $M_\alpha$ are the masses of the two particles of particle pair $\alpha$, and $m$ and $M$ are the masses of the incident particle and target nuclide, respectively.
 
-Appropriate formulae\footnote{To avoid ambiguity, it should be stated that below the channel threshold, that is, for $(E-\Xi_\alpha)<0$, SAMMY uses the convention of Lane-Thomas, namely setting $P_c=0$ and $S_c=\text{Re}(L_c)=L_c$, instead of using an analytical continuation of the shift and penetrability function in the complex plane by computing the expressions in Table \ref{tab:penetrabilities} as a function of an imaginary $\rho$ for which $iP_c(\rho)$ becomes real-valued, but separate from $S_c(\rho)$.} for $P$, $S$, and $\phi$ in the non-Coulomb case are shown in Table II.A.1 \ref{tab:penetrabilities}. For two charged particles, formulae for the penetrabilities are given in Section II.C.4 \ref{subsec:charged-particle-conventions}. The energy dependence of fission and capture widths is negligible over the energy range of these calculations. Therefore, a penetrability of unity may be used.
+Appropriate formulae\footnote{To avoid ambiguity, it should be stated that below the channel threshold, that is, for $(E-\Xi_\alpha)<0$, SAMMY uses the convention of Lane-Thomas, namely setting $P_c=0$ and $S_c=\text{Re}(L_c)=L_c$, instead of using an analytical continuation of the shift and penetrability function in the complex plane by computing the expressions in Table \ref{tab:penetrabilities} as a function of an imaginary $\rho$ for which $iP_c(\rho)$ becomes real-valued, but separate from $S_c(\rho)$.} for $P$, $S$, and $\varphi$ in the non-Coulomb case are shown in Table II.A.1 \ref{tab:penetrabilities}. For two charged particles, formulae for the penetrabilities are given in Section II.C.4 \ref{subsec:charged-particle-conventions}. The energy dependence of fission and capture widths is negligible over the energy range of these calculations. Therefore, a penetrability of unity may be used.
 
 % multiline cell: \begin{tabular}{@{}c@{}} line1 \\ line2 \end{tabular}
 \begin{threeparttable}
 \centering
-\caption{Hard-sphere penetrability (penetration factor) $P$, level shift factor $S$, and potential-scattering phase shift $\phi$ for orbital angular momentum $l$, wave number $k$, and channel radius $a_c$, with $\rho=ka_c$.} \label{tab:penetrabilities}
+\caption{Hard-sphere penetrability (penetration factor) $P$, level shift factor $S$, and potential-scattering phase shift $\varphi$ for orbital angular momentum $l$, wave number $k$, and channel radius $a_c$, with $\rho=ka_c$.} \label{tab:penetrabilities}
 \begin{tabular}{ l c c c }
     \hline\hline
-    $\mathbf{l}$ & $\mathbf{P_l}$                                                                   & $\mathbf{S_l}$                                                                                       & $\mathbf{\phi_l}$   \cr
+    $\mathbf{l}$ & $\mathbf{P_l}$                                                                   & $\mathbf{S_l}$                                                                                       & $\mathbf{\varphi_l}$   \cr
     \hline\hline
              0   & $\rho$                                                                            &  0                                                                                                   & $\rho$                                                          \cr
              1   & $\rho^3/(1+\rho^2)$                                                               & $-1/(1+\rho^2)$                                                                                      & $\rho-\tan^{-1}\rho$                                             \cr
              2   & $\rho^5/(9+3\rho^2+\rho^4)$                                                       & $-(18+3\rho^2)/(9+3\rho^2+\rho^4)$                                                                   & $\rho-\tan^{-1}\left[3\rho/(3-\rho^2)\right]$                    \cr
              3   & \begin{tabular}{@{}c@{}}$\rho^7/$ \\ $(225+45\rho^2+6\rho^4+\rho^6)$\end{tabular} & \begin{tabular}{@{}c@{}} $-(675+90\rho^2+6\rho^4)/$ \\ $(225+45\rho^2+6\rho^4+\rho^6)$ \end{tabular} & $\rho-\tan^{-1}\left[\rho(15-\rho^2)/(15-6\rho^2)\right]$        \cr
              \hline
-             $l$ & $\frac{\rho^2P_{l-1}}{(1-S_{l-1})^2+P_{l-1}^2}$                                   & $\frac{\rho^2(l-S_{l-1})}{(1-S_{l-1})^2+P_{l-1}^2}-l$                                                & $\phi_{l-1} - \tan^{-1}\left(P_{l-1}/(l-S_{l-1})\right)$\tnote{\textdagger} \cr
+             $l$ & $\frac{\rho^2P_{l-1}}{(1-S_{l-1})^2+P_{l-1}^2}$                                   & $\frac{\rho^2(l-S_{l-1})}{(1-S_{l-1})^2+P_{l-1}^2}-l$                                                & $\varphi_{l-1} - \tan^{-1}\left(P_{l-1}/(l-S_{l-1})\right)$\tnote{\textdagger} \cr
 
     \hline\hline
 
 \end{tabular}
 \begin{tablenotes}
-    \item[\textdagger] \footnotesize{The iterative formula for $\phi_l$ could also be defined by $B_l = (B_{l-1}+X_l)/(1-B_{l-1}X_l)$ where $B_l=\tan(\rho-\phi_l)$ and $X_l=P_{l-1}/(l-S_{l-1})$}
+    \item[\textdagger] \footnotesize{The iterative formula for $\varphi_l$ could also be defined by $B_l = (B_{l-1}+X_l)/(1-B_{l-1}X_l)$ where $B_l=\tan(\rho-\varphi_l)$ and $X_l=P_{l-1}/(l-S_{l-1})$}
 \end{tablenotes}
 \end{threeparttable}\\
 
@@ -660,14 +660,14 @@ For spinless particles, $I_l^* = O_l$, so that
 and
 
 \begin{equation}
-    \frac{I_l}{O_l} = \frac{O_l^*}{O_l} = \frac{|O|e^{-i\phi}}{|O|e^{i\phi}} = e^{-2i\phi}.
+    \frac{I_l}{O_l} = \frac{O_l^*}{O_l} = \frac{|O|e^{-i\varphi}}{|O|e^{i\varphi}} = e^{-2i\varphi}.
 \end{equation}
 
 \noindent
 Therefore Eq. (II A2.34)\ref{eq:u-l-i-l-matching} becomes
 
 \begin{equation}
-    U_l = e^{-2i\phi}\frac{1-R_l(L_l^*-B_l)}{1-R_l(L_l-B_l)},
+    U_l = e^{-2i\varphi}\frac{1-R_l(L_l^*-B_l)}{1-R_l(L_l-B_l)},
 \end{equation}
 
 \noindent
@@ -701,8 +701,8 @@ For angle-integrated cross sections, the equation found by inserting Eq. (II A2
 \begin{equation}
     \begin{aligned}
         \sigma =& \int \left[-\frac{1}{2ik}\sum_l(2l+1)\left[U_l^*-1\right]P_l(\cos\theta)\right] \\
-        & \times\left[\frac{1}{2ik}\sum_{l'}(2l'+1)\left[U_{l'}-1\right]P_{l'}(\cos\theta)\right]d(\cos\theta)d\phi \\
-        = & \frac{1}{4k^2}\sum_{ll'}(2l+1)(2l'+1)[U_l^*-1][U_{l'}-1]\int_0^{2\pi}d\phi\int_{-1}^1P_l(\cos\theta)P_{l'}(\cos\theta)d(\cos\theta) \\
+        & \times\left[\frac{1}{2ik}\sum_{l'}(2l'+1)\left[U_{l'}-1\right]P_{l'}(\cos\theta)\right]d(\cos\theta)d\varphi \\
+        = & \frac{1}{4k^2}\sum_{ll'}(2l+1)(2l'+1)[U_l^*-1][U_{l'}-1]\int_0^{2\pi}d\varphi\int_{-1}^1P_l(\cos\theta)P_{l'}(\cos\theta)d(\cos\theta) \\
         = & \frac{1}{4k^2}\sum_{ll'}(2l+1)(2l'+1)[U_l^*-1][U_{l'}-1]2\pi\frac{2}{2l+1}\delta_{ll'} \\
         = & \frac{\pi}{k^2}\sum_l(2l+1)|U_l-1|^2 .
     \end{aligned}
@@ -821,8 +821,8 @@ The observable cross sections are found in terms of $X$ by first substituting Eq
 
 \begin{equation}\label{eq:sigma-in-terms-of-X}
     \begin{aligned}
-        \sigma_{\alpha,\alpha'}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \Biggl[ \bigl(\sin^2\phi_c(1-2X_{cc}^i) & - X_{cc}^r\sin(2\phi_c)\bigr)\delta_{\alpha,\alpha'} \\
-        & + \sum_{c'}\left({X^i_{cc'}}^2 + {X^r_{cc'}}^2\right) \Biggr].
+        \sigma_{\alpha,\alpha'}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \Biggl[ \bigl(\sin^2\varphi_c(1-2X_{cc}^i) & - X_{cc}^r\sin(2\varphi_c)\bigr)\delta_{\alpha,\alpha'} \\
+        & + \sum_{c'}\left ( {X^i_{cc'} }^2 + {X^r_{cc'}}^2 \right ) \Biggr].
     \end{aligned}
 \end{equation}
 
@@ -834,7 +834,7 @@ 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}\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],
+    \sigma_{total}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \left [ \left ( \sin^2\varphi_c + X^i_{cc}\cos(2\varphi_c)- X_{cc}^r\sin(2\varphi_c) \right ) \right ],
 \end{equation}
 
 \noindent
@@ -844,8 +844,8 @@ The angle integrated elastic cross section is given by
 
 \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].
+        \sigma_{elastic}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \Biggl[ \sin^2\varphi_c(1-2X^i_{cc}) &-X^r_{cc}\sin(2\varphi_c) \\
+        & +\sum_{c'}\left ( {X^i_{cc'}}^2 + {X^r_{cc'}}^2 \right ) \Biggr].
     \end{aligned}
 \end{equation}
 
@@ -855,7 +855,7 @@ 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}\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].
+    \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 $\alpha'$.
@@ -863,7 +863,7 @@ Here $c$ is restricted to those channels of the $J^\pi$ spin group from which th
 The absorption cross section has the form
 
 \begin{equation}
-    \sigma_{absorption}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c\left[X^i_{cc} -\sum_{c'} \left({X^i_{cc'}}^2+{X^r_{cc'}}^2\right)\right].
+    \sigma_{absorption}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_c \left [ X^i_{cc} -\sum_{c'} \left ( {X^i_{cc'}}^2 + {X^r_{cc'}}^2 \right ) \right ].
 \end{equation}
 
 \noindent
@@ -872,7 +872,7 @@ Here both the sum over $c$ and the sum over $c'$ include all incident particle c
 The capture cross section for the eliminated radiation channels can be calculated directly as
 
 \begin{equation}\label{eq:sig-cap-in-terms-of-X}
-    \sigma_{capture}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_{inc\; c}\left[X^i_{cc} -\sum_{all\; c'} \left({X^i_{cc'}}^2+{X^r_{cc'}}^2\right)\right].
+    \sigma_{capture}(E) = \frac{4\pi}{k_\alpha^2}\sum_J g_{J\alpha} \sum_{inc\; c}\left [ X^i_{cc} -\sum_{all\; c'} \left ( {X^i_{cc'}}^2 + {X^r_{cc'}}^2 \right ) \right ].
 \end{equation}
 
 \noindent
@@ -996,8 +996,8 @@ In this form, $X$ can be substituted into the equations for the various cross se
 
 \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],
+        \sigma_{\text {total }}(E) &=\frac{4 \pi}{k_{\alpha}^{2}} g_{J}\left[\sin ^{2} \varphi_{c}+\frac{\Gamma \Gamma_{1}}{4 d} \cos \left(2 \varphi_{c}\right)-\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{2 d} \sin \left(2 \varphi_{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 \varphi_{c}\right)-\frac{\left(E-E_{\lambda}\right) \Gamma_{1}}{d} \sin \left(2 \varphi_{c}\right)\right],
     \end{aligned}
 \end{equation}
 
@@ -1005,8 +1005,8 @@ in which $d$ has been used to represent $|E_\lambda-E-i\Gamma/2|^2=(E-E_\lambda)
 
 \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]
+        \sigma_{\text {elastic }}(E)=& \frac{4 \pi}{k_{\alpha}^{2}} \sum_{J} g_{J \alpha} \sum_{c}\left[\sin ^{2} \varphi_{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 \varphi_{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}
 
@@ -1015,8 +1015,8 @@ 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].
+        \sigma_{\text {elastic }}(E)=\frac{2 \pi}{k_{\alpha}^{2}} \sum_{J} g_{J \alpha} \sum_{c} & {\left[1-\cos 2 \varphi_{c}\left(1-\frac{\Gamma \Gamma_{1}}{2 d}\right)\right.} \\
+        &\left.-\sin 2 \varphi_{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}
 
@@ -1235,8 +1235,8 @@ external resonances to the tail of the R-matrix. A logarithmic parameterization
 
 \begin{equation}
     \begin{aligned}
-        R_{c}^{e x t}(E)=\left.\bar{R}\right|_{\text {con, },} &+\bar{R}_{\text {lin, },} E+\bar{R}_{q, c} E^{2}-s_{\text {lin, },}\left(E_{c}^{u p}-E_{c}^{\text {down }}\right) \\
-        &-\left(s_{\text {con, }, c}+s_{\text {lin, },} E\right) \ln \left[\frac{E_{c}^{u p}-E}{E-E_{c}^{\text {down }}}\right] .
+        R_{c}^{e x t}(E)=\bar{R}_{\text {con},c} &+\bar{R}_{\text {lin},c} E+\bar{R}_{q, c} E^{2}-s_{\text {lin},c}\left(E_{c}^{u p}-E_{c}^{\text {down }}\right) \\
+        &-\left(s_{\text {con}, c}+s_{\text {lin },c} E\right) \ln \left[\frac{E_{c}^{u p}-E}{E-E_{c}^{\text {down }}}\right] .
     \end{aligned}
 \end{equation}
 
@@ -1246,9 +1246,12 @@ The {\em u}-parameters (i.e., the parameters on which Bayes' equations will oper
 
 \begin{equation}
     \begin{aligned}
-        &u\left(E_{c}^{\text {down }}\right)=E_{c}^{\text {down }} \quad u\left(E_{c}^{u p}\right)=E_{c}^{u p}\\
-        &u\left(\bar{R}_{\text {con }, c}\right)=\bar{R}_{\text {con }, c} \quad u\left(\bar{R}_{l i n, c}\right)=\bar{R}_{l i n, c} u\left(\bar{R}_{q, c}\right)=\bar{R}_{q, c}\\
-        &u\left(s_{c o n, c}\right)=\sqrt{s_{c o n, c}} \quad u\left(s_{l i n, c}\right)=s_{l i n, c}
+        &u\left(E_{c}^{\text {down }}\right)=E_{c}^{\text {down }} 
+        \quad u\left(E_{c}^{u p}\right)=E_{c}^{u p}\\
+        &u\left(\bar{R}_{\text {con }, c}\right)=\bar{R}_{\text {con }, c} 
+        \quad u\left(\bar{R}_{l i n, c}\right)=\bar{R}_{l i n, c} u\left(\bar{R}_{q, c}\right)=\bar{R}_{q, c}\\
+        &u\left(s_{c o n, c}\right)=\sqrt{s_{c o n, c}} 
+        \quad u\left(s_{l i n, c}\right)=s_{l i n, c}
     \end{aligned}
 \end{equation}
 
@@ -1301,6 +1304,7 @@ Test case tr110 shows an artificial but extreme example of a situation in which
     \label{fig:pseudo-full-r-matrix}
 \end{figure}
 
+\clearpage
 \noindent
 \textbf{Different treatments for different capture channels}
 
@@ -1319,7 +1323,7 @@ contribution from the aggregate channels.
 The formula used to calculate the capture cross section is similar to Eq. (II B1 a.6)\ref{eq:sig-cap-in-terms-of-X}, with only the non-capture exit channels included in the summation over $c'$:
 
 \begin{equation}
-    \sigma_{\text {capture }}(E)=\frac{4 \pi}{k_{\alpha}^{2}} \sum_{J} g_{J \alpha} \sum_{c}\left[X_{c c}^{i}-\sum_{\substack{c^{\prime}=\text { non-capture } \\ \text { exit channels }}}\left\{X_{c c^{\prime}}^{i}+X_{c c^{\prime}}^{r}{ }^{2}\right\}\right].
+    \sigma_{\text {capture }}(E)=\frac{4 \pi}{k_{\alpha}^{2}} \sum_{J} g_{J \alpha} \sum_{c}\left[X_{c c}^{i}-\sum_{\substack{c^{\prime}=\text { non-capture } \\ \text { exit channels }}}\left\{ {X_{c c^{\prime}}^{i}}^2 + X_{c c^{\prime}}^{r}{ }^{2}\right\}\right].
 \end{equation}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1344,7 +1348,7 @@ The MLBW elastic (scattering) cross section may be written in the form
 
 \begin{equation}\label{eq:mlbw-elastic}
     \begin{aligned}
-        \sigma^{\text {elastic }}=& \frac{\pi}{k^{2}} \sum_{J} g_{J} \sum_{c}\left\{(1-\cos 2 \varphi)\left(2-\sum_{\lambda} \Gamma_{\lambda c} \Gamma_{\lambda} / d_{\lambda}\right)\right.\\
+        \sigma_{\text {elastic }}=& \frac{\pi}{k^{2}} \sum_{J} g_{J} \sum_{c}\left\{(1-\cos 2 \varphi)\left(2-\sum_{\lambda} \Gamma_{\lambda c} \Gamma_{\lambda} / d_{\lambda}\right)\right.\\
         &+2 \sin 2 \varphi \sum_{\lambda} \Gamma_{\lambda c}\left(E-E_{\lambda}\right) / d_{\lambda} \\
         &\left.+\left(\sum_{\lambda} \Gamma_{\lambda c}\left(E-E_{\lambda}\right) / d_{\lambda}\right)^{2}+\left(\sum_{\lambda} \Gamma_{\lambda c} \Gamma_{\lambda} / 2 d_{\lambda}\right)^{2}\right\},
     \end{aligned}
@@ -1380,12 +1384,12 @@ in which the sum over $c$ includes all particle channels (i.e., over all channel
 For both MLBW and SLBW, the fission cross section is given by
 
 \begin{equation}\label{eq:mlbw-fission}
-    \sigma^{fission} = \frac{\pi}{k^2}\sum_Jg_J\sum_c\sum_{c'}\sum_\lambda \frac{\Gamma_{\lambda_c}\Gamma_{\lambda c'}}{d_\lambda},
+    \sigma_\text{fission} = \frac{\pi}{k^2}\sum_Jg_J\sum_c\sum_{c'}\sum_\lambda \frac{\Gamma_{\lambda_c}\Gamma_{\lambda c'}}{d_\lambda},
 \end{equation}
 
 \noindent
 % >>> removed reference to section for indiv. reax types
-in which the sum over $c$ includes only incident (neutron) channels, $d_\lambda$ is again given by Eq. (II B3 a.4)\ref{eq:d-lambda}, and the sum over $c'$ includes all exit channels. Caution: In principle, Eq. (II B3 a.5)\ref{eq:mlbw-fission} could be used to describe any reaction channel, where term ``reaction'' encompasses any non-elastic, non-capture channel. However, the only reaction channel permitted in ENDF is fission; for SLBW only one fission channel is permitted, and for MLBW two fission channels may be used. In addition, ENDF allows only one neutron channel (i.e., only one entrance channel). Because SAMMY's Breit-Wigner options were created solely for use with ENDF evaluations (for comparison purposes), similar restrictions apply to the use of the Breit-Wigner approximations in SAMMY. (For the more general case involving other reactions such as inelastic, (n,p), (n,$\alpha$), or fission with more than two channels, use the Reich-Moore approximation as discussed in Section \ref{subsec:r-matrix-RM} ``Specifying individual reaction types'')
+in which the sum over $c$ includes only incident (neutron) channels, $d_\lambda$ is again given by Eq. (II B3 a.4)\ref{eq:d-lambda}, and the sum over $c'$ includes all exit channels. Caution: In principle, Eq. (II B3 a.5)\ref{eq:mlbw-fission} could be used to describe any reaction channel, where term ``reaction'' encompasses any non-elastic, non-capture channel. However, the only reaction channel permitted in ENDF (LRF=2) is fission. In addition, ENDF allows only one neutron channel (i.e., only one entrance channel). Because SAMMY's Breit-Wigner options were created solely for use with ENDF evaluations (for comparison purposes), similar restrictions apply to the use of the Breit-Wigner approximations in SAMMY. (For the more general case involving other reactions such as inelastic, (n,p), (n,$\alpha$), or fission with more than two channels, use the Reich-Moore approximation as discussed in Section \ref{subsec:r-matrix-RM} ``Specifying individual reaction types'')
 
 The Breit-Wigner form for the capture cross section is
 
@@ -1397,14 +1401,14 @@ The Breit-Wigner form for the capture cross section is
 where, again, the sum over $c$ includes only incident (neutron) channels. Total and absorption cross sections are given by the appropriate sums of the other three cross sections,
 
 \begin{equation}
-    \sigma^{total} = \sigma^{elastic} + \sigma^{fission} + \sigma^{capture}
+    \sigma_\text{total} = \sigma_\text{elastic} + \sigma_\text{fission} + \sigma_\text{capture}
 \end{equation}
 
 \noindent
 and
 
 \begin{equation}
-    \sigma^{absorption} = \sigma^{fission} + \sigma^{capture}.
+    \sigma_\text{absorption} = \sigma_\text{fission} + \sigma_\text{capture}.
 \end{equation}
 
 As noted in Section IV.C\ref{sec:construct-par-set}, it is the $u$-parameters on which Bayes' equations operate. The uparameters associated with the MLBW and SLBW resonances are defined similarly to those for Reich-Moore resonances: