From 1822967a2e11135e0b08a5e6870dc291cb2d10b2 Mon Sep 17 00:00:00 2001
From: Brown <2mx@ornl.gov>
Date: Wed, 1 Dec 2021 18:01:53 -0500
Subject: [PATCH] add scattering r-matrix eqtns
---
docs/tex/scattering-theory.tex | 142 ++++++++++++++++++++++++++++++---
1 file changed, 133 insertions(+), 9 deletions(-)
diff --git a/docs/tex/scattering-theory.tex b/docs/tex/scattering-theory.tex
index 6eda9f18a..9af811010 100644
--- a/docs/tex/scattering-theory.tex
+++ b/docs/tex/scattering-theory.tex
@@ -72,21 +72,21 @@ In all formulae given below, spin quantum numbers (e.g., $J$ ) are implicitly as
Let the angle-integrated cross sections from entrance channel $c$ to exit channel $c'$ with total angular momentum $J$ be represented by $\sigma_{cc'}$. This cross section is given in terms of the scattering matrix $U_{cc'}$ as
\begin{equation}\label{eq:sigma-ccprime}
-\sigma_{cc'} = \frac{\pi}{k_\alpha^2}g_{J\alpha}\left|e^{2iw_c}\delta_{cc'}-U_{cc'}\right|^2\delta_{JJ'} \:,
+ \sigma_{cc'} = \frac{\pi}{k_\alpha^2}g_{J\alpha}\left|e^{2iw_c}\delta_{cc'}-U_{cc'}\right|^2\delta_{JJ'} \:,
\end{equation}
\noindent
where $k_\alpha$ is the wave number (and $K_\alpha = \hbar k_\alpha =$ center-of-mass momentum) associated with incident particle pair $\alpha$, $g_{J\alpha}$ is the spin statistical factor, and $w_c$ is the Coulomb phase-shift difference. Note that $w_c$ is zero for non-Coulomb channels. (Details for the charged-particle case are presented in Section II.C.4.\ref{}) The spin statistical factor $g_{J\alpha}$ is given by
\begin{equation}\label{eq:spin-stat-factor}
-g_{J\alpha} = \frac{2J+1}{(2i+1)(2I+1)} \:,
+ g_{J\alpha} = \frac{2J+1}{(2i+1)(2I+1)} \:,
\end{equation}
\noindent
and center-of-mass momentum $K_\alpha$ by
\begin{equation}\label{eq:center-of-mass-momentum}
-K_\alpha^2 = (\hbar k_\alpha)^2 = \frac{2mM^2}{(m+M)^2}E \:.
+ K_\alpha^2 = (\hbar k_\alpha)^2 = \frac{2mM^2}{(m+M)^2}E \:.
\end{equation}
\noindent
@@ -95,28 +95,28 @@ Here $E$ is the \textbf{laboratory} kinetic energy of the incident (moving) part
The scattering matrix $U$ can be written in terms of matrix $W$ as
\begin{equation}\label{eq:scat-matrix}
-U_{cc'} = \Omega_cW_{cc'}\Omega_{c'} \:,
+ U_{cc'} = \Omega_cW_{cc'}\Omega_{c'} \:,
\end{equation}
\noindent
where $\Omega$ is given by
\begin{equation}\label{eq:omega}
-\Omega_c = e^{i(w_c-\phi_c)} \:.
+ \Omega_c = e^{i(w_c-\phi_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{}. 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} \:.
+ W = P^{1/2}\left(I-RL\right)^{-1}\left(I-RL^*\right)P^{-1/2} \:.
\end{equation}
\noindent
The quantity $I$ in this equation represents the identity matrix, and superscript $*$ indicates a complex conjugate. The form of the R-matrix is given in Section IIA.1\ref{} in general Section II.B\ref{} for the versions used in SAMMY. The quantity $L$ in Eq. \ref{eq:W-matrix} is given by
\begin{equation}\label{eq:L-matrix}
-L = (S-B) + iP \:,
+ L = (S-B) + iP \:,
\end{equation}
\noindent
@@ -137,7 +137,7 @@ where $\rho$ is related to the center-of-mass momentum which in turn is related
as shown in Section II.C.2\ref{}. 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 for $P$, $S$, and $\phi$ in the non-Coulomb case are shown in Table IIA.1. For two charged particles, formulae for the penetrabilities are given in Section II.C.4\ref{}.
+Appropriate formulae for $P$, $S$, and $\phi$ in the non-Coulomb case are shown in Table IIA.1. For two charged particles, formulae for the penetrabilities are given in Section II.C.4\ref{tab:penetrabilities}.
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.
@@ -160,10 +160,134 @@ The energy dependence of fission and capture widths is negligible over the energ
\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 $X_l=P_{l-1}/(l-S_l)$}
+ \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 $X_l=P_{l-1}/(l-S_{l-1})$}
\end{tablenotes}
\end{threeparttable}
+Formulae for a particular cross section type can be derived by summing over the terms in Eq. (II A.1)\ref{eq:sigma-ccprime}. For the total cross section, the sum over all possible exit channels and all spin groups gives
+
+\begin{equation}\label{eq:sigma-tot}
+ \begin{aligned}
+ \sigma^{total} & = \sum_{\substack{incident \\ channels \\ c }} \sum_{\substack{all \\ channels \\ c' }} \sum_J \frac{\pi}{k_\alpha^2}g_\alpha|\delta_{cc'}-U_{cc'}|^2 \\
+ & = \frac{\pi}{k_\alpha^2} \sum_{J}g_J \sum_{\substack{incident \\ channels \\ c }} \sum_{\substack{all \\ channels \\ c' }} \left( \delta_{cc'}-U_{cc'}\delta_{cc'}-U^*_{cc'}\delta_{cc'}+\left|U_{cc'}\right|^2 \right) \\
+ & = \frac{2\pi}{k_\alpha^2} \sum_J g_J \sum_{\substack{incident \\ channels \\ c }} \left( 1-\text{Re}\left(U_{cc}\right) \right).
+ \end{aligned}
+\end{equation}
+
+For non-charged incident particles, the elastic (or scattering) cross section is given by
+
+\begin{equation}\label{eq:sigma-scat}
+ \sigma_{\alpha\alpha} = \frac{\pi}{k_\alpha^2} \sum_J g_J \sum_{\substack{ c=incident \\ channel }} \left( 1-2\text{Re}\left(U_{cc}\right) + \sum_{\substack{c'=incident \\ channel }} \left|U_{cc'}\right|^2 \right).
+\end{equation}
+
+Similarly, the cross section for any non-elastic reaction can be written
+
+\begin{equation}\label{eq:sigma-reaction}
+ \sigma_\alpha^{reaction} = \frac{\pi}{k_\alpha^2} \sum_J g_J \sum_{\substack{ c=incident \\ channel }} \sum_{\substack{ c'=reaction \\ channel }} \left| U_{cc'} \right|^2.
+\end{equation}
+
+In particular, the capture cross section could be written as the difference between the total and all
+other cross sections,
+
+\begin{equation}\label{eq:sigma-capture}
+ \sigma_\alpha^{reaction} = \frac{\pi}{k_\alpha^2} \sum_J g_J \sum_{\substack{ c=incident \\ channel }} \left( 1 - \sum_{\substack{ c'=all\: channels \\ except\: capture }} \left| U_{cc'} \right|^2 \right).
+\end{equation}
+
+(This form will be used later, in Section II.B.1.a, when the capture channels are treated in an
+approximate fashion.)
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{R-Matrix and A-Matrix Equations}\label{subsec:r-matrix-a-matrix}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+The R-matrix was introduced in Eq. \ref{eq:W-matrix} as
+
+\begin{equation} % don't ref, give \ref{eq:W-matrix}
+ W = P^{1/2}\left(I-RL\right)^{-1}\left(I-RL^*\right)P^{-1/2} \:,
+\end{equation}
+
+but the formula for the R-matrix was not given there. If $\lambda$ represents a particular resonance (or energy level), then the general form for the R-matrix is
+
+\begin{equation}\label{eq:r-matrix}
+ R_{cc'} = \sum_\lambda \frac{ \gamma_{\lambda c}\gamma_{\lambda c'} }{ E_\lambda - E } \delta_{J,J'} \:,
+\end{equation}
+
+where $E_\lambda$ represents the energy of the resonance, and the reduced width amplitude $\gamma$ is related to the partial width $\Gamma$ by
+
+\begin{equation}\label{eq:reduced-widths}
+ \Gamma_{\lambda c} = 2 \bf{P_c}\gamma_{\lambda c}^2.
+\end{equation}
+
+The sum in Eq. (II A1.2)\ref{eq:r-matrix} contains an infinite number of levels. All channels, including the ``gamma channel'' for which one of the particles is a photon, are represented by the channel indices.
+
+The R-matrix is not the only possibility for parameterization of the scattering matrix. In the R-matrix formulation, equations are expressed in terms of channel-channel interactions. It is also possible to formulate scattering theory in terms of level-level interactions; this formulation uses what is called the A-matrix, which is defined as
+
+\begin{equation}\label{eq:a-matrix}
+ A_{\mu\lambda}^{-1} = \left( E_\lambda - E \right)\delta_{\mu\lambda} - \sum_c \gamma_{\mu c}L_c\gamma_{\lambda c} .
+\end{equation}
+
+\noindent
+To see the relationship of the A-matrix to the R-matrix, we begin by multiplying both
+sides of Eq. (II A1.4) by $A$ and summing over $\lambda$:
+
+\begin{equation}
+ \begin{aligned}
+ \sum_\lambda A_{\mu\lambda}^{-1}A_{\lambda\nu} & = \sum_\lambda \left(E_\lambda-E\right)\delta_{\mu\lambda}A_{\lambda\nu} - \sum_c \gamma_{\mu c} L_c \gamma_{\lambda c} A_{\lambda \nu}, \\
+ & \text{or} \\
+ \delta_{\mu\nu} & = \left(E_\mu-E\right)A_{\mu\nu} - \sum_c \gamma_{\mu c}L_c \sum_\lambda \gamma_{\lambda c}A_{\lambda\nu} .
+ \end{aligned}
+\end{equation}
+
+\noindent
+Dividing by $(E_\mu - E)$, multiplying on the left by $\gamma_{\mu c'}$ and on the right by $\gamma_{\nu c''}$, and summing over $\mu$ puts this equation into the form
+
+\begin{equation}
+ \begin{aligned}
+ \sum_\mu \gamma_{\mu c'}\left(E_{\mu}-E\right)^{-1}\delta_{\mu\nu}\gamma_{\nu c''} = & \sum_\mu \gamma_{\mu c'}\left(E_\mu-E\right)^{-1}\left(E_\mu-E\right)A_{\mu\nu}\gamma_{\nu c''} \\
+ & - \sum_\mu \gamma_{\mu c'} \left(E_{\mu}-E\right)^{-1} \sum_c \gamma_{\mu c} L_c \sum_\lambda \gamma_{\lambda c} A_{\lambda \nu}\gamma_{\nu c''} ,
+ \end{aligned}
+\end{equation}
+
+\noindent
+which can be reduced to
+
+\begin{equation}
+ \begin{aligned}
+ \gamma_{\nu c'}\left(E_{\nu}-E\right)^{-1}\gamma_{\nu c''} = & \sum_{\mu} \gamma_{\mu c'} A_{\mu\nu}\gamma_{\nu c''} \\
+ & - \sum_c \left[ \sum_\mu \gamma_{\mu c'}\left(E_{\mu}-E\right)^{-1}\gamma_{\mu c} \right] L_c \sum_\lambda \gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''}.
+ \end{aligned}
+\end{equation}
+
+\noindent
+Summing over $\nu$ puts this into the form
+
+\begin{equation}
+ \begin{aligned}
+ \left[ \sum_\nu \gamma_{\nu c'}\left(E_{\nu}-E\right)^{-1}\gamma_{\nu c''} \right] = & \sum_{\mu\nu} \gamma_{\mu c'} A_{\mu\nu}\gamma_{\nu c''} \\
+ & - \sum_c \left[ \sum_\mu \gamma_{\mu c'}\left(E_{\mu}-E\right)^{-1}\gamma_{\mu c} \right] L_c \sum_{\lambda\nu} \gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''},
+ \end{aligned}
+\end{equation}
+
+\noindent
+in which we can replace the quantities in square brackets by the R-matrix, giving
+
+\begin{equation}
+ \begin{aligned}
+ R_{c'c''} = & \sum_{\mu\nu} \gamma_{\mu c'} A_{\mu\nu}\gamma_{\nu c''} - \sum_c R_{c'c} L_c \sum_{\lambda\nu} \gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''}, \\
+ = & \sum_c \left[ \delta_{c'c} - R_{c'c}L_c \right] \sum_{\lambda\nu}\gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''}.
+ \end{aligned}
+\end{equation}
+
+\noindent
+Solving for the summation, this equation can be rewritten as
+
+\begin{equation}
+ \left[(I-RL)^{-1}R\right]_{cc''} = \sum_{\lambda\nu} \gamma_{\lambda c} A_{\lambda\nu} \gamma_{\nu c''}.
+\end{equation}
+
+
+
+
--
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