From 33fcee312fc58e5698d1a94230ff4a4c46e5da08 Mon Sep 17 00:00:00 2001
From: Brown <2mx@ornl.gov>
Date: Mon, 13 Dec 2021 17:35:54 -0500
Subject: [PATCH] make CoM equations more explicit
---
docs/tex/sammy_main.pdf | 4 +-
docs/tex/scattering-theory.tex | 71 +++++++++++++++++++++-------------
2 files changed, 47 insertions(+), 28 deletions(-)
diff --git a/docs/tex/sammy_main.pdf b/docs/tex/sammy_main.pdf
index 95b28a404..c5665c693 100644
--- a/docs/tex/sammy_main.pdf
+++ b/docs/tex/sammy_main.pdf
@@ -1,3 +1,3 @@
version https://git-lfs.github.com/spec/v1
-oid sha256:83ea9c422948f4959672c9b7e6bf728ed7092a2eed3b5c4dc8fd02d02f34746e
-size 463011
+oid sha256:5c20e643d710f8b1e991b154513aca20d5a2b0760f3c85366a2dee1c3e1feb07
+size 474945
diff --git a/docs/tex/scattering-theory.tex b/docs/tex/scattering-theory.tex
index fdd6076e2..a38956587 100644
--- a/docs/tex/scattering-theory.tex
+++ b/docs/tex/scattering-theory.tex
@@ -65,6 +65,7 @@ vector sum of the spins of the two particles of the pair: $\vec{s} = \vec{i} + \
of $l$ and $s$: $\vec{J} = \vec{l} + \vec{s}$.
\end{itemize}
+\noindent
Only $J$ and its associated parity $\pi$ are conserved for any given interaction. The other quantum numbers may differ from channel to channel, as long as the sum rules for spin and parity are obeyed. Within this document and within the SAMMY code, the set of all channels with the same $J$ and $\pi$ are called a ``spin group.''
In all formulae given below, spin quantum numbers (e.g., $J$ ) are implicitly assumed to include the associated parity. Quantized vector sum rules are implicitly assumed to be obeyed. Readers unfamiliar with these sum rules are referred to Section II.C.1.a\ref{} for a mini-tutorial on the subject.
@@ -135,11 +136,10 @@ where $\rho$ is related to the center-of-mass momentum which in turn is related
\rho = k_\alpha a_c = \frac{1}{\hbar} \sqrt{\frac{2m_\alpha M_\alpha}{m_\alpha+M_\alpha} \frac{M}{m+M}} \sqrt{(E-\Xi_\alpha)}\: a_c \:,
\end{equation}
+\noindent
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{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.
+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.
% multiline cell: \begin{tabular}{@{}c@{}} line1 \\ line2 \end{tabular}
\begin{threeparttable}
@@ -174,18 +174,21 @@ Formulae for a particular cross section type can be derived by summing over the
\end{aligned}
\end{equation}
+\noindent
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}
+\noindent
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}
+\noindent
In particular, the capture cross section could be written as the difference between the total and all
other cross sections,
@@ -193,6 +196,7 @@ other cross sections,
\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}
+\noindent
(This form will be used later, in Section II.B.1.a, when the capture channels are treated in an
approximate fashion.)
@@ -206,19 +210,22 @@ The R-matrix was introduced in Eq. \ref{eq:W-matrix} as
W = P^{1/2}\left(I-RL\right)^{-1}\left(I-RL^*\right)P^{-1/2} \:,
\end{equation}
+\noindent
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}
+\noindent
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.
+\noindent
+Note that in Eq. \ref{eq:r-matrix} that energies and widths are given in laboratory frame of reference, while the derivation in \S \ref{subsec:derivation-of-scat-theory} is in center-of-mass (please see Eq. \ref{eq:lab-to-com-params} for the relationship of laboratory and center-of-mass parameters). 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
@@ -348,11 +355,11 @@ Many authors have given derivations of the equations for the scattering matrix i
The Schr\"{o}dinger equation with a complex potential is
\begin{equation}
- \left( \frac{-\hbar^2}{2m}\nabla^2 + V + iW \right)\psi = E\psi,
+ \left( \frac{-\hbar^2}{2\mu}\nabla^2 + V + iW \right)\psi = E^{\text{(CoM)}}\psi,
\end{equation}
\noindent
-in which one can consider that $V$ causes scattering and $W$ causes absorption. The wave function
+in which one can consider that $V$ causes scattering and $W$ causes absorption and the reduced mass $\mu = \frac{mM}{m+M}$ ($m$ and $M$ defined as before). \textbf{Note that energies in \S \ref{subsec:derivation-of-scat-theory} are in the center-of-mass (CoM) frame.} The wave function
can be expanded in the usual fashion,
\begin{equation}
@@ -363,7 +370,7 @@ can be expanded in the usual fashion,
for which the radial portion obeys the equation
\begin{equation}\label{eq:radial-wave-equation}
- \frac{d^2u_l}{dr^2} + \left[ k^2-\frac{2m}{\hbar^2}(V+iW)-\frac{l(l+1)}{r^2} \right]u_l = 0,
+ \frac{d^2u_l}{dr^2} + \left[ k^2-\frac{2\mu}{\hbar^2}(V+iW)-\frac{l(l+1)}{r^2} \right]u_l = 0,
\end{equation}
\noindent
@@ -398,14 +405,14 @@ Our goal is to determine an appropriate analytic form for $U_l$.
For the interior region $r<a$ , we define eigenfunctions $w_{\lambda l}(r)$ and eigenvalues $E_\lambda$,
\begin{equation}
- E_\lambda = \frac{\hbar^2k_\lambda^2}{2m},
+ E_\lambda^\text{(CoM)} = \frac{\hbar^2k_\lambda^2}{2\mu},
\end{equation}
\noindent
for the wave equation without absorption $(W = 0)$,
\begin{equation}\label{eq:wave-func-without-abs}
- \frac{d^2w_{\lambda l}}{dr^2} + \left[k_\lambda^2-\frac{2m}{\hbar^2}V-\frac{l(l+1)}{r^2}\right]w_{\lambda l} = 0,
+ \frac{d^2w_{\lambda l}}{dr^2} + \left[k_\lambda^2-\frac{2\mu}{\hbar^2}V-\frac{l(l+1)}{r^2}\right]w_{\lambda l} = 0,
\end{equation}
\noindent
@@ -432,7 +439,7 @@ in which both equations of (II A2.9)\ref{eq:schrod-bound-conditions} have been i
\begin{equation}\label{eq:reform-orthog}
\begin{aligned}
- \int_0^a\left( \frac{d^2w_{\lambda l}}{dr^2}w_{\mu l} - w_{\lambda l}\frac{d^2w_{\mu l}}{dr^2} \right) dr & = \int_0^a\left( \left[-k_\lambda^2-\frac{2mV}{\hbar^2}\right]w_{\lambda l}w_{\mu l} - w_{\lambda l}\left[-k_\lambda^2-\frac{2mV}{\hbar^2}\right]w_{\lambda l} \right)dr \\
+ \int_0^a\left( \frac{d^2w_{\lambda l}}{dr^2}w_{\mu l} - w_{\lambda l}\frac{d^2w_{\mu l}}{dr^2} \right) dr & = \int_0^a\left( \left[-k_\lambda^2-\frac{2\mu V}{\hbar^2}\right]w_{\lambda l}w_{\mu l} - w_{\lambda l}\left[-k_\lambda^2-\frac{2\mu V}{\hbar^2}\right]w_{\lambda l} \right)dr \\
& = \int_0^a\left( -k_\lambda^2w_{\lambda l}w_{\mu l} + k_\mu^2w_{\lambda l}w_{\mu l} \right)dr \\
& = -(k_\lambda^2-k_\mu^2)\int_0^a w_{\lambda l}w_{\mu l}dr .
\end{aligned}
@@ -492,8 +499,8 @@ which can be expanded by use of Eqs. (II A2.3)\ref{eq:radial-wave-equation} and
\begin{equation}\label{eq:schroding-plugin}
\begin{aligned}
& \int_0^a \left( \frac{d^2u_{l}}{dr^2}w_{\lambda l} - u_{l}\frac{d^2w_{\lambda l}}{dr^2} \right) dr \\
- & = \int_0^a\left( \left[k^2-\frac{2m}{\hbar^2}(V+iW)-\frac{l(l+1)}{r^2}\right]u_lw_{\lambda l} + u_l\left[k_\lambda^2-\frac{2m}{\hbar^2}V-\frac{l(l+1)}{r^2}\right]w_{\lambda l} \right)dr \\
- & = (k_\lambda^2 - k^2) \int_0^a u_lw_{\lambda l}dr + \frac{2m}{\hbar}\int_0^aWu_lw_{\lambda l}dr.
+ & = \int_0^a\left( \left[k^2-\frac{2\mu}{\hbar^2}(V+iW)-\frac{l(l+1)}{r^2}\right]u_lw_{\lambda l} + u_l\left[k_\lambda^2-\frac{2\mu}{\hbar^2}V-\frac{l(l+1)}{r^2}\right]w_{\lambda l} \right)dr \\
+ & = (k_\lambda^2 - k^2) \int_0^a u_lw_{\lambda l}dr + \frac{2\mu}{\hbar}\int_0^aWu_lw_{\lambda l}dr.
\end{aligned}
\end{equation}
@@ -508,7 +515,7 @@ Defining $\overline{W}_{\lambda l}$ as
permits rewriting Eq. (II A2.18)\ref{eq:schroding-plugin} in the form
\begin{equation}\label{eq:schroding-plugin-reduced}
- \int_0^a \left( \frac{d^2u_{l}}{dr^2}w_{\lambda l} - u_{l}\frac{d^2w_{\lambda l}}{dr^2} \right) dr = \left( k_\lambda^2 - k^2 + i\frac{2m}{\hbar^2}\overline{W}_{\lambda l} \right)\int_0^a u_lw_{\lambda l}dr.
+ \int_0^a \left( \frac{d^2u_{l}}{dr^2}w_{\lambda l} - u_{l}\frac{d^2w_{\lambda l}}{dr^2} \right) dr = \left( k_\lambda^2 - k^2 + i\frac{2\mu}{\hbar^2}\overline{W}_{\lambda l} \right)\int_0^a u_lw_{\lambda l}dr.
\end{equation}
\noindent
@@ -525,7 +532,7 @@ Integrating the left-hand side of this equation gives
in which we have again made use of the boundary condition of Eq. (II A2.9)\ref{eq:schrod-bound-conditions}. Integrating the right-hand side of Eq. (II A2.20)\ref{eq:schroding-plugin-reduced} by applying Eq. (II A2.16)\ref{eq:c-lambda-l} gives
\begin{equation}\label{eq:schroding-right}
- \left( k_\lambda^2 - k^2 + i\frac{2m}{\hbar^2}\overline{W}_{\lambda l} \right)\int_0^a u_lw_{\lambda l}dr = \left( k_\lambda^2 - k^2 + i\frac{2m}{\hbar^2}\overline{W}_{\lambda l} \right)c_{\lambda l}.
+ \left( k_\lambda^2 - k^2 + i\frac{2\mu}{\hbar^2}\overline{W}_{\lambda l} \right)\int_0^a u_lw_{\lambda l}dr = \left( k_\lambda^2 - k^2 + i\frac{2\mu}{\hbar^2}\overline{W}_{\lambda l} \right)c_{\lambda l}.
\end{equation}
\noindent
@@ -533,8 +540,8 @@ Equating Eqs. (II A2.21)\ref{eq:schroding-left} and (II A2.22)\ref{eq:schroding-
\begin{equation}
\begin{aligned}
- \left[ a\frac{du_l}{dr}-u_lB_l \right]_{r=a}\frac{w_{\lambda l}}{a} & = \left( k_\lambda^2 - k^2 + i\frac{2m}{\hbar^2}\overline{W}_{\lambda l} \right)c_{\lambda l}, \\
- \left[ a\frac{du_l}{dr}-u_lB_l \right]_{r=a}\frac{w_{\lambda l}}{a} & = \left( E_\lambda - E + i\overline{W}_{\lambda l} \right)\frac{2mc_{\lambda l}}{\hbar^2},
+ \left[ a\frac{du_l}{dr}-u_lB_l \right]_{r=a}\frac{w_{\lambda l}}{a} & = \left( k_\lambda^2 - k^2 + i\frac{2\mu }{\hbar^2}\overline{W}_{\lambda l} \right)c_{\lambda l}, \\
+ \left[ a\frac{du_l}{dr}-u_lB_l \right]_{r=a}\frac{w_{\lambda l}}{a} & = \left( E^\text{(CoM)}_\lambda - E^\text{(CoM)} + i\overline{W}_{\lambda l} \right)\frac{2\mu c_{\lambda l}}{\hbar^2},
\end{aligned}
\end{equation}
@@ -542,21 +549,21 @@ Equating Eqs. (II A2.21)\ref{eq:schroding-left} and (II A2.22)\ref{eq:schroding-
or
\begin{equation}
- c_{\lambda l} = \frac{\hbar^2w_{\lambda l}(a)}{2ma\left(E_\lambda-E-i\overline{W}_{\lambda l}\right)} \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a}.
+ c_{\lambda l} = \frac{\hbar^2w_{\lambda l}(a)}{2\mu a\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{W}_{\lambda l}\right)} \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a}.
\end{equation}
\noindent
Inserting this into Eq. (II A2.15)\ref{eq:internal-wave-func} gives
\begin{equation}
- u_l(r) = \sum_{\lambda} w_{\lambda l}(r)\frac{\hbar^2w_{\lambda l}(a)}{2ma\left(E_\lambda-E-i\overline{W}_{\lambda l}\right)} \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a},
+ u_l(r) = \sum_{\lambda} w_{\lambda l}(r)\frac{\hbar^2w_{\lambda l}(a)}{2\mu a\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{W}_{\lambda l}\right)} \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a},
\end{equation}
\noindent
which when evaluated at $r = a$ , becomes
\begin{equation}
- u_l(a) = \sum_{\lambda} \frac{\hbar^2w_{\lambda l}^2(a)}{2ma\left(E_\lambda-E-i\overline{W}_{\lambda l}\right)} \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a},
+ u_l(a) = \sum_{\lambda} \frac{\hbar^2w_{\lambda l}^2(a)}{2\mu a\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{W}_{\lambda l}\right)} \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a},
\end{equation}
\noindent
@@ -564,8 +571,8 @@ Rearranging, this becomes
\begin{equation}\label{eq:u-l-eval-at-bound}
\begin{aligned}
- u_l(a) & = \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a} \sum_{\lambda} \frac{\hbar^2w_{\lambda l}^2(a)/2ma}{\left(E_\lambda-E-i\overline{W}_{\lambda l}\right)}
- & = \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a} \sum_{\lambda} \frac{\gamma_{\lambda l}^2}{\left(E_\lambda-E-i\overline{\Gamma}_{\lambda l}/2\right)},
+ u_l(a) & = \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a} \sum_{\lambda} \frac{\hbar^2w_{\lambda l}^2(a)/2\mu a}{\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{W}_{\lambda l}\right)} \\
+ & = \left[a\frac{du_l}{dr}-u_lB_l\right]_{r=a} \sum_{\lambda} \frac{\gamma_{\lambda l}^2}{\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{\Gamma}_{\lambda l}/2\right)},
\end{aligned}
\end{equation}
@@ -573,7 +580,7 @@ Rearranging, this becomes
in which the decay amplitude $\gamma_{\lambda l}$ is defined as
\begin{equation}
- \gamma_{\lambda l} \equiv \sqrt{\frac{\hbar^2w_{\lambda l}^2(a)}{2ma}}
+ \gamma_{\lambda l} \equiv \sqrt{\frac{\hbar^2w_{\lambda l}^2(a)}{2\mu a}}
\end{equation}
\noindent
@@ -585,10 +592,10 @@ and the absorption width $\Gamma_{\lambda l}$ as
\end{equation}
\noindent
-If we then define the R-function as
+If we then define the $R$-function as
-\begin{equation}
- R_l = \sum_\lambda \frac{\gamma_{\lambda l}^2}{\left(E_\lambda-E-i\overline{\Gamma}_{\lambda l}/2\right)},
+\begin{equation}\label{eq:R-func-center-of-mass}
+ R_l = \sum_\lambda \frac{\gamma_{\lambda l}^2}{\left(E^\text{(CoM)}_\lambda-E^\text{(CoM)}-i\overline{\Gamma}_{\lambda l}/2\right)},
\end{equation}
\noindent
@@ -599,7 +606,19 @@ then Eq. (II A2.27)\ref{eq:u-l-eval-at-bound} can be written in the form
\end{equation}
\noindent
-in which everything is evaluated at the matching radius a.
+in which everything is evaluated at the matching radius $a$. Note that the form of Eq. \ref{eq:R-func-center-of-mass} (which is in the CoM frame) is the same as if it were in the laboratory frame of reference. This is because of canceling terms in the numerator and denominator of the R-matrix, e.g.
+
+\begin{equation} \label{eq:lab-to-com-params}
+ \begin{aligned}
+ E & \equiv E^{\text{(lab)}} = \frac{M}{m+M}E^{\text{(CoM)}}, \\
+ E_\lambda & \equiv E^\text{(lab)}_{\lambda} = \frac{M}{m+M}E^\text{(CoM)}_{\lambda}, \\
+ \gamma_{\lambda,l}^2 & \equiv (\gamma^\text{(lab)}_{\lambda,l})^2 = \frac{M}{m+M}(\gamma^\text{(CoM)}_{\lambda,l})^2, \text{and} \\
+ \Gamma_{\gamma,l} & \equiv \Gamma^\text{(lab)}_{\gamma,l} = \frac{M}{m+M} \Gamma^\text{(CoM)}_{\gamma,l}.
+ \end{aligned}
+\end{equation}
+
+\noindent
+Due to this relationship, Eq. \ref{eq:R-func-center-of-mass} can be used for CoM or laboratory frame parameters. Conventional SAMMY parameterization is to use the laboratory frame.
\noindent
\textbf{Scattering matrix in terms of R-matrix (neutrons only)}
--
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