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add more equations

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docs/tex/sammyrefs.bib
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......@@ -1234,5 +1234,11 @@ abstract = "The response of the RPI 16-segment NaI(Tl) multiplicity detector sys
howpublished = {{lecture notes on R-matrix theory, presented at the SAMMY workshop of February 18-22, 2002, at the NEA Headquarters in Paris, France (2002)}}
}
@techreport{frohner1980applied,
title={Applied neutron resonance theory},
author={Fr{\"o}hner, Fritz H},
year={1980},
institution={reprinted from \textit{Nuclear Theory for Applications}, p. 59 ff., International Centre for Theoretical Physics, Trieste (1980). Available as KFK 2669 (1978).}
}
@Comment{jabref-meta: databaseType:bibtex;}
docs/tex/scattering-theory.tex
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View file @ ee1b6379
......@@ -320,16 +320,16 @@ interference is to be neglected; that is, only the interactions within one level
gamma channels, $L=S+iP$ reduces to $L=i$, so Eq. (II A1.13)\ref{eq:a-matrix-rm-approx} becomes
\begin{equation}\label{eq:a-matrix-rm-reduced}
A_{\mu\lambda}^{-1} \approx \left(E_\lambda-E-i\bar{\Gamma}_{\lambda\gamma}/2\right)\delta_{\mu\lambda} - \sum_{\substack{c=particle \\ channels}} \gamma_{\mu c}L_c\gamma_{\lambda c} .
A_{\mu\lambda}^{-1} \approx \left(E_\lambda-E-i\overline{\Gamma}_{\lambda\gamma}/2\right)\delta_{\mu\lambda} - \sum_{\substack{c=particle \\ channels}} \gamma_{\mu c}L_c\gamma_{\lambda c} .
\end{equation}
\noindent
The bar over $\bar{\Gamma}_{\lambda\gamma}$ is used to indicate the special treatment for this channel.
The bar over $\overline{\Gamma}_{\lambda\gamma}$ is used to indicate the special treatment for this channel.
In this form, our expression for $A$ is analogous to the exact expression in Eq. (II A1.4)\ref{a-matrix} with two modifications: the additional imaginary term is added to the energy difference, and the sum over the channels includes only the ``particle channels'' (non-eliminated channels). It is therefore possible to immediately write the R-matrix formula for the eliminated-channel approximation as
In this form, our expression for $A$ is analogous to the exact expression in Eq. (II A1.4)\ref{eq:a-matrix} with two modifications: the additional imaginary term is added to the energy difference, and the sum over the channels includes only the ``particle channels'' (non-eliminated channels). It is therefore possible to immediately write the R-matrix formula for the eliminated-channel approximation as
\begin{equation}
R_{cc'} = \sum_{\lambda}\frac{\gamma_{\lambda c}\gamma_{\lambda c'}}{E_\lambda-E-i\bar{\Gamma}_{\lambda\gamma}/2} \delta_{JJ'} ,
R_{cc'} = \sum_{\lambda}\frac{\gamma_{\lambda c}\gamma_{\lambda c'}}{E_\lambda-E-i\overline{\Gamma}_{\lambda\gamma}/2} \delta_{JJ'} ,
\end{equation}
where the channel indices c and c ' refer only to particle channels, not to the gamma channels.
......@@ -362,7 +362,7 @@ can be expanded in the usual fashion,
\noindent
for which the radial portion obeys the equation
\begin{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,
\end{equation}
......@@ -376,7 +376,7 @@ subject to the conditions that $|\psi|^2$ is everywhere finite and that
In the external region, $r>a$, the nuclear forces are zero $(V=W=0)$, so the solution has
the form
\begin{equation}
\begin{equation}\label{eq:in-out-wave-funcs}
u_l(r) = I_l(r) - U_l O_l(r) .
\end{equation}
......@@ -472,7 +472,7 @@ expanded in terms of the eigenfunctions as
\noindent
with
\begin{equation}
\begin{equation}\label{eq:c-lambda-l}
c_{\lambda l} = \int_0^a u_l w_{\lambda l} dr.
\end{equation}
......@@ -483,10 +483,259 @@ This equation for $c_{\lambda l}$ is derived by multiplying Eq. (II A2.15)\ref{e
Consider now the integral
\begin{equation}
\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( \frac{d^2u_{l}}{dr^2}w_{\lambda l} - u_{l}\frac{d^2w_{\lambda l}}{dr^2} \right) dr,
\end{equation}
\noindent
which can be expanded by use of Eqs. (II A2.3)\ref{eq:radial-wave-equation} and (II A2.8)\ref{eq:wave-func-without-abs} to give
\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.
\end{aligned}
\end{equation}
\noindent
Defining $\overline{W}_{\lambda l}$ as
\begin{equation}
\overline{W}_{\lambda l} = \frac{ \int_0^a Wu_lw_{\lambda l}dr }{ \int_0^a u_lw_{\lambda l}dr }
\end{equation}
\noindent
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.
\end{equation}
\noindent
Integrating the left-hand side of this equation gives
\begin{equation}\label{eq:schroding-left}
\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 & = \left[ \frac{du_{l}}{dr}w_{\lambda l} - u_{l}\frac{dw_{\lambda l}}{dr} \right]_0^a = \left[ \frac{du_{l}}{dr}w_{\lambda l} - u_{l}\frac{dw_{\lambda l}}{dr} \right]_{r=a} \\
& = \left[ \frac{du_{l}}{dr}w_{\lambda l} - u_{l}\frac{B_l}{a}w_{\lambda l} \right]_{r=a} = \left[ a\frac{du_{l}}{dr} - u_{l}B_l \right]_{r=a} \frac{w_{\lambda l(a)}}{a},
\end{aligned}
\end{equation}
\noindent
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}.
\end{equation}
\noindent
Equating Eqs. (II A2.21)\ref{eq:schroding-left} and (II A2.22)\ref{eq:schroding-right} therefore gives
\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},
\end{aligned}
\end{equation}
\noindent
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}.
\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},
\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},
\end{equation}
\noindent
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)},
\end{aligned}
\end{equation}
\noindent
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}}
\end{equation}
\noindent
and the absorption width $\Gamma_{\lambda l}$ as
% NOTE: I'm changing from Nancy's formula here with overline{W}
\begin{equation}
\Gamma_{\lambda l} \equiv 2 \overline{W}_{\lambda l} .
\end{equation}
\noindent
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)},
\end{equation}
\noindent
then Eq. (II A2.27)\ref{eq:u-l-eval-at-bound} can be written in the form
\begin{equation}\label{eq:u-l-r-matrix}
u_l = \left(a\frac{du_l}{dr}-u_lB_l\right)R_l,
\end{equation}
\noindent
in which everything is evaluated at the matching radius a.
\noindent
\textbf{Scattering matrix in terms of R-matrix (neutrons only)}
Equation (II A2.31)\ref{eq:u-l-r-matrix} can be converted into the usual R-matrix formulae by inserting Eq. (II A2.5)\ref{eq:in-out-wave-funcs},
\begin{equation}
u_l = I_l - U_l O_l,
\end{equation}
\noindent
yielding
\begin{equation}\label{eq:u-l-expanded}
I_l - U_l O_l = \left[a\left(\frac{dI_l}{dr}-U_l\frac{dO_l}{dr}\right)-B_l(I_l-U_lO_l)\right]R_l,
\end{equation}
which can be expanded by use of Eqs. (II A2.3) and (II A2.8) to give
\noindent
in which everything is again evaluated at the matching radius $a$. Solving Eq. (II A2.33)\ref{eq:u-l-expanded} for $U$ gives
\begin{equation}\label{eq:u-l-i-l-matching}
U_l\left[-O_l+R_l\left(a\frac{dO_l}{dr}-B_lO_l\right)\right] = I_l-R_l\left(a\frac{dI_l}{dr}-B_lI_l\right),
\end{equation}
\noindent
or
\begin{equation}
U_l = \frac{ I_l-R_l\left(a\frac{dI_l}{dr}-B_lI_l\right) }{ \left[-O_l+R_l\left(a\frac{dO_l}{dr}-B_lO_l\right)\right] } = \frac{I_l}{O_l}\frac{ 1-R_l\left(\frac{a}{I_l}\frac{dI_l}{dr}-B_l\right) }{ 1-R_l\left(\frac{a}{O_l}\frac{dO_l}{dr}-B_l\right) }.
\end{equation}
\noindent
We define $L_l$ as
\begin{equation}
L_l \equiv \frac{a}{O_l(a)}\frac{dO_l}{dr}\Bigr|_{r=a} \equiv S_l + iP_l.
\end{equation}
\noindent
For spinless particles, $I_l^* = O_l$, so that
\begin{equation}
\frac{a}{I_l(a)} \frac{dI_l}{dr}\Bigr|_{r=a} = L_l^* = S_l-iP_l
\end{equation}
\noindent
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}.
\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)},
\end{equation}
\noindent
which is the usual form for the scattering matrix in terms of the R-matrix in this simple case.
\noindent
\textbf{Relating the scattering matrix to the cross sections}
The relationship between the scattering matrix $U$ and the cross section $\sigma$ is also described by many authors; see, for example, [AF71]\cite{foderaro_1971}. Here we provide a summary for the simplest case. The wave function for a spinless particle far from the scattering source can be written as
\begin{equation}
\psi(r,\theta) = e^{ikz} + \frac{e^{ikr}}{r}f(\theta),
\end{equation}
\noindent
where $f$ has the form
\begin{equation}\label{f-theta}
f(\theta) = \frac{1}{2ik}\sum_l(2l+1)\left[U_l-1\right]P_l(\cos\theta).
\end{equation}
\noindent
The cross section is then given by
\begin{equation}\label{eq:angle-diff-xs-to-f-theta}
\frac{d\sigma}{d\Omega} = \left|f(\theta)\right|^2.
\end{equation}
For angle-integrated cross sections, the equation found by inserting Eq. (II A2 a.2)\ref{f-theta} into Eq. (II A2 a.3)\ref{eq:angle-diff-xs-to-f-theta} can be integrated to give
\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) \\
= & \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}
\end{equation}
This is analogous to the ``standard'' scattering theory equation shown in Eq. (II A.1)\ref{eq:sigma-ccprime}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Versions of Multilevel R-matrix Theory} \label{sec:versions-of-r-matrix}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Many representations of multilevel R-matrix theory have been developed over the years.
For a summary of the more common versions, the reader is referred to the works of Fr\"{o}hner
[FF80, FF00]\cite{frohner1980applied,jeff18_frohner}.
Four versions of R-matrix theory are available in SAMMY: the Reich-Moore
approximation (Section II.B.1)\ref{subsec:r-matrix-RM}, the single-level (SLBW) and multilevel Breit-Wigner (MLBW) approximations (Section II.B.3)\ref{subsec:r-matrix-BW}, and a variant on the Reich Moore which mimics the full R-matrix (Section II.B.2)\ref{subsec:r-matrix-full}. An option to include a direct capture component is also provided (Section II.B.4)\ref{subsec:direct-capture}.
The Reich-Moore approximation is the preferred method for most modern evaluations; it
is the default formalism for SAMMY runs. Fr\"{o}hner, in fact, suggests that the Reich-Moore approximation is universally applicable to all cases: ``Experience has shown that with this approximation [Reich Moore] all resonance cross section data can be described in detail, in the windows as well as in the peaks, even the weirdest multilevel interference patterns...It works equally well for light, medium-mass and heavy nuclei, fissile and nonfissile.'' \cite{jeff18_frohner}(pg. 60). For most purposes, Reich Moore is indeed indistinguishable from the exact formulation. Notable exceptions are situations where interference effects exist between capture and other channels. For those cases, small modifications to the SAMMY input will permit the user to mimic the effect of the non-approximated R-matrix; see Section II.B.2\ref{subsec:r-matrix-full} for details. Occasionally it is not possible to properly describe a cross section within the confines of R-matrix theory, because the reaction includes a direct component. SAMMY has provisions for the user to provide a numerical description of this component; see Section II.B.4\ref{subsec:direct-capture} for details.
Also available within SAMMY are both the SLBW and the MLBW formulations (Section II.B.3)\ref{subsec:r-matrix-BW}; these are included for the sake of completeness, for comparison purposes, and because many of the evaluations in the nuclear data files were performed with Breit-Wigner formulae. However, it is strongly recommended that only Reich Moore be used for new evaluations, for several reasons: MLBW is often inadequate; SLBW is almost always inadequate. When it is correct, MLBW gives identical results to Reich Moore. ``Ease of Programming'' is no longer a valid excuse for using MLBW, since the programming has already been accomplished. Similarly, a slow computer is no longer a legitimate excuse, since modern computers can readily handle the more rigorous formulae.
Finally, it should be noted that SAMMY's implementation of MLBW does not correspond to the usual definition of MLBW. Instead, SAMMY uses the ENDF [ENDF-102]\cite{endf8} convention in which only the elastic cross section is truly multilevel, and all other types of cross section are single level.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Reich-Moore Approximation to Multilevel R-Matrix Theory} \label{subsec:r-matrix-RM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simulation of Full R-Matrix} \label{subsec:r-matrix-full}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Breit-Wigner Approximations} \label{subsec:r-matrix-BW}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Direct Capture Component} \label{subsec:direct-capture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
......
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