scattering-theory.tex

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\chapter{Scattering Theory}\label{ch:scattering_theory}

Details of scattering theory have been well understood since the middle of the previous century, when they were summarized in a review article by Lane and Thomas [AL58]\cite{lane_thomas_1958}. A wealth of additional reference material is available to the student of scattering theory; only a few are listed here. The text by Foderaro [AF71]\cite{foderaro_1971} provides a more elementary introduction to the subject. One publication by Fr\"{o}hner [FF80]\cite{frohner_1978} is based on lectures presented at the International Centre for Theoretical Physics (ICTP) Winter Courses on Nuclear Physics and Reactors, 1978; this is a comprehensive and useful guide to applied neutron resonance theory. It includes a variety of topics, including preparation of data, various approximations to scattering theory, Doppler broadening, experimental complications, data-fitting procedures, and statistical tests. Another Fr\"{o}hner paper [FF00]\cite{jeff18_frohner} is somewhat more theoretical, and covers many aspects of data fitting in the resonance region.

The particular aspect of scattering theory with which we are concerned is the R-matrix formalism. A summary of the underlying principles is given here.

R-matrix theory is a mathematically rigorous phenomenological description of what is actually seen in an experiment (i.e., the measured cross section). The theory is not a model of neutron-nucleus interaction, in the sense that it makes no assumptions about the underlying physics of the interaction. Instead it parameterizes the measurement in terms of quantities such as the interaction radii and boundary conditions, resonance energies and widths, and quantum numbers; values for these parameters may be determined by fitting theoretical calculations to observed data. The theory is mathematically correct, in that it is analytic, unitary, and rigorous; nevertheless, in practical applications, the theory is always approximated in some fashion.

R-matrix theory is based on the following assumptions\footnote{In practical applications two of these four assumptions may be violated in one degree or another: (1) The theory may be used for relativistic neutron energies, and corrected for relativistic effects; nevertheless, non-relativistic quantum mechanics is assumed. (2) A fission experiment with more than two final products is treated as a two-step process. That is, the immediate result of the neutron-nuclide interaction is assumed to be limited to two final products, at least one of which decays prior to detection.}: 

(1) the applicability of non-relativistic quantum mechanics; 

(2) the absence or unimportance of all processes in which more than two product nuclei are formed; 

(3) the absence or unimportance of all processes of creation or destruction; and 

(4) the existence of a finite radial separation beyond which no nuclear interactions occur, although Coulomb interactions are given special treatment.

R-matrix theory is expressed in terms of channels, where a channel is defined as a pair of (incoming or outgoing) particles, plus specific information relevant to the interaction between the two particles. A schematic depicting entrance and exit channels is shown in Fig. \ref{scattering-theory_rmatrix_channel_diagram}. Note that entrance channels can also occur as exit channels, but some exit channels (e.g., fission channels) do not occur as entrance channels. Two interacting particles are shown in the portion of the figure that is labeled ``Interior Region''; here the particles are separated by less than the interaction radius $a$.

\begin{figure}
    \centering
    \includegraphics[width=0.4\textwidth]{figures/rmatrix_inc_channel.pdf} \\[\smallskipamount]
    \includegraphics[width=0.4\textwidth]{figures/rmatrix_interior_region.pdf}\hfill
    \includegraphics[width=0.4\textwidth]{figures/rmatrix_exit_channel.pdf}
    \caption{Schematic of entrance and exit channels as used in scattering theory. For the interior region (with separation distance $r < a$), no assumptions are made about the nature of the interaction. In the figure, $m$, $i$, and $z$ refer to the mass, spin, and charge of the incident particle while $M$, $I$ and $Z$ refer to the target particle. Orbital angular momentum is denoted by $l$ and velocity by $v$. Primes are used for post-collision quantities.}
    \label{scattering-theory_rmatrix_channel_diagram}
\end{figure}

In Section \ref{sec:equations-for-scattering-theory}, general equations of scattering theory are presented and their derivations discussed. The fundamental R-matrix equations are presented. Section II.A.1\ref{} gives a detailed derivation of the equations for a simple case. Section II.A.2\ref{} shows the relationship between the R-matrix and the A-matrix, which is another common representation of scattering theory.

The approximations to R-matrix theory available in the SAMMY code are detailed in Section II.B\ref{}. The recommended choice for most applications is the Reich-Moore approximation, described in Section II.B.1\ref{}. For some applications, the Reich-Moore approximation is inadequate; for those cases, a method for using SAMMY's Reich-Moore approximation to mimic the full (exact) R-matrix is presented Section II.B.2\ref{}. Two historically useful but now obsolete approximations are single-level and multilevel Breit Wigner (SLBW and MLBW), discussed in Section II.B.3\ref{}. Provisions for including non-compound (direct) effects are discussed in Section II.B.4\ref{}.

In Section II.C\ref{}, details are given for the SAMMY nomenclature and other conventions,
for transformations to the center-of-momentum system, and for the calculation of penetrability,
shift factors, and hard-sphere phase shifts in both Coulomb and non-Coulomb cases.

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\section{Equations For Scattering theory} \label{sec:equations-for-scattering-theory}
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In this section, equations for scattering theory are presented but not derived. Specifics for
the R-matrix formulation of scattering theory are presented in Section II.A.1\ref{}, which provides a
discussion of an alternative formulation (the A-matrix). Readers interested in the derivation of
the equations for scattering theory are referred to the Lane and Thomas article \cite{lane_thomas_1958} for a
detailed derivation in the general case, or to Section II.A.2\ref{} of this document for a simplified
version.

In scattering theory, a channel may be defined by $c = (\alpha, l, s, J)$, where the following
definitions apply:

\begin{itemize}
\item $\alpha$ represents the two particles making up the channel; $\alpha$ includes mass ($m$ and $M$), charge ($z$ and $Z$), spin ($i$ and $I$ ) with associated parities, and all other quantum numbers for each of the two particles, plus the Q-value (equivalent to the negative of the threshold energy in the
center of momentum system).
\item $l$ is the orbital angular momentum of the pair, and the associated parity is given by $(-1)^l$.
\item $s$ represents the channel spin (including the associated parity); that is, $s$ is the quantized
vector sum of the spins of the two particles of the pair: $\vec{s} = \vec{i} + \vec{I}$
\item $J$ is the total angular momentum (and associated parity); that is, $J$ is the quantized vector sum
of $l$ and $s$: $\vec{J} = \vec{l} + \vec{s}$.
\end{itemize}

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.

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'} \:,
\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)} \:, 
\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 \:. 
\end{equation}

\noindent
Here $E$ is the \textbf{laboratory} kinetic energy of the incident (moving) particle. A derivation of this value for $K_\alpha$ is given in Section II.C.2\ref{}.

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'} \:,
\end{equation}

\noindent
where $\Omega$ is given by

\begin{equation}\label{eq:omega}
    \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} \:. 
\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 \:, 
\end{equation}

\noindent
with $P$ being the penetration factor (penetrability) $S$ the shift factor, and $B$ the arbitrary boundary constant at the channel radius $a_c$. $P$ and $S$ are functions of energy $E$, and also depend on the orbital angular momentum $l$ and the channel radius $a_c$. Formulae for $P$ and $S$ are found in many references (see, for example Eq. (2.9) in \cite{lynn_1968}).

For non-Coulomb interactions, the penetrability and shift factor have the form 

\begin{equation}\label{eq:pen-shift-func-of-rho}
    P\rightarrow P_l(\rho) \qquad \text{and} \qquad S \rightarrow S_l(\rho) \:,
\end{equation}

\noindent
where $\rho$ is related to the center-of-mass momentum which in turn is related to the laboratory energy of the incident particle $(E)$. For arbitrary channel $c$ with a particle pair $\alpha$, orbital angular momentum $l$, and channel radius $a_c$, $\rho$ has the form

\begin{equation}\label{eq:de-Broglie-radius}
    \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}

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.

% 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}
\begin{tabular}{ l c c c }
    \hline\hline
    $\mathbf{l}$ & $\mathbf{P_l}$                                                                   & $\mathbf{S_l}$                                                                                       & $\mathbf{\phi_l}$   \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

    \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 $X_l=P_{l-1}/(l-S_{l-1})$}
\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}