@@ -764,21 +764,151 @@ where $P_c$ is the penetrability, whose value is a function of the type of parti
...
@@ -764,21 +764,151 @@ where $P_c$ is the penetrability, whose value is a function of the type of parti
\noindent
\noindent
that is, the penetrability is effectively 1. (Note: In this manual, the reduced-width amplitude for the eliminated-channel capture width will be denoted by a bar above the symbol $\Gamma$.)
that is, the penetrability is effectively 1. (Note: In this manual, the reduced-width amplitude for the eliminated-channel capture width will be denoted by a bar above the symbol $\Gamma$.)
% CHANGE: I'm capitalizing gamma
% >>> CHANGE: I'm capitalizing gamma
Cross sections may be calculated by using the above expressions for $R$, with $L$ given by Eq. (II A.7)\ref{eq:L-matrix}, to generate $W$, and from there calculating $U$ and, ultimately, $\sigma$. However, while Eq. (II A.6)\ref{eq:W-matrix} for $W$ is correct, an equivalent form that is computationally more stable \cite{larson_1993} is
Cross sections may be calculated by using the above expressions for $R$, with $L$ given by Eq. (II A.7)\ref{eq:L-matrix}, to generate $W$, and from there calculating $U$ and, ultimately, $\sigma$. However, while Eq. (II A.6)\ref{eq:W-matrix} for $W$ is correct, an equivalent form that is computationally more stable \cite{larson_1993} is
\begin{equation}
\begin{equation}\label{eq:W-in-terms-of-X}
W = I + 2iX,
W = I + 2iX,
\end{equation}
\end{equation}
\noindent
\noindent
where $X$ is given in matrix notation by
where $X$ is given in matrix notation by
\begin{equation}
\begin{equation}\label{eq:X-matrix}
X = P^{1/2}L^{-1}\left(L^{-1}-R\right)^{-1}RP^{1/2}.
X = P^{1/2}L^{-1}\left(L^{-1}-R\right)^{-1}RP^{1/2}.
\end{equation}
\end{equation}
\noindent
When the suppressed indices and implied summations are inserted, the expression for $X$ becomes
The various cross sections are then written in terms of X.
All calculations internally within SAMMY are expressed in terms of so-called ``u-parameters,'' as distinguished from ``p-parameters,'' which are the input quantities. The u-parameters associated with the resonance p-parameters are as follows:
-\sqrt{\frac{|\Gamma_{\lambda c}|}{2P_l(|E_\lambda-\Xi_c|)}}\qquad&\text{if}\qquad\Gamma_{\lambda c} < 0 \qquad\text{in the PARameter file},\qquad
\end{aligned}
\end{equation}
\noindent
in which $\Xi_c$ is the energy threshold for the channel (Section II.C.2)\ref{subsec:kinematic-conventions}. It is important to note that the partial-width parameter $\Gamma_{\lambda c}$ is always a positive quantity, while the reduced-width amplitude $\gamma_{\lambda c}$ can be either positive or negative. Nevertheless, in the original SAMMY input or output PARameter file (and also in the ENDF File 2 formats\ref{endf8}), partial widths may appear with negative signs. The convention is that the sign given in those files is associated with the amplitude $\gamma_{\lambda c}$ rather than with the partial width $\Gamma_{\lambda c}$.
% >>> THIS NEEDS TO BE UPDATED on release of a new manual (but it's still true)
As of Revision 8 of this document and Release sammy-8.0.0 of the code, the reduced-width amplitudes and square root of resonance energy may be used as input to SAMMY; see Table VI B.2\ref{} for details. To use this option, include the command ``\texttt{REDUCED WIDTH AMPLITudes are used for input}'' in card set 2 of the INPut file. An output file SAMMY.RED is created in this format whenever output file SAMMY.PAR is created.
% >>> CHANGE: I'm not sub-dividing sections any further; just putting headers
\noindent
\textbf{Energy-differential cross sections}
The observable cross sections are found in terms of $X$ by first substituting Eqs. (II A.4\ref{eq:scat-matrix}, II A.5\ref{eq:omega}, and II B1.3\ref{eq:W-in-terms-of-X}) into Eq. (II A.1)\ref{eq:sigma-ccprime}, summing over spin groups (i.e., over $J^\pi$ ), and then summing over all channels corresponding to those particle pairs and spin groups. If $X^r$ represents the real part and $X^i$ the imaginary part of $X$, then the angle-integrated (but energy-differential) cross section for the interaction that leads from particle pair $\alpha$ to particle pair $\alpha'$ has the form
(This formula is accurate only for cases in which one of particles in $\alpha$ is a neutron; however, both particles in $\alpha'$ may be charged.)
In Eq. (II B1 a.1)\ref{eq:sigma-in-terms-of-X} the summations are over those channels $c$ and $c'$ (of the spin group defined by $J^\pi$) for which the particle pairs are, respectively, $\alpha$ and $\alpha'$. More than one ``incident channel'' $c =(\alpha ,l, s, J)$ can contribute to this cross section, for example when both $l =0$ and $l =2$ are possible, or when, in the case of incident neutrons and non-zero spin target nuclei, both channel spins are allowed. Similarly, there may be several ``exit channels'' $c' =(\alpha',l', s', J')$, depending on the particular reaction being calculated (e.g., elastic, inelastic, fission).
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
In this case, both $c$ and $c'$ are limited to those channels of the $J^\pi$ spin group for which the particle-pair is $\alpha$; again, there may be more than one such channel for a given spin group.
Similarly, the reaction cross section from particle pair $\alpha$ to particle pair $\alpha'$ (where $\alpha'$ is not equal to $\alpha$) is
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 $\alpha'$.
Here both the sum over $c$ and the sum over $c'$ include all incident particle channels (i.e., particle pair $\alpha$ only) for the $J^\pi$ spin group.
The capture cross section for the eliminated radiation channels can be calculated directly as
or may be found by subtracting the sum of all reaction cross sections from the absorption cross section. In Eq. (II B1 a.6)\ref{eq:sig-cap-in-terms-of-X}, the sum over $c$ includes all incident particle channels for the $J^\pi$ spin group, and the sum over $c'$ includes \textbf{all} particle channels, both incident and exit, for that spin group.
\noindent
\textbf{One-level two-channel case}
For a simple one-level, two-channel case for which the shift factor is set to zero, the various cross sections defined directly above can easily be expressed in terms of resonance parameters. Users are reminded that SAMMY is by no means restricted to this simple case and can be used with as many levels and as many channels as are needed to describe the particular physical situation. Nevertheless, it is useful to examine the cross section equations for this simple case: while these equations are a crude over-simplification for most physical situations, there is often physical insight to be gained by examination of these equations.
For this simple case, the X matrix of Eq. (II B1.4)\ref{eq:X-matrix} takes the form
in which the subscript on the penetrabilities denotes the channel number (not the angular momentum), the symbol $D$ has been used for $E_\lambda-E-i\overline{\gamma}^2_{\lambda\gamma}$, and the subscript $\lambda$ has been omitted from the reduced-width amplitudes for simplicity's sake. This equation can be rewritten as
