% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% The Project Gutenberg EBook of A Fortran Program for Elastic Scattering %
% Analyses with the Nuclear Optical Model, by Michel A. Melkanoff         %
% and David S. Saxon and John S. Nodvik and David G. Cantor               %
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
% with this eBook or online at www.gutenberg.org                          %
%                                                                         %
%                                                                         %
% Title: A Fortran Program for Elastic Scattering Analyses with the       %
%        Nuclear Optical Model                                            %
%                                                                         %
% Author: Michel A. Melkanoff                                             %
%         David S. Saxon                                                  %
%         John S. Nodvik                                                  %
%         David G. Cantor                                                 %
%                                                                         %
% Release Date: August 24, 2009 [EBook #29784]                            %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: ISO-8859-1                                      %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK ELASTIC SCATTERING ANALYSES ***
%                                                                         %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %

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%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%

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\begin{PGtext}
The Project Gutenberg EBook of A Fortran Program for Elastic Scattering
Analyses with the Nuclear Optical Model, by Michel A. Melkanoff
and David S. Saxon and John S. Nodvik and David G. Cantor

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org


Title: A Fortran Program for Elastic Scattering Analyses with the
       Nuclear Optical Model

Author: Michel A. Melkanoff
        David S. Saxon
        John S. Nodvik
        David G. Cantor

Release Date: August 24, 2009 [EBook #29784]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK ELASTIC SCATTERING ANALYSES ***
\end{PGtext}
\end{minipage}
\end{center}

\clearpage

%%%% Credits and transcriber's note %%%%
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\begin{PGtext}
Produced by David Starner, Andrew D. Hwang, and the Online
Distributed Proofreading Team at http://www.pgdp.net
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%% -----File: 001.png---Folio xx-------

\clearpage

\pdfbookmark[0]{Front Matter}{Front Matter}

\begin{center}
\begin{tabular}{l}
\textbf{\Huge A FORTRAN Program} \\[0.25in]
\textbf{\Huge for Elastic Scattering Analyses} \\[0.25in]
\textbf{\Huge with the Nuclear Optical Model} \\[0.375in]
\end{tabular}

\begin{tabular}{ll}
\Author{MICHEL A. MELKANOFF}{University of California, Los Angeles}\qquad &
\Author{DAVID S. SAXON}{University of California, Los Angeles} \\[0.25in]
\Author{JOHN S. NODVIK}{University of Southern California} &
\Author{DAVID G. CANTOR}{University of California, Los Angeles}
\end{tabular}
\vfill

{\Large UNIVERSITY\quad OF\quad CALIFORNIA\quad PRESS}\\[0.125in]
{\small BERKELEY AND LOS ANGELES 1961}
\end{center}

\phantomsection
\pdfbookmark[1]{Title Page}{Title Page}

%% -----File: 002.png---Folio xx-------

\clearpage
\null\vfill
\begin{center}
{\small UNIVERSITY OF CALIFORNIA PUBLICATIONS IN AUTOMATIC COMPUTATION}\\[0.125in]

{\small\textsc{Number 1}}\\[0.125in]

\begin{minipage}{4.125in}
\small
This publication was prepared partly under the sponsorship of
the Office of Naval Research. Reproduction in whole or in part
is permitted for any purpose of the United States Government.
\end{minipage}\\[0.25in]

{\footnotesize
\textsc{university of california press}, Berkeley and Los Angeles, California \\
\textsc{cambridge university press}, London, England}\\[0.25in]
\$4.50\\[0.25in]
\textit{Second Printing, 1961}
\vfill\vfill

{\scriptsize PRINTED IN THE UNITED STATES OF AMERICA}
\end{center}

%% -----File: 003.png---Folio xx-------

\clearpage
\phantomsection
\pdfbookmark[1]{Acknowledgements}{Acknowledgements}

\section*{\SecHeading{Acknowledgements}}
\ChapSkip

The authors would like to express their sincere appreciation
to the Western Data Processing Center, Graduate School of Business
Administration, UCLA, for the use of their \Acro{IBM~709} computer. Special
thanks are due to Mrs.\ Lisa Greenstadt and Mrs.\ Lois Holloway who have
worked intensively and skillfully to prepare the program.

This program is largely based on experience gained on the
SWAC, and the authors recall this with gratitude to Numerical Analysis
Research, Department of Mathematics, UCLA\@.

Finally the authors would like to express their appreciation
to the National Science Foundation and the Office of Naval Research for
financial support.

%% -----File: 004.png---Folio xx-------

\TableofContents
\phantomsection
\pdfbookmark[1]{Table of Contents}{Table of Contents}

\iffalse%%%% COMMENTED SCANNED TEXT %%%%
TABLE OF CONTENTS

Chapter Page

I.  INTRODUCTION

II. MATHEMATICAL DESCRIPTION
    A. General Formulation
       1. Uncharged Incident Particles
       2. Charged Incident Particles
    B. Optical Model Potential
       1. Diffuse Surface Optical Model with Volume
          Absorption and Coulomb Spin-Orbit
       2. Nuclear Form Factors
       3. Final Formulation for Machine Calculation
       4. Numerical Integration
       5. Coulomb Functions
       6. Phase Shifts
       7. Cross Section and Polarization
       8. Chi Square Deviation
       9. Normalization

III. PROGRAM DESCRIPTION
     A. General Description
       1. Machine Specifications
       2. General Program Description
       3. Use of the WDPC Load-and-Go System
       4. Error Indications
     B. Detailed Descriptions of the Specific Routines
        of the Program

%% -----File: 005.png---Folio xx-------

IV. DESCRIPTION OF INPUT DATA

V.  GLOSSARY AND DESCRIPTION OF SYMBOLIC VARIABLES
    APPEARING IN COMMON AND DIMENSION STATEMENTS

VI. SYMBOLIC LISTING OF THE PROGRAM

VII. TYPICAL INPUT AND OUTPUT
     A. Input Data for Protons Against Copper at 9.75 Mev
     B. Output Listing

VIII. FURTHER SUBROUTINES AND PROGRAMS IN PREPARATION BIBLIOGRAPHY
\fi%%%% END OF COMMENTED TEXT %%%%
%% -----File: 006.png---Folio 1-------

\mainmatter
\pagenumbering{arabic}
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\fancyhead{}
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\setlength{\headheight}{14.5pt}
\fancyhead[C]{--\ \thepage\ --}

\Chapter{I.}{Introduction}

The purpose of the present report is to describe in complete detail
a \FORTRAN\ code named Program \SCAT{4} written by the UCLA group in order to
analyze elastic scattering of various particles against complex nuclei
by means of the diffuse surface optical model of the nucleus.

While a number of similar programs have been prepared and used by
other groups, there have been many requests for the UCLA program because of
its flexibility and the availability of \Acro{IBM~704} and~\Acro{709} computers for which
the program is written.

The present program still contains some undesirable features and the
UCLA group is constantly modifying it to make it more efficient and flexible.
However, a ``final'' program will probably never be reached and it was decided
to release Program \SCAT{4} without further delay; as they develop, modifications
and additions will be described in later reports.

Other laboratories will probably add further modifications and the
UCLA group will be grateful for description of such modifications as well as
for any suggestions in this regard. Modifications and additions deemed
worthwhile will be passed on to other users of the program but while the
UCLA group is willing to serve partially as a central clearing house, the
entire clerical responsibility cannot be assumed by the UCLA group.

It should also be noted that, while every effort has been made to
check out the program, the UCLA group cannot guarantee its complete correctness.

Program \SCAT{4} is available on a symbolic deck and will be mailed on
request. Air mailing will require prepaid postage by requesting parties.

Potential users of program \SCAT{4} may find it useful to follow these
suggestions in reading the present report:

% [** PP: Added periods after items (a), (c), and (d)]
\begin{itemize}
\item[1)] If the potential user is only interested in analyses with standard
potentials he may proceed as follows:
\begin{itemize}
\item[a)] Read the introduction to the mathematical description. %** period

\item[b)] Consider the fundamental equations: \Eqno{34}, \Eqno{35}, \Eqno{51}, \Eqno{78} through
\Eqno{85}, \Eqno{132}, \Eqno{137} through \Eqno{139} in \ChapRef{II}.
% [** PP: Original reads Chapter II, section 1; most aren't in section 1.]

\item[c)] Read \ChapRef{III}, section~A and the general flow chart. %** period

\item[d)] Read the description of subroutines \Code{INPT4} and
\Code{OUTPT4} in \ChapRef{III}, section~B. %** period

\item[e)] Read \ChapRef{IV} and~\hyperref[chapter:VII.]{VII}.
\end{itemize}

%% -----File: 007.png---Folio 2-------

\item[2)] If the potential user is interested in all the features of the program,
then a perusal of the whole report is advisable. The mathematical description
of \ChapRef{II} is a brief review of the theory and the basic equations
are all listed there. Symbolic \FORTRAN\ variables are indicated in capital
letters and may be looked up in the glossary making up \ChapRef{V}.
\end{itemize}

Note that the program may be used for incident neutral particle by
letting $ZZ'= 0$.

%% -----File: 008.png---Folio 3-------


\Chapter{II.}{Mathematical Description}

Program \SCAT{4} calculates in the center-of-mass system the differential
elastic scattering cross sections~$\sigma(\theta)$, the polarization~$P(\theta)$, and the total
reaction cross section~$\sigma_R$ for particles of spin~$0$ or~$1/2$ having any mass,
charge and (non-relativistic) energy scattered by spinless nuclei of any mass
and charge for various sets of diffuse surface optical model parameters. The
incident and target particles are assumed to interact through a two-body
potential consisting of a complex nuclear potential which includes spin-orbit
interaction and whose shape can be specified by input parameters. When the
incident particle is charged, the two body potential contains, in addition, the
coulomb potential between an incident point charge and an extended, constant
charge density target.

The calculations include numerical integrations of the radial
Schroedinger equations for the effective partial waves. The complex phase
shifts are obtained as usual by matching the logarithmic derivatives of the
numerically obtained nuclear wave functions to that of the coulomb (or
spherical Bessel) functions. The phase shifts are then used to compute
polarizations and cross sections which may be compared to the experimental
values by means of the $\chi^2$~test.


\Section{A.}{General Formulation}

We begin with a brief review of the basic theory relating to the scattering
of spin~$1/2$ particles by a zero spin target\footnotemark.
\footnotetext{See J. Lepore, Phys.\ Rev.~\textbf{79}, 137 (1950).}%
We shall first consider
the case of an uncharged incident particle and indicate later the modifications
necessary if the incident particle is charged.

The interaction is assumed to be of the form
\[
V_T = V_1 + V_2\, \vec{S} · \vec{L}
\Tag{1}
\]
where $V_1$ and~$V_2$ are complex quantities depending only on the distance~$r$ between
the incident particle and the target particle. In terms of the Pauli spin
operator~$\vec\sigma$, the spin operator of the incident particle,~$\vec{S}$, is given by
\[
\vec{S} = \frac{1}{2} \hbar \vec{\sigma}
\Tag{2}
\]
%% -----File: 009.png---Folio 4-------
and the (relative) orbital angular momentum operator is given by
\[
\vec{L} = \vec{r} × \left(\frac{\hbar}{i}\vec{\nabla}\right).
\Tag{3}
\]
The Schroedinger equation is then
\[
\left[-\frac{\hbar^2}{2\mu} \vec{\nabla}^2
  + V_1(r) + V_2(r)\, \vec{S} · \vec{L} \right] \Psi = E \Psi
\Tag{4}
\]
where
\[
\mu = \frac{m_i\, m_b}{m_i + m_b}
\Tag{5}
\]
is the reduced mass, $m_i$ and~$m_b$ being respectively the masses of the incident
and target particles in atomic mass units.
\[
E = \frac{m_b}{m_i + m_b} E_\LAB
\Tag{6}
\]
is the energy in the center of mass system, $E_\LAB$ being the lab energy of the
incident particle in~\MeV.


\Subsection{1.}{Uncharged Incident Particles}

The wave function corresponding to a wave incident in the positive~$z$
direction and normalized to one incident particle per unit time per unit
area is
\[
\Psi_\inc = \frac{1}{\sqrt{v}} e^{ikz} \chi_\inc
\Tag{7}
\]
where $v$ is the relative velocity, the wave number~$k$ is given by
\[
k = \sqrt{\frac{2\mu E}{\hbar^2}}
  = 0.2195376 \sqrt{\mu E}~\text{fermi}^{-1}
\Tag{8}
\]
%% -----File: 010.png---Folio 5-------
and the incident spin function is
\[
\chi_\inc = a_{1/2}\alpha + a_{-1/2}\beta
\Tag{9}
\]
where $\alpha$ and~$\beta$ are normalized spin eigenfunctions of~$S_z$
and~$a_{1/2}$, $a_{-1/2}$ the corresponding amplitudes.

The partial wave expansion corresponding to~\Eqno{7} is given by:
\[
\Psi_\inc = \frac{1}{\sqrt{v}} \sum_{\ell=0}^\infty
  (2\ell + 1) i^\ell j_\ell(kr) \sqrt{\frac{4 \pi}{2\ell + 1}}
  Y_\ell^0(\theta, \varphi) \left[a_{1/2}\alpha + a_{-1/2}\beta\right]
\Tag{10}
\]
where $j_\ell(kr)$ is the regular spherical Bessel function of order~$\ell$ and the
normalized spherical harmonics are defined as
\[
Y_\ell^m(\theta,\varphi) = (-1)^\frac{m+|m|}{2}
  \sqrt{\frac{2\ell + 1}{4\pi}}
  \sqrt{\frac{(\ell - |m|)!}{(\ell + |m|)!}}
  P_\ell^{|m|}(\cos\theta) e^{im\varphi}
\Tag{11}
\]
where $P_\ell^{|m|}(\cos\theta)$ are the associated Legendre polynomials.

The product functions $Y_\ell^0\alpha$ and $Y_\ell^0\beta$ which appear in~\Eqno{10} are simultaneous
eigenfunctions of the operators $\vec{L}^2$, $L_z$, $\vec{S}^2$, and $S_z$ but not of the
operator $\vec{L}·\vec{S}$ which appears in the spin-orbit interaction. This may be
remedied by introducing functions $\Y_{j\ell s}^{m_j}$ which are simultaneous eigenfunctions
of $\vec{L}^2$, $\vec{S}^2$, $\vec{J}^2$, and $J_z$ and thus of
$\vec{L}·\vec{S}$ where $\vec{J}$ is the total angular momentum,
\[
\vec{J} = \vec{L} + \vec{S}.
\Tag{12}
\]
Since $s=1/2$, the possible values of $j$ are $j=\ell+1/2$ and $j=\ell-1/2$; the
corresponding eigenfunctions are given by
{\small
\[
\left.
\begin{aligned}
  \Y_{\ell+1/2, \ell, s}^{m_j}
  &= \Neg
     \sqrt{\frac{\ell + m_j + 1/2}{2\ell + 1}} Y_\ell^{m_j-1/2}\alpha
   + \sqrt{\frac{\ell - m_j + 1/2}{2\ell + 1}} Y_\ell^{m_j+1/2}\beta,
   \text{ for $j = \ell+1/2$} \\
%
  \Y_{\ell-1/2, \ell, s}^{m_j} %[** PP: Changed denominator to 2\ell + 1]
  &= -\sqrt{\frac{\ell - m_j + 1/2}{2\ell + 1}}Y_\ell^{m_j-1/2}\alpha
    + \sqrt{\frac{\ell + m_j + 1/2}{2\ell + 1}}Y_\ell^{m_j+1/2}\beta,
   \text{ for $j = \ell-1/2$}
\end{aligned}
\right\}
\Tag{13}
\]}% [** PP: End of \small]
%% -----File: 011.png---Folio 6-------
The incident wave function may now be written as
\[
\begin{aligned}
  \Psi_\inc
  &= \sqrt{\frac{4\pi}{V}}
     \sum_{\ell=0}^{\infty} \sqrt{\ell + 1}\, i^{\ell}\, j_{\ell}(kr)
     \left[ \PadTo{-a}{a}_{1/2} \Y_{\ell + 1/2, \ell, 1/2}^{1/2}
         + a_{-1/2} \Y_{\ell + 1/2, \ell, 1/2}^{-1/2} \right] \\
  %
  &+ \sqrt{\frac{4\pi}{V}}
     \sum_{\ell=0}^{\infty} \PadTo{\sqrt{\ell + 1}}{\sqrt{\ell}\ }\,
       i^{\ell}\, j_{\ell}(kr)
     \left[ -a_{1/2} \Y_{\ell - 1/2, \ell, 1/2}^{1/2}
          + a_{-1/2} \Y_{\ell - 1/2, \ell, 1/2}^{-1/2} \right]
%[** PP: No period]
\end{aligned}
\Tag{14}
\]
The total wave function can be written in a form similar to~\Eqno{14}:
\begin{align*}
  \Psi_\total
  &= \Psi_\inc + \Psi_\scatt \\
%
  &= \sqrt{\frac{4\pi}{V}} \sum_{\ell=0}^{\infty}
       \sqrt{\ell + 1}\, i^{\ell}\, \frac{\Psi_\ell^+ (r)}{kr}
       \left[\PadTo{-a}{a}_{1/2} \Y_{\ell + 1/2, \ell, 1/2}^{1/2}
           + a_{-1/2} \Y_{\ell + 1/2, \ell, 1/2}^{-1/2} \right] \\
%
  &+ \sqrt{\frac{4\pi}{V}} \sum_{\ell=0}^{\infty}
       \PadTo{\sqrt{\ell + 1}}{\sqrt{\ell}\ }\,
                         i^{\ell}\, \frac{\Psi_\ell^- (r)}{kr}
       \left[ -a_{1/2} \Y_{\ell - 1/2, \ell, 1/2}^{1/2}
            + a_{-1/2} \Y_{\ell - 1/2, \ell, 1/2}^{-1/2} \right]
\Tag{15}% belongs in between lines
\end{align*}
where $\Psi_\ell^+$ is the radial function associated with $j=\ell+1/2$ and~$\Psi_\ell^-$ is associated
with $j=\ell-1/2$.

The terms appearing in~\Eqno{15} are not coupled by the spin-orbit interaction,
and substitution into the Schroedinger \Eqref{4} yields the
following radial equations:
\[
\frac{d^2\Psi_\ell^±}{dr^2}
  + \left\{ k^2 - \frac{2\mu}{\hbar^2}
      \left[ V_1 + \frac{\hbar^2}{2} \ellAlt V_2 \right]
      - \frac{\ell(\ell + 1)}{r^2} \right\} \Psi_\ell^± = 0
\Tag{16}
\]
where the quantity~$\ell$ appears in the equation for~$\Psi_\ell^±$ and~$-\ell-1$ appears in
the equation for~$\Psi_\ell^-$.

The radial wave function~$\Psi_\ell^±$ must reduce to that of the incident wave,
$kr\, j_\ell(kr)$, when there is no interaction and must be such that only the outgoing
wave is modified by the interaction. These conditions are satisfied
by the asymptotic expression
\begin{align*}
\Psi_\ell^± &\cong kr\, j_\ell (kr)
  + C_\ell^± \left[ -y_\ell (kr) + i\, j_\ell (kr)\right]
\Tag{17} \\
\intertext{which reduces to}
\Psi_\ell^± &\cong kr\, j_\ell (kr)
  + C_\ell^± \, e^{i (kr - \ell \pi/2)}
\Tag{18} \\
%% -----File: 012.png---Folio 7-------
\intertext{or equivalently}
\Psi_\ell^± &\cong % [** PP: Oversize fraction]
  \sin(kr - \frac{\ell\pi}{2}) + C_\ell^± e^{i(kr - \ell\pi/2)}
\Tag{19}
\end{align*}
as may be seen by applying the asymptotic expression for the regular and
irregular spherical Bessel functions:
\[
\left.
\begin{aligned}
kr\, j_\ell(kr) &\cong \Neg\sin(kr - \ell \pi/2) \\
kr\, y_\ell(kr) &\cong    -\cos(kr - \ell \pi/2).
\end{aligned}
\right\}
\Tag{20}
\]
On the other hand, in terms of complex phase shifts~$\delta_\ell^±$, \Eqno{19} must be of
the form
\[
\Psi_\ell^±
  \cong A_\ell^± \sin(kr - \ell \pi/2 + \delta_\ell^±)
\Tag{21}
\]
Comparison of the coefficients of~$e^{ikr}$ and~$e^{-ikr}$ in \Eqrefs[abbr]{21} and~\Eqno{19}
yields
\begin{align*}
C_\ell^± &= \frac{1}{2i}(e^{2i\delta_\ell^±} - 1)
\Tag{22} \\
A_\ell^± &= e^{i\delta_\ell^±}
\Tag{23}
\end{align*}

Substituting \Eqno{18} into~\Eqno{15} and subtracting $\Psi_\inc$ as given by~\Eqno{14},
yields for $\Psi_\scatt$ the asymptotic form:
\[
\Psi_\scatt \cong \frac{1}{\sqrt{V}} \frac{e^{ikr}}{r}
  \left\{
      A(\theta) \left[ a_{1/2}\alpha + a_{-1/2}\beta \right]
  + i B(\theta) \left[ a_{-1/2} e^{-i\varphi}\alpha
                      - a_{1/2}  e^{i\varphi}\beta \right]
   \right\}
\Tag{24}
\]
where
\[
\left.
\begin{aligned}
A(\theta) &= \Neg\frac{1}{k}  \sum_{\ell=0}^\infty
    \left[ (\ell + 1) C_\ell^{+} + \ell C_\ell^{-} \right]
    P_\ell(\cos \theta) \\
B(\theta) &= -\frac{i}{k} \sum_{\ell=0}^\infty
    \left[ C_\ell^{+} - C_\ell^{-} \right]
    P_\ell^1(\cos \theta)
\end{aligned}
\right\}
\Tag{25}
\]

The wave function of the scattered wave can more conveniently be
expressed in terms of~$\vec{\sigma}$ and~$\vec{n}$, the unit vector normal to the scattering
plane defined by
\[
\vec{n} \sin\theta = \vec{k_1} × \vec{k_0}
\Tag{26}
\]
%% -----File: 013.png---Folio 8-------
where $\vec{k}_0$ and $\vec{k}_1$ are unit vectors in the direction of propagation before and
after scattering; thus
\[
\Psi_\scatt \cong \frac{1}{\sqrt{V}} \frac{e^{ikr}}{r}
  \left[ A(\theta) + B(\theta) \vec{\sigma} · \vec{n} \right] \chi_\inc
  = \frac{1}{\sqrt{V}} \frac{e^{ikr}}{r} f(\theta) \chi_\inc
\Tag{27}
\]
where $f(\theta)$ is the operator
\[
f(\theta) = A(\theta) + B(\theta) \vec{\sigma} · \vec{n}.
\Tag{28}
\]

The differential elastic scattering cross section and polarization
vector which are given by
\begin{align*}
\sigma(\theta)
  &= \left\langle \left[ f(\theta) \chi_\inc \right]^{\dagger}
                  \left[ f(\theta) \chi_\inc \right] \right\rangle
\Tag{29} \\
%
\vec{P}(\theta)
  &= \frac{\left\langle
                  \left[ f(\theta) \chi_\inc \right]^{\dagger}
                  \left[ f(\theta) \chi_\inc \right] \right\rangle}
          {\sigma(\theta)}
\Tag{30}
\end{align*}
thus become
\begin{align*}
\sigma(\theta)
  &= |A|^2 + |B|^2 + (A^*B + AB^*) \vec{n} · \vec{P}_0
\Tag{31} \\
%
\vec{P}(\theta)
  &= \frac{(|A|^2 - |B|^2) \vec{P}_0
           + \left[ A^*B + AB^*
           + 2|B|^2 \vec{P}_0 · \vec{n} \right] \vec{n}
           + i (A^*B - AB^*) \vec{n} × \vec{P}_0}
          {|A|^2 + |B|^2 + (A^*B + AB^*) \vec{P}_0 · \vec{n}}
\Tag{32}
\end{align*}
where the incident polarization vector $\vec{P}_0$, is given by
\[
\vec{P}_0 = \left\langle \chi_\inc^{\dag}\ \vec{\sigma} \chi_\inc \right\rangle
\Tag{33}
\]

If the incident beam is unpolarized, i.e., $\vec{P}_0=0$, the scattered beam
is polarized along the direction~$\vec{n}$, perpendicular to the scattering plane and
\begin{align*}
\sigma(\theta)  &= |A|^2 + |B|^2
\Tag{34} \\
\vec{P}(\theta) &= P(\theta) \vec{n}
  = \frac{(A^*B + AB^*)}{|A|^2 + |B|^2} \vec{n}
\Tag{35}
\end{align*}

%% -----File: 014.png---Folio 9-------

Experimentally, the polarization is sometimes obtained from a double
scattering experiment in the same plane wherein the polarization in the
first scattering is known\footnotemark.
\footnotetext{L.~Rosen, Proceedings of the International Conference on the Nuclear Optical
Model, Florida State University, Tallahassee, 1959, pp.~72--90.} The differential elastic scattering cross section
for the second scattering may then be obtained from~\Eqno{31} and~\Eqno{35}:
\[
\begin{aligned}
  \sigma_2(\theta)
  &= (|A|^2 + |B|^2) \left[
    1 + \frac{A^*B + AB^*}{|A|^2 + |B|^2} \vec{n}_2 · \vec{P}_1
    \right]\\
%
  &= (|A|^2 + |B|^2) (1 + \vec{P}_2 · \vec{P}_1).
\end{aligned}
\Tag{36}
\]

\begin{figure*}
\begin{center}
\input{./images/014a.eepic}
\phantomsection\label{figure:1}
\end{center}
\end{figure*}
%[** Illustration: Fig.1]

Referring to \textsc{Figure~\FigRef{1}}, it is clear that
\[
\vec{n}_1 = \vec{n}_2^r = -\vec{n}_2^{\ell},
\Tag{37}
\]
so that the differential scattering cross sections along the~$r$ and~$\ell$ beams
are as follows:
\[
\left.
\begin{aligned}
  \sigma_2^r(\theta)      &= (|A|^2 + |B|^2)(1 + P_2 P_1) \\
  \sigma_2^{\ell}(\theta) &= (|A|^2 + |B|^2)(1 - P_2 P_1),
\end{aligned}
\right\}
\Tag{38}
\]
the ratio of the scattering intensities becomes
\[
\frac{\sigma_2^{\ell}(\theta)}
     {\sigma_2^r(\theta)}
  = \frac{1 - P_2 P_1}{1 + P_2 P_1},
\Tag{39}
\]
%% -----File: 015.png---Folio 10-------
and solving for $P_2$:
\[
P_2 = \frac{1}{P_1}\,
      \frac{\sigma_2^\ell - \sigma_2^r}
           {\sigma_2^\ell + \sigma_2^r}
\Tag{40}
\]
which reduces when $P_1 = 1$ to
\[
P_2 = \frac{\sigma_2^\ell - \sigma_2^r}
           {\sigma_2^\ell + \sigma_2^r}
\Tag{41}
\]


\Subsection{2.}{Charged Incident Particles}

We next consider the case in which the incident particle has charge~$Ze$
and the target particle has charge~$Z'e$. The potential~$V(r)$ must now
include a term~$V_c(r)$ which describes the coulomb interaction. For small
values of~$r$, $V_c$ will depend on the assumed charge distribution, while for
large values of~$r$, we must have
\[
V_c = \frac{ZZ'e^2}{r} \quad\text{($r$ large)}.
\Tag{42}
\]
It is convenient to introduce the parameter~$\eta$,
\[
\eta = \frac{\mu\ ZZ'e^2}{\hbar^2 k}
  = 0.15805086\, ZZ'\, \sqrt{\frac{m_i}{E_\LAB}}
\Tag{43}
\]

For the ``incident wave'' we take $\Psi_c(r)\chi_\inc$, where $\Psi_c$ is the solution
to the Schroedinger equation
\[
-\frac{\hbar}{2\mu}\, \vec{\nabla}^2\Psi_c
   + \frac{ZZ'e^2}{r}\, \Psi_c = E\Psi_c
\Tag{44}
\]
corresponding to the scattering of two point charges.

It is well known that in that case % [** PP: No indent in original]
\[
\Psi_c = \frac{1}{\sqrt{V}}\,
         \Gamma(1 + i\eta) e^{-1/2\eta\pi}\,
         e^{ikz} F(-i\eta, 1, ik\xi)
\Tag{45}
\]
where $\xi=r-z$ and $F$ is the confluent hypergeometric function.

It is important to note that $\Psi_c$ includes a distorted incoming wave
\emph{plus} a scattered wave due to the point charge potential, and as such is not
%% -----File: 016.png---Folio 11-------
strictly an incident wave.

The asymptotic form of~$\Psi_c$ is given by
\[
\begin{aligned}
  \Psi_c &\cong \frac{1}{\sqrt{V}}
    \left\{ e^{i \left[ kz\ - \eta\, \ell n\, k(r-z) \right]}
    \left( 1 - \frac{\eta^2}{ik(r-z)} \right) \right. \\
    %
  &+ \left. \frac{1}{r} f_c(\theta)\, e^{i(kr - \eta\, \ell n\, 2kr)}
  \right\}
\end{aligned}
\Tag{46}
\]
where
\[
f_c(\theta) = - \frac{\eta}{2k \sin^2 \theta/2}\,
  e^{-i \eta\, \ell n (\sin^2 \theta/2) + 2i\ \sigma_0}
\Tag{47}
\]
is the Rutherford scattering amplitude and $\sigma_0$ is given by \Eqref{49},
below, with $\ell = 0$.

The partial wave expansion of~$\Psi_c$ is given by
\[
\Psi_c = \frac{1}{\sqrt{V}}
  \sum_{\ell=0}^\infty (2\ell+1)\, i^\ell\, e^{i\sigma_\ell}\,
    \frac{F_\ell(\eta, kr)}{kr} \sqrt{\frac{4\pi}{2\ell + 1}}
    Y_\ell^0(\theta, \varphi)
\Tag{48}
\]
where $F_\ell(\eta, kr)$ is the regular coulomb function and $\sigma_\ell$ is the usual coulomb
phase shift given by
\[
\sigma_\ell = \arg \Gamma(\ell + 1 + i \eta)
\Tag{49}
\]

Comparing \Eqref{48} with~\Eqno{10} we see that in \Eqref{14} it is
necessary to replace $j_\ell(kr)$ by $e^{i\sigma_\ell} \frac{F_\ell(\eta, kr)}{kr}$; thus, in this case,
\begin{align*}
  \Psi_\inc &= \sqrt{\frac{4\pi}{V}} \sum_{\ell=0}^\infty
    \sqrt{\ell+1}\, i^\ell\, e^{i \sigma_\ell}\,
    \frac{F_\ell(\eta, kr)}{kr}
    \left[ \PadTo{-a}{a}_{1/2}  \Y_{\ell + 1/2, \ell, 1/2}^1/2 +
           a_{-1/2} \Y_{\ell + 1/2, \ell, 1/2}^{-1/2} \right] \\
    %
&
\Tag{50} \\
  &+ \sqrt{\frac{4\pi}{V}} \sum_{\ell=0}^\infty
    \PadTo{\sqrt{\ell+1}}{\sqrt{\ell}\ }\, i^\ell\, e^{i\sigma_\ell}\,
    \frac{F_\ell(\eta, kr)}{kr}
    \left[ -a_{1/2}  \Y_{\ell - 1/2, \ell, 1/2}^1/2 +
            a_{-1/2} \Y_{\ell - 1/2, \ell, 1/2}^{-1/2} \right]
\end{align*}

The total wave function can be written as a sum of the ``incident'' wave,
$\Psi_\inc$, plus a ``scattered'' wave, $\Psi_\scatt$, where $\Psi_\scatt$ now includes only interference
terms and deviations from pure Rutherford scattering:
%% -----File: 017.png---Folio 12-------
\begin{align*}
\Psi_\total &= \Psi_\inc + \Psi_\scatt \\
%
&= \sqrt{\frac{4\pi}{V}} \sum_{\ell=0}^\infty
   \sqrt{\ell+1}\, i^\ell \, e^{i\sigma_\ell}\,
   \frac{\Psi_\ell^+(r)}{kr}
   \left[ \PadTo{-a}{a}_{1/2}  \Y_{\ell + 1/2, \ell, 1/2}^1/2 +
          a_{-1/2} \Y_{\ell + 1/2, \ell, 1/2}^{-1/2} \right] \\
&
\Tag{51} \\
   %
&+ \sqrt{\frac{4\pi}{V}} \sum_{\ell=0}^\infty
   \PadTo{\sqrt{\ell+1}}{\sqrt{\ell}\ }\, i^\ell\, e^{i\sigma_\ell}\,
   \frac{\Psi_\ell^-(r)}{kr}
   \left[ -a_{1/2}  \Y_{\ell - 1/2, \ell, 1/2}^1/2 +
           a_{-1/2} \Y_{\ell - 1/2, \ell, 1/2}^{-1/2} \right]
\end{align*}
This wave function, $\Psi_\total$, % [** PP: Typo \Psi_tot]
is formally almost identical to the expression given
by \Eqref{15} and the radial wave functions~$\Psi_\ell^±$ obey an equation which is
formally identical to \Eqref{16} except that $V_1(r)$ must now include the
coulomb potential~$V_c(r)$ which may differ from a point charge potential at
close distances.

The radial wave function $\Psi_\ell^±$ must now reduce to the ``incident'' wave,
$F_\ell(\eta, kr)$, when the potential becomes a coulomb point charge potential, and
must be such that only the outgoing wave is modified by the non-coulomb
interaction. These conditions are satisfied by the asymptotic expression:
\begin{align*}
\Psi_\ell^±
  &\cong F_\ell(\eta, kr)
  + C_\ell^± \left[ G_\ell(\eta, kr) + i F_\ell(\eta, kr) \right]
\Tag{52} \\
\intertext{which reduces to}
\Psi_\ell^±
  &\cong F_\ell(\eta, kr)
  + C_\ell^± e^{i(kr - \eta\, \ell n\, 2kr - \ell\pi/2 + \sigma_\ell)}
\Tag{53}
\end{align*}
or equivalently
\[
\Psi_\ell^±
\cong \sin(kr - \eta\, \ell n\, 2kr - \ell \pi/2 + \sigma_\ell) +
  C_\ell^±
      e^{i(kr - \eta\, \ell n\, 2kr - \ell \pi/2 + \sigma_\ell)}
\Tag{54}
\]
as may be seen by introducing the asymptotic expressions for the regular and
irregular coulomb functions:
\[
\left.
\begin{aligned}
  F_\ell(\eta, kr)
  &\cong \sin(kr - \eta\, \ell n\, 2kr - \ell \pi/2 + \sigma_\ell) \\
  G_\ell(\eta, kr)
  &\cong \cos(kr - \eta\, \ell n\, 2kr - \ell \pi/2 + \sigma_\ell)
\end{aligned}
\right\}
\Tag{55}
\]

%% -----File: 018.png---Folio 13-------

In this case, the ``nuclear phase shift'' $\delta_{\ell}^{±}$ is taken to be such
that the asymptotic form of~$\Psi_{\ell}^{±}$ is given by
\[
\Psi_{\ell}^{±} \cong A_{\ell}^{±}
  \sin (kr - \eta\, \ell n\, 2kr - \ell \pi/2 + \sigma_\ell + \delta_\ell^±)
\Tag{56}
\]
Comparison of the coefficients of $e^{i(kr - \eta\, \ell n\, 2kr)}$ and $e^{-i(kr - \eta\, \ell n\, kr)}$
in \Eqrefs{54} and~\Eqno{56} yields
\begin{align*}
C_{\ell}^{±} &= \frac{1}{2i} \left[ e^{2i\, \delta_{\ell}^{±}}-1 \right]
\Tag{57} \\
A_{\ell}^{±} &= e^{i\, \delta_{\ell}^{±}}
\Tag{58}
\end{align*}
Substituting~\Eqno{53} into~\Eqno{51} and making use of~\Eqno{46} and~\Eqno{50} we obtain for
the asymptotic form of the total wave function
\begin{align*}
  \Psi_\total &\cong \frac{1}{\sqrt{V}}
    \left\{ e^{i[kz - \eta\, \ell n\, k(r-z)]}
    \left[ 1 - \frac{\eta^2}{ik(r-z)} \right] \right\} \chi_\inc \\[-12pt]
&\Tag{59} \\
%
    &+ \frac{1}{\sqrt{V}} \frac{e^{i(kr - \eta\, \ell n\, 2kr)}}{r}
    \left\{ A(\theta) \left[ a_{1/2} \alpha + a_{-1/2} \beta \right] +
           i B(\theta) \left[ a_{-1/2} e^{-i\varphi} \alpha
                            - a_{1/2} e^{i\varphi} \beta \right] \right\}
\end{align*}
where
\[
\begin{aligned}
  A(\theta) &= f_c(\theta)
    + \frac{1}{k} \sum_{\ell=0}^\infty e^{2i \sigma_\ell}
      \left[ (\ell+1) C_{\ell}^{+} + \ell C_{\ell}^{-} \right]
      P_{\ell} (\cos\theta) \\
  B(\theta) &= -\frac{i}{k} \sum_{\ell=0}^\infty e^{2i\sigma_\ell}
    \left[ C_{\ell}^{+} - C_{\ell}^{-} \right]
      P_{\ell}^1 (\cos\theta)
\end{aligned}
\Tag{60}
\]
and $f_c(\theta)$ is given by \Eqref{47}.

From this point, the formulation follows through as in the case of
uncharged particles.


\Section{B.}{Optical Model Potential}

\Subsection[Diffuse Surface Optical Model with Volume Absorption and
Coulomb Spin-Orbit.]%
{1.}{Diffuse Surface Optical Model with Volume Absorption and \hfill\break
Coulomb Spin-Orbit.}

The interaction~\Eqno{1} is assumed to have the form
\[
V_T = V_\text{CN} + V_\text{SO} + V_\text{Coul} + V_\text{Coul SO}
\Tag{61}
\]
%% -----File: 019.png---Folio 14-------
where the terms appearing in \Eqref{61} are respectively the central
nuclear, spin-orbit nuclear, coulomb, and coulomb spin-orbit potentials.

We shall first consider the case for which the real and imaginary
parts of the central potential have a special common form factor (corresponding
to volume absorption), and the spin-orbit potential is of the
Thomas type. This particular central potential form factor has been used
extensively and will be referred to as the standard form factor. We shall
then discuss other form factors available in the program.


\Subsubsection{(a)}{Central nuclear potential}
\[
V_\text{CN} = (-V - iW) \frac{1}{(1 + e^{(r - R_N)/a})}
\Tag{62}
\]
where~$V$ and~$W$ are respectively the depths of the real and imaginary part of
the nuclear potential in \MeV\ ($V$ and~$W$ are positive for an attractive, absorbing
potential), and a common volume absorption form factor is assumed, where
\[
R_N = R_\text{ON} m_b^{1/3} × 10^{-13}~\text{cm}
\Tag{63}
\]
$R_\text{ON}$ being the nuclear radius constant and~$a$ is the rounding parameter in $10^{-13}~\text{cm}$.


\Subsubsection{(b)}{Nuclear spin-orbit potential}

The nuclear spin-orbit potential is often written in the Thomas form
\[
V_\text{SO} = \lambda \frac{1}{2 M_p^2 c^2}
  \left\{ \frac{1}{r}\, \frac{d}{dr}
    \left[ \frac{-V}{1 + e^{(r-R_N)/a}} \right] \right\} \vec{S} · \vec{L}
\Tag{64}
\]
where $M_p$ is the proton test mass and $c$ the velocity of light. If~$\lambda$ were~$1$,
the spin-orbit term would be that predicted by the Dirac equation. To
provide more freedom in the model one writes
\[
\lambda = 4 \left( \frac{M_p}{M_\pi} \right)^2 \frac{V_S + iW_S}{V}
\Tag{65}
\]
%% -----File: 020.png---Folio 15-------
where $M_\pi$ is the pion rest mass and~$V_S$ and~$W_S$ are respectively the strengths
of the real and imaginary parts of the nuclear spin-orbit potential in~\MeV.

It may be noted that a negative value of the real part of~$\lambda$ would be
in accordance with the shell model of the nucleus where a (real) negative
spin-orbit term is required to give the proper level sequence in contra-distinction
to the atomic case.


\Subsubsection{(c)}{Coulomb potential}

The coulomb potential is taken here to correspond to a constant
charge density within the nucleus extending to a distance~$R_c$ given by
\[
R_c = R_\text{oc} m_b^{1/3} × 10^{-13}~\text{cm}
\Tag{66}
\]
where $R_\text{oc}$ is the coulomb radius constant; thus
\[
\begin{aligned}
V_\text{Coul}
  &= (ZZ'e^2/2R_c) (3 - r^2/R_c^2) && \quad\text{for $r \leq R_c$} \\
  &= ZZ'e^2/r                      && \quad\text{for $r \geq R_c$}
\end{aligned}
\Tag{67}
\]


\Subsubsection{(d)}{Coulomb spin-orbit potential}

The coulomb spin-orbit term is assumed to have the form\footnote
  {W.~Heckrotte, Phys.\ Rev.\ \textbf{101}, l406 (1956).}
\[
V_\text{Coul SO}
  = (\mu_P - \tfrac{1}{2}) \frac{1}{M_P^2 c^2}
    \left[ \frac{1}{r}\, \frac{d}{dr} V_\text{Coul} \right] \vec{S} · \vec{L}
\Tag{68}
\]
where $\mu_P$ is the proton magnetic moment in nuclear magnetons. It may be
noted that the coulomb spin-orbit term is negligible except at very high
energies.

Substituting \Eqrefs{62}, \Eqno{64}, \Eqno{67}, and~\Eqno{68} into \Eqref{16}
and transforming to the dimensionless variable
\[
\rho = kr
\Tag{69}
\]
%% -----File: 021.png---Folio 16-------
we find
\begin{multline}
  \left\{\vphantom{\Bigg|} -\frac{d^2}{d\rho^2}
          +\frac{\ell(\ell + 1)}{\rho^2}
          - \left( \frac{V + iW}{E} \right)
            \left( \frac{1}{1 + e^{(\rho - \bar{\rho}_N)/ka}} \right)
  \right. \\
%
  + \left( \frac{\hbar}{M_\pi c} \right)^2
    \left( \frac{V_S + iW_S}{E} \right) k^2
    \left[ - \frac{1}{\rho}\, \frac{d}{d\rho}
      \left( \frac{1}{1 + e^{(\rho - \bar{\rho}_N)/ka}} \right) \right]
    \ellAlt
\Tag{70} \\
%
\left. + U_\text{Coul} + U_\text{Coul SO} - 1
  \vphantom{\Bigg|}\right\} \Psi_\ell^±(\rho) = 0
\end{multline}
where
\[
\begin{aligned}
  U_\text{Coul}
  &= \frac{\eta}{\bar{\rho}_c} \left( 3 - \frac{\rho^2}{\bar{\rho}_c^2} \right)
  && \text{for $\rho \leq \bar{\rho}_c$} \\
  %
  &= 2 \eta/\rho && \text{for $\rho \geq \bar{\rho}_c$}
\end{aligned}
\Tag{71}
\]
\[
\begin{aligned}
  U_\text{Coul SO}
  &= -\frac{1}{2} \left( \frac{\hbar}{M_P c} \right)^2
      (\mu_P - \tfrac{1}{2}) (2\eta) \left( k^2 / \bar{\rho}_c^3 \right)
      \ellAlt && \text{for $\rho \leq \bar{\rho}_c$} \\
%
  &= -\frac{1}{2} \left( \frac{\hbar}{M_P c} \right)^2
      (\mu_P - \tfrac{1}{2}) (2\eta) \left( k^2 / \rho^3 \right)
      \ellAlt && \text{for $\rho \geq \bar{\rho}_c$}
\end{aligned}
\Tag{72}
\]
and where
\begin{align*}
\bar{\rho}_N &= k R_N
\Tag{73} \\
\bar{\rho}_c &= k R_c.
\Tag{74}
\end{align*}
Substituting now % [** PP: Typo substituting]
\begin{align*}
&\left( \frac{\hbar}{M_\pi c} \right)^2 = 2.00 × 10^{-26}~\text{cm}^2
\Tag{75} \\
%
&2\eta\, k^2 · \tfrac{1}{2} \left( \frac{\hbar}{M_P c} \right)^2
\cong 2\eta \left( \frac{E}{M_P c^2} \right)
= 2\eta \, \frac{E}{931}
\Tag{76} \\
%
&\mu_P - \tfrac{1}{2} = 2.7934 - 0.5 = 2.2934
\Tag{77}
\end{align*}
%% -----File: 022.png---Folio 17-------
into \Eqref{70} yields:
{\small
\begin{align*}
  &\frac{d^2}{d\rho^2} \Psi_\ell^± (\rho)
  = \left\{ -1 + \frac{\ell(\ell + 1)}{\rho^2}
   - \left( \frac{V + iW}{E} \right)
     \left( \frac{1}{1 + e^{(\rho - \bar{\rho}_N)}/ka} \right)
   + \frac{\eta}{\bar{\rho}_c}
     \left( 3 - \frac{\rho^2}{\bar{\rho}_c^2} \right) \right. \\
%
  &+ \left. \left[ 2 \left( \frac{V_S + iW_S}{E} \right)
       \left( \frac{k}{a} \right)
       \left( \frac{1}{\rho}\,
              \frac{e^{(\rho - \bar{\rho}_N)/ka}}
                   {(1 + e^{(\rho - \bar{\rho}_N)/ka})^2} \right)
   - 0.004926\, \frac{\eta E}{\bar{\rho}_c^3} \right]
     \ellAlt \right\}
  \Psi_\ell^±(\rho), \text{ for $\rho \leq \bar{\rho}_c$} \\
%
  &\qquad= \left\{ -1 + \frac{\ell(\ell + 1)}{\rho^2}
   - \left( \frac{V + iW}{E} \right)
     \left( \frac{1}{1 + e^{(\rho - \bar{\rho}_N)/ka}} \right)
   + \frac{2\eta}{\rho} \right.
\Tag{78} \\
%
  &+ \left. \left[ 2 \left( \frac{V_S + iW_S}{E} \right)
       \left( \frac{k}{a} \right)
       \left( \frac{1}{\rho}\,
              \frac{e^{(\rho - \bar{\rho}_N)/ka}}
                   {(1 + e^{(\rho - \bar{\rho}_N)/ka})^2} \right)
   - 0.004926\, \frac{\eta E}{\rho^3} \right]
     \ellAlt \right\}
  \Psi_\ell^±(\rho),\text{ for $\rho \geq \bar{\rho}_c$}
\end{align*}}% [** PP: End of \small]


\Subsection{2.}{Nuclear Form Factors}

\Eqref[cap]{78} may be rewritten in such a way as to display explicitly
the various nuclear form factors:
\begin{align*}
  &\frac{d^2}{d\rho^2} \Psi_\ell^±(\rho)
  = \left\{ -1 + \frac{\ell(\ell+1)}{\rho^2}
     - \frac{V}{E}\, f_\CR(\rho)
   - i \frac{W}{E}\, f_\CI(\rho)
   + \frac{\eta}{\bar{\rho}_c}
     \left( 3 - \frac{\rho^2}{\bar{\rho}_c^2} \right) \right. \\
%
  &+ \left. \left[ \frac{V_S}{E}\, \frac{2k}{a}\, f_\SR(\rho)
       + i \frac{W_S}{E}\, \frac{2k}{a}\, f_\SI(\rho)
       - 0.004926 \frac{\eta E}{\bar{\rho}_c^3} \right]
     \ellAlt \right\}
     \Psi_\ell^±(\rho), \text{ for $\rho \leq \bar{\rho}_c$} \\
%
  &\qquad= \left\{ -1 + \frac{\ell(\ell+1)}{\rho^2}
     -   \frac{V}{E}\, f_\CR(\rho)
     - i \frac{W}{E}\, f_\CI(\rho)
     + \frac{2 \eta}{\rho} \right.
\Tag{79} \\
%
  &+ \left. \left[ \frac{V_S}{E}\, \frac{2k}{a}\, f_\SR(\rho)
     + i \frac{W_S}{E}\, \frac{2k}{a}\, f_\SI(\rho)
     - 0.004926 \frac{\eta E}{\rho^3} \right]
    \ellAlt \right\}
    \Psi_\ell^±(\rho), \text{ for $\rho \geq \bar{\rho}_c$}
\end{align*}

Three basic nuclear form factors and some special modifications of them
are presently available in the program. In addition the coulomb spin-orbit
term may be excluded at will. The required form factors may be chosen by
assigning the proper values to the symbolic quantities \Code{KTRL} as described on
% [** PP: ``pages 41 and 42'' in original]
\hyperref[page:41]{pages~\pageref{page:41}}~\textit{ff}.

%% -----File: 023.png---Folio 18-------

\Subsubsection{(a)}{Basic Form Factors}

\Subsubsubsection{(i)}{Volume absorption}
(\Code{KTRL(I) = 0, I = 1, 7, 8, 9, 10})
\begin{align*}
f_\CR(\rho)
  &= f_\CI(\rho) %[** PP: Slant fraction in original]
   = \frac{1}{(1 + e^{(\rho-\bar{\rho}_N)/ka})}
\Tag{80} \\
f_\SR(\rho)
  &= f_\SI(\rho)
   = \frac{1}{\rho}\, \frac{e^{(\rho - \bar{\rho}_N)/ka}}
                      {(1 + e^{(\rho - \bar{\rho}_N)/ka})^2}
\Tag{81}
\end{align*}


\Subsubsubsection{(ii)}{Gaussian absorption}
(\Code{KTRL(1) = 1})

$f_\CR$ is given by~\Eqno{80}, $f_\SR$ and~$f_\SI$ are given by~\Eqno{81} and
\begin{align*}
f_\CI(\rho) &= e^{-[(\rho - \bar{\rho}_G)/kb]^2}
\Tag{82} \\
\intertext{where}
\bar{\rho}_G &= k R_\text{OG} m_b^{1/3},
\Tag{83}
\end{align*}
$R_\text{OG}$ being the nuclear Gaussian radius constant, %[** PP: Typo contant]
and~$b$ determines the Gaussian width.


\Subsubsubsection{(iii)}{Square well}
(\Code{KTRL(1) = 2})
\begin{align*}
  f_\CR(\rho) = f_\CI(\rho)
    &= 1 \qquad \text{for $\rho \leq \bar{\rho}_N$} \\[-6pt]
\Tag{84} \\[-6pt]
    &= 0 \qquad \text{for $\rho \geq \bar{\rho}_N$} \\
f_\SR(\rho) = f_\SI(\rho) &= 0. % [** PP: Added period]
\Tag{85}
\end{align*}


\Subsubsection{(b)}{Special Central Nuclear Form Factors\footnotemark}
\footnotetext{J.S.~Nodvik, Proceedings of the International Conference on the Nuclear
Model, Florida State University, Tallahassee, 1959, pp.~16--23.}
(\Code{KTRL(1) = 0})

The purpose of these form factors is to allow one to modify the
knee or tail of the potential curve and produce central rises or depressions
in the real and/or imaginary parts of the central nuclear potential, as
specified by proper choice of the \Code{KTRL}'s.

%% -----File: 024.png---Folio 19-------

\Subsubsubsection{(i)}{Form~A}
(\Code{KTRL(7) = 1} for real part, \Code{KTRL(8) = 1} for imaginary part).
\[
\left.
\begin{aligned}
  f_\CR(\rho) \text{ and/or } f_\CI(\rho)
  &= \left[1 + h_A(\rho)\right] f_{nA_1}(\rho) &
    0 &< \rho \leq \rho_{m_A} \\
  &= f_{nA_1}(\rho) &
    \rho_{m_A} &\leq \rho \leq \bar{\rho}_N \\
  &= f_{nA_2}(\rho) &
    \bar{\rho}_N &\leq \rho \leq \rho_\text{max}
\end{aligned}
\right\}
\Tag{86}
\]


\Subsubsubsection{(ii)}{Form~B}
(\Code{KTRL(7) = 2} for real part, \Code{KTRL(8) = 2} for imaginary part).
\[
\left.
\begin{aligned}
  f_\CR(\rho) \text{ and/or } f_\CI(\rho)
  &= \left[1 + h_B(\rho)\right] f_{nB_1}(\rho) &
    0 &< \rho \leq \rho_{m_B} \\
  &=\quad f_{nB_1}(\rho) &
    \rho_{m_B} &\leq \rho \leq \bar{\rho}_N \\
  &=\quad f_{nB_2}(\rho) &
    \bar{\rho}_N &\leq \rho \leq \rho_\text{max}
\end{aligned}
\right\}
\Tag{87}
\]

% [** PP: Changed subscript o to 0 throughout]
The presence of forms~$A$ and~$B$ allows distinct form factors in the real
and imaginary parts. The presence of $A_1$, $A_2$ and~$B_1$, $B_2$ allows distinct shapes
in the knee and tail of the form factors. Letting~$x$ be either $A$ or~$B$, and~$n$
be either $nA_1$, $nA_2$, $nB_1$, or~$nB_2$,
\begin{align*}
  h_x(\rho)
  &= h_{0x} \left[ 2 \left( \frac{\rho}{\rho_{m_x}} \right)^3
   - 3 \left( \frac{\rho}{\rho_{m_x}} \right)^2 + 1 \right]
   = h_{0x} \left( 1 - \frac{\rho}{\rho_{m_x}} \right)^2
            \left( 1 + \frac{\rho}{\rho_{m_x}} \right)
\Tag{88} \\
f_n(\rho)
  &= \frac{1}{1 + g_n(\rho)}
\Tag{89} \\
\intertext{where}
g_n(\rho)
  &= \exp \left\{ \frac{1}{n}\, \left( \frac{\bar{\rho}_N}{ka} \right)
     \left[ \left( \frac{\rho}{\bar{\rho}_N} \right)^n - 1 \right] \right\}
\Tag{90}
\end{align*}
where $h_{0A}$, $h_{0B}$, $nA_1$, $nA_2$, $nB_1$, $nB_2$, $\rho_{m_A}$, $\rho_{m_B}$ are selected constants. (The $n$'s
are always taken as $\geq 0$.)

\Note{1} If $h_{0x}$ is taken to be zero and $nx_1$, $nx_2$ are taken to be~$1$, forms $A$
and~$B$ reduce to the volume absorption form.

\Note{2} The three curves defined by \Eqrefs{86} and~\Eqno{87} join smoothly
with continuous derivatives as long as $\rho_{m_x}$ is chosen less than~$\bar{\rho}_N$.

\Note{3} Positive values of $h_{0x}$ will produce central rises in the form
factors while negative values will produce a central depression.

%% -----File: 025.png---Folio 20-------

\Note{4} If $nx_1 > 1$, the knee of the potential will be sharper than for the
usual volume absorption case, while $0 \leq nx_1 \leq 1$ will soften the knee of the
curve.

\Note{5} If $nx_2 > 1$, this will shorten the potential tail while $0 \leq nx_2 \leq 1$
will extend it.

Some typical shapes are presented in \textsc{Figures~\FigRef{2}}, \FigRef{3}, and~\FigRef{4}.


\Subsubsection{(c)}{Special Nuclear Spin-Orbit Form Factors
(\Code{KTRL(1) = 0})} % [** PP: Omitted period]
%****\footnote Error, or refers to previous footnote?

% [** PP: A and B are labels, not math]
Two special nuclear spin-orbit form factors are available. They
can be applied to the real and/or imaginary parts of the nuclear spin-orbit % [** PP: Added hyphen]
potential. The first of these form factors corresponds to the Thomas term
applied to form~A in the central nuclear potential, while the second uses
form~B itself; this permits one to study the result of deviations from the
Thomas form.


\Subsubsubsection{(i)}{Derivative form factor~A}
(\Code{KTRL(9) = 1} for real part, \Code{KTRL(1O) = 1}
for imaginary part)
{\small% [** PP: Hard-coded size change to make everything fit]
\[
\left.
\begin{aligned}
  f_\SR(\rho) \text{ and/or } f_\SI(\rho)
  &= (ka) \left[
    -\frac{1}{\rho}\, \frac{d}{d\rho}\, (\text{form factor~A})
    \right] \\
  &= (ka) \left[ -\left( \frac{1}{\rho}\, \frac{dh_A(\rho)}{d\rho} \right)
     f_{nA_1}(\rho)
   - \left( 1 + h_A(\rho) \right)
     \left( \frac{1}{\rho}\, \frac{df_{nA_1}(\rho)}{d\rho} \right)
     \right] && \\
  &&&\makebox[-12pt][r]{\text{for $0 \leq \rho \leq \rho_{m_a}$}} \\
  %
  &= (ka) \left[ - \frac{1}{\rho}\, \frac{df_{nA_1}(\rho)}{d\rho}
     \right] && \makebox[-12pt][r]{\text{for $\rho_{m_a} \leq \rho \leq \overline{\rho}_N$}} \\
  %
  &= (ka) \left[ - \frac{1}{\rho}\, \frac{df_{nA_2}(\rho)}{d\rho}
     \right] && \makebox[-12pt][r]{\text{for $\bar{\rho}_N \leq \rho \leq \rho_\text{max}$}}
\end{aligned}
\hspace*{-12pt}\right\}
\Tag{91}
\]
}% [** PP: End of \small]
where
\begin{align*}
-\frac{1}{\rho}\, \frac{dh_A(\rho)}{d\rho}
  &= \frac{6h_{0A}}{\rho_{m_A}^2} \left( 1 - \frac{\rho}{\rho_{m_A}} \right)
\Tag{92} \\
-\frac{1}{\rho}\, \frac{df_n{\rho}}{d\rho}
  &= \left( \frac{\bar{\rho}_N}{ka} \right)
  \frac{1}{\rho^2}\, \left( \frac{\rho}{\bar{\rho}_N} \right)^n
  g_n(\rho) \left[ f_n(\rho) \right]^2
\Tag{93}
\end{align*}
and $f_n(\rho)$ and $g_n(\rho)$ are given by \Eqrefs{89} and~\Eqno{90}.

%% -----File: 026.png---Folio 21-------

\Figure{2}{026a}
%[Illustration: Fig. 2. - Central form factor, tail variations]

\iffalse %% Fig. 2
%\begin{center}
%  \underline{p - Cu \quad $9.75$~\MeV}
%  \bigskip
%\end{center}
%
%$\mathrm{RO} = 1.20$, $\mathrm{A} = 0.52$, $\mathrm{HA} = 0$,
%$\mathrm{FN1A} = 1$, $\rho_{m_A} = \bar{\rho}_N = 3.24$
%
%\begin{tabular}{cccl}
%  \textcircled{1} & $\mathrm{FN2A}$ & = & $0.01$  \\
%  \textcircled{2} & $\mathrm{FN2A}$ & = & $1$ \quad (standard \ form \ factor)\\
%  \textcircled{3} & $\mathrm{FN2A}$ & = & $3$     \\
%  \textcircled{4} & $\mathrm{FN2A}$ & = & $10$
%\end{tabular}
\fi%% End of Fig. 2

%% -----File: 027.png---Folio 22-------

\Figure{3}{027a}
%[Illustration: Fig.~3. - Central form factors; central dips and rises]

\iffalse %% Fig. 3
%\begin{center}
%  \underline{p - Ag \quad $135$~\MeV}
%  \bigskip
%\end{center}
%
%$\mathrm{RO} = 1.25$, $\mathrm{A} = 0.65$, $\mathrm{FN1A} = 1$, $\mathrm{FN2A} = 1$
%
%\begin{tabular}{ccccccc}
%  \textcircled{1} & $\mathrm{HA}=1$ & $\mathrm{PMA}=1$, & $\rho_{m_A}=$
%  & $\bar{\rho}_N=$ & $15.04$ & (standard form factor)\\
  %
%  \textcircled{2} & $\mathrm{HA}=1$ & " & " & " & " &\\
  %
%  \textcircled{3} & $\mathrm{HA}=-1$ & " & " & " & " &\\
  %
%  \textcircled{4} & $\mathrm{HA}=2$, & $\mathrm{PMA}=0.7$, &
%  $\rho_{m_A}=$ & $0.7\bar{\rho}_N=$ & $10.53$ &
%\end{tabular}
\fi%% End of Fig. 3

%% -----File: 028.png---Folio 23-------

\Figure{4}{028a}
%[Illustration: Fig.~4. Central form factors, knee variation]

\iffalse %% Fig. 4
%\begin{center}
%  \underline{p - Cu \quad $9.75$~\MeV}
%  \bigskip
%\end{center}
%
%$\mathrm{RO} = 1.20$, $\mathrm{A} = 0.52$, $\mathrm{FN2A} = 1$,
%$\mathrm{PMA} = 1$, $\rho_{m_A}=\bar{\rho}_N=3.24$
%
%\begin{tabular}{cllc}
% 1. & $\mathrm{FN1A}=0.01$, & $\mathrm{HA}=0$ & \\
% 2. & $\mathrm{FN1A}=1$, & $\mathrm{HA}=0$ & (standard form factor)\\
% 3. & $\mathrm{FN1A}=3$, & $\mathrm{HA}=0.046$ &
%\end{tabular}
\fi%% End of Fig. 4

%% -----File: 029.png---Folio 24-------


% [** PP: Not capitalizing factor, for consistency]
\Subsubsubsection{(ii)}{Form factor~B}
(\Code{KTRL(9) = 2} for real part, \Code{KTRL(10) = 2} for
imaginary part)
\[
\text{$f_\SR(\rho)$ and/or $f_\SI(\rho) = \frac{1}{2}$. [form factor~B as per \Eqref{87}]}
\Tag{94}
\]

\Note{} If $h_{0A}$ is taken to be zero while $nA_1$ and~$nA_2$ are taken to be~$1$, the
derivative form factor in~\Eqno{91} becomes identical to the usual spin-orbit
form factor~\Eqno{81}.

Some typical shapes are presented in \textsc{Figures~\FigRef{5}}, \FigRef{6}, and~\FigRef{7}.


\Subsection{3.}{Final Formulation for Machine Calculation}

The complex radial wave function $\Psi_\ell^±(\rho)$ may be written as
\[
\Psi_\ell^±(\rho) = x_\ell^±(\rho) + i y_\ell^±(\rho)
\Tag{95}
\]
and \Eqref{79} for $\vec{\sigma} · \vec{\ell} = \ell$ or
$-\ell-1$ can now be separated into two real
coupled differential equations, and dropping the subscripts and superscripts
for convenience:
\[
\left.
\begin{aligned}
  \frac{d^2 x}{d \rho^2} &= px - qy \\
  \frac{d^2 y}{d \rho^2} &= qx + py
\end{aligned}
\right\}
\Tag{96}
\]
where
\[
\left.
\begin{aligned}
  p &= U_\CR + U_\SR \ellAlt + \frac{\ell(\ell+1)}{\rho^2} \\
  q &= U_\CI + U_\SI \ellAlt
\end{aligned}
\right\}
\Tag{97}
\]
Formulas~\Eqno{97} are convenient for programming purposes as the $U$'s are now
independent of~$\ell$, indeed:
\[
\left.
\begin{aligned}
  U_\CR
  &= -1 - \frac{V}{E}\, f_\CR
    + \frac{\eta}{\bar{\rho}_c}
      \left( 3 - \frac{\rho^2}{\bar{\rho}_c^2} \right)
      && \text{for $\rho \leq \bar{\rho}_c$} \\
%
  &= -1 - \frac{V}{E}\, f_\CR
    + \frac{2\eta}{\rho}
      && \text{for $\rho \geq \bar{\rho}_c$}
\end{aligned}
\right\}
\Tag{98}
\]

%% -----File: 030.png---Folio 25-------

\Figure{5}{030a}
%[Illustration: Fig. 5. - Spin-orbit form factor(derivative form). Tail variation.]

\iffalse %% Fig. 5
%\begin{center}
%  \underline{p - Cu \quad $9.75$~\MeV}
%  \bigskip
%\end{center}
%
%$\mathrm{RO} = 1.20$, $\mathrm{A} = 0.52$, $\mathrm{HA}=0$, $\mathrm{FN1A} = 1$,
%$\mathrm{PMA} = 1$, $\rho_{m_A}=\bar{\rho}_N=3.24$
%
%\begin{tabular}{cll}
%  \textcircled{1} & $\mathrm{FN2A} = 0.01$ &\\
%  \textcircled{2} & $\mathrm{FN2A} = 1$    & (standard derivative form factor)\\
%  \textcircled{3} & $\mathrm{FN2A} = 3$    &\\
%  \textcircled{4} & $\mathrm{FN2A} = 10$   &
%\end{tabular}
\fi%% End of Fig. 5

%% -----File: 031.png---Folio 26-------

\Figure{6}{031a}
%[Illustration: Fig.~6. - Spin-orbit form factors (derivative form). Effect
%of dips and rises in central form factor]

\iffalse %% Fig. 6
%\begin{center}
%  \underline{p - Ag \quad $135$~\MeV}
%  \bigskip
%\end{center}
%
%$\mathrm{RO} = 1.25$, $\mathrm{A} = 0.65$, $\mathrm{FN1A} = 1$, $\mathrm{FN2A} = 1$
%
%\begin{tabular}{clccccc}
%  \textcircled{1} & $\mathrm{HA}=0$ & $\mathrm{PMA}=1$, & $\rho_{m_A}=$
%  & $\bar{\rho}_N=$ & $15.04$ & (standard derivative form factor)\\
  %
%  \textcircled{2} & $\mathrm{HA}=1$ & " & " & " & " &\\
  %
%  \textcircled{3} & $\mathrm{HA}=-1$ & " & " & " & " &\\
  %
%  \textcircled{4} & $\mathrm{HA}=2$, & $\mathrm{PMA}=0.7$, &
%  $\rho_{m_A}=0.7$ & $\bar{\rho}_N=$ & $10.53$ &
%\end{tabular}
\fi%% End of Fig. 6

%% -----File: 032.png---Folio 27-------

\Figure{7}{032a}
%[Illustration: Fig.7. Spin-orbit form factor (derivative form).
%Knee variation.]

\iffalse %% Fig. 7
%\begin{center}
%  \underline{p - Cu \quad $9.75$~\MeV}
%  \bigskip
%\end{center}
%
%$\mathrm{RO} = 1.20$, $\mathrm{A} = 0.52$, $\mathrm{FN2A} = 1$,
%$\mathrm{PMA} = 1$, $\rho_{m_A}=1$, $\bar{\rho}_N=3.24$
%
%\begin{tabular}{cllc}
% 1. & $\mathrm{FN1A}=0.01$, & $\mathrm{HA}=0$ & \\
% 2. & $\mathrm{FN1A}=1$, & $\mathrm{HA}=0$ & (standard derivative form factor)\\
% 3. & $\mathrm{FN1A}=3$, & $\mathrm{HA}=0.046$ &
%\end{tabular}
\fi%% End of Fig. 7

%% -----File: 033.png---Folio 28-------

\begin{align*}
  &U_\CI = -\frac{W}{E}\, f_\CI
\Tag{99} \\
%
&\left.
\begin{aligned}
  U_\SR
  &= \frac{V_S}{E}\, \frac{2k}{a}\, f_\SR
   - 0.004926 \frac{\eta E}{\bar{\rho}_c^3}
     && \text{for $\rho \leq \bar{\rho}_c$} \\
   %
  &= \frac{V_S}{E}\, \frac{2k}{a}\, f_\SR
   - 0.004926 \frac{\eta E}{\rho^3}
     && \text{for $\rho \geq \bar{\rho}_c$}
\end{aligned}
\right\}
\Tag{100} \\
%
&U_\SI = \frac{W_S}{E}\, \frac{2k}{a}\, f_\SI
\Tag{101}
\end{align*}


\Subsection{4.}{Numerical Integration}

\Eqrefs[cap]{96} must be integrated numerically twice for each $\ell=0$
to $\ell_\text{max}$ where $\ell_{\text{max}+1}$ corresponds to a
partial wave negligibly disturbed by
the scattering.

The method chosen for numerical integration is the 3-point Runge-Kutta
method: it lends itself to easy starting, permits one to change the
interval quite easily and gives excellent accuracy with relatively large
steps.

Given $x_{i1}$, $y_{i1}$, $\dot{x}_{i1}$, $\dot{y}_{i1}$, at~$\rho_i$,
where $\dot{x}_{i1} \equiv \left( \frac{dx}{d\rho} \right)_{i,1}$ etc.
\begin{align*}
  \ddot{x}_{i1} &= f(x_{i1},y_{i1},\rho_i);\quad
  \ddot{y}_{i1}  = g(x_{i1},y_{i1},\rho_i)
\Tag{102} \\
%
  x_{i2} &= x_{i1} + \dot{x}_{i1} \frac{\Delta\rho}{2};\quad
  y_{i2}  = y_{i1} + \dot{y}_{i1} \frac{\Delta\rho}{2}
\Tag{103} \\
%
  \ddot{x}_{i2} &= f(x_{i2},y_{i2},\rho_i+\frac{\Delta\rho}{2});\quad
  \ddot{y}_{i2}  = g(x_{i2},y_{i2},\rho_i+\frac{\Delta\rho}{2})
\Tag{104} \\
%
  x_{i3} &= x_{i2} + \ddot{x}_{i1} \frac{(\Delta\rho)^2}{4};\quad
  y_{i3}  = y_{i2} + \ddot{y}_{i1} \frac{(\Delta\rho)^2}{4}
\Tag{105} \\
%
  \ddot{x}_{i3} &= f(x_{i3},y_{i3},\rho_i+\frac{\Delta\rho}{2});\quad
  \ddot{y}_{i3}  = g(x_{i3},y_{i3},\rho_i+\frac{\Delta\rho}{2})
\Tag{106} \\
%
  x_{i4} &= x_{i2} + \dot{x}_{i1} \frac{\Delta\rho}{2}
                   + \ddot{x}_{i2} \frac{(\Delta\rho)^2}{2};\quad
  y_{i4}  = y_{i2} + \dot{y}_{i1} \frac{\Delta\rho}{2}
                   + \ddot{y}_{i2} \frac{(\Delta\rho)^2}{2}
\Tag{107} \\
%
  \ddot{x}_{i4} &= f(x_{i4},y_{i4},\rho_i+\Delta\rho);\quad
  \ddot{y}_{i4}  = g(x_{i4},y_{i4},\rho_i+\Delta\rho)
\Tag{108}
\end{align*}
%% -----File: 034.png---Folio 29-------
and finally
\begin{align*}
  x_{i+1,1} &= x_{i1} + \Delta x_i
    = x_{i1} + \frac{(\Delta\rho)^2}{6}
      ( \ddot{x}_{i1} + \ddot{x}_{i2} + \ddot{x}_{i3})
      + \Delta\rho \; \dot{x}_{i1}
\Tag{109} \\
%
  \dot{x}_{i+1,1} &= \dot{x}_{i1} + \Delta\dot{x}_i
    = \dot{x}_{i1} + \frac{\Delta\rho}{6}
      ( \ddot{x}_{i1} + 2\ddot{x}_{i2} + 2\ddot{x}_{i3} + \ddot{x}_{i4})
\Tag{110} \\
%
  y_{i+1,1} &= y_{i1} + \Delta y_i
    = y_{i1} + \frac{(\Delta\rho)^2}{6}
      ( \ddot{y}_{i1} + \ddot{y}_{i2} + \ddot{y}_{i3})
      + \Delta\rho \dot{y}_{i1}
\Tag{111} \\
%
  \dot{y}_{i+1,1} &= \dot{y}_{i1} + \Delta\dot{y}_i
    = \dot{y}_{i1} + \frac{\Delta\rho}{6}
      ( \ddot{y}_{i1} + 2\ddot{y}_{i2} + 2\ddot{y}_{i3} + \ddot{y}_{i4})
\Tag{112}
\end{align*}

The process is continued until the nuclear potential becomes negligible
at which time the wave functions and their first derivatives must be saved for
later matching with those of the coulomb function.

% [** PP: Heading on separate line in original]
\paragraph{Starting values:}
If $\rho_\initial$ is very small, the following starting values may be used:
\[
\left.
\begin{aligned}
  x_\ell (\rho = \rho_\initial) = (\Delta\rho_1)^{\ell+1};\quad
  \dot{x}_\ell (\rho = \rho_\initial) = (\ell+1)(\Delta\rho_1)^\ell \\
  %
  y_\ell (\rho = \rho_\initial) = 0;\quad
  \dot{y}_\ell (\rho = \rho_\initial) = 0
\end{aligned}
\right\}
\Tag{113}
\]


\Subsection{5.}{Coulomb Functions}

The regular and irregular coulomb functions are given by the following
asymptotic formulas which may be used successfully for large values of~$\rho$:
% [** PP: right-align braces in following groups]
\begin{align*}
\left.
\begin{aligned}
  F_0 &\sim \sin[\Real(\varphi_0)] e^{-\Imag(\varphi_0)} \\
  F_1 &\sim \sin[\Real(\varphi_1)] e^{-\Imag(\varphi_1)} \\
  G_0 &\sim \cos[\Real(\varphi_0)] e^{-\Imag(\varphi_0)} \\
  G_1 &\sim \cos[\Real(\varphi_1)] e^{-\Imag(\varphi_1)}
\end{aligned}
\right\} & % Top-level alignment stop
\Tag{114} \\
\intertext{where}
\left.
\begin{aligned}
  \varphi_0 &= \rho - \eta\, \ell n\, 2\rho + \sigma_0
    + \sum_{k=2}^\infty \frac{a_k}{\rho^{k-1}} \left( \frac{1}{1-k} \right)\\
    %
  \varphi_1 &= \rho - \eta\, \ell n\, 2\rho + \sigma_1 - \frac{\pi}{2}
    + \sum_{k=2}^\infty \frac{b_k}{\rho^{k-1}} \left( \frac{1}{1-k} \right)
\end{aligned}
\right\} & % Top-level alignment stop
\Tag{115} \\
%% -----File: 035.png---Folio 30-------
\intertext{and where}
\left.
\begin{gathered}
  a_1 = -\eta, \quad a_2 =  \frac{ -\eta^2}{2} + i\eta \\
  b_1 = -\eta, \quad b_2 = -\frac{2+\eta^2}{2} + i \frac{\eta}{2} \\
  a_k = -\left( \frac{1}{2} \sum_{m=1}^{k-1} a_m\, a_{k-m} \right)
        - i \frac{k-1}{2}\, a_{k-1}
\end{gathered}
\right\} & % Top-level alignment stop
\Tag{116} \\
\intertext{with a similar recurrence formula holding for~$b_k$}
\left.
\begin{aligned}
  \sigma_0 &= \arg \Gamma(1 + i\eta) \\
  \sigma_1 &= \sigma_0 + \tan^{-1}\eta
\end{aligned}
\right\} & % Top-level alignment stop
\Tag{117}
\end{align*}
Furthermore the quantity~$\sigma_0$ may be successfully approximated over the whole
range of~$\eta$ by the following formula:
\begin{align*}
  \sigma_0
  &= -\eta + \left( \frac{\eta}{2} \right) \ln(\eta^2+16)
     + \frac{7}{2} \tan^{-1} \left( \frac{\eta}{4} \right)
     - \left[ \tan^{-1}\eta + \tan^{-1} \left( \frac{\eta}{2} \right)
       + \tan^{-1} \left( \frac{\eta}{3} \right) \right] \\
&
\Tag{118} \\
     %
  &\quad - \frac{\eta}{12(\eta^2 + 16)}
     \left[ 1 + \frac{1}{30}\, \frac{\eta^2 - 48}{(\eta^2 + 16)^2}
       + \frac{1}{105}\, \frac{\eta^4 - 160 \eta^2 + 1280}{(16 + \eta^2)^4}
  \right].
\end{align*}

The above formulas which can of course be generalized for any value of~$\ell$ are
equivalent though not formally identical to the formulas listed by Abramowitz\footnote
  {Tables of Coulomb Wave Functions, Vol.~I, National Bureau of Standards,
  Applied Mathematics Series~17, Washington, 1952, p.~XV.}
and by Fröberg\footnote
  {C.~E.\ Fröberg, Rev.\ Mod.\ Phys. \textbf{27}, 399 (1955).}.

Rather than use these formulas for obtaining $F_\ell$ and~$G_\ell$ for any value of
$\ell > 1$, it is preferable to make use of recurrence formulas.

The following upward recurrence formula is suitable for finding~$G_\ell$:
\[
G_{\ell+1}
= \frac{(2\ell + 1) \left[ \eta + \frac{\ell(\ell + 1)}{\rho} \right] G_\ell
  - (\ell + 1) \left[ \ell^2 + \eta^2 \right]^{1/2} G_{\ell-1}}
       {\ell \left[ (\ell + 1)^2 + \eta^2 \right]^{1/2}}.
\Tag{119}
\]

%% -----File: 036.png---Folio 31-------

A similar recurrence relation can only be used for downward recurrence
on the~$F_\ell$'s, otherwise results rapidly lose all significance. This may be
done by means of a method due to Stegun and Abramowitz\footnote
  {Stegun and Abramowitz, Phys.\ Rev.\ \textbf{98}, 1851 (1955).} and which is essentially
as follows.

Let it be required to compute~$F_\ell$ from $\ell=0$ to $\ell=\ell_\text{max}$.
\begin{itemize}
\item[(1)] Let $\ell^{(1)} = \ell_\text{max} + 10$

(The number~10 is arbitrary but has found satisfactory from practical
experience)

Let $F_{\ell^{(1)}+1}^{(1)} = 0$ and $F_{\ell^{(1)}}^{(1)} =0.1$. Successive values of~$F_\ell^{(1)}$ can be computed
from $\ell=0$ to $\ell=\ell^{(1)}-1$ by means of the downward recurrence formula:
\[
F_{\ell-1}^{(1)}
  = \frac{(2\ell + 1) \left[ \eta + \frac{\ell(\ell+1)}{\rho} \right] F_\ell^{(1)}
    - \ell \left[ (\ell + 1)^2 + \eta^2 \right]^{1/2} F_{\ell+1}^{(1)}}
         {(\ell + 1)\left[\ell^2 + \eta^2\right]^{1/2}}.
\Tag{120}
\]
Letting the constant
\[
\alpha = (F_0^{(1)} G_1 - F_1^{(1)} G_0)(1 + \eta^2)^{1/2}
\Tag{121}
\]
one may compute successively
\[
F_\ell = F_\ell^{(1)}\alpha^{-1}
\Tag{122}
\]
for $\ell = \ell_\text{max} + 1$ to $\ell=0$.

\item[(2)] To verify the accuracy of the~$F_\ell$'s obtained above one may compute as above
a new set of functions $F_\ell^{(2)}$ starting perhaps from $\ell^{(2)} = \ell^{(1)} + 5$ (again the
number~5 is obtained from practical experience) and letting now $F_{\ell^{(2)}+1}^{(2)} = 0$,
$F_{\ell^{(2)}}^{(2)} = 0.1$. This yields a new set of~$F_\ell$'s.

\item[(3)] Comparison of the two sets of~$F_\ell$'s obtained in~(1) and~(2) above indicates
the accuracy of the computation. If this proves insufficient, let $\ell^{(3)} =\ell^{(2)} +5$
and starting from $F_{\ell^{(3)}+1}^{(3)} = 0$, $F_{\ell^{(3)}}^{(3)} = 0.1$ one may obtain a third
set set of~$F_\ell$'s which is to be compared with the second set.

This procedure may be continued until two successive sets of~$F_\ell$'s are
found to agree. The derivatives of the coulomb functions may be obtained
from the formula
\[
Y_\ell^{'}
  = \frac{ \left[ \frac{(\ell+1)^2}{\rho} + \eta \right] Y_\ell
    - \left[ (\ell + 1)^2 + \eta^2 \right]^{1/2} Y_{\ell+1}}{(\ell+1)}
\Tag{123}
\]
%% -----File: 037.png---Folio 32-------
where $Y_\ell$ stands for either $F_\ell$ or~$G_\ell$.
\end{itemize}


\Subsection{6.}{Phase Shifts}

The phase shifts are obtained in the usual fashion by matching
the logarithmic derivatives of the coulomb functions with those of the
numerically integrated functions at a value of~$\rho$ sufficiently large so
that the nuclear potential becomes negligible.

Matching the logarithmic derivative of the nuclear function
$\Psi_\ell = x_\ell + i y_\ell$ with that of its asymptotic form
\[
F_\ell + (G_\ell + i F_\ell) C_\ell
\]
yields
\[
\frac{\Psi_\ell^{'}}{\Psi_\ell}
  = \frac{F_\ell^{'} + (G_\ell^{'} + i F_\ell^{'}) C_\ell}
         {F_\ell     + (G_\ell     + i F_\ell) C_\ell}
\Tag{124}
\]
which lead to
\[
C_\ell^±
  = \frac{\Psi_\ell^± F_\ell^{'} - \Psi_\ell^{±'} F_\ell}
         {\Psi_\ell^{± '} G_\ell - \Psi_\ell^± G_\ell^{'}
           + i (\Psi_\ell^{± '} F_\ell - \Psi_\ell^± F_\ell^{'})}
\Tag{125}
\]
the quantities $C_\ell$ being related to the complex phase shifts through \Eqref{57}.


\Subsection{7.}{Cross Section and Polarization}

The differential elastic scattering cross section~$\sigma(\theta)$ and the
polarization~$P(\theta)$ for an unpolarized incident beam are obtained % [** PP: Typo obtain]
from
\Eqrefs{34} and~\Eqno{35} while the reaction cross section may be obtained % [** PP: Typo obtain]
as follows.
\[
\sigma_R = \frac{N_\abs}{N_\inc}
\Tag{126}
\]
where $N_\abs$ is the absorbed flux, and $N_\inc$ is the incident flux which was
assumed to be~$1$ (see \Eqref{7}). By definition,
\[
N_\abs = - \frac{\hbar}{2i\mu}
  \int\left[ \Psi_\total^\dagger \frac{\partial\Psi_\total}{\partial r}
           - \Psi_\total \frac{\partial\Psi_\total^\dagger}{\partial r} \right]
  r_0^2 \sin\theta\, d\theta\, d\varphi
\Tag{127}
\]
where the integral is taken over the surface of a large sphere of radius
$r = r_0$. Substituting \Eqref{51} for $\Psi_\total$ into \Eqref{127} and
%% -----File: 038.png---Folio 33-------
making use of the orthonormality of the $\Y^{m_j}_{j,\ell,s}$'s and of the relation
\[
\left| a_{1/2} \right|^2 + \left| a_{-1/2} \right|^2 - 1,
\Tag{128}
\]
yields after carrying out the surface integration:
\begin{align*}
\sigma_R = N_\abs
  &= \frac{4\pi}{V} \sum^\infty_{\ell=0} (\ell + 1)
     \left\{ r^2 \left( -\frac{\hbar}{2i\mu} \right)
     \left[ \frac{\Psi_\ell^{+*}}{kr}
       \frac{\partial}{\partial r} \left( \frac{\Psi_\ell^{+}}{kr} \right)
          - \frac{\Psi_\ell^{+}}{kr}
       \frac{\partial}{\partial r} \left( \frac{\Psi_\ell^{+*}}{kr} \right)
     \right] \right\}_{r=r_0} \\
    %
& \Tag{129} \\
  &- \frac{4\pi}{V} \sum^\infty_{\ell=0} \ell
     \left\{ r^2 \left( -\frac{\hbar}{2i\mu} \right)
     \left[ \frac{\Psi_\ell^{-*}}{kr}
       \frac{\partial}{\partial r} \left( \frac{\Psi_\ell^{-}}{kr} \right)
          - \frac{\Psi_\ell^{-}}{kr}
       \frac{\partial}{\partial r} \left( \frac{\Psi_\ell^{-*}}{kr} \right)
     \right] \right\}_{r=r_0}
\end{align*}
Now substituting the asymptotic form~\Eqno{52} for~$\Psi_\ell^±$ and making use of the
Wronskian relations
\[
G_\ell F'_\ell - F_\ell G'_\ell = 1
\Tag{130}
\]
we are led to the following:
{\small
\[
\frac{4\pi}{V} \left\{ r^2 \left( -\frac{\hbar}{2i\mu} \right)
  \left[ \frac{\Psi_\ell^{±*}}{kr}
    \frac{\partial}{\partial r} \left( \frac{\Psi_\ell^±}{kr} \right)
       - \frac{\Psi_\ell^{±}}{kr}
    \frac{\partial}{\partial r} \left( \frac{\Psi_\ell^{±*}}{kr} \right)
  \right] \right\}_{r=r_0}
  = \frac{4\pi}{k^2} \left[ \Imag(C_\ell^±) - |C_\ell^±|^2 \right].
\Tag{131}
\]}% [** PP: End of \small]
Finally, substitution of~\Eqno{131} into~\Eqno{129} yields
\begin{align*}
\sigma = \frac{4\pi}{k^2} \sum^\infty_{\ell=0} \biggl\{
  &(\ell + 1)
  \Bigl[
   \Imag(C_\ell^+) - \bigl(\Imag(C_\ell^+)\bigr)^2
                   - \bigl(\Real(C_\ell^+)\bigr)^2 \Bigr] \\
  &\qquad+ \ell
  \Bigl[
   \Imag(C_\ell^-) - \bigl(\Imag(C_\ell^-)\bigr)^2
                   - \bigl(\Real(C_\ell^-)\bigr)^2 \Bigr] \biggr\}.
\Tag{132}
\end{align*}

\Note{} The quantities $e^{2i\sigma_\ell}$ appearing in \Eqref{60} may be obtained by the
following recurrence formulas:
\[
\begin{aligned}
\Real(e^{2i\sigma_{\ell + 1}})
  &= \cos 2\sigma_{\ell + 1}
   = \left[ \frac{(\ell + 1)^2 - \eta^2}{(\ell + 1)^2 + \eta^2}
       \cos 2\sigma_\ell \right]
   - \left[ \frac{2\eta(\ell + 1)}{(\ell + 1)^2 + \eta^2}
       \sin 2\sigma_\ell \right] \\
%
\Imag(e^{2i\sigma_{\ell + 1}})
  &= \sin 2\sigma_{\ell + 1}
   = \left[ \frac{(\ell + 1)^2 - \eta^2}{(\ell + 1)^2 + \eta^2}
       \sin 2\sigma_\ell \right]
   + \left[ \frac{2\eta(\ell + 1)}{(\ell + 1)^2 + \eta^2}
       \cos 2\sigma_\ell \right]
\end{aligned}
\Tag{133}
\]
while the Legendre polynomials obey the usual relations
%% -----File: 039.png---Folio 34-------
\begin{gather*}
P_0 (\cos\theta) = 1,\quad P_1 (\cos\theta) = \cos\theta \\
%
P_{\ell+1}(\cos\theta)
  = \frac{1}{\ell+1} \left[ (2\ell+1) \cos\theta P_\ell (\cos\theta)
    - \ell P_{\ell-1}(\cos\theta) \right]
\Tag{134} \\
%
P_\ell^{(1)}(\cos\theta)
  = \frac{\ell+1}{\sin\theta} \left[ \cos\theta P_\ell(\cos\theta)
  - P_{\ell+1}(\cos\theta) \right].
\Tag{135}
\end{gather*}
One may also compute the Rutherford scattering cross section:
\[
\sigma_c(\theta) = \left| f_c(\theta) \right|^2.
\Tag{136}
\]


\Subsection{8.}{Chi Square Deviation}

Experimental and theoretical quantities may be compared by means of
the chi square deviation:
\[
\chi_T^2 = \chi_\sigma^2 + \chi_P^2
\Tag{137}
\]
where
\begin{align*}
\chi_\sigma^2 &= \sum_\theta \chi_\sigma^2(\theta)
  = \sum_\theta \left[
      \frac{\sigma^\text{th}(\theta) - \sigma^\text{ex}(\theta)}
           {\Delta\sigma^\text{ex}(\theta)}
  \right]^2
\Tag{138} \\
%
\chi_P^2 &= \sum_\theta \chi_P^2(\theta)
  = \sum_\theta \left[
      \frac{P^\text{th}(\theta) - P^\text{ex}(\theta)}
           {\Delta P^\text{ex}(\theta)}
  \right]^2
\Tag{139}
\end{align*}
where the $\sigma^\text{th}(\theta)$ and~$P^\text{th}(\theta)$ are the theoretically obtained cross sections and
polarizations while $\sigma^\text{ex}(\theta)$, $\Delta\sigma^\text{ex}(\theta)$, $P^\text{ex}(\theta)$,
$\Delta P^\text{ex}(\theta)$ are respectively the
experimentally given cross sections, standard deviations in the cross sections,
polarization and standard deviations in the polarization.

It should be noted that the constants were chosen such that the
differential and reaction cross section will be obtained in units of
$10^{-26}~\text{cm}^2$. The polarizations are of course dimensionless ratios.


\Subsection{9.}{Normalization}

The radial wave functions~$\Psi_\ell^{±}$ and their derivatives obtained from
numerical integration of the radial Schroedinger equation contain an arbitrary
normalization factor,~$1/M_\ell^±$. This factor however does not affect the
cross section and polarization since these are obtained from the phase
%% -----File: 040.png---Folio 35-------
shifts which in turn are obtained from ratios of logarithmic derivatives
(see \Eqref{125}) wherein the~$M_\ell$'s cancel out. If on the other hand the
normalized radial wave functions and their derivatives are required, the
normalization terms may be obtained as follows:

The asymptotic form of~$\Psi_\ell^±$ must obey \Eqref{52} but improper
normalization results in the fact that the calculated wave functions are
actually given by
\[
x_\ell^±(\rho) + iy_\ell^±(\rho)
  = M_\ell^± \left\{ F_\ell(\eta, \rho) +
    C_\ell^± \left[  G_\ell(\eta, \rho)
                 + i F_\ell(\eta, \rho) \right] \right\}
\Tag{140}
\]
Now, for $\rho \leq \rho_\text{max}$ the nuclear potentials are negligible and \Eqref{52}
represents the exact solution; in particular, at $\rho = \rho_\text{max}$, we must have
\[
x_\ell^±(\rho_\text{max}) + iy_\ell^±(\rho_\text{max})
  = M_\ell^± \left\{ F_\ell(\eta, \rho_\text{max})
  + C_\ell^± \left [ G_\ell(\eta, \rho_\text{max})
                 + i F_\ell(\eta, \rho_\text{max}) \right] \right\}
\Tag{141}
\]
whereby
\[
M_\ell^±
  = \frac{x_\ell^±(\rho_\text{max}) + i y_\ell^±(\rho_\text{max})}
         {F_\ell(\eta, \rho_\text{max})
        + C_\ell^± \left[ G_\ell(\eta, \rho_\text{max})
                      + i F_\ell(\eta, \rho_\text{max}) \right]}
\Tag{142}
\]
and the normalized radial wave functions and their derivatives are given by
\[
\left.
\begin{aligned}
\Psi_\ell^±(\rho)
  &= \frac{1}{M_\ell^±}
     \left[ x_\ell^±(\rho) + iy_\ell^±(\rho) \right] \\
% [** PP: Typo d\Psi_\ell(\rho) below?]
\frac{d\Psi_\ell^±(\rho)}{d\rho}
  &= \frac{1}{M_\ell^±}
     \left[ \dot{x}_\ell^±(\rho) + i\dot{y}_\ell^±(\rho) \right]
\end{aligned}
\right\}
\Tag{143}
\]
and the complete normalized wave function is given in \Eqref{51} with
$\Psi_\ell^±$ as above in \Eqref{143}.

\Note{} During the numerical integration the program may renormalize the wave
functions and their derivatives at any value of~$\rho$ for which overflow takes
place by dividing the functions and their derivatives by the largest of these.
This is accompanied by an explicit printout as explained %[** PP: Typo expained]
in the description
of subroutine \Code{RKINT}. Such occasional internal renormalization must of course
be taken into account if correctly normalized functions are required.

%% -----File: 041.png---Folio 36-------


\Chapter{III.}{Program Description}

\Section{A.}{General Description}

\Subsection{1.}{Machine Specifications}

Program \SCAT{4} has been written for an \Acro{IBM~704} with floating point
traps or an \Acro{IBM~709}, with a 32,768~words memory, no drum and a minimum of
two tape units.

The program can probably be modified for a 16K~memory by reducing
the number of~$\theta$'s (up to 75 allowed here) and the number of~$\ell$'s (up to 50
allowed here). A large part of the memory (7500~words) is occupied by the
Legendre polynomials and this may also be reduced by computing the polynomials
as required. Furthermore, the program contains a large number of
printouts % [** PP: Removed hyphen]
which may be abbreviated to save storage space.


\Subsection{2.}{General Program Description}

The program was designed to compute cross sections, polarizations
and chi square deviations at a number of specified points in the space of
the optical model parameters \Code{V}, \Code{W}, \Code{A}, \Code{VS}, \Code{WS}, and if needed~\Code{BG} (\Code{RO}, \Code{RC} and
\Code{RG} are kept fixed), for a given set of input data.

The time to carry out a run for a single set of parameters depends
of course upon the maximum values of~$\ell$ and~$\rho$; for p-Cu at 10~\MeV\ ($\ell_\text{max} = 10$,
$\rho = 0.0625$ $(.0625)\, 0.50$ $(0.25)\,10.0$) % [** PP: Parallel construction?]
a run takes about 45~seconds including
about 15~seconds for maximum output to tape.

The program has been written in the form of subroutines to allow
easy checking and modification. Some of these subroutines are not yet
available, but some provision have been made to include them in the future.
The following subroutines written in \FORTRAN\ are specific (sub)routines of
the program:
\begin{center}
\begin{tabular}{lcp{1in}>{\qquad}lcp{1in}}
  Main routine & -- & \Code{MAIN4}  & & & \\
  Subroutine   & -- & \Code{CTRL4}  & & & \\
  Subroutine   & -- & \Code{INPT4}  & Subroutine & -- & \Code{PGEN4} \\
  Subroutine   & -- & \Code{POT1CH} & Subroutine & -- & \Code{INTCTR}\\
  Subroutine   & -- & \Code{POP1}   & Subroutine & -- & \Code{RKINT} \\
  Subroutine   & -- & \Code{SIGZRO} & Subroutine & -- & \Code{CSUBL} \\
  Subroutine   & -- & \Code{FSUBC}  & Subroutine & -- & \Code{AB}    \\
  Subroutine   & -- & \Code{EXSGML} & Subroutine & -- & \Code{SGSGCP}\\
  Subroutine   & -- & \Code{RHOTB}  & Subroutine & -- & \Code{SIGMAR}\\
  Subroutine   & -- & \Code{COULFN} & Subroutine & -- & \Code{CHISQ} \\
  Subroutine   & -- & \Code{RMXINC} & Subroutine & -- & \Code{OUTPT4}\\
\end{tabular}
\end{center}
%% -----File: 042.png---Folio 37-------
The following subroutines are general utility routines used by the program:
\begin{itemize}
\item[]Subroutine - \Code{SKIP} written in \FORTRAN\
\item[]Subroutine - \Code{LEAVE} written in \FORTRAN\
\item[]Subroutine - \Code{SPILL} written in \Acro{FAP}
\end{itemize}
The following subroutines are used in conjunction with the Load-and-Go
system in use at WDPC (Western Data Processing Center, UCLA). The effect
of using this system is described in \hyperref[ref:III.A.3]{section~III-A-3} below.
\begin{itemize}
\item[]Subroutine - \Code{SAVE}
\item[]Subroutine - \Code{PDUMP}
\item[]Subroutine - \Code{EXIT}
\end{itemize}
\bigskip
The program assumes the presence of the following Fortran elementary function
subroutines:
\begin{center}
\begin{tabular}{lcl}
\Code{LOGF } & -- & (natural logarithm) \\
\Code{SINF } & -- & (sine)              \\
\Code{COSF } & -- & (cosine)            \\
\Code{EXPF } & -- & (exponential)       \\
\Code{SQRTF} & -- & (square root)       \\
\Code{ATANF} & -- & (arc tangent)       \\
\end{tabular}
\end{center}


\Subsection{3.}{Use of the WDPC Load-and-Go System}
\phantomsection\label{ref:III.A.3}% ad hoc

Program \SCAT{4} has been written for the Load-and-Go system in use at
the WDPC, UCLA\@. This \emph{only} affects it as follows:

\Subsubsubsection{(i)}{Special subroutines of the load-and-go system.}

\Subroutine{\Code{SAVE}}

The purpose of this subroutine is to allow the operator to interrupt
the calculation without loss. The program is normally run with Sense Switch~1
off; turning on Sense Switch~1 will cause the program to call \Code{SAVE} after
completing the innermost \Code{DO} loop of subroutine \Code{CTRL4}. \Code{SAVE} then writes
on tape the content of the core memory as well as all other information
required to continue the computation such as the contents of the AC, MQ,
index registers, etc\ldots. % ** Registers

A restart routine will then later reload the core memory, reset
all registers etc\ldots, and return right after the \Code{CALL SAVE} statement.
The following statements up to statement number~66 are then required to
properly position the input data tape as the latter was probably rewound
when the computation was interrupted.

%% -----File: 043.png---Folio 38-------

To eliminate the use of subroutine \Code{SAVE}, remove from subroutine \Code{CTRL4}
all statements from statement number~118 to statement number~66 inclusive.

\Subroutine{\Code{PDUMP}($\alpha$,$\beta$)}

The purpose of this subroutine is to provide a partial core dump of all
quantities between the location of the arguments in the call statement. Subroutine
\Code{PDUMP} is called by subroutine \Code{LEAVE} whenever difficulties such as
overflow or division by zero take place.

To eliminate subroutine \Code{PDUMP}, replace in subroutine \Code{LEAVE} the statement
\Code{CALL} \Code{PDUMP(A,ZZ)} by whatever statements will cause the required core dump.

\Subroutine{\Code{EXIT}}

This subroutine terminates the program.

To eliminate subroutine \Code{EXIT}, replace statement number~151 in subroutine
\Code{INPT4} by whatever statement will be used to terminate the program.


\Subsubsubsection{(ii)}{\Code{END} Statements.}

The usual \FORTRAN\ \Code{END} statements do not appear in the program as
the load-and-go system provides them automatically.


\Subsubsubsection{(iii)}{Input and Output Statements.}

In conjunction with the load-and-go system, the program is input
from tape, while the input data is brought in from tape~7 and all the output
is to % [** PP: sic, no auxiliary verb]
tape~6.

All these particular features can of course be easily modified to
use the program either directly or in conjunction with any other system.


\Subsection{4.}{Error Indications:}

\Subsubsubsection{(i)}{Division by zero.}

Every division which could conceivably have a zero divisor
either because of the range of numbers used or because of an error in the
input data is followed by an \Code{IF DIVIDE CHECK}. Detection of a zero denominator is
then followed by an explicit print out and a \Code{CALL LEAVE} statement which leads
to the next set of input data. In order to be sure that no division by zero
remains undetected, every subroutine which contains an \Code{IF DIVIDE CHECK} statement
also begins with an \Code{IF DIVIDE CHECK} to verify that the trigger is off at the
start of the subroutine; if the divide check trigger is found on at the start,
there is an explicit printout % [** PP: Removed hyphen]
to that effect followed by a \Code{CALL LEAVE} statement.

%% -----File: 044.png---Folio 39-------

\Subsubsubsection{(ii)}{Overflow. Underflow.}
\phantomsection\label{ref:III.A.4.ii}% ad hoc

Overflow and underflow are monitored by subroutine \Code{SPILL} (\Code{JSPILL},
\Code{ISPILL}, $x$, $y$) which needs only be called once by \Code{MAIN4}. When \Code{SPILL} is
called, it replaces the quantities \Code{JSPILL} and % [** PP: Typo AND]
\Code{ISPILL} by zeros. Thereafter,
in case of overflow (underflow) the subroutine replaces the overflowed
(underflowed) quantity with $x$ ($y$) % [** PP: Added space]
and places into \Code{JSPILL} (\Code{ISPILL}) the
address of the command which caused overflow (underflow) to occur for
the first time. Program \SCAT{4} uses $x=y=0$.

Every subroutine in which computations are carried out starts by
setting \Code{ISPILL} and \Code{JSPILL} equal to zero to insure correct identification
of possible subsequent overflow or underflow. The subroutine then ends
with a check of \Code{ISPILL} and \Code{JSPILL}. If either of these is not zero, there
is an explicit printout % [** PP: Removed hyphen]
describing the overflow or underflow. Underflow
results therefore in substituting zero for the underflowed quantity, but
the computation proceeds. Overflow on the other hand results in substituting
zero for the overflowed quantity and leads to a \Code{CALL LEAVE} statement to
stop the computation.


\Section{B.}{Detailed Descriptions of the Specific Routines of the Program}

\Routine{\Code{MAIN4}}

The main routine which is only used at the start of the program carries out
the following steps:

\begin{itemize}
  \item[1)] Calls \Code{SPILL} which controls overflow and underflow (see \hyperref[ref:III.A.4.ii]{III-A-4-ii}).
  One such call statement is sufficient to put \Code{SPILL} in permanent
  control for all subroutines.

  \item[2)] Sets up \Code{EPS1}, \Code{EPS2}, \Code{EPS3}, which are constants used to control the
  accuracy of the Coulomb functions computations, and \Code{EPS4} which is
  used in subroutine \Code{POT1CH}.

  \item[3)] Inputs identification and program numbers.

  \item[4)] Calls \Code{CTRL4}.
\end{itemize}

\Routine{\Code{CTRL4} (Control~4)}

This subroutine controls the whole flow of the program. It was coded as a
subroutine to allow it to be called by subroutine \Code{LEAVE}. It carries out the
following steps:

\begin{itemize}
  \item[1)] Advances group identification and resets run identification numbers.

  \item[2)] Call \Code{INPT4}.

%% -----File: 045.png---Folio 40-------

  \item[3)] Calls \Code{POT1CH}.

  \item[4)]%
   {\setlength{\tabcolsep}{0pt}\begin{tabular}[t]{l}
     If \Code{KTRL(5) = 1}, calls \Code{POP1} \\
     if \Code{KTRL(5) = 0}, proceeds. \\
   \end{tabular}}

  \item[5)] Calls \Code{SIGZRO}, \Code{FSUBC}, \Code{EXSGML}.

  \item[6)] Sets up five (or six) nested \Code{DO} loops for varying successively
  $V$, $W$, $a$, $V_s$, $W_s$ (and $b$ for a surface absorption potential). The
  following steps are always done within the innermost \Code{DO} loop:

  \begin{itemize}
    \item[a)]%
    {\setlength{\tabcolsep}{0pt}\begin{tabular}[t]{l}
      If Sense Switch~1 is on, calls \Code{SAVE} \\
      if Sense Switch~1 is off, proceeds. \\
    \end{tabular}}

    \item[b)] Advances run identification number.

% [** PP: Overfull but not visually jarring]
    \item[c)] Calls \Code{RHOTB}, \Code{COULFN}, \Code{RMXINC}, \Code{PGEN4}, \Code{INTCTR}, \Code{CSUBL}, \Code{AB}, \Code{SGSGCP}, \Code{SIGMAR}.

    \item[d)]%
    {\setlength{\tabcolsep}{0pt}\begin{tabular}[t]{l}
      If \Code{KTRL(2) = 0}, proceeds \\
      if \Code{KTRL(2) = 1}, calls \Code{CHISQ}. \\
    \end{tabular}}

    \item[e)] Calls \Code{OUTPT4}.
  \end{itemize}

\item[7)] When all the \Code{DO} loops have been completed, returns to step~1.
\end{itemize}

\Routine{\Code{INPT4} (Input~4)}

\begin{itemize}
  \item[1)] Inputs \Code{KTRL(1)};
  {\setlength{\tabcolsep}{0pt}\begin{tabular}[t]{l}
    if \Code{KTRL(1) = 100}, calls \Code{EXIT} \\
    if \Code{KTRL(1)} $\neq$ \Code{100}, proceeds. \\
  \end{tabular}}

  \item[2)] Inputs \Code{KTRL(I)}, \Code{I = 2 to 13}.

  \item[3)] Inputs \Code{FMI}, \Code{FMB}, \Code{ELAB}, \Code{ZZ}, \Code{RC}, \Code{V}, \Code{W}, \Code{RO}, \Code{A}, \Code{VS}, \Code{WS}, \Code{RG}, \Code{BG}, \Code{DV}, \Code{DW},
    \Code{DA}, \Code{DVS}, \Code{DWS}, \Code{DBG}, \Code{HA}, \Code{PMA}, \Code{FN1A}, \Code{FN2A}, \Code{HB}, \Code{PMB}, \Code{FN1B}, \Code{FN2B}, \Code{NVMAX},
    \Code{NWMAX}, \Code{NAMAX}, \Code{NVSMAX}, \Code{NWSMAX}, \Code{NBGMAX}.

  \item[4)] Sets up \Code{TV = V} to \Code{TBG = BG} (starting values of the parameters).

  \item[5)] Inputs \Code{NMAX}, forms \Code{NMAXP = NMAX-1}.

  \item[6)] Inputs \Code{RHOIN(I)}, \Code{I = 1 to NMAX} and \Code{DRHOIN(I), I = 1 to NMAXP}.

  \item[7)] Computes \Code{FMU} as per \Eqref{5}  \\
    Computes \Code{ECM} as per \Eqref{6}  \\
    Computes \Code{FKAY} as per \Eqref{8} \\
    Computes \Code{RHOBN} as per \Eqref{73} \\
    Computes \Code{RMA} and \Code{RMB}
      (see \hyperref[chapter:V.]{Glossary}, under \Code{PMA}, \Code{PMB}) \\
    Computes \Code{RHOBC} as per \Eqref{74} \\
    Computes \Code{ETA} as per \Eqref{43}.

  \item[8)] Inputs \Code{LMAXM}, forms \Code{IMAX = LMAXM + 1}.

  \item[9)] Sets \Code{IIN(J) = 1, J = 1 to LMAX} (see description of subroutine \Code{INTCTR})

%% -----File: 046.png---Folio 41-------

  \item[10)]%
  {\setlength{\tabcolsep}{0pt}\begin{tabular}[t]{ll}
    If \Code{KTRL(5) = 0}, & proceeds \\
    if \Code{KTRL(5)} $\neq$ \Code{0}: &
    \begin{tabular}[t]{l}
     a) inputs \Code{JMAX} \\
     b) inputs \Code{THETAD(I), I = 1 to JMAX} \\
     c) computes \Code{THETA(I), I = 1 to JMAX}. \\
    \end{tabular} \\
  \end{tabular}}
  \phantomsection\label{page:41}% ad hoc

  \item[11)]%
  {\setlength{\tabcolsep}{0pt}\begin{tabular}[t]{l}
    If \Code{KTRL(2) = 0} and/or \Code{KTRL(3) = 0}, proceeds, \\
    if \Code{KTRL(2)} $\neq$ \Code{0} and \Code{KTRL(3)} $\neq$ \Code{0}, inputs \\
   \quad\Code{SGMARX(I)}, \Code{DSGMEX(I)}, \Code{POLEX(I)}, \Code{DPOLEX(I)}, \Code{I = 1 to JMAX}. \\
  \end{tabular}}

  \item[12)] Returns to \Code{CTRL4}.
\end{itemize}

\Routine{\Code{POT1CH} (potential~1 check)} % [** PP: Typo POTICH]

The purpose of this subroutine is to check whether $\ell_\text{max}$ is sufficiently
large so that all the partial waves sensibly affected by the potential are
included and to check whether $\rho_\text{max}$ (the point at which the coulomb functions
will be matched to the nuclear wave functions) is sufficiently large to insure
that the non-coulomb part of the potential is negligible. If $\ell_\text{max}$ and/or
$\rho_\text{max}$ are too small, the subroutine increases them, and sets \Code{IIN}($\ell_\text{max}$)\Code{ = 1}.
The quantities $\rho_\text{max}$ and $\ell_\text{max}$ may be checked or not according to the value
assigned to \Code{KTRL(13)}:

\Code{KTRL(13) = 1}: check both $\ell_\text{max}$ and $\rho_\text{max}$

\Code{KTRL(13) = 2}: check $\rho_\text{max}$ only

\Code{KTRL(13) = 3}: check $\ell_\text{max}$ only

\Code{KTRL(13) = 4}: do not check either.
\smallskip

%% [** PP: Marked up lists below semantically to extent possible]
\noindent
$\rho_\text{max}$ and $\ell_\text{max}$ are checked in various ways depending upon the potential form.
The routine operates as follows:
\begin{itemize}
\item[1)] The routine first calculates the maximum values of \Code{V}, \Code{W}, \Code{A}, \Code{VS}, \Code{WS},
  and, in the case of a Gaussian absorption, of~\Code{BG} over the specified grid of
  these parameters.

\item[2)] If \Code{KTRL(1) = 0}, standard potential (or variation thereof), the routine
  checks, if required, that:
  \begin{itemize}
  \item[a)] $\rho_\text{max}$ is sufficiently large so that
  \[
  \frac{(V^2 + W^2)^{1/2}}{E}\,
  \frac{1}{(1 + e^{(\rho_\text{max} - \bar{\rho}_N)/ka})} \leq \epsilon_4.
% [** PP: Change ; to . and start new sentence]
\Tag{144}
  \]
  If this condition is not met, $\rho_\text{max}$ is increased by the last value of
  $\Delta\rho$ and the check is repeated. This is accompanied by the print out: \\
%% -----File: 047.png---Folio 42-------
\qquad\begin{tabular}{l}% [** PP: Reformatted messages here, below]
  \Code{RHOIN(NMAX) = }
    (value of old $\rho_\text{max}$)
    \Code{ + } (last value of \Code{DRHOIN}) \\
  \Code{RHOIN(NMAX) IS TOO SMALL IN NUCLEAR POTENTIAL}. \\
\end{tabular}

  \item[b)] The routine also checks, if required, that $\ell_\text{max}$ is sufficiently large
  so that
  \[
  \frac{\sqrt{V^2 + W^2}}{E}\,
  \frac{1}{(1 + e^{(\ell_\text{max} - \bar{\rho}_N)/ka})} \leq \epsilon_4.
% [** PP: Change ; to . and start new sentence]
\Tag{145}
  \]
  If this condition is not met, $\ell_\text{max}$ is increased by~$1$ and the check is repeated;
  this is accompanied by the following printout: \\
\qquad\begin{tabular}{l}
  \Code{LMAXM = } (value of old \Code{LMAXM}) \Code{ + 1} \\
  \Code{LMAXM TOO SMALL BECAUSE OF CENTRAL POTENTIAL}. \\
\end{tabular}

  The routine then checks that $\ell_\text{max}$ is sufficiently large so that
  \[
  2 k^2 \frac{\sqrt{V_S^2 + W_S^2}}{E}\,
  \frac{1}{(1 + e^{(\ell_\text{max} - \bar{\rho}_N)/ka})} \leq \epsilon_4.
% [** PP: Change ; to . and start new sentence]
\Tag{146}
  \]
  If this condition is not met, $\ell_\text{max}$ is increased by~$1$ and the check is repeated;
  this is accompanied by the following printout: \\
\qquad\begin{tabular}{l}
  \Code{LMAXM = } (value of old \Code{LMAXM}) \Code{ + 1} \\
  \Code{LMAXM TOO SMALL BECAUSE OF SPIN ORBIT POTENTIAL}. \\
\end{tabular}
\end{itemize}

\item[3)] If \Code{KTRL(1) = 1}, Gaussian absorption,
  \begin{itemize}
  \item[a)] The check on $\rho_\text{max}$ is as follows:
  \begin{align*}
  &\frac{V}{E}\,
   \frac{1}{(1 + e^{(\rho_\text{max} - \bar{\rho}_N)/ka})} \leq \epsilon_4;
\Tag{147} \\
  \intertext{and}
  &\frac{W}{E}\,
   e^{-(\rho_\text{max} - \bar{\rho}_G/kb)^2} \leq \epsilon_4.
% [** PP: Change ; to . and start new sentence]
\Tag{148}
  \end{align*}
  If these conditions are not met $\rho_\text{max}$ is increased as before and the checks are
  repeated; this is accompanied by the same printout as above.  %[**varioref]

  \item[b)] The check on $\ell_\text{max}$ is as follows:
  \begin{align*}
  &\frac{V}{E}\,
   \frac{1}{(1 + e^{(\ell_\text{max} - \bar{\rho}_N)/ka})} \leq \epsilon_4;
\Tag{149} \\
  \intertext{and}
  &\frac{W}{E}\,
   e^{-(\ell_\text{max} - \bar{\rho}_G/kb)^2} \leq \epsilon_4
\Tag{150}
  \end{align*}
  and as in \Eqref{146}.

%% -----File: 048.png---Folio 43-------

  If these conditions are not met $\ell_\text{max}$ is increased by~$1$ and the checks
  repeated. The prints-out are given on the previous page. %[**varioref]
\end{itemize}

\item[4)] If \Code{KTRL(1) = 2}, Square well
  \begin{itemize}
    \item[a)] The check on $\rho_\text{max}$ is as follows
    \[
    \rho_\text{max} > \bar{\rho}_N
\Tag{151}
    \]

    \item[b)] The check on $\ell_\text{max}$ is as follows
    \[
    \ell_\text{max} > \bar{\rho}_N + 3.
\Tag{152}
    \]
  \end{itemize}

Failure to meet these conditions leads % [** PP: Typo lead]
to increases in $\rho_\text{max}$ and/or $\ell_\text{max}$
accompanied by the same printouts % [** PP: Removed hyphen]
as given above, after which the checks
are repeated.
\end{itemize}

The program uses \Code{EPS4 = 0.001}. %% [** PP: Changed ; to .]
This quantity is specified in the
\Code{MAIN4} routine.

The checks described above are based on a rough estimate of the phase
shifts using a WKB expression.


\Routine{\Code{POP1}}

Computes \Code{P(L,J)}, \Code{PP(L,J)}, \Code{L = 1 to LMAXP}, \Code{J = 1 to JMAX} as per \Eqrefs{134}
and \Eqno{135} and returns to \Code{CTRL4}.


\Routine{\Code{SIGZRO} (Sigma zero)}

Computes \Code{SIGMA0} and \Code{SIGMA1} as per \Eqrefs{117} and~\Eqno{118} and returns to \Code{CTRL4}.


\Routine{\Code{FSUBC}}

Computes \Code{FCR(J)} and \Code{FCI(J)}, \Code{J = 1 to JMAX} as per \Eqref{47} and returns to \Code{CTRL4}.


\Routine{\Code{EXSGML} (Exponential sigma $\ell$)}

Computes \Code{EXSGMR(J)}, \Code{EXSGMI(J)} for \Code{J = 1 to LMAX} as per \Eqref{133} and returns to \Code{CTRL4}.


\Routine{\Code{RHOTB} (Rho tabulation)}
\phantomsection\label{ref:RHOTB}

The purpose of this subroutine is to construct a table of~$\rho$'s and~$\Delta \rho$'s
corresponding to each step of the numerical integration. This table is
%% -----File: 049.png---Folio 44-------
formed from the arrays of \Code{RHOIN(I)} and \Code{DRHOIN(I)} which are input by subroutine
\Code{INPT4}
\begin{center}
{\footnotesize
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{\textsc{Input Arrays}} \\
\hline
\Code{RHOIN(I)} & \Code{DRHOIN(I)} \\
\hline
\Code{RHOIN(1)} & \Code{DRHOIN(1)} \\
\Code{RHOIN(2)} & \Code{DRHOIN(2)} \\
   .            &  .               \\
   .            &  .               \\
   .            &  .               \\
   .            &  .               \\
\Code{RHOIN(NMAX-1)} & \Code{DRHOIN(NMAX-1)} \\
\Code{RHOIN(NMAX)}\hfill\break &   \\
\hline
\end{tabular}
\hfill
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{\textsc{Computed Tables}} \\
\hline
\Code{RHO(I)} & \Code{DRHO(I)}  \\
\hline
\Code{RHO(1)} & \Code{DRHO(1)}  \\
\Code{RHO(2)} & \Code{DRHO(2)}  \\
   .            &  .            \\
   .            &  .            \\
   .            &  .            \\
   .            &  .            \\
\Code{RHO(ILAST-1)} & \Code{DRHO(ILAST-1)} \\
\Code{RHO(ILAST)}\hfill\break & \\
\hline
\end{tabular}}% [** PP: Removed 2 ditto rows to make tables the same height]
\end{center}
$\rho$\Code{ = RHOIN(1) (DRHOIN(1)) RHOIN(2) } \ldots \Code{ (DRHOIN(NMAX-1)) RHOIN(NMAX)} \\
\Code{RHO(I+1) = RHO(I) + DRHO(I)} \\
\Code{DRHO(1) = DRHO(2) = } $\cdots$ \Code{ = DRHO(I) = DRHOIN(1)} \\
\Code{   up to RHO(I) = RHOIN(2), etc}\ldots. \\
\Code{RHO(1) = RHOIN(1); RHO(ILAST) = RHO(NMAX)} \\
\Code{ILAST }$\geq$\Code{ NMAX}.

If \Code{RHOIN(NMAX)} is given in such a way that it cannot be reached by
an integral number of \Code{DRHO(I)}'s, the last interval is shortened (up to~50\%)
or lengthened (by no more than~50\%) so that \Code{RHO(ILAST) = RHOIN(NMAX)}.


\Routine{\Code{COULFN} (Coulomb functions)}

This is the most complex subroutine of the program. It computes
the regular and irregular coulomb functions and their derivatives for
\Code{L = 1 to LMAXM} at $\rho$\Code{ = RHOMAX} by means of asymptotic formulas. The main
steps are as follows:
\begin{itemize}
\item[1)] The $a$ and~$b$ series appearing in \Eqref{115} are calculated
according to \Eqrefs{116} and are cut off when either:
%% -----File: 050.png---Folio 45-------
\begin{itemize}
\item[(a)] The term $N_a$ (or~$N_b$) is such that the next term exceeds in magnitude
the previous one, i.e., when
\[
[\Real (U_{N_a} + 1)]^2 + [\Imag (U_{N_a} + 1)]^2
\geq [\Real (U_{N_a})]^2 + [\Imag (U_{N_a})]^2
\Tag{153}
\]
where
\[
U_k = \frac{a_k}{(k-1) \rho_\text{max}^{k-1}}
\Tag{154}
\]
and similarly for the~$b$ series.

\item[(b)] The contributions of both the real and imaginary terms give undetectable
contributions to the real and imaginary parts of~$\varphi_0$ (and similarly
for~$\varphi_1$). During these computations, the value of $\rho_\text{max}$ may be increased by
addition of the last value of \Code{DRHOIN} and the computation starts all over again
under the following condition:
\begin{itemize}
\item[a)] The $a$ or~$b$ series is identically equal to zero. This is accompanied
by the printout: \\
\begin{tabular}{l} % [** PP: Re-formatting]
\Code{SERIES IN PHI0 OR PHI1 IS ZERO, CHECK DATA, IF OK} \\
\Code{INCREASE RHOMAX = }
  (value of old \Code{RHOMAX}) \Code{ + } (value of last \Code{DRHOIN}) \\
\end{tabular}

\item[b)] Either of the two series diverges too quickly, i.e., the $N_a$-th
(or $N_b$-th) term still gives a non-negligible contribution to the series
obtained so far, viz.\
\[
\left| \frac{\left[ \Real (U_{N_a}) \right]^2
           + \left[ \Imag (U_{N_a}) \right]^2}
            {\left[ \Real \left(\sum_{k=2}^{N_a-1} U_k\right) \right]^2
           + \left[ \Imag \left(\sum_{k=2}^{N_a-1} U_k\right) \right]^2}
\right| \geq EPS3
\Tag{155}
\]
(\Code{EPS3} is given the value $0.00001$ in the \Code{MAIN4} routine.)
% [** PP: Added period]

This is accompanied by the printout: \\
\begin{tabular}{l}
\Code{IF OK A OR B SERIES DIVERGES TOO QUICKLY} \\
\Code{INCREASE RHOMAX = }
  (value of old \Code{RHOMAX}) \Code{ + }(value of last \Code{DRHOIN}). \\
\end{tabular}

\item[c)] Over 48~terms are required in either the $a$ or~$b$ series. This is
accompanied by the printout: \\
\begin{tabular}{l}
\Code{INCREASE RHOMAX = }
  (value of old \Code{RHOMAX}) \Code{ + } (value of last \Code{DRHOIN}) \\
\Code{A OR B SERIES CONVERGES TOO SLOWLY}. \\
\end{tabular}
\end{itemize}
\end{itemize}

\item[2)] The quantities $\varphi_0$, $\varphi_1$, $F_0$, $F_1$, $G_0$, $G_1$ are formed according to
%% -----File: 051.png---Folio 46-------
\Eqrefs{114} and \Eqno{115}, and the Wronskian is checked for accuracy
requiring that
\[
\left| \mathscr{W} - \left[ 1 + \eta^2 \right]^{-1/2} \right|
= \left| F_0 G_1 - F_1 G_0 - \left[ 1 + \eta^2 \right]^{-1/2} \right|
\leq EPS1
\Tag{156}
\]
(\Code{EPS1} is given the value $0.00001$ in the \Code{MAIN4} routine.)
% [** PP: Added period]

If this condition is violated $\rho_\text{max}$ is increased and the computation starts
all over again; this is accompanied by the following printout: \\
\begin{tabular}{l}
\Code{INCREASE RHOMAX = }
  (old value of \Code{RHOMAX}) \Code{ + } (last value of \Code{DRHOIN}) \\
\Code{BAD INITIAL WRONSKIAN}.
\end{tabular}

\item[3)] The regular coulomb functions are formed by downward recurrence
as per \Eqrefs{120} and \Eqno{122} according to the accompanying description.

Agreement between successive sets of $F_\ell$'s is verified by checking
that
\[
\left| (F_\ell^{(n)} / F_\ell^{(n+1)}) - 1 \right| \leq EPS2
\Tag{157}
\]
(\Code{EPS2} is given the value $0.00001$ in the \Code{MAIN4} routine) for $\ell = 0$ to $\ell_\text{max}$.

% [** PP: Added paragraph break; ambiguous non-indent in original.]
During this computation the value of $\rho_\text{max}$ is increased and the computation
starts all over if it turns out that $\ell_{(1)} > \ell_\text{max} + 40$. This is
accompanied by the printout: \\
\begin{tabular}{l}
\Code{INCREASE RHOMAX = }
  (old value of \Code{RHOMAX}) \Code{ + } (last value of \Code{DRHOIN}) \\
\Code{L TOO LARGE IN FBAR(L)}.
\end{tabular}

\item[4)] The irregular coulomb functions are formed by upward recurrence
as per \Eqref{119} and the Wronskian for every $\ell = 0$ to $\ell_\text{max} + 1$ is
checked for accuracy requiring that
\[
\left| F_\ell G_{\ell + 1} - F_{\ell + 1} G_\ell
  - \frac{\ell + 1}{\left[ (\ell + 1)^2 + \eta^2\right]^{1/2}} \right|
\leq EPS1
\Tag{158}
\]
(\Code{EPS1} is given the value $0.00001$ in the \Code{MAIN4} routine.)
% [** PP: Added period]

If this condition is violated the value of $\rho_\text{max}$ is increased and the computation
starts all over again; this is accompanied by the printout: \\
\begin{tabular}{l}
\Code{INCREASE RHOMAX = }
  (old value of \Code{RHOMAX}) \Code{ + } (last value of \Code{DRHOIN}) \\
\Code{BAD WRONSKIAN FOR L = }
  (value of $\ell+1$ for which \Eqref{158} failed).
\end{tabular}

\item[5)] Finally the derivatives of the coulomb functions for $\ell = 0$ to $\ell_\text{max}$
are formed as per \Eqref{123}.
\end{itemize}

%% -----File: 052.png---Folio 47-------

\Routine{\Code{RMXINC} (Rho max increase)}

The purpose of this subroutine is to extend the table of \Code{RHO(I)} and
\Code{DRHO(I)} by increments of the last value \Code{DRHOIN} until the final value of
\Code{RHO(I)} equals \Code{RHOMAX} which may have been increased by the subroutine \Code{COULFN}.


\Routine{\Code{PGEN4} (Potential generator 4)}

The purpose of this subroutine is to form tables of the $\ell$-independent
parts of the potential corresponding to the \Code{RHO(I)} tables and suitable for
using in the numerical integrations.

These include:

\Code{UCRB(I)}, \Code{UCIB(I)}, \Code{USRB(I)}, \Code{USIB(I)} for \Code{I = 1 to ILAST} and corresponding
to the values at the beginning of an interval of integration; a corresponding
table of form factors is also formed:

\Code{FFCR(I)}, \Code{FFCI(I)}, \Code{FFSR(I)}, \Code{FFSI(I)}, \\
and

\Code{UCRM(I)}, \Code{UCIM(I)}, \Code{USRM(I)}, \Code{USIM(I)}, \\
and

\Code{FFCRM(I)}, \Code{FFCIM(I)}, \Code{FFSRM(I)}, \Code{FFSIM(I)} for \Code{I = 1 to ILAST - 1} corresponding % [** PP: Added space in ILAST -1]
to the values in the middle of an interval of integration.

The original and tightest part of the subroutine corresponds to a
standard form factor; modifications have been added to permit use of a
variety of form factors briefly described earlier.

The subroutine operates as follows: The \Code{UCR-}'s are calculated as per \Eqref{98},
the \Code{UCI-}'s as per \Eqref{99}, the \Code{USR-}'s as per \Eqref{100} and
the \Code{USI-}'s as per \Eqref{101}, wherein:


\Subsubsubsection{(i)}{\Code{KTRL(I) = 0}: Volume absorption or special nuclear form factor:}

\begin{tabular}{rll@{\ }l}
If \Code{KTRL(7) = 0},
 & $f_\CR$ is computed as per \Eqref{80};\quad
 & \Code{[FFCR]}\footnotemark & $= f_\CR$ \\
\Code{ = 1},
 & $f_\CR$ is computed as per \Eqref{86};
 & \Code{[FFCR]} & $= f_\CR$ \\
\Code{ = 2},
 & $f_\CR$ is computed as per \Eqref{87};
 & \Code{[FFCR]} & $= f_\CR$ \\[1ex]
%
If \Code{KTRL(8) = 0},
 & $f_\CI$ is computed as per \Eqref{80};
 & \Code{[FFCI]} & $= f_\CI$ \\
\Code{ = 1},
 & $f_\CI$ is computed as per \Eqref{86};
 & \Code{[FFCI]} & $= f_\CI$ \\
\Code{= 2},
 & $f_\CI$ is computed as per \Eqref{87};
 & \Code{[FFCI]} & $= f_\CI$ \\[1ex]
%
%% -----File: 053.png---Folio 48-------
If \Code{KTRL(9) = 0},
 & $f_\SR$ is computed as per \Eqref{81};
 & \Code{[FFSR]} & $= f_\SR$ \\
\Code{ = 1},
 & $f_\SR$ is computed as per \Eqref{91};
 & \Code{[FFSR]} & $= f_\SR/ka$ \\
\Code{= 2},
 & $f_\SR$ is computed as per \Eqref{94};
 & \Code{[FFSR]} & $= f_\SR/2$ \\[1ex]
%
If \Code{KTRL(10) = 0},
 & $f_\SI$ is computed as per \Eqref{81};
 & \Code{[FFSI]} & $= f_\SI$ \\
\Code{= 1},
 & $f_\SR$ is computed as per \Eqref{91};
 & \Code{[FFSI]} & $= f_\SI/ka$ \\
\Code{= 2},
 & $f_\SR$ is computed as per \Eqref{94};
 & \Code{[FFSI]} & $= f_\SI/2$ \\
\end{tabular}
% [** PP: \footnotemark on previous page]
\footnotetext{\Code{FFCR} refers to the symbolic variables \Code{FFCR(I)} and~\Code{FFCRM(I)} appearing in
the program (see \hyperref[chapter:V.]{glossary of symbols}), similarly for \Code{FFCI}, \Code{FFSR}, and~\Code{FFSI}.}


\Subsubsubsection{(ii)}{\Code{KTRL(1) = 1}: Gaussian absorption}

\begin{tabular}{rll@{\ }l}
$\phantom{If \Code{KTRL(9)} = 0,}$\quad
 & $f_\CR$ is computed as per \Eqref{80};\quad
 & \Code{[FFCR]} & $= f_\CR$ \\
 & $f_\CI$ is computed as per \Eqref{82};
 & \Code{[FFCI]} & $= f_\CI$ \\
 & $f_\SR$ is computed as per \Eqref{81};
 & \Code{[FFSR]} & $= f_\SR$ \\
 & $f_\SI$ is computed as per \Eqref{81};
 & \Code{[FFSI]} & $= f_\SI$ \\
\end{tabular}


\Subsubsubsection{(iii)}{\Code{KTRL(1) = 2}: Square well}

\begin{tabular}{rll@{\ }l}
$\phantom{If \Code{KTRL(9)} = 0,}$\quad
 & $f_\CR$ is computed as per \Eqref{84};
 & \Code{[FFCR]} & $= f_\CR$ \\
 & $f_\CI$ is computed as per \Eqref{84};
 & \Code{[FFCI]} & $= f_\CI$ \\
 & $f_\SR$ and $f_\CI$ are taken to be zero. & & \\
\end{tabular}\\
Furthermore,

If \Code{KTRL(11) = 1}, \Code{USR-} are computed as per \Eqref{100} \emph{including}
the coulomb spin-orbit term.

If \Code{KTRL(11) = 0}, \Code{USR-} are computed as per \Eqref{100} \emph{excluding} the
coulomb spin-orbit term, i.e, the second term on the right hand side. \Code{KTRL(7)}
to~\Code{KTRL(11)} can of course be given any combination of permitted values.


\Routine{\Code{INTCTR} (Integration Control)}

For each value of \Code{L = 1 to LMAX} this subroutine carries out the
following steps:
\begin{itemize}
\item[1)] Sets up starting values for the numerical integration as per \Eqref{113}.
The quantities \Code{IIN(L)} are not especially useful at the present time,
but they have been included in order to permit start of the numerical
integration at various values of~$\rho$ depending on~$\ell$ and thus permitting considerable
time saving by foreshortening the numerical integrations. A
study of this method is presently under way.

\item[2)] Calls \Code{RKINT} which performs the numerical integration.

\item[3)] Stores the final values of the functions and their derivatives at
the completion of each integration.
\end{itemize}

%% -----File: 054.png---Folio 49-------

\Routine{\Code{RKINT} (Runge-Kutta integration)}

This is the most crucial subroutine in the program as most of the
time is spent in numerical integration. Special efforts have therefore
been made to produce a rapid program.

The subroutine integrates numerically as per \Eqrefs{102} to~\Eqno{112}
the differential \Eqrefs{96} operating simultaneously on the two sets
corresponding to $\vec{\sigma}·\vec{\ell} = \ell$ and $-\ell - 1$.

Special provisions have been made to avoid overflow; this is accomplished
by dividing all the functions and their derivatives by the largest
of these at every step (\Code{RENORM}); whenever such renormalization is carried
out it is accompanied by the following printout: \\ % [** PP: Removed hyphen]
\Code{RENORMALIZATION FACTOR = }
  (value of \Code{RENORM}) \Code{IN RKINT FOR CODED} \\
\Code{L = }(value of $\ell + 1$) and \Code{RHO = }(value of $\rho$ at which renormalization took place).


\Routine{\Code{CSUBL}}

This subroutine computes $C_\ell^±$ as per \Eqref{125} for $\ell = 0$ to $\ell_\text{max}$.


\Routine{\Code{AB}}

This subroutine computes \Code{A(J)} and~\Code{B(J)} for \Code{J = 1 to JMAX} i.e., for
the various angles~$\theta$ % [** PP: Typo $\theta$'s]
required, as per \Eqref{60}.


\Routine{\Code{SQSGCP} (Sigma, sigma-coulomb, polarization)}

This subroutine computes $\sigma(\theta)$, $P(\theta)$, $\sigma_c(\theta)$, as per \Eqrefs{34},
\Eqno{35}; \Eqno{136} and finally $\sigma(\theta)/\sigma_c(\theta)$ for the various angles required.


\Routine{\Code{SIGMAR}}

This subroutine computes % [** PP: Typo compute]
$\sigma_R$ as per \Eqref{132}.


\Routine{\Code{CHISQ} (Chi Square)}

This subroutine computes $\chi_\sigma^2(\theta)$, $\chi_\sigma^2$, $\chi_P^2(\theta)$, $\chi_P^2$, $\chi_T^2$ as per \Eqrefs{137},
\Eqno{138} and~\Eqno{139}.

% [** PP: Changed exp to ex twice]
\Note{} The quantities $\Delta\sigma^\text{ex}(\theta)$ and $\Delta P^\text{ex}(\theta)$ are always assumed to be non-zero.
Thus to avoid including an unknown experimental quantity, the corresponding
standard deviation must be taken as very large.

%% -----File: 055.png---Folio 50-------

\Routine{\Code{OUTPT4} (Output 4)}

Several output formats are available:
\begin{itemize} % [** PP: Hard-coding ad hoc levels]
\item[(1)] \textbf{Minimum output} (\Code{KTRL(6) = 1}).
  \begin{itemize}
  \item[(a)] Basic quantities \\
    \Code{NUMPRG} \\
    \Code{KTRL(I)} for \Code{I = 1 to 13} \\
    \Code{FMI}, \Code{FMB}, \Code{ELAB}, \Code{ZZ}, \Code{V},
    \Code{W}, \Code{A}, \Code{RO}, \Code{VS}, \Code{WS}, \Code{RC},
    \Code{BG}, \Code{RG}
    \Code{RHOBN}, \Code{RHOBC}, \Code{RHOBNG}, \Code{ECM},% [** PP: Added space]
    \Code{ETA}, \Code{FKAY}, \Code{FKAYA}, \Code{FKAYB} \\
    and, if either \Code{KTRL(7)}, \Code{(8)}, \Code{(9)}, or
    \Code{(10)} is not zero, \\
    \Code{HA}, \Code{RMA}, \Code{FN1A}, \Code{FN2A}, \Code{PNA},
    \Code{HB}, \Code{RMB}, \Code{FN1B}, \Code{FN2B}, \Code{PMB}, \\
    then \Code{RHOMAX}, \Code{LMAXM}, \Code{NMAX}, \Code{RHOIN(I)
    for I = 1 to NMAX}, \\
    \Code{DRHOIN(I)} for \Code{I = 1 to NMAX-1, SGMRTH} \\
    and, if \Code{KTRL(2) = 1}, \Code{CHI2ST}, \Code{CHI2PT}, \Code{CHI2T}.

  \item[(b)] Basic Table \\
    \Code{THETAD(I)}, \Code{SGMATH(I)}, \Code{SRATIO(I)}, \Code{POLTH(I)}, \\
    and, if \Code{KTRL(2) = 1}, \Code{SGMAEX(I)}, \Code{POLEX(I)},
    for \Code{I = 1 to JMAX}.
  \end{itemize}

\item[(2)] \textbf{Normal output} (\Code{KTRL(6) = 0})
  \begin{itemize}
  \item[(a)] Basic quantities \\
    (See above)

  \item[(b)] Basic Table \\
    (See above)

  \item[(c)] Form factor table (output only if \Code{KTRL(12) = 1}) \\
    \Code{RHO(I)}, \Code{FFCR(I)}, \Code{FFCI(I)}, \Code{FFSR(I)},
    \Code{FFSI(I)}, \\
    for \Code{I = 1 to ILAST}.

  \item[(d)] Fitting table (output only if \Code{KTRL(2)=1}) \\
    \Code{THETAD(I)}, \Code{DSGMEX(I)}, \Code{DPOLEX(I)}, \Code{CHI2S(I)}, \\
    \Code{CHI2P(I)}, \Code{CHI2(I)} for \Code{I = 1 to JMAX}.

  \item[(e)] \Code{L} table \\
    \Code{L}, \Code{CR1(L)}, \Code{CI1(L)}, \Code{CR2(L)}, \Code{CI2(L)} for \Code{L = 1 to LMAXM}
    (corresponding to $\ell$ = 0 to $\ell_\text{max}$).
  \end{itemize}
\end{itemize}

%% -----File: 056.png---Folio 51-------

This output is made for \emph{every} run, and maybe preceded by underflow
descriptions which may be ignored, and by other comments referring to an
increase in $\rho_\text{max}$, $\ell_\text{max}$, renormalization, etc.

Every page of output is headed by the run number on the left and the
page number on the right. The number of lines per page is held to be less
than~50, otherwise the subroutine calls subroutine \Code{SKIP} which starts a new
page.


\Routine{\Code{SKIP}}

This subroutine increases the page number, resets~\Code{K}, the line counter,
and outputs the run and page number. Note that arguments giving the number
of lines, page and run numbers are required.


\Routine{\Code{LEAVE}}

This subroutine is called whenever a run gets into difficulty because
overflow, or division by zero occur. The subroutine calls \Code{PDUMP} to give a
partial core dump.

This subroutine was included so as to allow for various possible
requirements upon overflow and division by zero without having to change
every command where the difficulty might occur.

%% -----File: 057.png---Folio 52-------


\Chapter{IV.}{Description of Input Data}

All data is input from tape~7. The input data tape is prepared from \Acro{IBM}
cards which contain one piece of input data per card in either of the two following
formats:
\begin{center}
\input{./images/057a.eepic}
\end{center}
\iffalse
%     Columns  |            1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10| 11| 12|      13      | 14| 15|                               | 72|
%==============|==============|===|===|===|===|===|===|===|===|===|===|===|==============|===|===|===============================|===|
%Integers      |            x | x | x | x | x |<--------------------------- any Hollerith character -----------------------------|-->|
%Floating nos. | ± | 0 | . | x | x | x | x | x | x | x | x |   | ± | x | x |<-- any Hollerith character ---|-->|
\fi

\Note{} Any floating point format which uses 15~columns or less and is acceptable
to \FORTRAN\ may be used in place of the above.

\begin{itemize}
\item[(1)] The following identification data is input first: \\ % [** PP: Added :]
\begin{tabular}{lcl}
\Code{NUMRUN(1)} & : & month \\
\Code{NUMRUN(2)} & : & day   \\
\Code{NUMRUN(3)} & : & year  \\
\Code{NUMRUN(4)} & : & set number (put in 0 to start with 1) \\
\Code{NUMRUN(5)} & : & run number (put in 0 to start with 1) \\
\Code{NUMPRG}    & : & program number (we use 4).
\end{tabular}

\Note{} The identification which consists of the five quantities \Code{NUMRUN(I)}, \Code{I = 1 to 5},
is printed at the top left of every output sheet. \Code{NUMRUN(4)} is advanced every time
a new set of data is input, \Code{NUMRUN(5)} is advanced every time a run is made with a
new set of parameters.

\item[(2)] Then, for every set of run, i.e., for every set of input data:
\begin{itemize}
\item[(a)] \textbf{Controls} \\ % [ad hoc formatting]
\begin{tabular}{rcl}
\Code{KTRL(1) = 0} & : &
  Standard potential (possibly with generalized form factors) \\
\Code{= 1} & : & Gaussian absorption \\
\Code{= 2} & : & Square well\footnotemark \\
\Code{KTRL(2) = 0} & : & no $\chi^2$ required \\
\Code{= 1} & : & $\chi^2$ required \\
\end{tabular}
\footnotetext{The quantity~$A$ is eventually discarded but it must still be input as $~1/2$ to
avoid overflow in the early part of the program.}%

%% -----File: 058.png---Folio 53-------

\begin{tabular}{rcl}
\Code{KTRL(3) = 0} & : & same experimental values as in last set \\
        \Code{= 1} & : & new experimental values coming\footnotemark \\
\Code{KTRL(4)}$\phantom{=0}$ & : & not used in present program \\
\Code{KTRL(5) = 0} & : & same angles as in last set \\
       \Code{ = 1} & : & new angles coming \\
\Code{KTRL(6) = 0} & : & normal output \\
       \Code{ = 1} & : & minimum output \\
\Code{KTRL(7) = 0} & : & \Code{UCR} -- Standard form \\
       \Code{ = 1} & : & \Code{UCR} -- form~A \\
       \Code{ = 2} & : & \Code{UCR} -- form~B \\
\Code{KTRL(8) = 0} & : & \Code{UCI} -- Standard form \\
       \Code{ = 1} & : & \Code{UCI} -- form~A \\
       \Code{ = 2} & : & \Code{UCI} -- form~B \\
\Code{KTRL(9) = 0} & : & \Code{USR} -- derivative standard form \\
       \Code{ = 1} & : & \Code{USR} -- derivative form~A \\
       \Code{ = 2} & : & \Code{USR} -- form~B \\
\Code{KTRL(10)= 0} & : & \Code{USI} -- derivative standard form \\
       \Code{ = 1} & : & \Code{USI} -- derivative form~A \\
       \Code{ = 2} & : & \Code{USI} -- form~B \\
\Code{KTRL(11)= 0} & : & do not include coulomb spin-orbit \\
       \Code{ = 1} & : & do include coulomb spin-orbit \\
\Code{KTRL(12)= 0} & : & do not print out form factors \\
       \Code{ = 1} & : & do print out form factors \\
\Code{KTRL(13)= 1} & : & check $\rho_\text{max}$ and $\ell_\text{max}$ \\
       \Code{ = 2} & : & check $\rho_\text{max}$ only \\
       \Code{ = 3} & : & check $\ell_\text{max}$ only \\
       \Code{ = 4} & : & do not check $\rho_\text{max}$ nor $\ell_\text{max}$. \\
\end{tabular}
\footnotetext{\Code{KTRL(3) = 1} also requires \Code{KTRL(2) = 1} for proper operation.}

\item[(b)] \textbf{Basic data}
\begin{lstlisting}
FMI, FMB, ELAB, ZZ, RC, V, W, RO, A, VS, WS, RG, BG, DV, DW,
DA, DVS, DWS, DBG, HA, PMA, FN1A, FN2A, HB, PMB, FN1B, FN2B,
NVMAX, NWMAX, NAMAX, NVSMAX, NWSMAX, NBGMAX.
\end{lstlisting}

\item[(c)] \textbf{Integration data} \\
\Code{NMAX}, \Code{RHOIN(I)} for \Code{I = 1 to NMAX},
\Code{DRHOIN(I)} for \Code{I = 1 to NMAX - 1},

%% -----File: 059.png---Folio 54-------

\item[(d)] \Code{LMAXM}

\item[(e)] \textbf{Angles}: \\ % [** PP: Reformat next two items]
if \Code{KTRL(5) = 1} input:
\Code{JMAX}, \Code{THETAD(I)} for \Code{I = 1 to JMAX}

% [** PP: Item (d) in original]
\item[(f)] \textbf{Experimental data}: \\
if \Code{KTRL(2) = 1} and \Code{KTRL(3) = 1} input: \\
\begin{tabular}{rl}
\Code{SGMAEX(I)} & for \Code{I = 1 to JMAX} \\
\Code{DSGMEX(I)} & for \Code{I = 1 to JMAX} \\
 \Code{POLEX(I)} & for \Code{I = 1 to JMAX} \\
\Code{DPOLEX(I)} & for \Code{I = 1 to JMAX} \\
\end{tabular}
\end{itemize}% End of alph labels

\item[(3)] Final card: \\
\Code{KTRL(l) = 100}.
\end{itemize}% End of arabic labels

%% -----File: 060.png---Folio 55-------


\Chapter{V.}{Glossary and Description of Symbolic Variables
Appearing in Common and Dimension Statements}

\begin{center}
\begin{longtable}%
{>{\raggedright}p{1.25in}|>{\raggedright}p{1.75in}|p{2.75in}}
\FORTRAN\ Symbol & \qquad Math.\ Symbol & \qquad Description \\ \hline
\endfirsthead
\FORTRAN\ Symbol & \qquad Math.\ Symbol & \qquad Description \\ \hline
\endhead
\Code{A} & $a$ &
Rounding parameter appearing in standard potential, see \Eqref[abbr]{62} \\
%
\Code{AR(I)}, \Code{AI(I)} \Code{I = 1 to 75} &
$\Real\{a_i\}$, $\Imag\{a_i\}$ &
1) Real and imaginary parts of the
terms of the auxiliary % [** PP: Typo auxilliary]
series used
to calculate asymptotically % [** PP: Typo asympotically]
the
coulomb functions, see \Eqref[abbr]{116} \\
%
& $\Real\{A(\theta_i)\}$, $\Imag\{A(\theta_i)\}$ &
2) See \Eqref[abbr]{60} for definition \\
%
\Code{BR(I)}, \Code{BI(I)} \par\Code{I = 1 to 75} &
$\Real\{b_i\}$, $\Imag\{b_i\}$
$\Real\{B(\theta_i)\}$, $\Imag\{B(\theta_i)\}$ &
1) Ibid, see \Eqref[abbr]{116} \par
2) See \Eqref[abbr]{60} for definition \\
%
\Code{BG} & $b$ & Width parameter in Gaussian absorption
see \Eqref[abbr]{82} \\
%
\Code{CHI2(I)} \Code{I = 1 to 75} &
$\chi^2(\theta_i)$ & $=\chi_\sigma^2(\theta_i) + \chi_P^2(\theta_i)$ \\
%
\Code{CHI2P(I)} \Code{I = 1 to 75} &
$\chi_P^2(\theta_i)$ & See \Eqref[abbr]{139} \\
%
\Code{CHI2PT} & $\chi_P^2$ & See \Eqref[abbr]{139} \\
%
\Code{CHI2S(I)} \Code{I = 1 to 75} &
$\chi_\sigma^2(\theta_i)$ & See \Eqref[abbr]{138} \\
%
\Code{CHI2ST} & $\chi_\sigma^2$ & See \Eqref[abbr]{138} \\
%
\Code{CHI2T} & $\chi^2$ & $=\chi_\sigma^2 + \chi_P^2$ \\
%
\Code{CR1(L)}, \Code{CI1(L)} \Code{for L = 1 to 51} &
$\Real(C_\ell^{+})$, $\Imag(C_\ell^{+})$ &
See \Eqrefs[abbr]{57} and~\Eqno{125} \\
%
\Code{CR2(L)}, \Code{CI2(L)} &
$\Real(C_\ell^{-})$, $\Imag(C_\ell^{-})$ &
See \Eqrefs[abbr]{57} and~\Eqno{125} \\
%
\Code{DA}, \Code{DV}, \Code{DW}, \Code{DVS}, \Code{DWS}, \Code{DBG} & &
% [** PP: Removed trailing comma on prev line]
Amount by which $A$, $V$, $W$, $VS$, $WS$, $BG$
must be incremented for succeeding
runs (these increments may be input
as positive, zero or negative). \\
%
\Code{DPOLEX(I)} \Code{for I = 1 to 75} &
$\Delta P^\text{ex}(\theta_i)$ &
Standard deviation in the experimental
polarization (must \emph{never} be
input as~$0$) \\
%
\Code{DRHO(I)} \Code{for I = 1 to 250} &
$\Delta\rho_i$ & Interval of numerical integration
(see description of \hyperref[ref:RHOTB]{subroutine \Code{RHOTB}}) \\
%% -----File: 061.png---Folio 56-------
\Code{DRHOL} & &
Last interval to be used in the
numerical integration \\
%
\Code{DRHOIN(I)} \Code{I = 1 to 250} & &
Interval of numerical integration
specified by input for \Code{RHOIN(I)} $< \rho
\leq$ \Code{RHOIN(I+1)} (See description of
\hyperref[ref:RHOTB]{subroutine \Code{RHOTB}}) \\
%
\Code{DSGMEX(I)} \Code{I = 1 to 75} &
$\Delta\sigma^\text{ex}(\theta_i)$ &
Standard deviation in the experimental
differential elastic scattering
cross section in square
fermis/sterad, (must \emph{never} be
input as~$0$) \\
%
\Code{ECM} & $E$ & Incident energy in center-of-mass
system (\MeV) \\
%
\Code{ELAB} & $E_\LAB$ & Incident energy in laboratory system
(\MeV) \\
%
\Code{EPS1}, \Code{EPS2}, \Code{EPS3} &
$\epsilon_1$, $\epsilon_2$, $\epsilon_3$ &
Error thresholds appearing in various
parts of the calculation of the coulomb
functions. See \Eqrefs[abbr]{155} to \Eqno{158} \\
%
\Code{EPS4} & $\epsilon_4$ &
Error threshold used in \Code{POT1CH} subroutine,
see \Eqrefs[abbr]{144} to~\Eqno{150} \\
%
\Code{ETA} & $\eta$ & See \Eqref[abbr]{43} \\
%
\Code{ETA2} & $\eta^2$ & \\
%
\Code{EXSGMR(L)}, \Code{EXSGMI(L)} \Code{L = 1 to 51} &
$\Real\{e^{2i\sigma_\ell}\}$, $\Imag\{e^{2i\sigma_\ell}\}$ &
See \Eqref[abbr]{133} \\
%
\Code{F(L)}, \Code{L = 1 to 52} &
$F_\ell$ & See \Eqref[abbr]{114} and~\Eqno{122} \\
%
\Code{FBAR(L)}, \Code{L = 1 to 91} &
$F_\ell^{(n)}$ & See \Eqref[abbr]{120} \\
%
\Code{FCR(I)}, \Code{FCI(I)} \Code{I = 1 to 75} &
$\Real\{f_c(\theta_i)\}$, $\Imag\{f_c(\theta_i)\}$ &
See \Eqref[abbr]{47} \\
%
\Code{FFCR(I)}, \Code{FFCRM(I)} % [** PP: Typos FFCIM and ``, ,'']
\Code{I = 1 to 250} &
$f_\CR(\rho_i)$ \par $f_\CR(\rho_i + \frac{\Delta\rho_i}{2})$ &
Form factors for the real central part
of the potential at the beginning and
middle of an integration interval (See
\Eqrefs[abbr]{80}, \Eqno{84}, \Eqno{86}, \Eqno{87} and description
of subroutine \Code{PGEN4}) \\
%
\Code{FFCI(I)}, \Code{FFCIM(I)} \Code{I = 1 to 250} &
$f_\CI(\rho_i)$ \par $f_\CR(\rho_i + \frac{\Delta\rho_i}{2})$ &
As above for the imaginary central part
of the potential (See \Eqrefs[abbr]{80}, \Eqno{82},
\Eqno{84}, \Eqno{86}, \Eqno{87}, and description of
subroutine \Code{PGEN4}) \\
%% -----File: 062.png---Folio 57-------
\Code{FFSR(I)}, \Code{FFSRM(I)} \Code{I = 1 to 250} &
$f_\SR(\rho_i)$ $f_\SR(\rho_i + \frac{\Delta\rho_i}{2})$ &
As above for the real spin-orbit part
of the potential (See \Eqrefs[abbr]{81}, \Eqno{85},
\Eqno{91}, \Eqno{94} and description of subroutine
\Code{PGEN4}) \\
%
\Code{FFSI(I)}, \Code{FFSIM(I)} \Code{I = 1 to 250} &
$f_\SI(\rho_i)$ $f_\SI(\rho_i + \frac{\Delta\rho_i}{2})$ &
As above for the imaginary spin-orbit
part of the potential (See \Eqrefs[abbr]{81},
\Eqno{85}, \Eqno{91}, \Eqno{94}, and description of
subroutine \Code{PGEN4}) \\
%
\Code{FKAY} & $k$ & See \Eqref[abbr]{8} (inverse fermis) \\
%
\Code{FKAYA} & $ka$ & \\
%
\Code{FKAYB} & $kb$ & \\
%
\Code{FMB}   & $m_b$ &
Mass number of target nucleus (atomic
units) \\
%
\Code{FMI} & $m_i$ &
Mass number of incident particle
(atomic units) \\
%
\Code{FMU} & $\mu$ &
Reduced mass of incident particle
(atomic units (see \Eqref[abbr]{5}) \\
%
\Code{FN1A}, \Code{FN2A} & $nA_1$, $nA_2$ &
See \Eqref[abbr]{86} and following description \\
%
\Code{FN1B}, \Code{FN2B} & $nB_1$, $nB_2$ &
See \Eqref[abbr]{87} and following description \\
%
\Code{FF(L)}, \Code{L = 1 to 51} & $F_\ell'$ & See \Eqref[abbr]{123} \\
%
\Code{G(L)}, \Code{L = 1 to 52}  & $G_\ell$  &
See \Eqref[abbr]{114} and~\Eqno{119} \\
%
\Code{GP(L)}, \Code{L = 1 to 51} & $G_\ell'$ &
See \Eqref[abbr]{123} \\
%
\Code{HA}, \Code{HB} & $h_{0A}$, $h_{0B}$ & See \Eqref[abbr]{88} \\
%
\Code{IDATA} & &
Number of sets of data to be processed
after making use of subroutine \Code{SAVE} \\
%
\Code{IFIRST} & &
Initial value of~\Code{I}, the subscript
appearing in \Code{RHO(I)} \\
%
\Code{ILAST}  & &
Final value of~\Code{I}, the subscript
appearing in \Code{RHO(I)} \\
%
\Code{IIN(L)}, \Code{L = 1 to 51} & &
Originally designed to allow input of
any desired value of \Code{IFIRST} for various
\Code{L}'s in order to speed up the numerical
integration. In the present program
the \Code{IIN(L)} are all set equal to~$1$ by
subroutine \Code{INPT4} \\
%% -----File: 063.png---Folio 58-------
\Code{ISPILL}, \Code{JSPILL} & &
Underflow and overflow indicators used
in conjunction with subroutine \Code{SPILL} \\
%
\Code{JMAX} & &
Total number of angles input (\Code{JMAX }$\le$\Code{ 75}) \\
%
\Code{JMAXT} & &
Temporary storage for \Code{JMAX} used after
calling subroutine \Code{SAVE} \\
%
\Code{KTRL(I)} \Code{I = 1 to 13} & &
Controls used throughout the program
to specify the potential, input and
output type (see description of input
data) \\
%
\Code{KTRLT(I)} \Code{I = 1 to 13} & &
Temporary storage for \Code{KTRL(I)} used
after calling subroutine \Code{SAVE} \\
%
\Code{L}     & $\ell + 1$            & \\
%
\Code{LMAX}  & $\ell_\text{max} + 1$ & \\
%
\Code{LMAXM} & $\ell_\text{max}$     & \\
%
\Code{NA}, \Code{NV}, \Code{NW},
\Code{NVS}, \Code{NWS}, \Code{NBG} & &
\Code{DO} loop variables used in subroutine
\Code{CTRL4} to specify the number of times
the parameters have been incremented \\
%
\Code{NAMAX}, \Code{NVMAX},
\Code{NWMAX}, \Code{NVSMAX},
\Code{NWSMAX}, \Code{NBGMAX} & &
Total number of incrementations of
the parameters specified as input
data (${} \geq 1$) \\
%
\Code{NINPUT} & &
\Code{DO} loop variable used after calling
subroutine \Code{SAVE} in order to count the
number of sets of processed input data \\
%
\Code{NMAX} & &
Total number of input values of \Code{RHOIN(I)}
specified in input \\
%
\Code{NMAXT} & &
Temporary storage for \Code{NMAX} used after
calling subroutine \Code{SAVE} \\
%
\Code{NMAXP} & & \Code{= NMAX - 1} \\
%
\Code{NUMPRG} & &
Program number (see description of input
data)\\
%
\Code{NUMRUN(I)} \Code{I = 1 to 5} & &
Identification (see description of input
data) \\
%
\Code{POLEX(I)} \Code{I = 1 to 75} & $P^\text{ex}(\theta_{i})$ &
Experimental value of the polarization \\
%
\Code{POLTH(I)} \Code{I = 1 to 75} & $P^\text{th}(\theta_{i})$ &
Calculated value of the polarization
See \Eqref[abbr]{35} \\
%% -----File: 064.png---Folio 59-------
\Code{P(L,J)} \Code{L = 1 to 51} \Code{J = 1 to 75} &
$P_{\ell}(\theta_j)$ &
Legendre polynomial, see \Eqref[abbr]{134} \\
%
\Code{PP(L,J)} \Code{L = 1 to 50} \Code{J = 1 to 75} &
$P^{(l)}_{\ell}(\theta_j)$ &
Associated Legendre polynomial, see
\Eqref[abbr]{135} \\
%
\Code{PMA}, \Code{PMB} & % [** PP: Moved next entry into middle column]
$\rho_{m_{A}}/\bar{\rho_{N}}$ and $\rho_{m_{B}}/\bar{\rho_{N}}$ &
These are the
quantities specified
by the input
as they are more
convenient than
\Code{RMA} and \Code{RMB}. \\
%
\Code{RO} & $R_\text{ON}$ &
Nuclear radius constant (fermis), see
\Eqref[abbr]{63} \\
%
\Code{RC} & $R_\text{OC}$ &
Charge radius constant (fermis) see
\Eqref[abbr]{66} \\
%
\Code{RG} & $R_\text{OG}$ &
Gaussian radius constant (fermis) see
\Eqref[abbr]{83} \\
%
\Code{RHOBC} & $\bar{\rho}_{C}$ &
Value of $\rho$ at which the uniform charge
density ends, see \Eqref[abbr]{74} \\
%
\Code{RHOBN} & $\bar{\rho}_{N}$ &
Value of $\rho$ at which the standard
potential falls to half of its initial
value, see \Eqref[abbr]{73} \\
%
\Code{RHOBNG} & $\bar{\rho}_{G}$ &
Value of $\rho$ at which the Gaussian absorption
is centered \\
%
\Code{RHOIN(I)} \Code{I = 1 to 250} & &
Input values of $\rho$ for which the integration
interval must change from
\Code{DRHOIN(I-1)} to \Code{DRHOIN(I)}. See description
of subroutine \Code{RHOTB}) \\
%
\Code{ROMAX} & &
Final value of $\rho$ in the numerical
integration \\
%
\Code{RHO(I)} \Code{I = 1 to 250} & $\rho_{i}$ &
Value of $\rho$ at the $i$-th interval of
integration, see \Eqref[abbr]{14} \\
%
\Code{RMA}, \Code{RMB} & $\rho_{m_{A}}, \rho_{m_{B}}$ &
Values of $\rho$ at which special form
factors are matched to standard form
factors, see \Eqrefs[abbr]{86} and~\Eqno{87} \\
%
\Code{SGMAC(I)} \Code{I = 1 to 75} & $\sigma_{c}(\theta_{i})$ &
See \Eqref[abbr]{136} (square fermis/sterad) \\
%
\Code{SGMAEX(I)} \Code{I = 1 to 75} & $\sigma^\text{ex}(\theta_{i})$ &
Experimental values of the differential
elastic scattering cross section (square
fermis/sterad) \\
%% -----File: 065.png---Folio 60-------
\Code{SGMATH(I)} \Code{I = 1 to 75} & $\sigma^\text{th}(\theta_1)$ &
Calculated values of the differential
elastic scattering cross section
(square fermis/sterad), see \Eqref[abbr]{34} \\
%
\Code{SGMRTH} & $\sigma_R$ &
Calculated value of the reaction cross
section (square fermis) see \Eqref[abbr]{132} \\
%
\Code{SIGMA0} & $\sigma_0$ & See \Eqrefs[abbr]{117} and~\Eqno{118} \\
%
\Code{SIGMA1} & $\sigma_1$ & See \Eqref[abbr]{117} \\
%
\Code{SRATIO(I)} \Code{I = 1 to 75} &
$\sigma(\theta_i) / \sigma_c(\theta_i)$ &
Ratio of calculated to Rutherford cross
section \\
%
\Code{TA}, \Code{TV}, \Code{TW}, \Code{TVS},
\Code{TWS}, \Code{TBG}, & &
Storage for initial values input for
the parameters \\
%
\Code{THETAD(I)} \Code{I = 1 to 75} & $\theta_i$ &
Scattering angle in center-of-mass
system (degrees) \\
%
\Code{THETA(I)} \Code{I = 1 to 75} & $\theta_i$ &
As above (radians) \\
%
\Code{UCRB(I)}, \Code{UCRM(I)} \Code{I = 1 to 250} &
$U_\CR(\rho_i)$  $U_\CR(\rho_i + \frac{\Delta\rho_i}{2})$ &
$L$-independent part of the real central
potential at the beginning and in the
middle of the $i$-th interval of integration,
see \Eqref[abbr]{98} \\
%
\Code{UCIB(I)}, \Code{UCIM(I)} \Code{I = 1 to 250} &
$U_\CI(\rho_i)$ $U_\CI(\rho_i + \frac{\Delta\rho_i}{2})$ &
As above for the imaginary central
potential, see \Eqref[abbr]{99} \\
%
\Code{USRB(I)}, \Code{USRM(I)} \Code{I = 1 to 250} &
$U_\SR(\rho_i)$ $U_\SR(\rho_i + \frac{\Delta\rho_i}{2})$ &
As above for the real spin-orbit
potential, see \Eqref[abbr]{100} \\
%
\Code{USIB(I)}, \Code{USIM(I)} \Code{I = 1 to 250} &
$U_\SI(\rho_i)$  $U_\SI(\rho_i + \frac{\Delta\rho_i}{2})$ &
As above for the imaginary spin-orbit
potential, see \Eqref[abbr]{101} \\
%
\Code{V} & $V$ & Depth of real central potential (\MeV) \\
%
\Code{W} & $W$ & Depth of imaginary central potential (\MeV) \\
%
\Code{VS} & $V_S$ & Real part of spin-orbit potential depth (\MeV) \\
%
\Code{WS} & $W_S$ & Imaginary part of spin-orbit potential depth (\MeV) \\
%% -----File: 066.png---Folio 61-------
\Code{XC1}, \Code{XCP1} &
$x_{\ell}^{+}(\rho)$,  $\dot{x}_{\ell}^{+}(\rho)$ &
Real part of the radial (unnormalized)
wave function and its first derivative
for the case $L + 1/2$ \\
%
\Code{YC1}, \Code{YCP1} &
$y_{\ell}^{+}(\rho)$,  $\dot{y}_{\ell}^{+}(\rho)$ &
As above for the imaginary part and the case $L + 1/2$ \\
%
\Code{XD1}, \Code{XDP1} &
$x_{\ell}^{-}(\rho)$,  $\dot{x}_{\ell}^{-}(\rho)$ &
As above for the real part and the case $L - 1/2$ \\
%
\Code{YD1}, \Code{YDP1} &
$y_{\ell}^{-}(\rho)$,  $\dot{y}_{\ell}^{-}(\rho)$ &
As above for the imaginary part and the case $L - 1/2$ \\
%
\Code{X1(L)}, \Code{X1P(L)} \Code{L = 1 to 51} &
$x_{\ell}^{+}(\rho_\text{max})$, $\dot{x}_{\ell}^{+}(\rho_\text{max})$ &
Real part of the radial (unnormalized)
wave function and its first derivative
for the case $L + 1/2$ at the end of a
numerical integration \\
%
\Code{Y1(L)}, \Code{Y1P(L)} \Code{L = 1 to 51} &
$y_{\ell}^{+}(\rho_\text{max})$, $\dot{y}_{\ell}^{+}(\rho_\text{max})$ &
As above for the imaginary part and the case $L + 1/2$ \\
%
\Code{X2(L)}, \Code{X2P(L)} \Code{L = 1 to 51} &
$x_{\ell}^{-}(\rho_\text{max})$, $\dot{x}_{\ell}^{-}(\rho_\text{max})$ &
As above for the real part and the case $L - 1/2$ \\
%
\Code{Y2(L)}, \Code{Y2P(L)} \Code{L = 1 to 51} &
$y_{\ell}^{-}(\rho_\text{max})$, $\dot{y}_{\ell}^{-}(\rho_\text{max})$ &
As above for the imaginary part and the case $L - 1/2$ \\
%
\Code{ZZ} & $ZZ'$ &
Product of the atomic numbers of the target
nucleus and the incident particle. \\
\end{longtable}
\end{center}

%% -----File: 067.png---Folio 62-------


\Chapter{VI.}{Symbolic Listing of the Program}

\begin{lstlisting}
      MAIN ROUTINE - SCAT 4
      COMMON A,AR,AI,
     1BR,BI,BG,
     2CHI2,CHI2P,CHI2PT,CHI2S,CHI2ST,CHI2T,CR1,CI1,CR2,CI2,
     3DPOLEX,DSGMEX,DRHO,DRHOIN,DRHOL,DV,DW,DA,DVS,DWS,DBG,
     4ECM,ELAB,EPS1,EPS2,EPS3,EPS4,ETA,ETA2,EXSGMR,EXSGMI,
     5F,FBAR,FCR,FCI,FFCR,FFCI,FFCRM,FFCIM,FFSR,FFSI,FFSRM,FFSIM,
     6FKAY,FMB,FMI,FMU,FN1A,FN2A,FN1B,FN2B,FP,FKAYA,FKAYB,
     7G,GP,
     8HA,HB,
     9IDATA,IFIRST,IIN,ILAST,ISPILL
      COMMON JMAX,JMAXT,JSPILL,
     1KTRL,KTRLT,
     2L,LMAX,LMAXM,
     3NMAX,NMAXP,NMAXT,NINPUT,NUMRUN,NUMPRG,NVMAX,NWMAX,NAMAX,NVSMAX,
     4NWSMAX,NV,NW,NA,NVS,NWS,NBGMAX,NBG,
     5P,PP,POLEX,POLTH,PMA,PMB,
     6RC,RO,RHO,RHOBC,RHOBN,RHOIN,RHOMAX,RMA,RMB,RG,RHOBNG,
     7SGMAC,SGMAEX,SGMATH,SGMRTH,SIGMA0,SIGMA1,SRATIO,
     8THETA,THETAD,TV,TW,TA,TVS,TWS,TBG,
     9UCRB,UCIB,UCRM,UCIM,USRB,USIB,USRM,USIM
      COMMON V,VS,
     1W,WS,
     2X1,X2,X1P,X2P,XC1,XCP1,XD1,XDP1,
     3Y1,Y2,Y1P,Y2P,YC1,YCP1,YD1,YDP1,
     4ZZ
      DIMENSION AR(75),AI(75),
     1BR(75),BI(75),
     2CHI2(75),CHI2P(75),CHI2S(75),CR1(51),CI1(51),CR2(51),CI2(51),
     3DPOLEX(75),DSGMEX(75),DRHO(250),DRHOIN(250),
     4EXSGMR(51),EXSGMI(51),
     5F(52),FBAR(91),FCR(75),FCI(75),FFCR(250),FFCI(250),FFCRM(250),
     6FFCIM(250),FFSR(250),FFSI(250),FFSRM(250),FFSIM(250),FP(51),
     7G(52),GP(51),
     8IIN(51),
     9KTRL(13),KTRLT(13)
      DIMENSION NUMRUN(5),
     1P(51,75),PP(50,75),POLEX(75),POLTH(75),
     2RHO(250),RHOIN(250),
     3SGMAC(75),SGMAEX(75),SGMATH(75),SRATIO(75),
     4THETA(75),THETAD(75),
     5UCRB(250),UCIB(250),UCRM(250),UCIM(250),USRB(250),USIB(250),
     6USRM(250),USIM(250),
     7X1(51),X2(51),X1P(51),X2P(51),
     8Y1(51),Y2(51),Y1P(51),Y2P(51)
      CALL SPILL(JSPILL,ISPILL,0.,0.)
      EPS1= 0.00001
      EPS2= 0.00001
      EPS3= 0.00001
      EPS4=0.001
\end{lstlisting}
%% -----File: 068.png---Folio 63-------
\begin{lstlisting}
      READ INPUT TAPE 7,10,(NUMRUN(I)),I=1,5)
      READ INPUT TAPE 7,10,NUMPRG
  10  FORMAT(I5)
      CALL CTRL4
      GO
\end{lstlisting}
\clearpage

%% -----File: 069.png---Folio 64-------

\begin{lstlisting}
      SUBROUTINE CTRL4
  3   NUMRUN(4)=NUMRUN(4)+1
      NUMRUN(5)=0
      CALL INPT4
      CALL POT1CH
 35   IF(KTRL(5)) 80,81,80
 80   CALL POP1
 81   CALL SIGZRO
      CALL FSUBC
      CALL EXSGML
      DO 20 NV=1,NVMAX
      IF (NV-1) 102,101,102
  101 V=TV
      GO TO 103
  102 V=V+DV
  103 DO 20 NW=1,NWMAX
      IF (NW-1) 105,104,105
  104 W=TW
      GO TO 109
  105 W=W+DW
  109 DO 20 NA=1,NAMAX
      IF (NA-1) 111,110,111
  110 A=TA
      GO TO 112
  111 A=A+DA
  112 DO 20 NVS=1,NVSMAX
      IF (NVS-1) 114,113,114
  113 VS=TVS
      GO TO 115
  114 VS=VS+DVS
  115 DO 20 NWS=1,NWSMAX
      IF (NWS-1) 117,116,117
  116 WS=TWS
      GO TO 118
  117 WS=WS+DWS
 118  DO 20 NBG=1,NBGMAX
      IF(NBG-1) 120,119,120
 119  BG=TBG
      GO TO 121
 120  BG=BG+DBG
 121  IF (SENSE SWITCH 1) 26,27
  26  REWIND 7
      CALL SAVE(8)
      READ INPUT TAPE 7,50,(LGAR,I=1,6)
      IDATA= NUMRUN(4)
      DO 66 NINPUT=1, IDATA
      READ INPUT TAPE 7,50,(KTRLT(I),I=1,13)
 50   FORMAT (I5)
 51   FORMAT (E15.9)
      READ INPUT TAPE 7,51,(GAR,I=1,27)
\end{lstlisting}
%% -----File: 070.png---Folio 65-------
\begin{lstlisting}
      READ INPUT TAPE 7,50,(LGAR,I=1,6), NMAXT
      NT=2*NMAXT-1
      READ INPUT TAPE 7,51,(GAR,I=1,NT)
      READ INPUT TAPE 7,51,LGAR
      IF (KTRLT(5)) 71,70,71
 71   READ INPUT TAPE 7,50, JMAXT
      READ INPUT TAPE 7,51,(GAR,I=1,JMAXT)
 70   IF (KTRLT(2)) 61,66,61
 61   IF(KTRLT(3)) 63,66,63
  63  NT=4*JMAXT
      READ INPUT TAPE 7,51,(GAR,I=1,NT)
 66   CONTINUE
 27   NUMRUN(5)= NUMRUN(5)+1
      CALL RHOTB
      CALL COULFN
      CALL RMXINC
      CALL PGEN4
      CALL INTCTR
      CALL CSUBL
      CALL AB
      CALL SGSGCP
      CALL SIGMAR
      IF (KTRL(2)) 33,100,33
  33  CALL CHISQ
  100 CALL OUTPT4
  20  CONTINUE
      GO TO 3
\end{lstlisting}
\clearpage

%% -----File: 071.png---Folio 66-------

\begin{lstlisting}
      SUBROUTINE INPT4
      IF DIVIDE CHECK 100,110
 100  WRITE OUTPUT TAPE 6,101
 101  FORMAT(59H DIVIDE CHECK TRIGGER FOUND ON AT START OF INPT4 SUBROUT
     1INE)
      CALL LEAVE
      STOP
 110  ISPILL=0
      JSPILL=0
      READ INPUT TAPE 7,10,KTRL(1)
      IF (KTRL(1)-100) 150,151,151
  151 CALL EXIT
      STOP
  150 READ INPUT TAPE 7,10,(KTRL(I),I=2,13)
 10   FORMAT (I5)
      READ INPUT TAPE 7,12,FMI,FMB,ELAB,ZZ,RC,V,W,RO,A,VS,WS,RG,BG,
     1DV,DW,DA,DVS,DWS,DBG
      READ INPUT TAPE 7,12,HA,PMA,FN1A,FN2A,HB,PMB,FN1B,FN2B
      READ INPUT TAPE 7,10,NVMAX,NWMAX,NAMAX,NVSMAX,NWSMAX,NBGMAX
 12   FORMAT (E15.9)
      TV= V
      TW=W
      TA=A
      TVS=VS
      TWS=WS
      TBG=BG
      READ INPUT TAPE 7,10,NMAX
      NMAXP=NMAX-1
      READ INPUT TAPE 7,12,(RHOIN(I),I=1,NMAX),(DRHOIN(I),I=1,NMAXP)
      CO2=FMI+FMB
      FMU=(FMI*FMB)/CO2
      ECM=ELAB*(FMB/CO2)
      FKAY= .2195376*SQRTF(FMU*ECM)
      T=FKAY*(FMB**.333333333)
      RHOBN= T*RO
      RHOBNG=T*RG
      RMA=PMA*RHOBN
      RMB=PMB*RHOBN
      RHOBC= T*RC
      ETA= .15805086*ZZ*SQRTF(FMI/ELAB)
      IF DIVIDE CHECK 200,47
 200  WRITE OUTPUT TAPE 6,201
 201  FORMAT(43H INPUT DIVISOR WAS ZERO IN INPT4 SUBROUTINE)
      CALL LEAVE
      STOP
 47   READ INPUT TAPE 7,10,LMAXM
      LMAX=LMAXM+1
      DO 147 J=1,LMAX
 147  IIN(J)=1
      IF (KTRL(5)) 48,50,48
\end{lstlisting}
%% -----File: 072.png---Folio 67-------
\begin{lstlisting}
 48   READ INPUT TAPE 7,10,JMAX
      READ INPUT TAPE 7,12,(THETAD(I),I=1,JMAX)
      DO 49 I=1,JMAX
 49   THETA(I)= 0.01745329252*THETAD(I)
 50   IF(KTRL(2)) 51,207,51
 51   IF(KTRL(3)) 53,207,53
 53   READ INPUT TAPE 7,12,(SGMAEX(I),I=1,JMAX),(DSGMEX(I),I=1,JMAX),
     1(POLEX(I),I=1,JMAX),(DPOLEX(I),I=1,JMAX)
 207  IF(ISPILL)202,204,202
 202  WRITE OUTPUT TAPE 6,203,ISPILL
  203 FORMAT(23H UNDERFLOW OCCURRED AT I5,20H IN INPT4 SUBROUTINE)
 204  IF(JSPILL)205,210,205
 205  WRITE OUTPUT TAPE 6,206,JSPILL
  206 FORMAT(22H OVERFLOW OCCURRED AT I5,20H IN INPT4 SUBROUTINE)
      CALL LEAVE
      STOP
 210  RETURN
\end{lstlisting}
\clearpage

%% -----File: 073.png---Folio 68-------

\begin{lstlisting}
      SUBROUTINE POT1CH
      IF DIVIDE CHECK 30,31
  30  WRITE OUTPUT TAPE 6,130
  130 FORMAT (60H DIVIDE CHECK TRIGGER FOUND ON AT START OF POT1CH SUBRO
     1UTINE)
      CALL LEAVE
      STOP
  31  ISPILL=0
      JSPILL=0
      IKTRL=KTRL(13)
      NMAX=NMAX
      NMAXP= NMAX-1
      AMAX=NAMAX-1
      TTA=MAX1F(A,((AMAX*DA)+A))
      VMAX=NVMAX-1
      TTV=MAX1F(V,((VMAX*DV)+V))
      WMAX=NWMAX-1
      TTW=MAX1F(W,((WMAX*DW)+W))
      VSWAX=NVSMAX-1
      TTVS=MAX1F(VS,((VSMAX*DVS)+VS))
      WSMAX=NWSMAX-1
      TTWS=MAX1F(WS,((WSMAX*DWS)+WS))
      BGMAX=NBGMAX-1
      TTBG=MAX1F(BG,((BGMAX*DBG)+BG))
      FKAYA=FKAY*TTA
      FKAYB=FKAY*TTBG
      T2=SQRTF(TTV**2+TTW**2)/ECM
      T7=TTV/ECM
      T8=TTW/ECM
      IF DIVIDE CHECK 60,61
 60   WRITE OUTPUT TAPE 6,160
 160  FORMAT(26H ECM IS ZERO IN POT1CH SUB)
      CALL LEAVE
      STOP
 61   GO TO (3,3,111,15),IKTRL
 3    IF(KTRL(1)-2) 24,25,24
 25   IF(RHOIN(NMAX)-RHOBN) 10,10,8
 24   T1=1./(1.+EXPF((RHOIN(NMAX)-RHOBN)/FKAYA))
      IF DIVIDE CHECK 50,28
 50   WRITE OUTPUT TAPE 6,150
 150  FORMAT(28H FKAYA IS ZERO IN POT1CH SUB)
      CALL LEAVE
      STOP
 28   IF(KTRL(1)-1) 40,41,40
 40   T3= T2*T1
      GO TO 43
 41   T3=T7*T1
 43   IF(T3-EPS4) 42,42,10
 10   WRITE OUTPUT TAPE 6,100, RHOIN(NMAX),DRHOIN(NMAXP)
 100  FORMAT(13H RHOIN(NMAX)=E16.9,2H+ E16.9,46H RHOIN(NMAX) IS TOO SMAL
\end{lstlisting}
%% -----File: 074.png---Folio 69-------
\begin{lstlisting}
     1L IN NUCLEAR POTENTIAL)
      RHOIN(NMAX)= RHOIN(NMAX)+DRHOIN(NMAXP)
      GO TO 3
 42   IF(KTRL(1)-1) 8,6,8
 6    T11= EXPF(-((RHOIN(NMAX)-RHOBNG)/FKAYB)**2)
      IF((T8*T11)-EPS4) 8,8,7
 7    WRITE OUTPUT TAPE 6,103,RHOIN(NMAX),DRHOIN(MMAXP)
 103  FORMAT(13H RHOIN(NMAX)=E16.9,2H+ E16.9,46H RHOIN(NMAX) IS TOO SMAL
     1L IN NUCLEAR POTENTIAL)
      RHOIN(NMAX)= RHOIN(NMAX)+DRHOIN(NMAXP)
      GO TO 6
 8    GO TO(111,15),IKTRL
 111  FLMAX=LMAXM
      IF(KTRL(1)-2) 29,300,29
 300  IF(FLMAX-(RHOBN+3.)) 12,12,15
 29   T4=1./(1.+EXPF((FLMAX-RHOBN)/FKAYA))
      IF(KTRL(1)-1) 33,32,33
 33   T5= T2*T4
      GO TO 310
 32   T5=T7*T4
 310  IF(T5-EPS4)13,13,12
   12  WRITE OUTPUT TAPE 6,101,LMAXM
  101 FORMAT (7H LMAXM=I5,3H +1,45H LMAXM TOO SMALL BECAUSE OF CENTRAL P
     1OTENTIAL)
      LMAX= LMAX+1
      LMAXM= LMAXM+1
      IIN(LMAX)=1
      GO TO 111
 13   IF(KTRL(1)-1) 17,19,17
 19   T4=EXPF(-((FLMAX-RHOBNG)/FKAYB)**2)
      IF((T8*T4)-EPS4) 17,17,20
 20   WRITE OUTPUT TAPE 6,200,LMAXM
 200  FORMAT (7H LMAXM=I5,3H +1,45H LMAXM TOO SMALL BECAUSE OF CENTRAL P
     1OTENTIAL)
      LMAX=LMAX+1
      LMAXM=LMAXM+1
      IIN(LMAX)=1
      GO TO 19
 17   T2=SQRTF(TTVS**2+TTWS**2)/ECM
 18   FLMAX=LMAXM
      T4=1./(1.+EXPF((FLMAX-RHOBN)/FKAYA))
 38   T6=2.*T2*T4*(FKAYW**2)
      IF(T6-EPS4) 15,15,14
  14  WRITE OUTPUT TAPE 6,102, LMAXM
  102  FORMAT (7H LMAXM=I5,3H +1,48H LMAXM TOO SMALL BECAUSE OF SPIN ORB
     1IT POTENTIAL)
      LMAX= LMAX+1
      LMAXM= LMAXM+1
      IIN(LMAX)=1
      GO TO 18
\end{lstlisting}
%% -----File: 075.png---Folio 70-------
\begin{lstlisting}
 15   IF(ISPILL)202,204,202
 202  WRITE OUTPUT TAPE 6,203,ISPILL
  203 FORMAT(23H UNDERFLOW OCCURRED AT I5,14H IN POT1CH SUB)
 204  IF(JSPILL)205,210,205
 205  WRITE OUTPUT TAPE 6,206,JSPILL
  206 FORMAT(22H OVERFLOW OCCURRED AT I5,14H IN POT1CH SUB)
      CALL LEAVE
      STOP
 210  RETURN
\end{lstlisting}
\clearpage

%% -----File: 076.png---Folio 71-------

\begin{lstlisting}
      SUBROUTINE POP1
      IF DIVIDE CHECK 1,2
  1   WRITE OUTPUT TAPE 6,101
 101  FORMAT (58H DIVIDE CHECK TRIGGER FOUND ON AT START OF POP1 SUBROUT
     1INE)
      CALL LEAVE
      STOP
  2   ISPILL=0
      JSPILL=0
      LMAXP=LWAX+1
      DO 20 J=1,JMAX
      SI2=1./SINF(THETA(J))
      IF DIVIDE CHECK 3,4
  3   WRITE OUTPUT TAPE 6,103, J
 103  FORMAT (71H DIVISOR SINF THETA IS ZERO IN FIRST DIVISION OF POP1 S
     1UBROUTINE FOR J=I3)
      CALL LEAVE
      STOP
  4   CO=COSF(THETA(J))
      P(1,J)=1.0
      P(2,J)=CO
      PP(1,J)=0.0
      TWOLP1=3.
      FL=1.
      DO 20 L=1,LMAXP
      TL=FL+1.
      P(L+2,J)=(TWOLP1*CO*P(L+1,J)-FL*P(L,J))/TL
      PP(L+1,J)=TL*SI2*(CO*P(L+1,J)-P(L+2,J))
      TWOLP1=TWOLP1+2.
  20  FL=TL
      IF (ISPILL) 30,31,30
  30  WRITE OUTPUT TAPE 6,130, ISPILL
  130 FORMAT(23H UNDERFLOW OCCURRED AT I6,19H IN POP1 SUBROUTINE)
  31  IF (JSPILL) 32,33,32
  32  WRITE OUTPUT TAPE 6,132, JSPILL
  132 FORMAT (22H OVERFLOW OCCURRED AT I6,19H IN POP1 SUBROUTINE)
      CALL LEAVE
      STOP
  33  RETURN
\end{lstlisting}
\clearpage

%% -----File: 077.png---Folio 72-------

\begin{lstlisting}
      SUBROUTINE SIGZRO
      IF DIVIDE CHECK 5,6
  5   WRITE OUTPUT TAPE 6,105
  105 FORMAT (60H DIVIDE CHECK TRIGGER FOUND ON AT START OF SIGZRO SUBRO
     1UTINE)
      CALL LEAVE
      STOP
  6   ISPILL = 0
      JSPILL = 0
      SIGMA0=-(ETA/(12.*(ETA**2+16.)))*(1.+(ETA**2-48.)/(30.*((ETA**2+16
     1.)**2))+(ETA**4-160.*(ETA**2)+1280.)/(((16.+ETA**2)**4)*105.))
      SIGMA0=SIGMA0-ETA+(ETA/2.)*LOGF(ETA**2+16.)+((7./2.)*ATANF(ETA/4.)
     1)-(ATANF(ETA)+ATANF(ETA/2.)+ATANF(ETA/3.))
      SIGMA1=SIGMA0+ATANF(ETA)
  15  IF (ISPILL) 30,31,30
   30 WRITE OUTPUT TAPE 6,130,ISPILL
  130 FORMAT (23H UNDERFLOW OCCURRED AT I6,21H IN SIGZRO SUBROUTINE)
  31  IF (JSPILL) 32,11,32
  32  WRITE OUTPUT TAPE 6,132,JSPILL
  132 FORMAT (22H OVERFLOW OCCURRED AT I6,21H IN SIGZRO SUBROUTINE)
      CALL LEAVE
      STOP
  11  RETURN
\end{lstlisting}
\clearpage

%% -----File: 078.png---Folio 73-------

\begin{lstlisting}
      SUBROUTINE FSUBC
      IF DIVIDE CHECK 20,21
  20  WRITE OUTPUT TAPE 6,120
 120  FORMAT (53H DIVIDE TRIGGER FOUND ON AT START OF FSUBC SUBROUTINE)
      CALL LEAVE
      STOP
  21  ISPILL=0
      JSPILL=0
      DO 10 J=1,JMAX
      SN=(SINF(THETA(J)/2.0))**2
      FLN=ETA*(LOGF(SN))-2.0*SIGMA0
      FNO=ETA/(2.0*FKAY*(SN))
      IF DIVIDE CHECK 22,23
  22  WRITE OUTPUT TAPE 6,122,J
 122  FORMAT (23H DIVISOR IS ZERO FOR J=I3,20H IN FSUBC SUBROUTINE)
      CALL LEAVE
      STOP
  23  FCR(J)=(-FNO*COSF(FLN))
  10  FCI(J)=(FNO*SINF(FLN))
      IF (ISPILL) 24,25,24
  24  WRITE OUTPUT TAPE 6,124, ISPILL
  124 FORMAT (23H UNDERFLOW OCCURRED AT I6,20H IN FSUBC SUBROUTINE)
  25  IF (JSPILL) 26,27,26
  26  WRITE OUTPUT TAPE 6,126, JSPILL
  126 FORMAT (22H OVERFLOW OCCURRED AT I6,20H IN FSUBC SUBROUTINE)
      CALL LEAVE
      STOP
  27  RETURN
\end{lstlisting}
\clearpage

%% -----File: 079.png---Folio 74-------

\begin{lstlisting}
      SUBROUTINE EXSGML
      IF DIVIDE CHECK 10,11
  10  WRITE OUTPUT TAPE 6,110
  110 FORMAT (60H DIVIDE CHECK TRIGGER FOUND ON AT START OF EXSGML SUBRO
     1UTINE)
      CALL LEAVE
      STOP
  11  ISPILL=0
      JSPILL=0
  1   FL=O.
      EXSGMR(1)=COSF(2.0*SIGMA0)
      EXSGMI(1)=SINF(2.0*SIGMA0)
      ETA2=ETA**2
      ETA2A=2.0*ETA
      DO 20 L=2,LMAX
      FL=FL+1.0
      TER0=FL**2
      TER1=TER0+ETA2
      TER2=(TER0-ETA2)/TER1
      TER3=(ETA2A*FL)/TER1
      IF DIVIDE CHECK 12,13
  12  WRITE OUTPUT TAPE 6,112,L
 112  FORMAT (44H DIVISOR IS ZERO IN EXSGML SUBROUTINE FOR L=I3)
      CALL LEAVE
      STOP
  13  EXSGMR(L)=(TER2*EXSGMR(L-1))-(TER3*EXSGMI(L-1))
  20  EXSGMI(L)=(TER2*EXSGMI(L-1))+(TER3*EXSGMR(L-1))
      IF (ISPILL) 14,15,14
  14  WRITE OUTPUT TAPE 6,114, ISPILL
  114 FORMAT(23H UNDERFLOW OCCURRED AT I6,21H IN EXSGML SUBROUTINE)
  15  IF (JSPILL) 16,17,16
  16  WRITE OUTPUT TAPE 6,116,JSPILL
 116  FORMAT(22H OVERFLOW OCCURRED AT I6,21H IN EXSGML SUBROUTINE)
      CALL LEAVE
      STOP
  17  RETURN
\end{lstlisting}
\clearpage

%% -----File: 080.png---Folio 75-------

\begin{lstlisting}
      SUBROUTINE RHOTB
      DRHO(1)=DRHOIN(1)
      RHO(1)=RHOIN(1)
      N=1
      I=1
  20  RHO(I+1)=RHO(I)+DRHOIN(N)
      IF (RHO(I+1)-RHOIN(NMAX))30,50,70
 30   IF(ABSF(RHO(I+1)-RHOIN(N+1))-.5*DRHOIN(N)) 35,35,40
   35    N=XMINOF(N+1,NMAX-1)
  40  DRHO(I+1)=DRHOIN(N)
      I=I+1
      GO TO 20
  50  ILAST=I+1
 60   RHO(ILAST)=RHOIN(NMAX)
      DRHO(ILAST-1)=RHO(ILAST)-RHO(ILAST-1)
      RHOMAX=RHOIN(NMAX)
      DRHOL=DRHOIN(NMAX-1)
      IF(ISPILL) 80,81,80
  80  WRITE OUTPUT TAPE 6,180,ISPILL
 180  FORMAT(23H UNDERFLOW OCCURRED AT I6,21H IN RHOTB SUBROUTINE)
 81   IF(JSPILL)82,83,82
  82  WRITE OUTPUT TAPE 6,182,JSPILL
 182  FORMAT(22H OVERFLOW OCCURRED AT I6,21H IN RHOTB SUBROUTINE)
      CALL LEAVE
      STOP
  83  RETURN
  70  IF((RHO(I+1)-RHOIN(NMAX))-.5*DRHOIN(N))50,50,75
  75  ILAST=I
      GO TO 60
\end{lstlisting}
\clearpage

%% -----File: 081.png---Folio 76-------

\begin{lstlisting}
      SUBROUTINE COULFN
      IF DIVIDE CHECK 50,51
  50  WRITE OUTPUT TAPE 6,150
 150  FORMAT (60H DIVIDE CHECK TRIGGER FOUND ON AT START OF COULFN SUBRO
     1UTINE)
      CALL LEAVE
      STOP
  51  ISPILL=0
      JSPILL=0
      IKTRL=KTRL(13)
      LMAX=LMAXM+1
      ETA2=ETA**2
      SQ=SQRTF(1.+ETA2)
  1   IJ = 1
      AR(1)=-ETA
      AI(1)=0.
      AR(2)=-.5*ETA2
      AI(2)=.5*ETA
  2   SI=0.
      SR=0.
      PR= RHOMAX
      DO 10  K=2,49
      T= PR*FLOATF(1-K)
      TR=AR(K)/T
      TI=AI(K)/T
      IF DIVIDE CHECK 52,53
  52  WRITE OUTPUT TAPE 6,152
  152 FORMAT(57H DIVISOR T IS ZERO IN FIRST DIVISION OF COULFN SUBROUTIN
     1E)
      CALL LEAVE
      STOP
  53  SQN=TR**2+TI**2
      IF(K-2) 4,4,3
  3   IF(SQN-SQO) 4,4,11
  4   TR=SR+TR
      TI=SI+TI
      IF(TR-SR) 6,5,6
  5   IF(TI-SI) 6,13,6
  6   SR=TR
      SI=TI
      AR(K+1)=0.
      AI(K+1)=0.
      KP=K/2
      DO 7 M=1,KP
      KM=K+1-M
      AR(K+1)=AR(K+1)-AR(M)*AR(KM)+AI(W)*AI(KM)
      AI(K+1)=AI(K+1)-AI(KM)*AR(M)-AI(M)*AR(KM)
      IF(K-2*KP) 8,9,8
      AR(K+1)=AR(K+1)-.5*(AR(KP+1)**2-AI(KP+1)**2)
      AI(K+1)=AI(K+1)-AR(KP+1)*AI(KP+1)
\end{lstlisting}
%% -----File: 082.png---Folio 77-------
\begin{lstlisting}
  9   FK=.5*FLOATF(K)
      AI(K+1)=AI(K+1)-FK*AR(K)
      AR(K+1)=AR(K+1)+FK*AI(K)
      PR= PR*RHOMAX
 10   SQO=SQN
      GO TO 101
  11  T=SR**2+SI**2
      IF(T) 105,105,12
  12  IF(ABSF(SQO/T)-EPS3) 13,13,106
  13  GO TO (14,15),IJ
  14  PAR=RHOMAX-ETA*LOGF(2.*RHOMAX)
      PHI0R=PAR+SIGMA0+SR
      PHI0I=SI
      AR(2)=-1.+AR(2)
      IJ=2
      GO TO 2
  15  PHI1R=PAR+SIGMA1-1.570796325+SR
      PHI1I=SI
  25  T1=EXPF(-PHI0I)
      T2=EXPF(-PHI1I)
      G(1)=T1*COSF(PHI0R)
      G(2)=T2*COSF(PHI1R)
      F1=T1*SINF(PHI0R)
      F2=T2*SINF(PHI1R)
      IF(ABSF(F1*G(2)-F2*G(1)-1./SQ)-EPS1)  31,31,102
  31  IDEC=11
  32  I=LMAX+IDEC
      FBAR(I)=.1
      FBAR(I+1)=0.
      LIMIT=LMAXM+IDEC
      FL=LMAX+11
      T1=SQRTF((FL+1.)**2+ETA2)
      IF(JSPILL) 139,133,139
 139  WRITE OUTPUT TAPE 6,1390,JSPILL
 1390 FORMAT(23H OVERFLOW2 OCCURRED AT I6,21H IN COULFN SUBROUTINE)
      CALL LEAVE
      STOP
 133  DO 33 I=1,LIMIT
      L=LMAX+IDEC-I
      FL=L
      T2=SQRTF(FL**2+ETA2)
      FBAR(L)=((2.*FL+1.)*(ETA+FL*(FL+1.)/RHOMAX)*FBAR(L+l)-FL*T1*FBAR(L
     1+2))/((FL+1.)*T2)
      IF DIVIDE CHECK 54,600
  54  WRITE OUTPUT TAPE 6,154
  154 FORMAT(56H DIVISOR IS ZERO IN SECOND DIVISION OF COULFN SUBROUTINE
     1)
      CALL LEAVE
      STOP
 600  IF(JSPILL) 601,33,601
\end{lstlisting}
%% -----File: 083.png---Folio 78-------
\begin{lstlisting}
 601  WRITE OUTPUT TAPE 6,1601,JSPILL
 1601 FORMAT(22H OVERFLOW OCCURRED AT I6,21H IN COULFN SUBROUTINE,24H MU
     1LTIPLY FBAR(I) BY 0.1)
      K=LMAX+IDEC
      FBAR(K)=FBAR(K)*0.1
      JSPILL=0
      GO TO 133
  33  T1=T2
      ALPHA=1./((FBAR(1)*G(2)-FBAR(2)*G(1))*SQ)
      IF DIVIDE CHECK 55,43
  55  WRITE OUTPUT TAPE 6,155
  155 FORMAT (55H DIVISOR IS ZERO IN THIRD DIVISION OF COULFN SUBROUTINE
     1)
      CALL LEAVE
      STOP
  43  LMAXP=LMAX+1
      DO 34 I=1,LMAXP
  34  FBAR(I)=ALPHA*FBAR(I)
      IF(IDEC-11) 371,35,371
  371 IF(ABSF(F1/FBAR(1)-1.)-EPS2) 37,37,35
  35  DO 36 I=1,LMAXP
  36  F(I)=FBAR(I)
      IDEC=IDEC+5
      IF (IDEC-40) 32,32,103
 37   DO 38 I=1,LMAXP
      IF(ABSF(F(I)/FBAR(I)-1.)-EPS2) 44,44,35
  44  IF DIVIDE CHECK 56,38
  56  WRITE OUTPUT TAPE 6,156,L,I
 156  FORMAT(74H DIVISOR FBAR(I)-1. IS ZERO IN FOURTH DIVISION OF COULFN
     1 SUBROUTINE FOR  L=I3,7H AND I=I3)
      CALL LEAVE
      STOP
 38   CONTINUE
      DO 381 I=1,MAXP
 381  F(I)=FBAR(I)
 382  T1=SQ
      DO 40 L=1,LMAX
      FL=L
      T2=SQRTF((FL+1.)**2+ETA2)
      G(L+2)=((2.*FL+1.)*(ETA+FL*(FL+1.)/RHOWAX)*G(L+1)-(FL+1.)*T1*G(L))
     1/(FL*T2)
      TS=FL/T1
      IF DIVIDE CHECK 57,45
 57   WRITE OUTPUT TAPE 6,157
 157  FORMAT(58H DIVISOR T1 IS ZERO IN FIFTH DIVISION OF COULFN SUBROUTI
     1NE)
      CALL LEAVE
      STOP
  45  IF(ABSF(F(L)*G(L+1)-F(L+1)*G(L)-TS)-EPS1) 40,40,104
  40  T1=T2
\end{lstlisting}
%% -----File: 084.png---Folio 79-------
\begin{lstlisting}
  41  DO 42 L=1,LMAX
      FL=L
      T=FL**2
      T1=T/RHOMAX+ETA
      IF DIVIDE CHECK 58,46
  58  WRITE OUTPUT TAPE 6,158
  158 FORMAT (62H DIVISOR RHOMAX IS ZERO IN SIXTH DIVISION OF COULFN SUB
     1ROUTINE)
      CALL LEAVE
      STOP
  46  T2=SQRTF(T+ETA2)
      FP(L)=(T1*F(L)-T2*F(L+1))/FL
  42  GP(L)=(T1*G(L)-T2*G(L+1))/FL
      IF DIVIDE CHECK 59,47
  59  WRITE OUTPUT TAPE 6,159
  159 FORMAT(60H DIVISOR FL IS ZERO IN SEVENTH DIVISION OF COULFN SUBROU
     1TINE)
      CALL LEAVE
      STOP
  47  IF(ISPILL) 60,61,60
  60  WRITE OUTPUT TAPE 6,160,ISPILL
 160  FORMAT(23H UNDERFLOW OCCURRED AT I6,21H IN COULFN SUBROUTINE)
  61  IF(JSPILL) 62,63,62
  62  WRITE OUTPUT TAPE 6,162,JSPILL
  162 FORMAT(22H OVERFLOW OCCURRED AT I6,21H IN COULFN SUBROUTINE)
      CALL LEAVE
      STOP
  63  RETURN
  101 WRITE OUTPUT TAPE 6,121,RHOMAX,DRHOL
      GO TO (110,110,109,109),IKTRL
 109  WRITE OUTPUT TAPE 6,114
      GO TO 13
  102 WRITE OUTPUT TAPE 6,122,RHOMAX,DRHOL
      GO TO(110,110,111,111),IKTRL
 111  WRITE OUTPUT TAPE 6,114
      GO TO 31
  103 WRITE OUTPUT TAPE 6,123,RHOMAX,DRHOL
      GO TO (110,110,112,112),IKTRL
 112  WRITE OUTPUT TAPE 6,114
      GO TO 382
  104 WRITE OUTPUT TAPE 6,124,RHOMAX,DRHOL ,L
      GO TO (110,110,113,113),IKTRL
 113  WRITE OUTPUT TAPE 6,114
      GO TO 40
  105 WRITE OUTPUT TAPE 6,125,RHOMAX,DRHOL
      GO TO (110,110,115,115),IKTRL
 115  WRITE OUTPUT TAPE 6,114
      GO TO 12
 106  WRITE OUTPUT TAPE 6,126,RHOMAX,DRHOL
      GO TO (110,110,116,116),IKTRL
\end{lstlisting}
%% -----File: 085.png---Folio 80-------
\begin{lstlisting}
 116  WRITE OUTPUT TAPE 6,114
      GO TO 13
  110 RHOMAX=RHOMAX+DRHOL
      GO TO 1
 121  FORMAT(18H INCREASE RHO MAX=E11.4,2H+ E11.4,35H A OR B SERIES CONV
     1ERGES TOO SLOWLY)
  122 FORMAT(18H INCREASE RHO MAX=E11.4,2H+ E11.4,22H BAD INITIAL WRONSK
     1IAN)
 123  FORMAT(18H INCREASE RHO MAX=E11.4,2H+ E11.4,24H L TOO LARGE IN FBA
     1R (L))
 124  FORMAT(18H INCREASE RHO MAX=E11.4,2H+ E11.4,21H BAD WRONSKIAN FOR
     1L=I3)
 125  FORMAT(67H SERIES IN PHI0 OR PHI1 IS ZERO, CHECK DATA, IF OK INCRE
     1ASE RHOMAX=E11.4,2H+ E11.4)
 126  FORMAT(52H A OR B SERIES DIVERGES TOO QUICKLY INCREASE RHOMAX=E11.
     14,2H+ E11.4)
 114  FORMAT(42H RHOMAX INCREASE NOT PERMITTED BY KTRL(13))





      SUBROUTINE RMXINC
  3   IF (RHOMAX-RHO(ILAST)) 1,2,1
  1   ILAST=ILAST+1
      RHO(ILAST)=RHO(ILAST-1)+DRHOL
      DRHO(ILAST-1)=DRHOL
      GO TO 3
  2   RETURN
\end{lstlisting}
\clearpage

%% -----File: 086.png---Folio 81-------

\begin{lstlisting}
      SUBROUTINE PGEN4
      IF DIVIDE CHECK 60,61
  60  WRITE OUTPUT TAPE 6,160
  160 FORMAT (59H DIVIDE CHECK TRIGGER FOUND ON AT START OF PGEN4 SUBROU
     1TINE)
      CALL LEAVE
      STOP
  61  ISPILL=0
      JSPILL=0
      IF(KTRL(1)) 3,4,3
 3    KTRL(7)=0
      KTRL(8)=0
      KTRL(9)=0
      KTRL(10)=0
 4    T1=V/ECM
      T2=W/ECM
      T10=VS/ECM
      T11=WS/ECM
      T12=FKAY*BG
      T3=2.*FKAY/A
      IF DIVIDE CHECK 62,65
  62  WRITE OUTPUT TAPE 6,162
  162 FORMAT (65H DIVISORS ECM OR A WERE WRONGLY INPUT AS ZERO IN PGEN4
     1SUBROUTINE)
      CALL LEAVE
      STOP
  65  T4=T10*T3
      T5=T11*T3
      T6=FKAY*A
      T7=ETA/RHOBC
      IF DIVIDE CHECK 63,64
  63  WRITE OUTPUT TAPE 6,163
  163 FORMAT(61H DIVISOR RHOBC IS ZERO IN SECOND DIVISION OF PGEN4 SUBRO
     1UTINE)
      CALL LEAVE
      STOP
  64  T8=RHOBC**2
      T9=ETA*2.
      I=1
  40  EX=EXPF((RHO(I)-RHOBN)/T6)
      IF DIVIDE CHECK 80,66
  80  WRITE OUTPUT TAPE 6,165
 165  FORMAT (58H QUANTITY T6 IS ZERO IN THIRD DIVISION OF PGEN4 SUBROUT
     1INE)
      CALL LEAVE
      STOP
  66  K=1
  41  IF(I-1) 42,43,42
  42  IF(DRHO(I)-DRHO(I-1)) 43,44,43
  43  HDRHO=DRHO(I)*.5
\end{lstlisting}
%% -----File: 087.png---Folio 82-------
\begin{lstlisting}
      DEX=EXPF(HDRHO/T6)
 44   IF(KTRL(1)-2)53,52,53
 52   IF(RHO(I)-RHOBN) 54,55,55
 54   S1=1.0
      GO TO 68
 55   S1=0.0
      GO TO 68
 53   S1=1./(1.+EX)
      IF DIVIDE CHECK 67,68
  67  WRITE OUTPUT TAPE 6,167
 167  FORMAT(60H DIVISOR 1.+EX IS ZERO IN FOURTH DIVISION OF PGEN4 SUBRO
     1UTINE)
      CALL LEAVE
      STOP
  68  S2=EX*(S1**2)
      S4=S2/RHO(I)
      IF DIVIDE CHECK 69,70
  69  WRITE OUTPUT TAPE 6,169,I
 169  FORMAT(58H DIVISOR RHO IS ZERO IN FIFTH DIVISION OF PGEN4 SUBROUTI
     1NE)
      CALL LEAVE
      STOP
  70  IF (RHO(I)-RHOBC) 9,9,10
  9   S3=T7*(3.-(RHO(I)**2)/T8)
      GO TO 11
  10  S3=T9/RHO(I)
 11   IF (KTRL(7)) 350,300,350
  300 UCRB(I)=-1.-T1*S1+S3
      FFCR(I)=S1
  301 IF (KTRL(8)) 355,302,355
 302  IF(KTRL(1)-1) 309,308,309
 308  S1=EXPF(-((RHO(I)-RHOBNG)/T12)**2)
      IF DIVIDE CHECK 82,309
 82   WRITE OUTPUT TAPE 6,182
 182  FORMAT(22H BG IS ZERO IN PGEN SR)
      CALL LEAVE
      STOP
 309  UCIB(I)=-T2*S1
      FFCI(I)=S1
  303 IF (KTRL(9)) 360,304,360
  304 USRB(I)=T4*S4
      FFSR(I)=S4
  305 IF (KTRL(11)) 501,500,501
  500 IF (KTRL(10))365,306,365
  306 USIB(I)=T5*S4
      FFSI(I)=S4
  307 IF (I-ILAST) 50,200,200
  350 ITT=1
      GO TO 340
  355 ITT=2
\end{lstlisting}
%% -----File: 088.png---Folio 83-------
\begin{lstlisting}
      GO TO 340
  340 ITQ=1
      IF(ITT-1) 380,380,381
  380 IF (KTRL(7)-1) 352,351,352
  351 TW=HA
      TRM=RMA
      TN1=FN1A
      TN2=FN2A
      GO TO 400
 352  TH=HB
      TRM=RMB
      TN1=FN1B
      TN2=FN2B
      GO TO 400
  381 IF (KTRL(8)-1) 352,351,352
  400 IF (RHO(I)-RHOBN) 410,410,411
  410 TTN=TN1
      GO TO 412
  411 TTN=TN2
 412  T20=RHO(I)/RHOBN
      IF (TTN*LOGF(T20)-80.) 403,403,409
  403 TQ=(T20**TTN-1.)*RHOBN/(TTN*FKAY*A)
      IF DIVIDE CHECK 405,406
 405  TG=T20**(RHOBN/(FKAY*A))
      GO TO 407
  406 IF (TQ-80.) 408,408,409
  408 TG=EXPF(TQ)
      GO TO 407
 409  TF=0.
      GO TO 422
  407 TFN=1./(1.+TG)
      IF (RHO(I)-TRM) 420,420,419
  419 TF=TFN
      GO TO 418
  420 T21=RHO(I)/TRM
      THH=TH*(1.+(2.*T21))*((1.-T21)**2)
      TF=TFN*(1.+THH)
  418 TFF=TF
  421 GO TO (422,423),ITQ
  422 GO TO (425,426,427,428),ITT
  425 FFCR(I)=TF
      UCRB(I)=-1.-T1*FFCR(I)+S3
      GO TO 301
  426 FFCI(I)=TF
      UCIB(I)=-T2*FFCI(I)
      GO TO 303
 427  FFSR(I)=TF
      IF (ITQ-1) 470,470,471
  471 USRB(I)=FKAY*A*T4*FFSR(I)
      GO TO 305
\end{lstlisting}
%% -----File: 089.png---Folio 84-------
\begin{lstlisting}
 470  USRB(I)=(T4/2.)*FFSR(I)
      GO TO 305
  428 FFSI(I)=TF
      IF(ITQ-1) 472,472,473
  473 USIB(I)=FKAY*A*T5*FFSI(I)
      GO TO 307
  360 ITT=3
      IF (KTRL(9)-1) 431,431,430
  430 ITQ=1
      GO TO 352
  365 ITT=4
      IF (KTRL(10)-1)  431,431,430
 472  USIB(I)=(T5/2.)*FFSI(I)
      GO TO 307
 431  ITQ=2
      GO TO 351
  423 T23=(RHOBN/(FKAY*A))*(T20**TTN)*TG*((TFN/RHO(I))**2)
      T25=T23
      IF(RHO(I)-TRM) 460,460,461
 460  T24=6.*TH*(1.-T21)/(TRM**2)
      T25=(T24*TFN)+((1.+THH)*T23)
 461  TF=T25
      IF(ITT-3) 427,427,428
 501  T30=0.004927*ETA*ECM
      IF(RHO(I)-RHOBC) 502,502,503
  502 SOCOUL=T30/(RHOBC**3)
      GO TO 504
  503 SOCOUL=T30/(RHO(I)**3)
  504 USRB(I)=USRB(I)+SOCOUL
      GO TO 500
  50  I=I+1
      EX=EX*DEX
      RHOM=RHO(I-1)+HDRHO
      IF(KTRL(1)-2) 153,152,153
 152  IF(RHOM-RHOBN)34,35,35
 34   S1=1.0
      GO TO 72
 35   S1=0.0
      GO TO 72
 153  S1=1./(1.+EX)
      IF DIVIDE CHECK 71,72
  71  WRITE OUTPUT TAPE 6,171
 171  FORMAT(54H DIVISOR 15 ZERO IN SIXTH DIVISION OF PGEN4 SUBROUTINE)
      CALL LEAVE
      STOP
  72  S2=EX*(S1**2)
      S4=S2/RHOM
      IF DIVIDE CHECK 73,74
  73  WRITE OUTPUT TAPE 6,173
  173 FORMAT (62H QUANTITY RHOM IS ZERO IN SEVENTH DIVISION OF PGEN4 SUB
\end{lstlisting}
%% -----File: 090.png---Folio 85-------
\begin{lstlisting}
     1ROUTINE)
      CALL LEAVE
      STOP
  74  IF(RHOM-RHOBC) 21,21,22
  21  S3=T7*(3.-(RHOM**2)/T8)
      GO TO 23
  22  S3=T9/RHOM
  23  IF (KTRL(7))1350,1300,1350
 1300 UCRM(I-1)=-1.-T1*S1+S3
      FFCRM(I-1)=51
 1301 IF (KTRL(8)) 1355,1302,1355
 1302 IF(KTRL(1)-1) 1309,1308,1309
 1308 S1=EXPF(-((RHOM-RHOBNG)/T12)**2)
 1309 UCIM(I-1)=-T2*S1
      FFCIM(I-1)=S1
 1303 IF (KTRL(9)) 1360,1304,1360
 1304 USRM(I-1)=T4*S4
      FFSRM(I-1)=S4
 1305 IF (KTRL(I1)) 1501,1500,1501
 1500 IF (KTRL(I0))1365,1306,1365
 1306 USIM(I-1)=T5*S4
      FFSIM(I-1)=S4
 1307 IF (K-10) 24,40,40
 1350 ITT=1
      GO TO 1340
 1355 ITT=2
      GO TO 1340
 1340 ITQ=1
      IF (ITT-1)1380,1380,1381
 1380 IF (KTRL(7)-1) 1352,1351,1352
 1351 TH=HA
      TRM=RMA
      TN1=FN1A
      TN2=FN2A
      GO TO 1400
 1352 TH=HB
      TRM=RMB
      TN1=FN1B
      TN2=FN2B
      GO TO 1400
 1381 IF (KTRL(8)-1) 1352,1351,1352
 1400 IF (RHOM-RHOBN) 1410,1410,1411
 1410 TTN=TN1
      GO TO 1412
 1411 TTN=TN2
 1412 T20=RHOM/RHOBN
      IF (TTN*LOGF(T20)-80.) 1403,1403,1409
 1403 TQ=(T20**TTN-1.)*RHOBN/(TTN*FKAY*A)
      IF DIVIDE CHECK 1405,1406
 1405 TG=T20**(RHOBN/(FKAY*A))
\end{lstlisting}
%% -----File: 091.png---Folio 86-------
\begin{lstlisting}
      GO TO 1407
 1406 IF (TQ-80.) 1408,1408,1409
 1408 TG=EXPF(TQ)
      GO TO 1407
 1409 TF=0.
      GO TO 1422
 1407 TFN=1./(1.+TG)
      IF (RHOM-TRM) 1420,1420,1419
 1419 TF=TFN
      GO TO 1418
 1420 T21=RHOM/TRM
      TRH=TH*(1.+(2.*T21))*((1.-T21)**2)
      TF=TFN*(1.+THH)
 1418 TFF=TF
 1421 GO TO (1422,1423),ITQ
 1422 GO TO (1425,1426,1427,1428),ITT
 1425 FFCRM(I-1)=TF
      UCRM(I-1)=-1.-T1*FFCRM(I-1)+S3
      GO TO 1301
 1426 FFCIM(I-1)=TF
      UCIM(I-1)=-T2*FFCIM(I-1)
      GO TO 1303
 1427 FFSRM(I-1)=TF
      IF (ITQ-1) 1470,1470,1471
 1471 USRM(I-1)=FKAY*A*T4*FFSRM(I-1)
      GO TO 1305
 1470 USRM(I-1)=(T4/2.)*FFSRM(I-1)
      GO TO 1305
 1428 FFSIM(I-1)=TF
      IF (ITQ-1) 1472,1472,1473
 1473 USIM(I-1)=FKAY*A*T5*FFSIM(I-1)
      GO TO 1307
 1360 ITT=3
      IF (KTRL(9)-1) 1431,1431,1430
 1430 ITQ=1
      GO TO 1352
 1365 IIT=4
      IF (KTRL(10)-1) 1431,1431,1430
 1472 USIM(I-1)=(T5/2.)*FFSIM(I-1)
      GO TO 1307
 1431 ITQ=2
      GO TO 1351
 1423 T23=(RHOBN/(FKAY*A))*(T20**TTN)*TG*((TFN/RHOM)**2)
      T25=T23
      IF(RHOM-TRM) 1460,1460,1461
 1460 T24=6.*TH*(1.-T21)/(TRM**2)
      T25=(T24*TFN)+((1.+THH)*T23)
 1461 TF=T25
      IF (ITT-3) 1427,1427,1428
 1501 T30=0.004927*ETA*ECM
\end{lstlisting}
%% -----File: 092.png---Folio 87-------
\begin{lstlisting}
      IF (RHOM-RHOBC) 1502,1502,1503
 1502 SOCOUL=T30/(RHOBC**3)
      GO TO 1504
 1503 SOCOUL=T30/(RHOM**3)
 1504 USRM(I-1)=USRM(I-1)+SOCOUL
      GO TO 1500
   24 K=K+1
      EX=EX*DEX
      GO TO 42
  200 IF(ISPILL) 75,76,75
   75 WRITE OUTPUT TAPE 6,175,ISPILL
  175 FORMAT(23H UNDERFLOW OCCURRED AT I6,20H IN PGEN4 SUBROUTINE)
   76 IF (JSPILL) 77,51,77
   77 WRITE OUTPUT TAPE 6,177, JSPILL
  177 FORMAT(22H OVERFLOW OCCURRED AT I6,20H IN PGEN4 SUBROUTINE)
      CALL LEAVE
      STOP
   51 RETURN
\end{lstlisting}
\clearpage

%% -----File: 093.png---Folio 88-------

\begin{lstlisting}
      SUBROUTINE INTCTR
      DO1 L=1,LMAX
      IFIRST=IIN(L)
      T=RHO(IFIRST)**(L-1)
      XC1=T*RHO(IFIRST)
      XD1=XC1
      FL=L
      XCP1=FL*T
      XDP1=XCP1
      YC1=0.
      YD1=0.
      YCP1=0.
      YDP1=0.
      CALL RKINT
      X1(L)=XC1
      X2(L)=XD1
      Y1(L)=YC1
      Y2(L)=YD1
      X1P(L)=XCP1
      X2P(L)=XDP1
      Y1P(L)=YCP1
    1 Y2P(L)=YDP1
      RETURN
\end{lstlisting}
\clearpage

%% -----File: 094.png---Folio 89-------

\begin{lstlisting}
      SUBROUTINE RKINT
      IF DIVIDE CHECK 10,11
 10   WRITE OUTPUT TAPE 6,110,L,I
  110 FORMAT(66H DIVIDE CHECK TRIGGER FOUND ON AT START OF RKINT SUBROUT
     1INE FOR L=I3,7H AND I=I3)
      CALL LEAVE
      STOP
  11  ISPILL=0
      JSPILL=0
   1  FL=L-1
      F2L=-1.-FL
      F3L=FL*(FL+1.)
      TB=UCRB(IFIRST)+F3L/(RHO(IFIRST)**2)
      IF DIVIDE CHECK 12,13
  12  WRITE OUTPUT TAPE 6,112,L,I
 112  FORMAT(76H DIVISOR RHO(IFIRST)**2 IS ZERO IN FIRST DIVISION OF RKI
     1NT SUBROUTINE FOR L=I3,7H AND I=I3)
      CALL LEAVE
      STOP
  13  PCB=TB+USRB(IFIRST)*FL
      PDB=TB+USRB(IFIRST)*F2L
      QCB=UCIB(IFIRST)+USIB(IFIRST)*FL
      QDB=UCIB(IFIRST)+USIB(IFIRST)*F2L
      IK=ILAST-1
      DO 6 I=IFIRST,IK
  2   HDRHO=.5*DRHO(I)
      DRHO2=(DRHO(I)**2)*.5
      RHOM=RHO(I)+HDRHO
      TM=UCRM(I)+F3L/(RHOM**2)
      IF DIVIDE CHECK 14,15
  14  WRITE OUTPUT TAPE 6,114,L,I
 114  FORMAT(70H DIVISOR RHOM**2 IS ZERO IN SECOND DIVISION OF RKINT SUB
     1ROUTINE FOR L=I3,7H AND I=I3)
      CALL LEAVE
      STOP
  15  PCM=TM+USRM(I)*FL
      PDM=TM+USRM(I)*F2L
      QCM=UCIM(I)+USIM(I)*FL
      QDM=UCIM(I)+USIM(I)*F2L
      XCPP1=PCB*XC1-QCB*YC1
      YCPP1=QCB*XC1+PCB*YC1
      XDPP1=PDB*XD1-QDB*YD1
      YDPP1=QDB*XD1+PDB*YD1
      XC2=XC1+XCP1*HDRHO
      YC2=YC1+YCP1*HDRHO
      XD2=XD1+XDP1*HDRHO
      YD2=YD1+YDP1*HDRHO
      XCPP2=PCM*XC2-QCM*YC2
      YCPP2=QCM*XC2+PCM*YC2
      XDPP2=PDM*XD2-QDM*YD2
\end{lstlisting}
%% -----File: 095.png---Folio 90-------
\begin{lstlisting}
      YDPP2=QDM*XD2+PDM*YD2
      DRHO4=.5*DRHO2
      SDRHO=.33333333*HDRHO
      XC3=XC2+XCPP1*DRHO4
      YC3=YC2+YCPP1*DRHO4
      XD3=XD2+XDPP1*DRHO4
      YD3=YD2+YDPP1*DRHO4
      XCPP3=PCM*XC3-QCW*YC3
      YCPP3=QCM*XC3+PCM*YC3
      XDPP3=PDM*XD3-QDM*YD3
      YDPP3=QDM*XD3+PDW*YD3
      XC4=XC2+XCPP2*DRHO2+XCP1*HDRHO
      YC4=YC2+YCPP2*DRHO2+YCP1*HDRHO
      XD4=XD2+XDPP2*DRHO2+XDP1*HDRHO
      YD4=YD2+YDPP2*DRHO2+YDP1*HDRHO
      TB=UCRB(I+1)+F3L/(RHO(I+1)**2)
      IF DIVIDE CHECK 16,17
  16  WRITE OUTPUT TAPE 6,116,L,I
 116  FORMAT(74H DIVISOR RHO(I+1)**2 IS ZERO IN THIRD DIVISION FOR RKINT
     1 SUBROUTINE FOR L=I3,7H AND I=I3)
      CALL LEAVE
      STOP
  17  PCB=TB+USRB(I+1)*FL
      PDB=TB+USRB(I+1)*F2L
      QCB=UCIB(I+1)+USIB(I+1)*FL
      QDB=UCI8(I+1)+USIB(I+1)*F2L
      XCPP4=PCB*XC4-QCB*YC4
      YCPP4=QCB*XC4+PCB*YC4
      XDPP4=PDB*XD4-QDB*YD4
      YDPP4=QDB*XD4+PDB*YD4
      SXC=XCPP2+XCPP3
      SYC=YCPP2+YCPP3
      SXD=XDPP2+XDPP3
      SYD=YDPP2+YDPP3
      TXC=SXC+XCPP1
      TYC=SYC+YCPP1
      TXD=SXD+XDPP1
      TYD=SYD+YDPP1
      TXC1=XC1+DRHO(I)*(XCP1+SDRHO*TXC)
      TYC1=YC1+DRHO(I)*(YCP1+SDRHO*TYC)
      TXD1=XD1+DRHO(I)*(XDP1+SDRHO*TXD)
      TYD1=YD1+DRWO(I)*(YDP1+SDRHO*TYD)
      TXCP1=XCP1+SDRHO*(TXC+SXC+XCPP4)
      TYCP1=YCP1+SDRHO*(TYC+SYC+YCPP4)
      TXDP1=XDP1+SDRHO*(TXD+SXD+XDPP4)
      TYDP1=YDP1+SDRHO*(TYD+SYD+YDPP4)
      IF (JSPILL) 20,21,20
  20  RENORM=MAX1F(ABSF(XC1),ABSF(YC1),ABSF(XCP1),ABSF(YCP1),ABSF(XD1),
     1ABSF(YD1),ABSF(XDP1),ABSF(YDP1))
      XC1=XC1/RENORM
\end{lstlisting}
%% -----File: 096.png---Folio 91-------
\begin{lstlisting}
      YC1=YC1/RENORM
      XCP1=XCP1/RENORM
      YCP1=YCP1/RENORM
      XD1=XD1/RENORM
      YD1=YD1/RENORM
      XDP1=XDP1/RENORM
      YDP1=YDP1/RENORM
      WRITE OUTPUT TAPE 6,200,RENORM,L,RHO(I)
  200 FORMAT(24H RENORMALIZATION FACTOR=E16.9,22H IN RKINT FOR CODED L=I
     13,9H AND RHO=E16.9)
      JSPILL=0
      GO TO2
  21  XC1=TXC1
      YC1=TYC1
      XD1=TXD1
      YD1=TYD1
      XCP1=TXCP1
      YCP1=TYCP1
      XDP1=TXDP1
      YDP1=TYDP1
  6   CONTINUE
      IF (ISPILL) 30,31,30
  30  WRITE OUTPUT TAPE 6,130, ISPILL,L,I
  130 FORMAT(23H UNDERFLOW OCCURRED AT I6,27H IN RKINT SUBROUTINE FOR L=
     1I3.7H AND I=I3)
  31  IF (JSPILL) 32,4,32
  32  WRITE OUTPUT TAPE 6,132, JSPILL,L,I
 132  FORMAT(22H OVERFLOW OCCURRED AT I6,27H IN RKINT SUBROUTINE FOR L=I
     13,7H AND I=I3)
      CALL LEAVE
      STOP
   4  RETURN
\end{lstlisting}
\clearpage

%% -----File: 097.png---Folio 92-------

\begin{lstlisting}
      SUBROUTINE CSUBL
      IF DIVIDE CHECK 50,51
  50  WRITE OUTPUT TAPE 6,150
 150  FORMAT (59H DIVIDE CHECK TRIGGER FOUND ON AT START OF CSUBL SUBROU
     1TINE)
      CALL LEAVE
      STOP
  51  ISPILL=0
      JSPILL=0
      DO 40 L=1,LMAX
      XNORM1=MAX1F(ABSF(X1(L))*ABSF(Y1(L)),ABSF(X1P(L)),ABSF(Y1P(L)))
      TX1L=N1(L)/XNORM1
      TY1L=Y1(L)/XNORM1
      TX1PL=N1P(L)/XNORM1
      TY1PL=Y1P(L)/XNORM1
      FNORM=MAX1F(F(L),G(L),FP(L),GP(L))
      TFL=F(L)/FNORM
      TGL=G(L)/FNORM
      TFPL=FP(L)/FNORM
      TGPL=GP(L)/FNORM
      CO1=TFL*TY1PL-TFPL*TY1L
      CO2=TFPL*TX1L-TFL*TX1PL
      CO3=TY1L*TGPL-TY1PL*TGL+TX1L*TFPL-TX1PL*TFL
      CO4=TX1PL*TGL-TX1L*TGPL+TY1L*TFPL-TY1PL*TFL
      CO7=1.0/(CO3**2+CO4**2)
      IF DIVIDE CHECK 52,53
  52  WRITE OUTPUT TAPE 6,152
  152 FORMAT(54H DIVISOR IS ZERO IN FIRST DIVISION OF CSUBL SUBROUTINE)
      CALL LEAVE
      STOP
  53  CR1(L)=(CO1*CO3+CO2*CO4)*CO7
      CI1(L)=(CO2*CO3-CO1*CO4)*CO7
      XNORM2=MAX1F(ABSF(X2(L)),ABSF(Y2(L)),ABSF(X2P(L)),ABSF(Y2P(L)))
      TX2L=N2(L)/XNORM2
      TY2L=Y2(L)/XNORM2
      TX2PL=N2P(L)/XNORM2
      TY2PL=Y2P(L)/XNORM2
      CO1=TFL*TY2PL-TFPL*TY2L
      CO2=TFPL*TX2L-TFL*TX2PL
      CO3=TY2L*TGPL-TY2PL*TGL+TX2L*TFPL-TX2PL*TFL
      CO4=TX2PL*TGL-TX2L*TGPL+TY2L*TFPL-TY2PL*TFL
      CO7=1.0/(CO3**2+CO4**2)
      IF DIVIDE CHECK 54,55
  54  WRITE OUTPUT TAPE 6,154
 154  FORMAT (55H DIVISOR IS ZERO IN SECOND DIVISION OF CSUBL SUBROUTINE
     1)
      CALL LEAVE
      STOP
 55   CR2(L)=(CO1*CO3+CO2*CO4)*CO7
 40   CI2(L)=(CO2*CO3-CO1*CO4)*CO7
\end{lstlisting}
%% -----File: 098.png---Folio 93-------
\begin{lstlisting}
      IF (ISPILL) 56,57,56
  56  WRITE OUTPUT TAPE 6,156,ISPILL,L
  156 FORMAT (23H UNDERFLOW OCCURRED AT I6,27H IN CSUBL SUBROUTINE FOR L
     1=I3)
  57  IF (JSPILL) 58,59,58
  58  WRITE OUTPUT TAPE 6,158, JSPILL, L
 158  FORMAT (22H OVERFLOW OCCURRED AT I6,27H IN CSUBL SUBROUTINE FOR L=
     1I3)
      CALL LEAVE
      STOP
  59  RETURN
\end{lstlisting}
\clearpage

%% -----File: 099.png---Folio 94-------

\begin{lstlisting}
      SUBROUTINE AB
      IF DIVIDE CHECK 1,2
  1   WRITE OUTPUT TAPE 6,101
 101  FORMAT (56H DIVIDE CHECK TRIGGER FOUND ON AT START OF AB SUBROUTIN
     1E)
      CALL LEAVE
      STOP
  2   ISPILL=0
      JSPILL=0
      FKAYD=1./FKAY
      IF DIVIDE CHECK 3,4
  3   WRITE OUTPUT TAPE 6,103
  103 FORMAT(38H DIVISOR FKAY IS ZERO IN AB SUBROUTINE)
      CALL LEAVE
      STOP
  4   DO 20 J=1,JMAX
      ASUMR=0.
      ASUMI=0.
      BSUMR=0.
      BSUMI=0.
      DO 10 L=1,LMAX
      FL=L
      ATR1=FL*CR1(L)+(FL-1.)*CR2(L)
      ATI1=FL*CI1(L)+(FL-1.)*CI2(L)
      BTR1=CR1(L)-CR2(L)
      BTI1=CI1(L)-CI2(L)
      ATR2=ATR1*EXSGMR(L)-(ATI1*EXSGMI(L))
      ATI2=ATR1*EXSGMI(L)+(ATI1*EXSGMR(L))
      BTR2=BTR1*EXSGMR(L)-(BTI1*EXSGMI(L))
      BTI2=BTR1*EXSGMI(L)+(BTI1*EXSGMR(L))
      ASUMR=ASUMR+(ATR2*P(L,J))
      ASUMI=ASUMI+(ATI2*P(L,J))
      BSUMR=BSUMR+(BTR2*PP(L,J))
  10  BSUMI=BSUMI+(BTI2*PP(L,J))
      AR(J)= FCR(J)+(FKAYD*ASUMR)
      AI(J)=FCI(J)+(FKAYD*ASUMI)
      BR(J)= FKAYD*BSUMI
  20  BI(J)= -FKAYD*BSUMR
      IF (ISPILL) 30,31,30
  30  WRITE OUTPUT TAPE 6,130, ISPILL
 130  FORMAT(23H UNDERFLOW OCCURRED AT I6,17H IN AB SUBROUTINE)
 31   IF (JSPILL) 32,33,32
  32  WRITE OUTPUT TAPE 6,132,JSPILL
 132  FORMAT (22H OVERFLOW OCCURRED AT I6,17H IN AB SUBROUTINE)
      CALL LEAVE
      STOP
  33  RETURN
\end{lstlisting}
\clearpage

%% -----File: 100.png---Folio 95-------

\begin{lstlisting}
      SUBROUTINE SGSGCP
      IF DIVIDE CHECK 10,11
  10  WRITE OUTPUT TAPE 6,110
 110  FORMAT (60H DIVIDE CHECK TRIGGER FOUND ON AT START OF SGSGCP SUBRO
     1UTINE)
      CALL LEAVE
      STOP
  11  ISPILL=0
      JSPILL=0
      DO 5 J=1,JMAX
      SGMATH(J)=AR(J)**2.+AI(J)**2.+BR(J)**2.+BI(J)**2.
      POLTH(J)= (2.*(AR(J)*BR(J)+AI(J)*BI(J)))/SGMATH(J)
      IF DIVIDE CHECK 12,13
  12  WRITE OUTPUT TAPE 6,112,J
 112  FORMAT(30H DIVISOR SGMATH IS ZERO FOR J=13,21H IN SGSGCP SUBROUTIN
     1E)
      CALL LEAVE
      STOP
  13  SGMAC(J)=FCR(J)**2.+FCI(J)**2.
      IF(ETA) 7,7,8
  8   SRATIO(J)=SGMATH(J)/SGMAC(J)
      IF DIVIDE CHECK 14,15
  14  WRITE OUTPUT TAPE 6,114,J
  114 FORMAT(29H DIVISOR SGMAC IS ZERO FOR J=13,21H IN SGSGCP SUBROUTINE
     1)
      CALL LEAVE
      STOP
  15  GO TO 5
  7   SRATIO(J)=0.
  5   CONTINUE
      IF (ISPILL) 16,17,16
  16  WRITE OUTPUT TAPE 6,116,ISPILL
 116  FORMAT (23H UNDERFLOW OCCURRED AT 16,21H IN SGSGCP SUBROUTINE)
  17  IF (JSPILL) 18,19,18
  18  WRITE OUTPUT TAPE 6,118,JSPILL
 118  FORMAT(22H OVERFLOW OCCURRED AT 16,21H IN SGSGCP SUBROUTINE)
      CALL LEAVE
      STOP
  19  RETURN
\end{lstlisting}
\clearpage

%% -----File: 101.png---Folio 96-------

\begin{lstlisting}
      SUBROUTINE SIGMAR
      ISPILL=0
      JSPILL=0
      FL=0.
      SGMRTH=0.
      CPI=(12.56637060)/(FKAY**2)
      DO 20 L=I,LMAX
      SGMRTH=SGMRTH+FL*(C12(L)-(C12(L))**2-(CR2(L))**2)
      FL=FL+1.0
  20  SGMRTH=SGMRTH+FL*(CI1(L)-(CI1(L))**2-(CR1(L))**2)
      SGMRTH=CPI*SGMRTH
      IF(ISPILL) 10,11,10
  10  WRITE OUTPUT TAPE 6,110,ISPILL
  110 FORMAT(23H UNDERFLOW OCCURRED AT 16,21H IN SIGMAR SUBROUTINE)
  11  IF(JSPILL) 12,13,12
  12  WRITE OUTPUT TAPE 6,112,JSPILL
  112 FORMAT(22H OVERFLOW OCCURRED AT 16,21H IN SIGMAR SUBROUTINE)
      CALL LEAVE
      STOP
  13  RETURN
\end{lstlisting}
\clearpage

%% -----File: 102.png---Folio 97-------

\begin{lstlisting}
      SUBROUTINE CHISQ
      IF DIVIDE CHECK 10,11
  10  WRITE OUTPUT TAPE 6,110
  110 FORMAT(59H DIVIDE CHECK TRIGGER FOUND ON AT START OF CHISQ SUBROUT
     1INE)
      CALL LEAVE
      STOP
  11  ISPILL=0
      JSPILL=0
      CHI2ST=0
      CHI2PT=0
      DO 20 J=1,JMAX
      CHI2S(J)= ((SGMATH(J)-SGMAEX(J))/DSGMEX(J))**2.
      CHI2P(J)= ((POLTH(J)-POLEX(J))/DPOLEX(J))**2.
      IF DIVIDE CHECK 14,15
  14  WRITE OUTPUT TAPE 6,114,J
  114 FORMAT(40H DIVISOR DSGMEX OR DPOLEX IS ZERO FOR J=13,20H IN CHISQ
     1SUBROUTINE)
      CALL LEAVE
      STOP
  15  CHI2ST=CHI2ST + CHI2S(J)
      CHI2(J)=CHI25(J)+CHI2P(J)
  20  CHI2PT=CHI2PT+CHI2P(J)
      CHI2T=CHI2ST+CHI2PT
      IF (ISPILL) 16,17,16
  16  WRITE OUTPUT TAPE 6,116, ISPILL
  116 FORMAT(23H UNDERFLOW OCCURRED AT 16,20H IN CHISQ SUBROUTINE)
  17  IF(ISPILL) 18,19,18
  18  WRITE OUTPUT TAPE 6,118,JSPILL
  118 FORMAT(22H OVERFLOW OCCURRED AT 16,20H IN CHISQ SUBROUTINE)
      CALL LEAVE
      STOP
 19   RETURN
\end{lstlisting}
\clearpage

%% -----File: 103.png---Folio 98-------

\begin{lstlisting}
      SUBROUTINE OUTPT4
      NPGS=0
      CALL SKIP(K,NPGS,NUMRUN)
      WRITE OUTPUT TAPE 6,245,NUMPRG
  245 FORMAT (16H0PROGRAM NUMBER I5)
      DO 8 I=1,13
      WRITE OUTPUT TAPE 6,250,I,(KTRL(I))
  250 FORMAT (6H KTRL(I2,2H)=I2)
  8   CONTINUE
      WRITE OUTPUT TAPE 6,12
  12  FORMAT (11H0BASIC DATA)
      FKAYA=FKAY*A
      FKAYB=FKAY*BG
      WRITE OUTPUT TAPE 6,14,FMI,FMB,ELAB,ZZ,V,W,A,RO,VS,WS,RC,BG,RG,
 14   FORMAT(7H0MSUBI=E16.9,10H    MSUBB=E16.9,10H     ELAB=E16.9,10H
     1   ZZP=E16.9/7H0    V=E16.9,10H        W=E16.9,10H        A=E16.9,
     210H       RO=E16.9/7H0   VS=E16.9,10H       WS=E16.9,36H
     3                       RC=E16.9/59H0
     4                         BG=E16.9,10H       RG=E16.9)
      WRITE OUTPUT TAPE 6,16,RHOBN,RHOBC,RHOBNG,ECM,ETA,FKAY,FKAYA,FKAYB
 16   FORMAT(7H0RHOBN=E16.9,10H    RHOBC=E16.9,10H   RHOBNG=E16.9,10H
     1   ECM=E16.9/7H0  ETA=E16.9,10H        K=E16.9/10H       KA=E16.9,
     210H       KB=E16.9)
      KT=KTRL(7)+KTRL(8)+KTRL(9)+KTRL(10)
      IF (KT) 13,1818,13
 13   WRITE OUTPUT TAPE 6,150,HA,RMA,FN1A,FN2A,PMA,HB,RMB,FN1B,FN2B,PMB
 150  FORMAT(7H0   HA=E16.9,7H   RMA=E16.9,7H   N1A=E16.9,7H   N2A=E16.9
     1,7H   PMA=E16.9/7H    HB=E16.9,7H   RMB=E16.9,7H   N1B=E16.9,7H
     2N2B=E16.9,7H   PMB=E16.9)
 1818 WRITE OUTPUT TAPE 6,18,RHOMAX,LMAXM
  18  FORMAT (17H0INTEGRATION DATA/8H0RHOMAX=E16.9,10H    LMAXM=I5)
      WRITE OUTPUT TAPE 6,220,NMAX
 220  FORMAT (6H0NMAX=I5)
      WRITE OUTPUT TAPE 6,24
  24  FORMAT (6H0RHOIN)
      NOLINE=50
      K=20
      DO  40  I=1, NMAX,6
      IF(K-NOLINE) 30,29,29
  29  CALL SKIP(K,NPGS,NUMRUN)
  30  M=XMINOF(I+5,NMAX)
      K=K+1
      WRITE OUTPUT TAPE 6,32,(RHOIN(J),J=I,M)
  32  FORMAT(1H E19.9,5E20.9)
  40  CONTINUE
      WRITE OUTPUT TAPE 6,41
  41  FORMAT (7H0DRHOIN)
      DO 60 I=1,NMAX,6
      IF(K-NOLINE) 45,43,43
  43  CALL SKIP(K,NPGS,NUMRUN)
\end{lstlisting}
%% -----File: 104.png---Folio 99-------
\begin{lstlisting}
  45  M=XMINOF(I+5,NMAX-1)
      K=K+1
      WRITE OUTPUT TAPE 6,32,(DRHOIN(J),J=I,M)
 60   CONTINUE
      WRITE OUTPUT TAPE 6,118,SGMRTH
 118  FORMAT(12H0SIGMAR(TH)=E16.9)
 15   IF(KTRL(2)-1) 1900,20,1900
 20   WRITE OUTPUT TAPE 6,119,CHI2ST,CHI2PT,CHI2T
  119 FORMAT (25H0SUM OF CHI SQUARE SIGMA=E16.9/23H0SUM OF CHI SQUARE PO
     1L=E16.9/25H0SUM OF CHI SQUARE TOTAL=E16.9)
 21   CALL SKIP(K,NPGS,NUMRUN)
      WRITE OUTPUT TAPE 6,200
 200  FORMAT (113H        THETA             SIGMATH             SIG-SIGC
     1             POL TH             SIGMA EX             POL EX)
      DO 90 I=1,JMAX
      IF(K-NOLINE)75,70,70
  70  CALL SKIP(K,NPGS,NUMRUN)
  75  K=K+1
      WRITE OUTPUT TAPE 6,32,THETAD(I),SGMATH(I),SRATIO(I),POLTH(I),
     1SGMAEX(I),POLEX(I)
  90  CONTINUE
      GO TO 299
 1900 CALL SKIP (K,NPGS,NUMRUN)
      WRITE OUTPUT TAPE 6,1905
 1905 FORMAT (120H             THETA                       SIGMATH
     1                 SIG-SIGC                       POL TH
     2)
      DO 1920 I=1,JMAX
      IF (K-NOLINE) 1910,1908,1908
 1908 CALL SKIP (K,NPGS,NUMRUN)
 1910 K=K+1
      WRITE OUTPUT TAPE 6,1919,THETAD(I),SGMATH(I),SRATIO(I),POLTH(I)
 1919 FORMAT (1H E20.9,3E30.9)
 1920 CONTINUE
 299  IF(KTRL(6)-1) 300,121,300
 300  IF(KTRL(12)-1) 25,1700,25
 1700 CALL SKIP(K,NPGS,NUMRUN)
      WRITE OUTPUT TAPE 6,1701
 1701 FORMAT  (92H         RHO(I)                FFCR               FFCI
     1               FFSR               FFSI)
      DO 1709 I=1,ILAST
      IF (K-NOLINE) 1703,1702,1702
 1702 CALL SKIP (K,NPGS,NUMRUN)
 1703 WRITE OUTPUT TAPE 6,158,RHO(I),FFCR(I),FFCI(I),FFSR(I),FFSI(I)
  158 FORMAT(1H 5E20.9)
 1709 CONTINUE
  25  IF(KTRL(2)-1) 23,22,23
  22  CALL SKIP (K,NPGS,NUMRUN)
      WRITE OUTPUT TAPE 6,95
  95  FORMAT(120H        THETA            DSIGMA EX            DPOL EX
\end{lstlisting}
%% -----File: 105.png---Folio 100-------
\begin{lstlisting}
     1       CHI SQUARE SIGMA      CHI SQUARE POL     CHI SQUARE TOTAL  )
      DO 120 J=1,JMAX
      IF(K-NOLINE) 97,96,96
  96  CALL SKIP(K,NPGS,NUMRUN)
  97  K=K+1
      WRITE OUTPUT TAPE 6,32,THETAD(J),DSGMEX(J),DPOLEX(J)*CHI2S(J),
     1CHI2P(J))CHI2(J)
 120  CONTINUE
 23   CALL SKIP(K,NPGS,NUMRUN)
 1623 WRITE OUTPUT TAPE 6,1150
 1150 FORMAT (120H           L               REAL C(L+l/2)            IMA
     1G C(L+1/2)             REAL C(L-1/2)            IMAG C(L-1/2)
     2)
      DO 160 L=1,LMAX
      IF (K-NOLINE) 155,153,153
 153  CALL SKIP (K,NPGS,NUMRUN)
  155 K=K+1
      L1=L-1
      WRITE OUTPUT TAPE 6,1156,L1,CR1(L),CI1(L),CR2(L),CI2(L)
 1156 FORMAT (1H I11,E30.9,3E25.9)
  160 CONTINUE
  121 RETURN
\end{lstlisting}
\clearpage

%% -----File: 106.png---Folio 101-------

\begin{lstlisting}
      SUBROUTINE SKIP(K,NPGS,NUMRUN)
      NPGS=NPGS+1
      WRITE OUTPUT TAPE 6,1510, (NUMRUN(I),I=1,5),NPGS
 1510 FORMAT(12H1RUN NUMBER=I2,1H-I2,1H-I4,3H  -I3,3H  -I3,79H
     1                                                                 PA
     2GE 15/)
      K=0
      RETURN





      SUBROUTINE LEAVE
      CALL PDUMP(A,ZZ)
      CALL CTRL4
      RETURN
\end{lstlisting}
\clearpage

%% -----File: 107.png---Folio 102-------

\begin{lstlisting}
*      CARDS   COLUMN
*      FAP
       COUNT   43
*SPILL SUBROUTINE
       ENTRY   SPILL
 SPILL STZ*    1,4          STORE ZERO IN JSPILL
       STZ*    2,4            STORE ZERO IN ISPILL
       STZ     0             STORE ZERO IN LOCATION 00000
       CAL     1,4
       STA     AA41                 SET ADDRESS AA41,
       STA     AA36                    AA36 TO JSPILL
       CAL     2,4                       SET ADDRESS AA31
       STA     AA31                      TO ISPILL
       CLA*    3,4                         SET COMMON STORAGE
       STO     AA45
       CLA*    4,4                        SET COMMON STORAGE
       STO     AA46
       CAL     AA47                PLACE TRANSFER
       SLW     8                 INSTRUCTION IN LOCATION 8
       TRA     5,4                        EXIT TO MAIN PROGRAM
 AA16  LDI     0                  ENTRY IN CASE OF OVER-OR UNDERFLOW
       LFT     4                  TEST FOR OVERFLOW
       TRA     AA36           TRANSFER IN CASE OF OVERFLOW
       LFT     16
       TRA     AA24                     TRANSFER IN CASE OF UNDERFLOW
       TRA*    0                       TRANSFER TO MAIN PROGRAM, NO UFLOW
 AA24  LNT     1                        TEST FOR UNDERFLOW
       TRA*    0                   UNDERFLOW IN AC ONLY
       CAL     0                 PLACE LOCATION AT WHICH
       SUB     AA35              UNDERFLOW OCCURRED IN AC
       LLS     18               SHIFT LEFT 18
 AA31  STD     AA31         STORE IN ISPILL
       CLA     AA46         SET AC, MQ WITH
       LDQ     AA46                  SPECIFIED CONSTANTS
       TRA*    0                  EXIT TO MAIN PROGRAM
 AA35  HTR     1                    CONSTANT
 AA36  CLA     AA36                    TEST IF JSPILL ZERO
       TNZ     AA42             TRANSFER IN CASE JSPILL NON-ZERO
       CAL     0             PLACE LOCATION AT WHICH OVERFLOW OCCURRED
       SUB     AA35                         IN AC
       LLS     18                        SHIFT LEFT 18
 AA41  STD     AA41                   STORE IN JSPILL
 AA42  CLA     AA45                  SET AC,MQ WITH SPECIFIED CONSTANTS
       LDQ     AA45
       TRA*    0                       EXIT TO MAIN PROGRAM
 AA45  HTR     0                       COMMON STORAGE
 AA46  HTR     0                       COMMON STORAGE
 AA47  TRA     AA16               INSTRUCTION TO BE INSERTED AT LOC.  8
       END
\end{lstlisting}

%% -----File: 108.png---Folio 103-------


\Chapter{VII.}{Typical Input and Output}

\Section{A.}{Input Data for Protons against Copper at
\texorpdfstring{$9.75$~\MeV}{9.75 MeV}}
\begin{multicols}{4}
\begin{lstlisting}[basicstyle=\WideDatasize]
    3
   22
 1960
    0
    0
    4
    0
    1
    1
    0
    1
    0
    0
    0
    0
    0
    0
    0
    1
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    1
    1
    1
    1
\end{lstlisting}
%% -----File: 109.png---Folio 104-------
\begin{lstlisting}[basicstyle=\WideDatasize]
    1
    1
    3
+0.62500000 -01
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   10
   32
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+0.13460000 +03
+0.00000000 +00
\end{lstlisting}
%% -----File: 110.png---Folio 105-------
\begin{lstlisting}[basicstyle=\WideDatasize]
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+0.15000000 +00
\end{lstlisting}
%% -----File: 111.png---Folio 106-------
\begin{lstlisting}[basicstyle=\WideDatasize]
+0.15400000 +00
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+0.30000000 -01
\end{lstlisting}
%% -----File: 112.png---Folio 107-------
\begin{lstlisting}[basicstyle=\WideDatasize]
+0.10000000 +30
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  100
\end{lstlisting}
\end{multicols}

%% -----File: 113.png---Folio 108-------


\Section{B.}{Output Listing}

\begin{lstlisting}[basicstyle=\VeryWideDatasize]
RUN NUMBER= 2-40-1961  -  1  - 1                                                                           PAGE   1

PROGRAM NUMBER     4

KTRL( 1)=0
KTRL( 2)=1
KTRL( 3)=1
KTRL( 4)=0
KTRL( 5)=1
KTRL( 6)=0
KTRL( 7)=0
KTRL( 8)=0
KTRL( 9)=0
KTRL(10)=0
KTRL(11)=0
KTRL(12)=0
KTRL(13)=1

BASIC DATA

MSUB1= 0.099999994E 01    MSUBB= 0.639999993E 02     ELAB= 0.974999994E 01     ZZP= 0.289999999E 02

    V= 0.619999997E 02        W= 0.849999994E 01        A= 0.519999996E 00      RO= 0.119999997E 01

   VS=-0.399999999E 01       WS= 0.                                             RC= 0.119999997E 01

          9                                            BG= 0.                   RG= 0.

RHOBN= 0.393980615E 01    RHOBC= 0.323980615E 01   RHOBNG= 0.                  ECM= 0.959999986E 01

  ETA= 0.146788672E 01        K= 0.674959674E 00       KA= 0.350979023E-00      KB= 0.

INTEGRATION DATA

RHOMAX= 0.099999994E 02    LMAXM=   10

NMAX=    3

RHOIN

    0.625000000E-01     0.500000000E 00     0.099999994E 02

DRHOIN

    0.625000000E-01     0.250000000E-00

SIGMAR(TH)= 0.668857820E 02

SUM OF CHI SQUARE SIGMA= 0.587550342E 02

SUM OF CHI SQUARE POL= 0.999665476E 02

SUM OF CHI SQUARE TOTAL= 0.158721581E 03
\end{lstlisting}

%% -----File: 114.png---Folio 109-------

\begin{landscape}
\begin{lstlisting}[basicstyle=\VeryWideDatasize]
RUN NUMBER= 2-40-1961  -  1  -  1                                                                          PAGE     2

       THETA             SIGMATH             SIG-SIGC             POL TH              SIGMA EX              POL EX
    0.151999995E 02     0.366688885E 04     0.948844409E 00     0.624454483E-03     0.386499993E 04     0.
    0.202999994E 02     0.107591100E 04     0.877576292E 00    -0.488765538E-02     0.973399989E 03     0.
    0.253999993E 02     0.437771246E 03     0.864875652E 00    -0.126092605E-01     0.424699992E 03     0.
    0.279999994E 02     0.302826010E 03     0.877252147E 00    -0.152099080E-01     0.                  0.199999996E-01
    0.303999998E 02     0.223364875E 03     0.892697871E 00    -0.161199562E-01     0.226899996E 03     0.
    0.329999998E 02     0.164800696E 03     0.906894624E 00    -0.153716959E-01     0.                 -0.999999985E-02
    0.354999997E 02     0.124848992E 03     0.912098765E 00    -0.130572930E-01     0.134599991E 03     0.
    0.389999993E 02     0.854579188E 02     0.897354133E 00    -0.757811405E-02     0.                  0.299999997E-01
    0.405999996E 02     0.718648233E 02     0.880526960E 00    -0.435241245E-02     0.829199985E 02     0.
    0.430000006E 02     0.552558191E 02     0.843160637E 00     0.115248807E-02     0.                  0.599999994E-01
    0.455999993E 02     0.412997656E 02     0.787652783E 00     0.776244439E-02     0.476599991E 02     0.
    0.469999999E 02     0.351879030E 02     0.752354726E 00     0.114611000E-01     0.                  0.999999993E-01
    0.506999992E 02     0.227685094E 02     0.647046342E 00     0.208112434E-01     0.228699997E 02     0.
    0.514999993E 02     0.206800543E 02     0.623049393E 00     0.225530863E-01     0.                  0.159999996E-00
    0.539999999E 02     0.152632877E 02     0.548359923E 00     0.265358075E-01     0.                  0.199999996E-00
    0.556999996E 02     0.124075061E 02     0.499775782E-00     0.272366613E-01     0.124100000E 02     0.
    0.569999993E 02     0.106054634E 02     0.464955918E-00     0.261134841E-01     0.                  0.169999994E-00
    0.599999994E 02     0.749187976E 01     0.396005623E-00     0.156014524E-01     0.                  0.169999994E-00
    0.607999995E 02     0.686953478E 01     0.380956881E-00     0.104624555E-01     0.645599991E 01     0.
    0.654999994E 02     0.452632517E 01     0.327856168E-00    -0.413961068E-01     0.                  0.999999993E-01
    0.657999992E 02     0.443769753E 01     0.326697305E-00    -0.457080074E-01     0.407499999E 01     0.
    0.689999998E 02     0.381219082E 01     0.331831254E-00    -0.927333571E-01     0.                 -0.999999985E-02
    0.707999989E 02     0.366038024E 01     0.348589554E-00    -0.115995258E-00     0.333899997E 01     0.
    0.754999995E 02     0.365827605E 01     0.434639670E-00    -0.151801050E-00     0.                 -0.199999996E-00
    0.758999996E 02     0.367384672E 01     0.444403417E-00    -0.153142102E-00     0.335599996E 01     0.
    0.808999993E 02     0.392783776E 01     0.588552453E 00    -0.153684869E-00     0.375699997E 01     0.
    0.858999990E 02     0.411561452E 01     0.750190347E 00    -0.136236615E-00     0.385699995E 01     0.
    0.859999999E 02     0.411754631E 01     0.753359631E 00    -0.135786600E-00     0.                 -0.129999995E-00
    0.908999994E 02     0.410156414E 01     0.894658349E 00    -0.110665120E-00     0.384599999E 01     0.
    0.954999998E 02     0.388636135E 01     0.986762404E 00    -0.826589502E-01     0.                 -0.699999988E-01
    0.958999991E 02     0.385919407E 01     0.992326975E 00    -0.800342456E-01     0.375699997E 01     0.
    0.099999994E 03     0.351563454E 01     0.102388248E 01    -0.512336008E-01     0.                  0.199999996E-01
\end{lstlisting}
\end{landscape}

%% -----File: 115.png---Folio 110-------

\begin{landscape}
\begin{lstlisting}[basicstyle=\VeryWideDatasize]
RUN NUMBER= 2-40-1961  -  1  -  1                                                                           PAGE    3
       THETA            DSIGMA EX            DPOL EX         CHI SQUARE SIGMA     CHI SQUARE POL     CHI SQUARE TOTAL
    0.151999995E 02     0.397999994E 03     0.099999994E 30     0.247771524E-00      0.                  0.247771524E-00
    0.202999994E 02     0.354999997E 02     0.099999994E 30     0.833843596E 01      0.                  0.833843596E 01
    0.253999993E 02     0.166999996E 02     0.099999994E 30     0.612634748E 00      0.                  0.612634748E 00
    0.279999994E 02     0.099999994E 30     0.299999997E-01     0.                   0.137748629E 01     0.137748629E 01
    0.303999998E 02     0.907999992E 01     0.099999994E 30     0.151578002E-00      0.                  0.151578002E-00
    0.329999998E 02     0.099999994E 30     0.399999991E-01     0.                   0.180344537E-01     0.180344537E-01
    0.354999997E 02     0.537999995E 01     0.099999994E 30     0.328498974E 01      0.                  0.328498974E 01
    0.389999993E 02     0.099999994E 30     0.299999997E-01     0.                   0.156901620E 01     0.156901620E 01
    0.405999996E 02     0.372999996E 01     0.099999994E 30     0.878443092E 01      0.                  0.878443092E 01
    0.430000000E 02     0.099999994E 30     0.299999997E-01     0.                   0.384781063E 01     0.384781063E 01
    0.455999993E 02     0.191000000E 01     0.099999994E 30     0.110886693E 02      0.                  0.110886693E 02
    0.469999999E 02     0.099999994E 30     0.299999997E-01     0.                   0.871015161E 01     0.871015161E 01
    0.506999992E 02     0.914999999E 00     0.099999994E 30     0.123029307E-01      0.                  0.123029307E-01
    0.514999993E 02     0.099999994E 30     0.399999991E-01     0.                   0.118072823E 02     0.118072823E 02
    0.539999999E 02     0.099999994E 30     0.399999991E-01     0.                   0.188061401E 02     0.188061401E 02
    0.556999996E 02     0.495999999E-00     0.099999994E 30     0.232838377E-04      0.                  0.232838377E-04
    0.569999993E 02     0.099999994E 30     0.399999991E-01     0.                   0.129395790E 02     0.129395790E 02
    0.599999994E 02     0.099999994E 30     0.299999997E-01     0.                   0.264876761E 02     0.264876761E 02
    0.607999995E 02     0.257999994E-00     0.099999994E 30     0.256912217E 01      0.                  0.256912217E 01
    0.654999994E 02     0.099999994E 30     0.499999993E-01     0.                   0.799714297E 01     0.799714297E 01
    0.657999992E 02     0.162999995E-00     0.099999994E 30     0.495124198E 01      0.                  0.495124198E 01
    0.689999998E 02     0.099999994E 30     0.399999991E-01     0.                   0.427800477E 01     0.427800477E 01
    0.707999989E 02     0.133999996E-00     0.099999994E 30     0.575213231E 01      0.                  0.575213231E 01
    0.754999995E 02     0.099999994E 30     0.599999994E-01     0.                   0.645316236E 00     0.645316236E 00
    0.758999996E 02     0.133999996E-00     0.099999994E 30     0.562634163E 01      0.                  0.562634163E 01
    0.808999993E 02     0.149999999E-00     0.099999994E 30     0.129713513E 01      0.                  0.129713513E 01
    0.858999990E 02     0.153999999E-00     0.099999994E 30     0.282009937E 01      0.                  0.282009937E 01
    0.859999999E 02     0.099999994E 30     0.599999994E-01     0.                   0.930132821E-02     0.930132821E-02
    0.908999994E 02     0.153999999E-00     0.099999994E 30     0.275396556E 01      0.                  0.275396556E 01
    0.954999998E 02     0.099999994E 30     0.499999993E-01     0.                   0.640996160E-01     0.640996180E-01
    0.958999991E 02     0.149999999E-00     0.099999994E 30     0.464161523E-00      0.                  0.464161523E-00
    0.099999994E 03     0.099999994E 30     0.599999994E-01     0.                   0.140950717E 01     0.140950717E 01
\end{lstlisting}
\end{landscape}

%% -----File: 116.png---Folio 111-------

\begin{landscape}
\begin{lstlisting}[basicstyle=\VeryWideDatasize]
RUN NUMBER= 2-40-1961    -  1  -  1                                                                        PAGE     4
         L                REAL C(L+1/2)            IMAG C(L+1/2)           REAL C(L-1/2)            IMAG C(L-1/2)
          0               -0.149473831E-00          0.621800341E 00        -0.139704145E-00          0.618552327E 00
          1               -0.890974633E-01          0.280818105E-00        -0.112762213E-00          0.266789824E-00
          2               -0.187045686E-00          0.241325634E-00        -0.174136B25E-00          0.274325125E-00
          3                0.670826085E-01          0.135792047E-00         0.546362303E-01          0.848520368E-01
          4                0.756759964E-02          0.149855547E-01         0.229124613E-02          0.170210496E-01
          5                0.503797509E-02          0.559476413E-02         0.384988777E-02          0.176129699E-02
          6                0.628714196E-03          0.123979807E-03         0.362633042E-03          0.807685591E-04
          7                0.627130263E-04          0.826003924E-05         0.370286375E-04          0.729130283E-05
          8                0.713366367E-05          0.780125931E-06         0.441690803E-05          0.750615060E-06
          9                0.130098701E-05          0.808378339E-07         0.101125993E-05          0.797655620E-07
         10                0.413292557E-06          0.833891876E-08         0.381933421E-06          0.844628319E-08
\end{lstlisting}
\end{landscape}

%% -----File: 117.png---Folio 112-------


\Chapter{VIII.}{Further Subroutines and Programs in Preparation}

The following subroutines are presently being prepared at UCLA:

\Subroutine{\Code{TV}}

This subroutine is designed to output on CRT and on film various required
curves such as $\sigma(\theta)$ vs~$\theta$, $\sigma(\theta)/\sigma_c(\theta)$ vs~$\theta$, $P(\theta)$ vs~$\theta$.

\Subroutine{\Code{RHOBEG}}

This subroutine will make use of the quantities \Code{IIN(L)} to allow the
numerical integrations to start at different values of~$\rho$ depending upon~$\ell$
in order to speed up the numerical integration.

\Subroutine{\Code{FLUX}}

This subroutine will if desired compute the normalized total wave functions,
the scattered flux~$\vec{j}$, the divergence and the curl of~$\vec{j}$ at specified
values of~$\rho$ and~$\theta$.

All the above subroutines will of course require some modification of
the basic program.

The following programs are presently being prepared at UCLA:

\Program{\SCAT{3}}

This program will be similar to program \SCAT{4} except that it will treat
incident and target particles of zero spin, thus speeding up the calculation
for that case.

\Program{\SCAT{5}}

This is a modified version of program \SCAT{4} offering a simplified input
and using only as many $\ell$'s as may be significant in the $C_\ell$'s calculations.

\Program{\SCAT{K}}

This is a modified version of program \SCAT{4} designed to analyze the scattering
of K-mesons against complex nuclei, including the use of an approximate
Klein-Gordon equation, relativistic kinematic corrections, and averaging of
the cross sections over angles, energies, and representative nuclei.

%% -----File: 118.png---Folio 113-------

\Program{\SCAT{6}}

This is a modified version of program \SCAT{4} designed to calculate cross
sections and polarization of spin~$1$ particles scattered by $0$~spin targets.

\Program{\Acro{SEEK~4}}

This is a program designed to search automatically the parameter space
so as to minimize $\chi^{2}$.

%% -----File: 119.png---Folio 114-------

\clearpage
\section*{\centering\normalsize\bfseries BIBLIOGRAPHY OF DIFFUSE SURFACE OPTICAL MODEL
ANALYSES BY MACHINE CALCULATIONS}
\phantomsection\pdfbookmark[0]{Bibliography}{Bibliography}

\begin{Biblio}
\item[] R.~D.~Albert, $(p,n)$ Cross Sections and Proton Optical-Model Parameters in
  4- to 5.5-\MeV\ Energy Region, UCRL-5488 (1959).

\item[] H.~J.~Amster, Optical Model Evidence for Surface Absorption of Neutrons,
  Phys.\ Rev.\ \textbf{113}, 911 (1959).

\item[] H.~J.~Amster and L.~M.~Culpepper, Surface Modified Nuclear Optical Model:
  Description of the \Code{SUMNUM} Code for the \Acro{NORC} Computer, WAPD-TM-87,
  Bettis Plant of the AEC, Pittsburgh, Pa.

\item[] Baker, Byfield, and Rainwater, Theoretical Calculations of the Scattering
  of $\pi$-Mesons by Complex Nuclei, Phys.\ Rev.\ \textbf{112}, 1773 (1958).

\item[] H.~R.~Beyster, Predictions of Fast Neutrons Scattering Data with a Diffuse
  Surface Potential Well, LA-2099 (1956).

\item[] Beyster, Walt and Salmi, Interaction of 1.0-, 2.5-, 3.25-, and 7.0-\MeV\
  Neutrons with Nuclei  Phys.\ Rev.\ \textbf{104}, 1319 (1956).

\item[] Bjorklund, Blandford and Fernbach, Analysis of Elastic Scattering and
  Polarization of 300-\MeV\ Protons, Phys.\ Rev.\ \textbf{108}, 795 (1957).

\item[] F.~Bjorklund and S.~Fernbach, Elastic Scattering of 7-\MeV\ Neutrons
  (Theoretical Curves), UCRL-4927-T (1957).

\item[] F.~Bjorklund and S.~Fernbach, Optical-Model Analysis of Scattering of 4.1-,
  7-, and 14-\MeV\ Neutrons by Complex Nuclei, Phys.\ Rev.\ \textbf{109}. 1295 (1958).

\item[] F.~Bjorklund and S.~Fernbach, Exact Phase-Shift Calculation for Nucleon-Nuclear
  Scattering, UCRL-5028 (1958).

\item[] Bjorklund, Fernbach and Sherman, Optical Model of Nucleus with Absorbing
  Surface, Phys.\ Rev.\ \textbf{101}, 1832 (1956).

\item[] W.~B.~Cheston and A.~E.~Glassgold, Elastic Scattering % [** PP: Typo Scatering]
  of Alpha-Particles with
  the Optical Model, Phys.\ Rev.\ \textbf{106}, 1215 (1957).

\item[] Culler, Fernbach and Sherman, Optical Model Analysis of Scattering of 14-\MeV\
  Neutrons, Phys.\ Rev.\ \textbf{101}, 1047 (1956).

\item[] Eisberg, Gugelot and Porter, Conference on the Statistical Aspects of the
  Nucleus, Brookhaven National Laboratory (1955).

\item[] W.~S.~Emmerich, Cross Section Calculations for Fast Neutron Scattering,
  Westinghouse Research Report 60-94511-6-R17 (1957).

\item[] W.~S.~Emmerich, Optical Model Theory of Neutron Scattering and Reactions,
  Westinghouse Research Report 6-94511-6-R20 (1958).

%% -----File: 120.png---Folio 115-------

\item[] Franklin, Margolis and Oberthal, Scattering of $\mu$-Mesons by Nuclei, Phys.\ Rev.\
  \textbf{111}, 296 (1958).

\item[] Glassgold, Cheston, Stein, Schuld and Erickson, Analysis of Proton-Nucleus
  Scattering at 9.8~\MeV, Phys.\ Rev.\ \textbf{106}, 1207 (1958).

\item[] A.~E.~Glassgold, Interaction of Antiprotons with Complex Nuclei, Phys.\ Rev.\
  \textbf{110}, 220 (1958).

\item[] A.~E.~Glassgold and P.~J.~Kellogg, Proton-Nucleus Scattering at 17~\MeV,
  Phys.\ Rev.\ \textbf{107}, 1372 (1957).

\item[] A.~E.~Glassgold and P.~J.~Kellogg, Nuclear Scattering of 40- and 95-\MeV\
  Protons, Phys.\ Rev.\ \textbf{109}, 1291 (1958).

\item[] Green, Porter and Saxon, Proceedings of the International Conference on the
  Nuclear Optical Model, Florida State University, Tallahassee (1959).

\item[] G.~Igo, Optical-Model Analysis of the Elastic Scattering of Alpha Particles,
  Phys.\ Rev.\ \textbf{106}, 126 (1957).

\item[] G.~Igo, Optical Model Potential at the Nuclear Surface for the Elastic
  Scattering of Alpha Particles, Phys.\ Rev.\ Let.\ \textbf{1}, 72 (1958).

\item[] G.~Igo, Optical Model Analysis of the Scattering of Alpha Particles from
  Helium (in press).

\item[] G.~Igo, Optical-Model Analysis of Excitation Function Data and Theoretical
  Reaction Cross Sections for Alpha Particles, Phys.\ Rev.\ \textbf{115}, 1665 (1959).

\item[] Igo, Ravenhall, Tiemann, Chupp, Goldhaber, Goldhaber, Lanutti and Thaler,
  The Scattering of $K^+$-Mesons in Emulsion, Phys.\ Rev.\ \textbf{109}, 2133 (1958).

\item[] R.~Jastrow and I.~Harris, Nuclear Cross Sections for the Scattering of
  Neutrons and Protons, Proceedings of the ONR Decennial Symposium (1957).

\item[] Lukyanov, Orlov and Turovstev, Optical Model of the Interaction between
  Intermediate Energy Neutrons and Nuclei, Nucl.\ Phys.\ \textbf{8}, 325 (1958).

\item[] I.~E.~McCarthy, Flux of Particles in the Optical Model, Nucl.\ Phys.\ \textbf{10},
  583 (1959).

\item[] Melkanoff, Moszkowski, Nodvik and Saxon, Energy Dependence of the Optical
  Model Parameters, Phys.\ Rev.\ \textbf{101}, 507 (1956).

\item[] Melkanoff, Nodvik and Saxon, Diffuse-Surface Optical Model Analysis of
  Elastic Scattering of 17- and 31.5-\MeV\ Protons, Phys.\ Rev.\ \textbf{106}, 793 (1957).

\item[] Melkanoff, Price, Stork and Ticho, Optical Model Analysis of Elastic
  Scattering of 125-\MeV\ $K^+$-Mesons in Nuclear Emulsions, Phys.\ Rev.\ \textbf{113},
  1303 (1959).

\item[] University of Minnesota Annual Progress Report 1956-1957, 1957-1958, 1958,
  University of Minnesota Linear Accelerator Laboratory, Minneapolis, Minn.

%% -----File: 121.png---Folio 116-------

\item[] J.~S.~Nodvik and D.~S.~Saxon, Analysis of Elastic Cross Sections and
  Polarization of 10 \MeV\ Protons (in press).

\item[] C.~E.~Porter, Nitrogen-Nitrogen Elastic Scattering, Phys.\ Rev.\ \textbf{112}, 1722
  (1958).

\item[] H.~M.~Shey, Scattering of Neutrons by Non-spherical Nuclei, Phys.\ Rev.\
  \textbf{113}, 900 (1959).

\item[] R.~D.~Woods and D.~S.~Saxon, Diffuse Surface Optical Model for Nucleon-Nuclei
  Scattering, Phys.\ Rev.\ \textbf{95}, 577 (1954).
\end{Biblio}

%%%%%%%%%%%%%%%%%%%%%%% BACK MATTER %%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage

\phantomsection
\pdfbookmark[0]{PG License}{Project Gutenberg License}
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Scattering Analyses with the Nuclear Optical Model, by Michel A. Melkanoff
and David S. Saxon and John S. Nodvik and David G. Cantor

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% End of the Project Gutenberg EBook of A Fortran Program for Elastic     %
% Scattering Analyses with the Nuclear Optical Model, by Michel A. Melkanoff
% and David S. Saxon and John S. Nodvik and David G. Cantor               %
%                                                                         %
% *** END OF THIS PROJECT GUTENBERG EBOOK ELASTIC SCATTERING ANALYSES *** %
%                                                                         %
% ***** This file should be named 29784-t.tex or 29784-t.zip *****        %
% This and all associated files of various formats will be found in:      %
%         http://www.gutenberg.org/2/9/7/8/29784/                         %
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[104] [105] (./29784-t.aux)

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listings.sty    2004/10/17 1.3b (Carsten Heinz)
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geometry.sty    2002/07/08 v3.2 Page Geometry
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hyperref.sty    2007/02/07 v6.75r Hypertext links for LaTeX
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hyperref.cfg    2002/06/06 v1.2 hyperref configuration of TeXLive
kvoptions.sty    2006/08/22 v2.4 Connects package keyval with LaTeX options (HO
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lstlang1.sty    2004/09/05 1.3 listings language file
lstlang1.sty    2004/09/05 1.3 listings language file
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supp-pdf.tex
 nameref.sty    2006/12/27 v2.28 Cross-referencing by name of section
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 29784-t.out
 29784-t.out
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