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diff --git a/29784-t/29784-t.tex b/29784-t/29784-t.tex new file mode 100644 index 0000000..3910bf8 --- /dev/null +++ b/29784-t/29784-t.tex @@ -0,0 +1,7804 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% 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] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK ELASTIC SCATTERING ANALYSES *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{29784} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Standard document class. Required. %% +%% %% +%% inputenc: Standard DP encoding. Required. %% +%% fontenc: Font encoding, for bold smallcaps. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: AMS mathematics symbols. Required. %% +%% mathrsfs: AMS script fonts. Required. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% longtable: Multi-page tables. Required. %% +%% array: Enhanced tabular environment. Required. %% +%% %% +%% multicol: Multi-column environment for index. Required. %% +%% lscape: For data tables in landscape orientation. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% textcomp: Ditto mark. Required. %% +%% %% +%% indentfirst: Indent units' first paragraph. Required. %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% listings: Prettified Fortran code. Required. %% +%% %% +%% rotating: Graphics rotation. Required. %% +%% tikz: Extended picture macros. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% %% +%% %% +%% Producers's Comments: The text block is fairly wide (6.3 in) to %% +%% accommodate wide displayed equations and program listings. %% +%% Nonetheless, the listing on page 93 gives five overfull hbox %% +%% warnings, the largest almost 91 pt. %% +%% %% +%% The original text is a typed manuscript. Every effort has %% +%% been made to capture the authors' semantic intent with %% +%% regard to sectioning. %% +%% %% +%% Things to Check: The book contains nine landscape pages -- six %% +%% plots, and three pages of program output -- which should %% +%% automatically float to good page breaks. No text runs off %% +%% the page. %% +%% %% +%% %% +%% Spellcheck: OK %% +%% Smoothreading pool: No %% +%% lacheck: OK, about 100 false positives %% +%% %% +%% Lprep/gutcheck: OK %% +%% PDF pages: 111 (if ForPrinting set to false) %% +%% PDF page size: US Letter (8.5" x 11") %% +%% PDF bookmarks: created, point to ToC entries %% +%% PDF document info: filled in %% +%% ToC page numbers: OK %% +%% Images: 8 eepic files written with tikz macros %% +%% %% +%% Summary of log file: %% +%% * 4 underfull hboxes %% +%% * 13 overfull hboxes (largest ~90pt) %% +%% * 20 underfull vboxes %% +%% %% +%% %% +%% Compile History: %% +%% %% +%% May, 2009: adhere (Andrew D. Hwang) %% +%% texlive2007, GNU/Linux %% +%% %% +%% Command block: %% +%% %% +%% pdflatex x2 %% +%% %% +%% %% +%% August 2009: pglatex. %% +%% Compile this project with: %% +%% pdflatex 29784-t.tex ..... TWO times %% +%% %% +%% pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% %%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\listfiles +\documentclass[12pt,letterpaper]{book}[2005/09/16] + +\usepackage[utf8]{inputenc}[2006/05/05] +\usepackage[T1]{fontenc} +\usepackage{amssymb}[2002/01/22] +\usepackage{amsmath}[2000/07/18] +\usepackage{mathrsfs}[1996/01/01] + +\usepackage{fancyhdr} +\usepackage{longtable}[2004/02/01] +\usepackage{array}[2005/08/23] + +\usepackage{multicol}[2006/05/18] +\usepackage{lscape}[2000/10/22] + +\usepackage{alltt}[1997/06/16] +\usepackage{textcomp}[2005/09/27] + +\usepackage{indentfirst}[1995/11/23] +\usepackage{ifthen}[2001/05/26] + +\usepackage{listings}[2004/10/17] + +\usepackage{rotating}[1997/09/26] +\usepackage{tikz}[2006/10/17] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Interlude: Set up PRINTING (default) or SCREEN VIEWING %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% ForPrinting=true (default) false +% Asymmetric margins Symmetric margins +% Black hyperlinks Blue hyperlinks +% Even-odd page headings Uniform page headings +% Start Preface, ToC, etc. recto No blank verso pages +% +\newboolean{ForPrinting} + +%% *** UNCOMMENT the next line for a PRINT-OPTIMIZED VERSION *** %% +%\setboolean{ForPrinting}{true} + +%% Initialize values to ForPrinting=false +\newcommand{\Margins}{hmarginratio=1:1} % Symmetric margins +\newcommand{\HLinkColor}{blue} % Hyperlink color +\newcommand{\PDFPageLayout}{SinglePage} +\newcommand{\TransNote}{Transcriber's Note} +\newcommand{\TransNoteCommon}{% + Minor typographical corrections, changes to the presentational + style, and regularizations of spelling and hyphenation have been + made without comment. +% All such changes are [** PP: Noted] in this source file + \bigskip + + Every effort has been made to remove OCR errors from the \FORTRAN\ + code. +} +\newcommand{\TransNoteText}{% + \TransNoteCommon + \bigskip + + This PDF file is formatted for screen viewing, but may be easily + recompiled for printing. Please see the preamble of the \LaTeX\ + source file for instructions. +} + +%% Re-set if ForPrinting=true +\ifthenelse{\boolean{ForPrinting}}{% + \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins + \renewcommand{\HLinkColor}{black} % Hyperlink color + \renewcommand{\PDFPageLayout}{TwoPageRight} + \renewcommand{\TransNote}{Transcriber's Note} + \renewcommand{\TransNoteText}{% + \TransNoteCommon + \bigskip + + This PDF file is formatted for printing, but may be easily + recompiled for screen viewing. Please see the preamble of the + \LaTeX\ source file for instructions. + } +}{% If ForPrinting=false, don't skip to recto + \renewcommand{\cleardoublepage}{\clearpage} +} +%%%% End of PRINTING/SCREEN VIEWING code; back to packages %%%% + + +% [** N.B. There are four line breaks tuned to 5in text width] +\usepackage[body={6.3in,9in},\Margins]{geometry}[2002/07/08] + +%%%% PDF and hyperref options %%%% +\providecommand{\ebook}{00000} % Overridden during white washing +\usepackage[pdftex, + hyperfootnotes=false, + pdftitle={The Project Gutenberg eBook \#\ebook: A Fortran program for elastic scattering analyses with the nuclear optical model}, + pdfauthor={Michel A. Melkanoff, John S. Nodvik, David S. Saxon, David G. Cantor}, + pdfkeywords={David Starner, Andrew D. Hwang, + Project Gutenberg Online Distributed Proofreading Team}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value; if true, see @@@@ below + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=1, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2007/02/07] + + +%%%% lstlistings setup %%%% +\lstloadlanguages{fortran} +\lstset{language=[77]fortran, + basicstyle=\footnotesize, + aboveskip=0pt, + belowskip=0pt} % Abut adjacent environments + +% listings basic size for program output +\newcommand{\WideDatasize}{\scriptsize} +\newcommand{\VeryWideDatasize}{\scriptsize} + +%%%% ToC setup %%%% +\newcommand{\SecHeading}[1]{\centering\Large\bfseries\scshape #1} + +\AtBeginDocument{\renewcommand{\contentsname}% + {\protect\SecHeading{Table of Contents}\protect\\[0pt]}} + +\newcommand{\ToCBox}[1]{\makebox[3em][l]{#1}} + +% No page numbers +\AtBeginDocument{\addtocontents{toc}{\protect\thispagestyle{empty}}} +\newcommand{\TableofContents}{{\small\tableofcontents}} + + +%%%% Sectional divisions %%%% +\newcommand{\ChapSkip}{\vspace*{0.25in}} + +% Must handle manual linebreak +\newcommand{\Chapter}[3][] + {\clearpage + \thispagestyle{empty} + \section*{\SecHeading{#2~#3}} + \ChapSkip + \label{chapter:#2}% + \ifthenelse{\equal{#1}{}}% + {\addcontentsline{toc}{chapter}% No optional argument + {\texorpdfstring{\protect\ToCBox{#2} #3}{#2 #3}}}% + {\addcontentsline{toc}{chapter}% Optional argument + {\texorpdfstring{\protect\ToCBox{#2} #3}{#2 #1}}}% +} + +\newcommand{\Section}[2] + {\section*{\normalfont\large\bfseries\scshape#1~#2} + \addcontentsline{toc}{section}% + {\texorpdfstring{#1~#2}{#1 #2}}% +} + +% Must handle manual linebreak +\newcommand{\Subsection}[3][] + {\subsection*{\normalfont\normalsize\bfseries\scshape#2~#3} + \ifthenelse{\equal{#1}{}}% + {\addcontentsline{toc}{subsection}% No optional argument + {\texorpdfstring{#2~#3}{#2 #3}}}% + {\addcontentsline{toc}{subsection}% Optional argument + {\texorpdfstring{#2~#3}{#2 #1}}}% +} + +\newcommand{\Subsubsection}[2] + {\subsubsection*{\normalfont\normalsize#1~\bfseries\scshape#2}} + +\makeatletter +\newcommand{\subsubsubsection} +{\@startsection{subsubsubsection}{4}{\parindent}% + {-0.5\baselineskip}{0pt}{\normalfont\normalsize}}% run-in heading + +\newcommand{\subsubsubsubsection} +{\@startsection{subsubsubsubsection}{4}{0pt}% + {0.25\baselineskip}{0.01pt}{\normalfont\normalsize}}% +\makeatother + +\newcommand{\Subsubsubsection}[2]{\subsubsubsection*{#1~\scshape#2}\quad} + +\newcommand{\Note}[1] + {\ifthenelse{\not\equal{#1}{}} + {\paragraph{Note~#1:}} + {\paragraph{Note:}}} + + +% Standard (sub-)list formatting parameters +\newcommand{\ListInit}{% + \setlength{\leftmargin}{\parindent}% + \setlength{\labelwidth}{0pt}% + \setlength{\labelsep}{0pt}% + \setlength{\itemindent}{-\parindent}% + \setlength{\itemsep}{0pt}% +} + +% General-purpose replacement for itemize (with bold labels) +\newenvironment{Biblio}{\begin{list}{}{\ListInit}}{\end{list}} + + +%%%% Miscellaneous formatting %%%% +%%%% Fixed-width environment to format PG boilerplate %%%% +% 9.2pt leaves no overfull hbox at 80 char line width +\newenvironment{PGtext}{% +\begin{alltt} +\fontsize{9.2}{10.5}\ttfamily\selectfont}% +{\end{alltt}} + +% For titlepage +\newcommand{\Author}[2] + {\begin{tabular}{l}% + \textbf{#1} \\ + {\footnotesize#2} + \end{tabular}} + +% For equation alignment; set #2 in a box as wide as #1 +\newlength{\TmpLen} +\newcommand{\PadTo}[2]% + {\settowidth{\TmpLen}{$#1$}% + \makebox[\TmpLen][r]{$#2$}} + +\newcommand{\Tag}[1]{\phantomsection\label{eqn:#1}\tag{#1}} +\newcommand{\Eqno}[1]{\hyperref[eqn:#1]{(#1)}} + +% \Eqref{n} - eqnation~(n); +% \Eqref[cap]{n} - Equation~(n); +% \Eqref[abbr]{n} - eq.~(n) +\newcommand{\Eqref}[2][]% +{\ifthenelse{\equal{#1}{}}% + {\hyperref[eqn:#2]{equation~(#2)}}% + {\ifthenelse{\equal{#1}{cap}}% + {\hyperref[eqn:#2]{Equation~(#2)}}% + {\ifthenelse{\equal{#1}{abbr}}% + {\hyperref[eqn:#2]{eq.~(#2)}}{}}}}% + +\newcommand{\Eqrefs}[2][]% +{\ifthenelse{\equal{#1}{}}% + {\hyperref[eqn:#2]{equations~(#2)}}% + {\ifthenelse{\equal{#1}{cap}}% + {\hyperref[eqn:#2]{Equations~(#2)}}% + {\ifthenelse{\equal{#1}{abbr}}% + {\hyperref[eqn:#2]{eqs.~(#2)}}{}}}}% + + +% \Figure{4}{028a}: Figure 4 made from 028a.eepic +\newcommand{\Figure}[2] +{\begin{figure*} +\null\vfill +\begin{center}\begin{minipage}{\linewidth} +\rotatebox{90}{\input{./images/#2.eepic}} +\end{minipage} +\phantomsection\label{figure:#1} +\end{center} +\vfill +\end{figure*} +} + +\newcommand{\ChapRef}[1]{\hyperref[chapter:#1.]{chapter~#1}} +\newcommand{\FigRef}[1]{\hyperref[figure:#1]{#1}} + +\newcommand{\Program}[1]{\subsubsubsubsection*{\bfseries Program~#1}} +\newcommand{\Routine}[1]{\subsubsubsubsection*{\bfseries #1}} +\newcommand{\Subroutine}[1]{\subsubsubsubsection*{\bfseries Subroutine~#1}} + + +%%%% Miscelaneous macros %%%% +\DeclareMathOperator{\Imag}{Im} +\DeclareMathOperator{\Real}{Re} + +\newcommand{\Acro}[1]{\texttt{#1}}% Acronyms +\newcommand{\Code}[1]{\text{\lstinline@#1@}} + +\newcommand{\Neg}{\phantom{-}} +\newcommand{\FORTRAN}{\Acro{FORTRAN}} +\newcommand{\SCAT}[1]{\Acro{SCAT~#1}} + +\newcommand{\abs}{\text{abs}} +\newcommand{\CI}{\text{CI}} +\newcommand{\CR}{\text{CR}} +\newcommand{\inc}{\text{inc}} +\newcommand{\initial}{\text{initial}} +\newcommand{\LAB}{\text{LAB}} +\newcommand{\scatt}{\text{scatt}} +\newcommand{\SI}{\text{SI}} +\newcommand{\SR}{\text{SR}} +\newcommand{\total}{\text{total}} + +\newcommand{\MeV}{MeV} +\newcommand{\Y}{\, \mathscr{Y}} + +\newcommand{\Ditto}{\textquotestraightdblbase} + +% \ell or -\ell-1; must be called in math mode +\newcommand{\ellAlt} + {\biggl({\begin{smallmatrix}%{c} + \ell \\ \text{or}\vphantom{\big|} \\ -\ell-1 + \end{smallmatrix}}\biggr)} + + +%% Globally adjust size of super- and sub-scripts +\makeatletter +\DeclareMathSizes{\@xiipt}{\@xiipt}{10}{8} +\makeatother + +\renewcommand{\arraystretch}{1.2} % loosen array rows +\renewcommand{\headrulewidth}{0pt} + +\DeclareInputMath{183}{\cdot} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagestyle{empty} +\pagenumbering{roman} + +%%%% PG BOILERPLATE %%%% +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\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] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK ELASTIC SCATTERING ANALYSES *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by David Starner, Andrew D. Hwang, and the Online +Distributed Proofreading Team at http://www.pgdp.net +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{center} +\begin{minipage}{0.66\textwidth} +\small +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% +\phantomsection +\pdfbookmark[1]{Transcriber's Note}{Transcriber's Note} + +\raggedright +\TransNoteText +\end{minipage} +\end{center} + + +%% -----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} +\pagestyle{fancy} +\fancyhead{} +\fancyfoot{} +\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 ++0.10000000 +01 ++0.64000000 +02 ++0.97500000 +01 ++0.29000000 +02 ++0.12000000 +01 ++0.62000000 +02 ++0.85000000 +01 ++0.12000000 +01 ++0.52000000 +00 +-0.40000000 +01 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 + 1 + 1 + 1 + 1 +\end{lstlisting} +%% -----File: 109.png---Folio 104------- +\begin{lstlisting}[basicstyle=\WideDatasize] + 1 + 1 + 3 ++0.62500000 -01 ++0.50000000 +00 ++0.10000000 +02 ++0.62500000 -01 ++0.25000000 +00 + 10 + 32 ++0.15200000 +02 ++0.20300000 +02 ++0.25400000 +02 ++0.28000000 +02 ++0.30400000 +02 ++0.33000000 +02 ++0.35500000 +02 ++0.39000000 +02 ++0.40600000 +02 ++0.43000000 +02 ++0.45600000 +02 ++0.47000000 +02 ++0.507000000+02 ++0.51500000 +02 ++0.54000000 +02 ++0.55700000 +02 ++0.57000000 +02 ++0.60000000 +02 ++0.60800000 +02 ++0.65500000 +02 ++0.65800000 +02 ++0.69000000 +02 ++0.70800000 +02 ++0.75500000 +02 ++0.75900000 +02 ++0.80900000 +02 ++0.85900000 +02 ++0.86000000 +02 ++0.90900000 +02 ++0.95500000 +02 ++0.95900000 +02 ++0.10000000 +03 ++0.38650000 +04 ++0.97340000 +03 ++0.42470000 +03 ++0.00000000 +00 ++0.22690000 +03 ++0.00000000 +00 ++0.13460000 +03 ++0.00000000 +00 +\end{lstlisting} +%% -----File: 110.png---Folio 105------- +\begin{lstlisting}[basicstyle=\WideDatasize] ++0.82920000 +02 ++0.00000000 +00 ++0.47660000 +02 ++0.00000000 +00 ++0.22870000 +02 ++0.00000000 +00 ++0.00000000 +00 ++0.12410000 +02 ++0.00000000 +00 ++0.00000000 +00 ++0.64560000 +01 ++0.00000000 +00 ++0.40750000 +01 ++0.00000000 +00 ++0.33390000 +01 ++0.00000000 +00 ++0.33560000 +01 ++0.37570000 +01 ++0.38570000 +01 ++0.00000000 +00 ++0.38460000 +01 ++0.00000000 +00 ++0.37570000 +01 ++0.00000000 +00 ++0.39800000 +03 ++0.35500000 +02 ++0.16700000 +02 ++0.10000000 +30 ++0.90800000 +01 ++0.10000000 +30 ++0.53800000 +01 ++0.10000000 +30 ++0.37300000 +01 ++0.10000000 +30 ++0.19100000 +01 ++0.10000000 +30 ++0.91500000 +00 ++0.10000000 +30 ++0.10000000 +30 ++0.49600000 +00 ++0.10000000 +30 ++0.10000000 +30 ++0.25800000 +00 ++0.10000000 +30 ++0.16300000 +00 ++0.10000000 +30 ++0.13400000 +00 ++0.10000000 +30 ++0.13400000 +00 ++0.15000000 +00 +\end{lstlisting} +%% -----File: 111.png---Folio 106------- +\begin{lstlisting}[basicstyle=\WideDatasize] ++0.15400000 +00 ++0.10000000 +30 ++0.15400000 +00 ++0.10000000 +30 ++0.15000000 +00 ++0.10000000 +30 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 +-0.20000000 -01 ++0.00000000 +00 ++0.10000000 -01 ++0.00000000 +00 +-0.30000000 -01 ++0.00000000 +00 +-0.60000000 -01 ++0.00000000 +00 +-0.10000000 +00 ++0.00000000 +00 +-0.16000000 +00 +-0.20000000 +00 ++0.00000000 +00 +-0.17000000 +00 +-0.17000000 +00 ++0.00000000 +00 +-0.10000000 +00 ++0.00000000 +00 ++0.10000000 -01 ++0.00000000 +00 ++0.20000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.00000000 +00 ++0.13000000 +00 ++0.00000000 +00 ++0.70000000 -01 ++0.00000000 +00 +-0.20000000 -01 ++0.10000000 +30 ++0.10000000 +30 ++0.10000000 +30 ++0.30000000 -01 ++0.10000000 +30 ++0.40000000 -01 ++0.10000000 +30 ++0.30000000 -01 ++0.10000000 +30 ++0.30000000 -01 ++0.10000000 +30 ++0.30000000 -01 +\end{lstlisting} +%% -----File: 112.png---Folio 107------- +\begin{lstlisting}[basicstyle=\WideDatasize] ++0.10000000 +30 ++0.40000000 -01 ++0.40000000 -01 ++0.10000000 +30 ++0.40000000 -01 ++0.30000000 -01 ++0.10000000 +30 ++0.50000000 -01 ++0.10000000 +30 ++0.40000000 -01 ++0.10000000 +30 ++0.60000000 -01 ++0.10000000 +30 ++0.10000000 +30 ++0.10000000 +30 ++0.60000000 -01 ++0.10000000 +30 ++0.50000000 -01 ++0.10000000 +30 ++0.60000000 -01 + 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 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