% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % 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. 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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. 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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 60-94511-6-R17 (1957). \item[] W.~S.~Emmerich, Optical Model Theory of Neutron Scattering and Reactions, Westinghouse Research Report 6-94511-6-R20 (1958). %% -----File: 120.png---Folio 115------- \item[] Franklin, Margolis and Oberthal, Scattering of $\mu$-Mesons by Nuclei, Phys.\ Rev.\ \textbf{111}, 296 (1958). \item[] Glassgold, Cheston, Stein, Schuld and Erickson, Analysis of Proton-Nucleus Scattering at 9.8~\MeV, Phys.\ Rev.\ \textbf{106}, 1207 (1958). \item[] A.~E.~Glassgold, Interaction of Antiprotons with Complex Nuclei, Phys.\ Rev.\ \textbf{110}, 220 (1958). \item[] A.~E.~Glassgold and P.~J.~Kellogg, Proton-Nucleus Scattering at 17~\MeV, Phys.\ Rev.\ \textbf{107}, 1372 (1957). \item[] A.~E.~Glassgold and P.~J.~Kellogg, Nuclear Scattering of 40- and 95-\MeV\ Protons, Phys.\ Rev.\ \textbf{109}, 1291 (1958). \item[] Green, Porter and Saxon, Proceedings of the International Conference on the Nuclear Optical Model, Florida State University, Tallahassee (1959). \item[] G.~Igo, Optical-Model Analysis of the Elastic Scattering of Alpha Particles, Phys.\ Rev.\ \textbf{106}, 126 (1957). \item[] G.~Igo, Optical Model Potential at the Nuclear Surface for the Elastic Scattering of Alpha Particles, Phys.\ Rev.\ Let.\ \textbf{1}, 72 (1958). \item[] G.~Igo, Optical Model Analysis of the Scattering of Alpha Particles from Helium (in press). \item[] G.~Igo, Optical-Model Analysis of Excitation Function Data and Theoretical Reaction Cross Sections for Alpha Particles, Phys.\ Rev.\ \textbf{115}, 1665 (1959). \item[] Igo, Ravenhall, Tiemann, Chupp, Goldhaber, Goldhaber, Lanutti and Thaler, The Scattering of $K^+$-Mesons in Emulsion, Phys.\ Rev.\ \textbf{109}, 2133 (1958). \item[] R.~Jastrow and I.~Harris, Nuclear Cross Sections for the Scattering of Neutrons and Protons, Proceedings of the ONR Decennial Symposium (1957). \item[] Lukyanov, Orlov and Turovstev, Optical Model of the Interaction between Intermediate Energy Neutrons and Nuclei, Nucl.\ Phys.\ \textbf{8}, 325 (1958). \item[] I.~E.~McCarthy, Flux of Particles in the Optical Model, Nucl.\ Phys.\ \textbf{10}, 583 (1959). \item[] Melkanoff, Moszkowski, Nodvik and Saxon, Energy Dependence of the Optical Model Parameters, Phys.\ Rev.\ \textbf{101}, 507 (1956). \item[] Melkanoff, Nodvik and Saxon, Diffuse-Surface Optical Model Analysis of Elastic Scattering of 17- and 31.5-\MeV\ Protons, Phys.\ Rev.\ \textbf{106}, 793 (1957). \item[] Melkanoff, Price, Stork and Ticho, Optical Model Analysis of Elastic Scattering of 125-\MeV\ $K^+$-Mesons in Nuclear Emulsions, Phys.\ Rev.\ \textbf{113}, 1303 (1959). \item[] University of Minnesota Annual Progress Report 1956-1957, 1957-1958, 1958, University of Minnesota Linear Accelerator Laboratory, Minneapolis, Minn. %% -----File: 121.png---Folio 116------- \item[] J.~S.~Nodvik and D.~S.~Saxon, Analysis of Elastic Cross Sections and Polarization of 10 \MeV\ Protons (in press). \item[] C.~E.~Porter, Nitrogen-Nitrogen Elastic Scattering, Phys.\ Rev.\ \textbf{112}, 1722 (1958). \item[] H.~M.~Shey, Scattering of Neutrons by Non-spherical Nuclei, Phys.\ Rev.\ \textbf{113}, 900 (1959). \item[] R.~D.~Woods and D.~S.~Saxon, Diffuse Surface Optical Model for Nucleon-Nuclei Scattering, Phys.\ Rev.\ \textbf{95}, 577 (1954). \end{Biblio} %%%%%%%%%%%%%%%%%%%%%%% BACK MATTER %%%%%%%%%%%%%%%%%%%%%%%%% \clearpage \phantomsection \pdfbookmark[0]{PG License}{Project Gutenberg License} \fancyhead[C]{\textit{LICENSING}} \begin{PGtext} End of the Project Gutenberg EBook of A Fortran Program for Elastic Scattering Analyses with the Nuclear Optical Model, by Michel A. Melkanoff and David S. Saxon and John S. Nodvik and David G. Cantor *** END OF THIS PROJECT GUTENBERG EBOOK ELASTIC SCATTERING ANALYSES *** ***** This file should be named 29784-pdf.pdf or 29784-pdf.zip ***** This and all associated files of various formats will be found in: http://www.gutenberg.org/2/9/7/8/29784/ Produced by David Starner, Andrew D. Hwang, and the Online Distributed Proofreading Team at http://www.pgdp.net Updated editions will replace the previous one--the old editions will be renamed. Creating the works from public domain print editions means that no one owns a United States copyright in these works, so the Foundation (and you!) can copy and distribute it in the United States without permission and without paying copyright royalties. Special rules, set forth in the General Terms of Use part of this license, apply to copying and distributing Project Gutenberg-tm electronic works to protect the PROJECT GUTENBERG-tm concept and trademark. 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