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\title{\LARGE Getting started with {\sc Fresco}}
\author{I.J. Thompson
%
\\Department of Physics, University of Surrey, Guildford GU2 7XH, England,\\
and\\
Nuclear Theory and Modelling, Livermore National Laboratory, Livermore CA 94551, USA
\\ %
\\Email: I.Thompson@fresco.org.uk}
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\begin{document}
\maketitle
\label{fresco-app}
\section{General structure}
\label{fresco-general}
\index{Fresco}
{\sc Fresco} is a general-purpose reaction code, created and
frequently updated by Ian Thompson.
The code calculates virtually any
nuclear reaction which can be expressed in a coupled-channel
form. There is a public version of the code which can be
downloaded from the website {\sf www.fresco.org.uk}. {\sc Fresco} is accompanied by
{\sc sfresco}, a wrapper code that calls {\sc Fresco} for data fitting, {\sc sumbins}
and {\sc sumxen}, two auxiliary codes for integrated cross sections.
Although we do not include it here, in the same site you can also
find {\sc xfresco} the front-end program to {\sc Fresco} for X-window displays.
Its original version was written in Fortran 77 but some important
sections were ported to Fortran 90. An important part concerns
the input, which now uses {\it namelist} format, making it much
easier to view the relevant variables. In this section we
will discuss the general namelist format of the {\sc Fresco}
input.
There are several different layers of output produced by {\sc Fresco}.
The default output contains the most important information
concerning the calculation, repeating the input information,
and the resulting observables but most detailed information is
contained in the generated {\sf fort} files, including files ready for
plotting purposes. At the end of this section we present the list
of output produced by {\sc Fresco}.
\subsection{Input file}
\label{fresco-input}
Input files contain five major namelists regarding different
aspects of the calculation: {\it fresco, partition, pot, overlap,
coupling}. The first is for general parameters, the second for
defining the properties intrinsic to the projectile and the
target, the third for potentials, the fourth for the radial overlap functions and the last
for the couplings to be included. Keep in mind that in some
inputs, you may not find all these namelists. As {\sc Fresco} can
calculate rather intricate processes, input files can sometimes
look daunting. However, with the namelist format, you do not need
to define all variables but only those that are relevant for your
example. Below we introduce the contents of each namelist and their
purpose. Detailed instructions are given in the {\sc Fresco} input manual on the website.
\subsubsection*{{\it heading}}
Every input file starts with a heading (80 characters) that should
describe and identify the reaction to be calculated, with perhaps some
detail of the method and states included. The following line begins with NAMELIST
to indicate the subsequent style of input.
\subsubsection*{\&{\it fresco}}
This section introduces the parameters involved in the numerical
calculations. It contains the radial information: the step with
which the coupled-channel equations are integrated ({\it hcm}) and
the radius at which the integrated wave function gets matched to
the asymptotic form ({\it rmatch}). Whenever non-local kernels are
involved a few more parameters are needed, ({\it rintp, hnl, rnl,
centre}), but these will be discussed in \ssectref{fresco-transfer}.%
\footnote{
It is often useful to introduce lower radial cutoffs in the
calculations, especially for scattering below the Coulomb barrier.
With {\it cutr} (fm) or {\it cutl} you introduce a
lower radial cutoff in the coupled-channel equations. Whereas {\it
cutr} is the same for all partial waves, {\it cutl} allows you to
define an $L$-dependent cutoff ($L$ is the total angular
momentum of the set). The code will use
max({\it cutl}*$L$*{\it hcm,cutr}). If {\it cutr} is negative, the lower cutoff is put at
that distance inside the Coulomb turning point. Finally, {\it cutc} (fm) removes off-diagonal
couplings inside the given radius.
}
In this namelist you also find general options for the
calculations and the desired observables. Cross sections are
calculated from the angular range {\it thmin--thmax} (in degrees)
in steps of {\it thinc}. This is also where you define the number
of partial waves in the calculation, by providing the initial and
final total angular momentum: {\it jtmin, jtmax}.%
\footnote{
To enable greater speed and flexibility, you can define a number of angular
momentum intervals {\it jump(i), i}=2,5 and the steps with which
you want to perform the calculation {\it jbord(i), i}=2,5. Note
that {\it jump(1)}=1 and {\it jbord(1)=jtmin}, so that the first
interval is calculated fully. The omitted J values are provided
by interpolation on the scattering amplitudes $A(m'M':mM; L)$
prior to calculating cross sections. {\it jsets} is a variable
that enables the calculation of positive parity ({\it jsets}=`P')
or negative parity only ({\it jsets}=`M' or `N'), for each
energy. If {\it jsets$=0$, ` ', or F}, no restriction is made.
}
\index{convergence}
{\it absend} controls the convergence. If in the interval
max(0,{\it jtmin}) $<$ J $<$ {\it jtmax} the absorption in the elastic
channel is smaller than {\it absend} mb for three consecutive
$J\pi$ sets, the calculation stops. When {\it absend}$<0$, it
takes the full J-interval.%
\footnote{Sometimes, for accurate
elastic-scattering cross sections it is only necessary to include the
elastic channel. This can be done with the option {\it
jtmin}$<$0. Then, in the range {\it J }$<$ abs({\it jtmin}), transfers and excited
states are ignored in the calculation.
}
There are many control variables which trace intermediate steps in
the calculation (starting from zero, increasing values will give
more in depth information). Here we shall mention just a few of
the most frequently used. For printing the coupled partial waves
for each $J^{\pi}$ (total angular momentum and parity) use {\it
chans}; for details of the coupling coefficients {\it listcc}, and
for S matrices use {\it smats} (absorption \& reaction cross
sections for successive partitions and excitations are output
when {\it smats}$\geq$ 1, and elastic S-matrix elements are
ouput when {\it smats}$\geq$ 2).
The variable {\it xstabl}$>$0 prints
the cross sections and tensor analyzing powers up to rank $k$={\it xstabl}
for all excitation levels in all partitions in Fortran
file 16 (usually called {\sf fort.16}).
Finally, and most importantly, this is the place where the beam
energy is specified through {\it elab}. If you want a calculation
at several energies, you can use the array {\it elab(i) i=1,4} and
{\it nlab(i), i=1,3} to define the boundary points and the number of intermediate
energy steps between {\it elab(i)} and {\it elab(i+1)}.
By default, the code assumes the elastic channel, the channel with
the incoming plane wave, is the first excitation of the first
partition. You can change this by using {\it pel$>$1} for the
partition number and {\it exl$>$1} for the excitation within that
partition. Also, {\it elab} refers by default to the energy of the
projectile {\it lin}=1, but you can easily change the calculation
to inverse kinematics by setting {\it lin}=2.
\subsubsection*{\&{\it partition}}
\index{partition}
In {\it partition} you introduce all the mass partitions and the
corresponding channels to be considered in the reaction. In the
simplest case, elastic scattering within the optical model, you
introduce just one partition, including the details of the
projectile ({\it namep,\,massp,\,zp}) and the details of the target
({\it namet,\,masst,\,zt}). The $Q$-value for the reaction is given
with {\it qval} (MeV) and the number of states that you want to
include in this partition is {\it nex}. Below defining each
partition, you have to introduce the {\it nex} associated pairs of states (at least
one). This is done through another namelist \&{\it states}.
\subsubsection*{\&{\it states}}
Each pair of states is a specific combination of one state of
the projectile and one state of the target. So this is the place
where you introduce the spin, parity and excitation energy of
these states: ({\it jp,\,ptyp,\,ep}) for projectile and ({\it
jt,\,ptyt,\,et}) for target. The variables {\it bandp} and {\it bandt} are synonyms
for {\it ptyp} and {\it ptyt} respectively.
The optical potential for the distorted wave for $p+t$
relative motion is
given by the index {\it cpot}, also defined here. This namelist is
repeated as many times as necessary, to introduce all the pairs of states
you wish to include in the calculation. When repeating the \&{\it
states} namelist, if one of the bodies stays in the same state,
you should not introduce spin, parity and excitation energy again,
but just set {\it copyp} or {\it copyt} to refer to the
\&{\it states} namelist in which the original state was first introduced.
\subsubsection*{\&{\it pot}}
\index{single particle!state in Fresco}
\index{potential!optical}
This namelist contains the parameters for the potentials to be
used in the reaction calculation, either for bound single-particle
states or optical potentials. The namelist is repeated for each
term in the potential. To identify the potential, there is an index
{\it kp}, and all the components with a given {\it kp} value are added together
to produce the potential used.
So, when calculating the distorted waves, {\it cpot}
will refer to one of the {\it kp}. The same will be used when
calculating the bound single-particle states with {\it kbpot} (in
\&{\it overlap}). Each term in the potential is characterized by a
type and a shape, followed by parameters {\it p(i), i} = 1,$\ldots$,6.
Traditionally, we define the Coulomb term first ({\it type}=0,
{\it shape}=0 for a charged sphere). Then {\it p(1)=ap} and {\it
p(2)=at} correspond to mass number of the projectile and the
target needed for the conversion of the reduced radii into
physical radii $R=r(ap^{1/3} + at^{1/3})$. The {\it p(3)=$r_c$} is the
reduced Coulomb radius. Note that the same mass factor $(ap^{1/3}
+ at^{1/3})$ is used in all terms of a given potential. {\it
type}=1 corresponds to the volume nuclear interaction, with {\it
shape}=0 for Woods-Saxon shape.\footnote{
A large number of standard
shapes are predefined, of which we mention {\it shape}=2 for the
Gaussian form $\exp[-(r-R_0)^2/b^2]$ (with {\it p(2)}=$r_0$ and {\it
p(3)}=$b$) as the most common alternative.
} The parameters for
the real part are {\it p(1)=$V_0$, p(2)=$r_0$, p(3)=$a_0$} (for
depth in MeV, the reduced radius in fm and the diffuseness in fm),
while the parameters for the imaginary part are {\it p(4)=$W_i$,
p(5)=$r_i$, p(6)=$a_i$}. A surface nuclear interaction is
introduced with {\it type}=2 and the spin-orbit with {\it type=3},
for the projectile, and {\it type=4} for the
target.\footnote{
Tensor interactions and projectile/target
deformation can be introduced with {\it type=5--11}. We will return
to this in \ssectref{fresco-inelastic}.
}
When the potential does not have an analytic form, it is useful to
read it in numerically. This can be achieved setting {\it
shape}=7,8,9 to read from file {\sf fort.4} the real or the imaginary
part of the potential, or the full complex potential, respectively.
\subsubsection*{\&{\it overlap}}
\index{overlap function}
Overlap functions are needed in single-particle excitation
calculations or in transfer calculations. The overlaps can refer
to bound states and scattering states, but we will leave the
latter for \ssectref{fresco-breakup}. Every \&{\it overlap} begins
with an index {\it kn1}. The overlap function tells us how the composite nucleus B
looks relative to its core A. The composite nucleus and the core
are in partition {\it ic1} and {\it ic2} respectively, and refer
to the projectile ({\it in}=1) or target ({\it in}=2). In the
simpler case {\it kind}=0, we ignore the spin of the core and take
$|(l,sn)j \rangle$ coupling.\footnote{
Multi-channel spin couplings are also available with {\it kind}=3.
} The overlap
has {\it nn} number of nodes (including the origin), {\it l}
relative angular momentum, {\it sn} for the spin of the additional
fragment (typically a neutron or proton {\it sn}=1/2), and total
angular momentum {\it j}. The potential used in the calculation of
the state is that indexed {\it kbpot}. You can also introduce the
binding energy {\it be} if you want the potential to be adjusted
to reproduce the binding energy ({\it isc}=1 for adjusting the
depth of the central part). If no rescaling is needed, set {\it isc}=0.
A spectroscopic amplitude for the overlap can be set to
the value $\sqrt{nam}$*{\it ampl} if both of these are non-zero. This amplitude can
also be introduced after \&{\it coupling} in the namelist \&{\it cfp}.
For printing more detailed information into the standard output,
there is a trace variable {\it ipc}. Its default value is zero,
and, as it increases, it provides more detailed information on the
overlap function.\footnote{
Radial wave functions of bound states can
be obtained by setting {\it ipc} odd, intermediate iterations with {\it ipc}$\ge$3, and
the final iteration with {\it ipc}$>$0.}
\subsubsection*{\&{\it coupling}}
\index{coupling, in fresco}
Couplings are calculated with the information given in this
namelist and include general spin transfer ({\it kind}=1); electromagnetic
couplings ({\it kind}=2), single particle excitations ({\it
kind}=3,\,4 for projectile and target respectively); transfer
couplings ({\it kind}=5,\,6,\,7,\,8 for zero-range, local energy
approximation, finite-range and non-orthogonality corrections
respectively). The coupling is from all states in partition {\it
icfrom} to all states in partition {\it icto}. Couplings are
included in the reverse direction unless {\it icto $<0$}. For
specific options of the coupling we use the parameters {\it
ip1,\,ip2,\,ip3} and for choices of the potentials in the operator
there are {\it p1,\,p2} parameters. More detail on this namelist
and others that follow will be given with specific examples.
\begin{table}
\index{Fresco!input and output files}
\caption{File allocation for the inputs and outputs for {\sc Fresco}.}
{\small
\begin{tabular}{|l | l | l|} \hline\hline
File & Routines & Use \\
\hline
%1 & SFRESCO& FRESCO input when searching\\
2 & SFRESCO &search specification file\\
3 & FREADF, FR& temporary namelist file\\
4 & INTER & input external KIND=1,2 form factors\\
& POTENT &input external potentials\\
5 & & standard input\\
6 & & standard output\\
7 & DISPX & S-matrix elements\\
%8 & FR,INTER & s/p wfs, channel wfs\\
%9 & Q/KERNEL & complex transfer multipoles\\
%10 & FR,CRISS & S-matrix elements \\
%11 & Q/KERNEL & real transfer multipoles\\
%12 & KERNEL&transfer kernels\\
13 & FR & total cross sections for each Elab\\
%14 & INTER/CPAIR &interaction potentials\\
16 & CRISS & tables of cross sections\\
17 & FR & output scattering waves\\
%18 & FR & wfns of 'best' iterate\\
%19 & FR & local couplings\\
& & \\
20--33& For users & (eg bound states, amplitudes)\\
& &\\
34 & POTENT & output potentials\\
35 & FR & astrophysical S-factors for $E_{\rm cm}$\\
36 & CRISS & scattering Legendre coefficients\\
37 & CRISS & scattering amplitudes\\
38 & DISPX & cross sections for each $J\pi$\\
39 & FR & 2 cross sections for each $E_{\rm cm}$\\
40 & FR & all cross sections for each $E_{\rm lab}$\\
41 & SOURCE & source terms at each iteration\\
42 & SOURCE & bin wave functions for each $E$\\
43 & INFORM & bin phase shifts as $k$ functions\\
44 & INFORM & bin phase shifts as $E$ functions\\
45 & ERWIN & scattering phase shift as $E$ functions\\
46 & INFORM & ANC ratios \& bound wave functions \\
47 & & reduced matrix elements \\
48 & FR & misc log file\\
55 & INFORM & Single-particle wave functions\\
56 & FR & $J$ fusion, reaction and nonelastic \\
57 & FR & output CDCC amplitudes\\
58 & INFORM & bound state wave functions \\
59 & INFORM & bound state vertex functions \\
60-62 & RMATRIX & trace of R-matrix calculations\\
66 & INTER & KIND=1 nonlocal formfactor\\
71 & FR & phase shifts as $E_{\rm lab}$ functions\\
75 & FR & astrophysical S-factors for Elab\\
89 & MULTIP& all coupling potentials\\
105 & FCN & $\chi^2$ progress during fitting\\
106 & FCN & parameter snapshots during fitting\\
200 & CRISS & elastic cross section if not {\sf fort.201} \\
201-210 & CRISS & cross sections (cf 16) of up to 10 states\\
301 & CDCIN & new Fresco input \\
%302 & CDCIN & New Fresco input (temp) \\
303 & SFRESCO & input search file \\
304 & SFRESCO & output plot file \\
305 & CDCIN & new input from CDCC, col format \\
306 & SFRESCO/FRXX0 & input Fresco file \\
307 & SFRESCO/FRXX0 & initial Output Fresco file \\
308 & SFRESCO/FRXX0 & main Output Fresco file \\
\hline\hline
\end{tabular}
}
\label{output-files}
\end{table}
\subsection{Output files}
\label{fresco-output}
The main output file ({\sf fort.6} or stdout) contains first of all a
representation of all the parameters read from the input file. It
will provide a summary of the calculation of the overlap functions
(including binding energy, depth of the adjusted potential, rms
radius and asymptotic normalization coefficient) and the coupling
matrix elements. For each beam energy, it provides some
information relative to the kinematic variables in the reaction
followed by the contribution to the cross section of each partial
wave. Integrated cross sections and angular distributions are
printed at the end of the file.
Also at the end of the standard output file, as a reminder to the
user, is a list of other files that were created during the run
with additional information. Here we mention a few: {\sf fort.16}
contains all angular distributions in a graphic format (to be read
by {\sc xmgr} or {\sc xmgrace}); {\sf fort.13} contains total cross
sections for each channel; {\sf fort.56} contains the total absorptive,
reaction and non-elastic cross section for each angular momentum.
Separate cross sections are included in files 201, 202, etc., in the
order they were specified. A full list of file allocations is
given in \tableref{output-files}.
\section{Learning through examples}
\label{fresco-examples}
\subsection{Elastic scattering}
\label{fresco-elastic}
\index{elastic scattering!in Fresco}
\index{astrophysical reactions}
As an elastic scattering example, we chose the proton scattering
on $^{78}$Ni within the optical model. This exotic nucleus is an
important waiting point in the r-process. The
input for our example is shown \boxref{Example-elastic}.
The calculations are
performed up to a radius of {\it rmatch}=60\,fm and partial waves
up to {\it jtmax}=50 are included. Three beam energies are
calculated. For this case, only one partition is needed with the
appropriate ground states specificied (the proton is spin 1/2 and
positive parity, and the $^{78}$Ni, being an even-even nucleus,
has $J^\pi=0^+$). The only remaining ingredient is the potential
between the proton and $^{78}$Ni (indexed {\it cpot=kp}=1) which
contains a Coulomb part and a nuclear real and imaginary part.
The results can be found in the standard output file, but it is
easier to plot the {\sf fort.16} file to obtain \fig{example-el}.
\begin{boxed}
%{\tiny
%\begin{verbatim}
%-------------------------------------------------------------------
%p+Ni78 Coulomb and Nuclear
% &FRESCO hcm=0.1 rmatch=60 jtmin=0.0 jtmax=50 absend= 0.0010
% thmin=0.00 thmax=180.00 thinc=1.00 chans=1 smats=2 xstabl=1
% elab(1:3)=6.9 11.00 49.350 nlab(1:3)=1 1 /
% &PARTITION namep='p' massp=1.00 zp=1 namet='Ni78' masst=78.0000 zt=28 qval=0.000 nex=1 /
% &STATES jp=0.5 bandp=1 ep=0.0000 cpot=1 jt=0.0 bandt=1 et=0.0000 /
% &partition /
% &POT kp=1 ap=1.000 at=78.000 rc=1.2 /
% &POT kp=1 type=1 p1=40.00 p2=1.2 p3=0.65 p4=10.0 p5=1.2 p6=0.500 /
% &pot /
% &overlap /
% &coupling /
%-------------------------------------------------------------------
%\end{verbatim}
%}
\centerline{\includegraphics[clip,width=1.0\textwidth]{figures/Example-elastic}}
\caption{{\sc Fresco} input for the elastic scattering of protons on $^{78}$Ni at several
beam energies}
\label{Example-elastic}
\end{boxed}
\begin{figure}
\includegraphics[width=0.8\textwidth]{figures/example-el}
\caption{Elastic scattering of protons on $^{78}$Ni at several
beam energies, calculated with input from \boxref{Example-elastic}.} \label{example-el}
\end{figure}
\subsection{Inelastic scattering}
\label{fresco-inelastic}
\index{inelastic!in Fresco}
\index{astrophysical reactions}
Inelastic scattering exciting collective states can be illustrated
with the example $^{12}$C$(\alpha,\alpha)^{12}$C$(2^+)$, where the
carbon nucleus gets excited into its first excited state. This
reaction can provide complementary information to one of the most
important reactions in astrophysics,
the $\alpha$-capture reaction
on $^{12}$C. In this type of inelastic reaction, only one
partition is needed, but it contains two states. The projectile
state is not changed ({\it copyp}=1) but the appropriate spin,
parity and excitation energy need to be introduced for the target.
The input is shown in \boxref{Example-inelastic}.
\index{deformation}
In order for the reaction to happen, the potential needs to
contain a tensor $Y_{20}$ part to enable the target transition
$0^+ \rightarrow 2^+$. This is done assuming a rotor model for the
target and, in the input, only a deformation length needs to be
introduced. For deforming a projectile {\it type}=10, while for
deforming a target, {\it type}=11. Here the deformation length
$\delta_2$ is {\it p(2)}=1.3 fm, as twice highlighted in \boxref{Example-inelastic}.
The optical potential introduced
includes a Coulomb term, the nuclear real term, and a nuclear
imaginary with a volume ({\it type}=1) and a surface part ({\it
type}=2). Each part needs to be deformed, and only the two nuclear parts are deformed.
If Coulomb
deformation were needed, an additional line after the Coulomb
potential would have to be introduced with the same format, except
that, instead of the deformation length, the reduced matrix
element should be given.\footnote{
If you do not want to assume a rotational
model, you can introduce these couplings (either deformation
length or matrix element) for each initial-to-final state through
{\it type}=12,\,13 for projectile and target respectively.
In this case, give a \&{\it step} namelist specifying
{\it ib,\,ia,\,k,\,str} for a coupling from state {\it
ib} to state {\it ia}, multipolarity {\it k} and strength {\it str}
({\it str} is the reduced matrix element for Coulomb transitions
and the reduced deformation length for nuclear transitions).
}
As the proton and
neutrons do not necessarily have the same spatial distribution,
the deformation parameters will, in general, not be the same.
\index{distorted wave Born approximation!in Fresco}
\index{coupled channels!in Fresco}
The example shows a DWBA calculation as {\it iter}=1. You
could check the validity of the DWBA by including higher-order terms in
your Born expansion (increasing {\it iter}) or performing a full
coupled-channels calculation ({\it iter}=0, {\it iblock}=2).
Results for the inelastic excitation of $^{12}$C are shown in
\fig{example-inel}
\begin{boxed}
%{\tiny
%\begin{verbatim}
%-------------------------------------------------------------------
%alpha+c12 -> alpha+c12* @ 100 MeV; nuc def NAMELIST
%&FRESCO
% hcm=0.050 rmatch=20.000 rintp=0.20 jtmin=0.0 jtmax=40 absend= 0.01
% thmin=0.00 thmax=180.00 thinc=1.00
% iter=1 ips=0.0 iblock=0 chans=1 smats=2 xstabl=1 elab(1)=100.0 /
% &PARTITION namep='alpha' massp=4.0000 zp=2 namet='12C' masst=12.000 zt=6 qval=0.000 nex=2 /
% &STATES jp=0. bandp=1 ep=0.0000 cpot=1 jt=0.0 bandt=1 et=0.0000 /
% &STATES copyp=1 cpot=1 jt=2.0 bandt=1 et=3.6000 /
% &partition /
% &POT kp=1 ap=4.000 at=12.000 rc=1.2 /
% &POT kp=1 type=1 p1=40.0 p2=1.2 p3=0.65 p4=10.0 p5=1.2 p6=0.500 /
% &POT kp=1 type=2 p1=0.00 p2=1.2 p3=0.65 p4=6.0 p5=1.2 p6=0.500 /
% &POT kp=1 type=11 p1=0.0 p2=1.3 /
% &pot /
% &overlap /
% &coupling /
%-------------------------------------------------------------------
%\end{verbatim}
%}
\centerline{\includegraphics[clip,width=1.0\textwidth]{figures4/Example-inelastic4}}
\caption{{\sc Fresco} input for the inelastic excitation of
$^{12}$C by $\alpha$ particles at 100 MeV}
\label{Example-inelastic}
\end{boxed}
\begin{figure}
\includegraphics[width=0.8\textwidth]{figures/example-inel2}
\caption{Inelastic angular distribution for the excitation of
$^{12}$C by $\alpha$ particles at 100 MeV obtained with the input
of \boxref{Example-inelastic}. }
\label{example-inel}
\end{figure}
\begin{boxed}[tbp]
%{\tiny
%%\fbox{
%%\begin{minipage}{0.95\textheight}
%%\caption{Breakup example: input} \label{example-br-box}
%\begin{verbatim}
%CDCC 8B+208Pb ; nuclear and coulomb s-wave breakup
%NAMELIST
% &Fresco hcm= 0.01 rmatch= -60.000 rintp= 0.15 rsp= 0.0 rasym= 1000.00 accrcy= 0.001
% jtmin= 0.0 jtmax= 9000.0 absend= -50.0000
% jump = 1 10 50 200
% jbord= 0.0 200.0 300.0 1000.0 9000.0
% thmin= 0.00 thmax= 20.00 thinc= 0.05 cutr=-20.00
% ips= 0.0000 it0= 0 iter= 0 iblock= 21 nnu= 24 smallchan= 1.00E-12 smallcoup= 1.00E-12
% elab= 656.0000 pel=1 exl=1 lab=1 lin=1 lex=1 chans= 1 smats= 2 xstabl= 1 cdcc= 1/
% &Partition namep='8B' massp= 8. zp= 5 nex= 21 pwf=T namet='208Pb' masst=208. zt= 82 qval= 0.1370/
% &States jp= 1.5 ptyp=-1 ep= 0.0000 cpot= 1 jt= 0.0 ptyt= 1 et= 0.0000/
% &States jp= 0.5 ptyp= 1 ep= 0.1583 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 0.2180 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 0.3260 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 0.4830 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 0.6889 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 0.9438 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 1.2478 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 1.6007 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 2.0027 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 2.4536 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 2.9536 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 3.5025 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 4.1005 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 4.7474 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 5.4434 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 6.1884 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 6.9824 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 7.8253 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 8.7173 cpot= 1 copyt= 1/
% &States jp= 0.5 ptyp= 1 ep= 9.6583 cpot= 1 copyt= 1/
% &Partition namep='7Be' massp= 7. zp= 4 nex= -1 pwf=T namet='209Pb' masst=209.0000 zt=83 qval= 0./
% &States jp= 0.0 ptyp= 1 ep= 0.0000 cpot= 2 jt= 0.0 ptyt= 1 et= 0.0000/
% &Partition /
% &Pot kp= 1 type= 0 shape= 0 p(1:3)= 1.0000 0.0000 2.6500 /
% &Pot kp= 2 type= 0 shape= 0 p(1:3)= 208.0000 0.0000 1.3000 /
% &Pot kp= 2 type= 1 shape= 0 p(1:6)= 114.2000 1.2860 0.8530 9.4400 1.7390 0.8090 /
% &Pot kp= 3 type= 0 shape= 0 p(1:3)= 208.0000 0.0000 1.3000 /
% &Pot kp= 3 type= 1 shape= 0 p(1:6)= 34.8190 1.1700 0.7500 15.3400 1.3200 0.6010 /
% &Pot kp= 4 type= 0 shape= 0 p(1:3)= 1.0000 0.0000 2.3910 /
% &Pot kp= 4 type= 1 shape= 0 p(1:3)= 44.6750 2.3910 0.4800 /
% &Pot kp= 4 type= 3 shape= 0 p(1:3)= 4.8980 2.3910 0.4800 /
% &Pot /
% &Overlap kn1= 1 ic1=1 ic2=2 in= 1 kind=0 nn= 1 l=1 sn=0.5 j= 1.5 nam=1 ampl= 1.00 kbpot= 4 be= 0.1370 isc= 1 ipc=0 /
% &Overlap kn1= 2 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -0.0182 isc=12 ipc=2 nk= 20 er= -0.0344 /
% &Overlap kn1= 3 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -0.0771 isc=12 ipc=2 nk= 20 er= -0.0834 /
% &Overlap kn1= 4 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -0.1850 isc=12 ipc=2 nk= 20 er= -0.1324 /
% &Overlap kn1= 5 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -0.3419 isc=12 ipc=2 nk= 20 er= -0.1814 /
% &Overlap kn1= 6 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -0.5479 isc=12 ipc=2 nk= 20 er= -0.2304 /
% &Overlap kn1= 7 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -0.8028 isc=12 ipc=2 nk= 20 er= -0.2794 /
% &Overlap kn1= 8 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -1.1067 isc=12 ipc=2 nk= 20 er= -0.3284 /
% &Overlap kn1= 9 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -1.4596 isc=12 ipc=2 nk= 20 er= -0.3774 /
% &Overlap kn1= 10 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -1.8616 isc=12 ipc=2 nk= 20 er= -0.4264 /
% &Overlap kn1= 11 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -2.3125 isc=12 ipc=2 nk= 20 er= -0.4754 /
% &Overlap kn1= 12 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -2.8125 isc=12 ipc=2 nk= 20 er= -0.5245 /
% &Overlap kn1= 13 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -3.3614 isc=12 ipc=2 nk= 20 er= -0.5735 /
% &Overlap kn1= 14 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -3.9594 isc=12 ipc=2 nk= 20 er= -0.6225 /
% &Overlap kn1= 15 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -4.6064 isc=12 ipc=2 nk= 20 er= -0.6715 /
% &Overlap kn1= 16 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -5.3023 isc=12 ipc=2 nk= 20 er= -0.7205 /
% &Overlap kn1= 17 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -6.0473 isc=12 ipc=2 nk= 20 er= -0.7695 /
% &Overlap kn1= 18 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -6.8413 isc=12 ipc=2 nk= 20 er= -0.8185 /
% &Overlap kn1= 19 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -7.6843 isc=12 ipc=2 nk= 20 er= -0.8675 /
% &Overlap kn1= 20 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -8.5763 isc=12 ipc=2 nk= 20 er= -0.9165 /
% &Overlap kn1= 21 ic1=1 ic2=2 in= 1 kind=0 l=0 sn=0.5 j= 0.5 nam=1 ampl= 1.00 kbpot= 4 be= -9.5173 isc=12 ipc=2 nk= 20 er= -0.9655 /
% &Overlap / !
% &Coupling icto= 1 icfrom= 2 kind=3 ip1= 2 ip2= 0 ip3= 0 p1= 3.0000 p2= 2.0000 /
% &Coupling /
%\end{verbatim}
%%\end{minipage}
%} %}
\centerline{\includegraphics[clip,width=1.0\textwidth]{figures/Example-breakup1}}
\caption{{\sc Fresco} input for the breakup of $^8$B on $^{208}$Pb at 82 MeV/u}
\label{Example-breakup1}
\end{boxed}
\subsection{Breakup}
\label{fresco-breakup}
\index{breakup!in Fresco}
\index{astrophysical reactions}
\index{CDCC!in Fresco}
Breakup calculations can be modeled as single-particle excitation
into the continuum. In this example we show a typical CDCC
calculation. It calculates the
breakup of $^8$B into p + $^7$Be, under the field of $^{208}$Pb at
intermediate energies. The input is shown in \boxref{Example-breakup1}. The
breakup of $^8$B has been measured many times with the aim of
extracting the proton capture rate on $^7$Be.
\begin{figure}
\includegraphics[width=0.49\textwidth]{figures/example-br-en2} ~~
\includegraphics[width=0.475\textwidth]{figures/examples-br-ang2}
\caption{Breakup of $^8$B on $^{208}$Pb at 82 MeV/u.
Left: p$-^7$Be relative energy distribution. Right: center-of-mass angular
distribution. Both are obtained with the input of \boxref{Example-breakup1}.}
\label{example-br}
\end{figure}
\index{capture!proton}
Several new ingredients need to be explained. First of all, due to
the long range of the Coulomb interaction it is very important to
include the effect of couplings out to large distances. Instead of
integrating the CDCC equations up to very large radii, we
introduce {\it rasym}. Setting {\it rmatch$<$0} tells the code
that the integration of the equations should be done up to {\it
rmatch}, numerically, but these should then be matched with
coupled-channel Coulomb functions up to {\it rasym}. Also
important are the partial waves. For these intermediate energies,
many partial waves need to be included and it is useful, instead
of calculating each single one, to interpolate between them. This
can be done with {\it jump} and {\it jbord}. In this example, we
start with {\it jtmin}=0 until $j=200$ in steps of $1$, for
$jt=200{-}300$ use steps of $10$, for $jt=300{-}1000$ use steps of
$50$, and for $jt=1000{-}9000$ use steps of $200$. With the
inclusion of so many partial waves, the strong repulsion at short
distances can introduce numerical problems. This is avoided with
a radial cutoff $cutr=-20$ fm, where the minus
sign puts the cutoff 20 fm inside the Coulomb turning point.
This example contains only $s$-waves in the continuum, sliced into
20 energy bins. Other partial waves ($p,d,f$ are needed for
convergence) are left out of this example to make it less time
consuming (beware, it will still take a few minutes in a desktop
computer!). Since in general there will be many channels involved,
it is convenient to drop off channels/couplings whenever they are
weak. This is done through {\it smallchan} and {\it smallcoup}. To
perform a full CDCC calculation, {\it iter}=0 and {\it iblock}=21.
The continuum of $^8$B is binned into discrete excited states of
positive energy, so under the first partition the namelist {\it
states} needs to be repeated for each bin, with appropriate
excitation energy and quantum numbers. Since in this example we
are not interested in the second partition, it does not get
printed with the option of negative {\it nex}. Several new
variables are needed when defining the bins: negative {\it be} provides bins with energy
relative to threshold $|be|$, with a width {\it er}, and an
amplitude $\sqrt{nam}$*{\it ampl}. To characterize the weight function of the
bin we use {\it isc} ({\it isc}=2 for non-resonant bins, and {\it isc}=4
for resonant bins). Note that here, the same potential is
used for the $^8$B bound and continuum states. This need not be
the case.
\begin{boxed}[tbp]
%{\tiny
%\begin{verbatim}
%-------------------------------------------------------------------
%8B+208Pb ; N+C q=0,1,2
%CDCC
% &CDCC
% hcm=0.01 rmatch=-60 rasym=1000 accrcy=0.001 absend=-50
% jbord= 0 200 300 1000 9000 jump = 1 10 50 200
% thmax=20 thinc=0.05 cutr=-20 smats=2 xstabl=1
% ncoul=0 reor=0 q=2 elab=656
% /
% &NUCLEUS part='Proj' name='8B' charge=5 mass=8 spin=1.5 parity=-1 be = 0.137 n=1 l=1 j=1.5 /
% &NUCLEUS part='Core' name='7Be' charge=4 mass=7 /
% &NUCLEUS part='Valence' name='proton' charge=1 mass=1 spin=0.5/
% &NUCLEUS part='Target' name='208Pb' charge=82 mass=208 spin=0 /
% &BIN spin=0.5 parity=+1 start=0.001 step=0.50 end=10. energy=F l=0 j=0.5/
% &BIN /
% &POTENTIAL part='Proj' a1=1 rc=2.65 /
% &POTENTIAL part='Core' a1=208 rc=1.3 v=114.2 vr0=1.286 a=0.853 w=9.44 wr0=1.739 aw=0.809 /
% &POTENTIAL part='Valence' a1=208 rc=1.3 v=34.819 vr0=1.17 a=0.75 w=15.340 wr0=1.32 aw=0.601/
% &POTENTIAL part='Gs' a1=1 rc=2.391 v=44.675 vr0=2.391 a=.48 vso=4.898 rso0=2.391 aso=0.48 /
%-------------------------------------------------------------------
%\end{verbatim}
%}
\centerline{\includegraphics[clip,width=1.0\textwidth]{figures/Example-breakup2}}
\caption{{\sc Fresco} input for the breakup of $^8$B on $^{208}$Pb at 82 MeV per
nucleon (short version)}
\label{Example-breakup2}
\end{boxed}
After defining the overlaps, coupling parameters are
introduced: {\it kind=3} stands for single-particle excitations of
the projectile ({\it kind=4} would be for the target), {\it ip1}
is the maximum multipole order in the expansion of the couplings
included, {\it ip2}=0,1,2 for Coulomb and nuclear,
nuclear only and Coulomb only, respectively, and {\it ip3} makes
specific selections of couplings with default {\it ip3}=0 when all
couplings are included.\footnote{
If {\it ip3}=1, there are no
reorientation couplings for all but the monopole, if {\it ip3}=2,
only reorientation couplings are included, and if {\it ip3}=3 it
includes only couplings to and from the ground state. More options
exist but are not presented here.
}
For the interactions in the
coupling matrix, the core-target is potential index {\it p1}=3 and
the valence-target is potential index {\it p2}=2.
Angular distributions of the cross sections for each energy bin
can be found in {\sf fort.16}. To obtain a total angular distribution
one needs to sum over all bins (use {\sf sumbins $<$ {\sf fort.16} $>$
xxx.xsum}). For the breakup example shown here, the resulting total
angular distribution is plotted in \fig{example-br}(left). If
you are interested in the energy distribution, {\sf fort.13} contains
all angular integrated cross sections for each bin. In general,
for each energy, a sum over all $\ell$ partial waves within the
projectile is necessary (use {\sf sumxen $<$ {\sf fort.13} $>$
xxx.xen}). In \fig{example-br}(right) we show the energy
distribution for the $^8$B breakup here considered. In addition,
it is useful to look at {\sf fort.56} (cross section per partial wave
$L$) \index{cross section!per partial wave} to ensure that enough partial waves are included in the
calculation.
%, and to save {\sf fort.57} (usually copied onto xxx.cdcc)
%which contains all the scattering amplitudes
%needed for calculating the three-body observables.
Defining a long list of bin states and overlaps can be easily
automated. The revised {\sc cdcc} style of input has been developed specifically for large
CDCC calculations, and transforms a simpler input into the
standard input we have just gone through. The simpler input would then
look like \boxref{Example-breakup2}.
\subsection{Transfer}
\label{fresco-transfer}
\index{transfer!in Fresco}
\index{astrophysical reactions}
\index{asymptotic normalization coefficient}
Transfer reactions are often used to extract structure information
to input in astrophysical simulations. Here we consider the
$^{14}$N$(^{17}$F$,^{18}$Ne$)^{13}$C transfer
reaction at 10 MeV per nucleon. This
reaction was measured with the aim of extracting the asymptotic
normalization coefficient of specific states in $^{18}$Ne which in
turn provides a significant part of the rate for
$^{17}$F(p,$\gamma)$. The proton capture reaction on $^{17}$F
appears in the $rp$-process in novae environments.
The ratio of the proton capture rate and the decay rate of $^{17}$F is also very
important for the understanding of galactic $^{17}$O, $^{18}$O and
$^{15}$N. The input for the transfer example is given in
\boxref{Example-transfer}.
\index{non-locality}
A few important new parameters need to be defined when performing
the transfer calculation. Because the process involves a non-local
kernel $V^\pp_{fi} (R',R)$,
in addition to the radial grids already
understood, we need to introduce {\it rintp, hnl, rnl, centre}.
The {\it rintp} is the step in $R$, {\it hnl, rnl} are the non-local step
and the non-local range in $R'-R$, respectively, and centered at {\it centre}. Gaussian
quadrature is used for the angular integrations in constructing the non-local kernels, and
{\it nnu} is the number of the Gaussian integration points to be
included.
In this example the core has non-zero spin, and in order to
generate the appropriate overlap of the composite nucleus
$^{14}$N, it is necessary to take into account, not only the
angular momentum of the neutron but also the spin of $^{13}$C.
This can be done with {\it kind}=3 in the overlap definition where
the spin of the core {\it ia} and of the composite {\it ib} need
to be specified. The coupling scheme is $|(l_n,s_n)j,I_A;I_B\rangle$.
\begin{boxed}
%{\tiny
%\begin{verbatim}
%-------------------------------------------------------------------
%n14(f17,ne18)c13 @ 170 MeV; NAMELIST
%-------------------------------------------------------------------
%\end{verbatim}
%}
\centerline{\includegraphics[clip,width=1.0\textwidth]{figures4/Example-transfer4}}
\caption{{\sc Fresco} input for the transfer reaction $^{14}$N($^{17}$F,$^{18}$Ne)$^{13}$C at 10 MeV/u.}
\label{Example-transfer}
\end{boxed}
\begin{figure}
\includegraphics[width=0.8\textwidth]{figures/example-tr}
\caption{Transfer cross section for
$^{14}$N($^{17}$F,$^{18}$Ne)$^{13}$C at 10 MeV/u calculated
with the input of \boxref{Example-transfer}.}
\label{example-tr}
\end{figure}
The only other new part of the input concerns the transfer
coupling itself, as all other parts (partitions, potentials and
overlaps) have already been previously presented. Transfer
couplings are defined in the namelist \&{\it coupling} by {\it
kind}=5,6,7 for zero-range, low energy approximation and finite
range, respectively. For finite-range transfers, {\it ip1}=0,1
stands for post or prior, {\it ip2}=$0,1,{-}1$ for no remnant, full
real remnant and full complex remnant respectively and {\it ip3}
denotes the index of the core-core optical potential. If {\it
ip3}=0 then it uses the optical potential for the first pair of
excited states in the partition of the projectile core.
Following the \&{\it coupling} namelist, we need to define the
amplitudes (coefficients of fractional parentage) of all the
overlaps to be included in the calculation. Here, this is done
with \&{\it cfp} where {\it in}=1,2 for projectile or target, {\it
ib/ia} corresponds to the state index of the composite/core and
{\it kn} is the index of the corresponding overlap function. So
the first \&{\it cfp} refers to the $\langle^{17}$F$|^{18}$Ne$\rangle$ overlap and
the second \&{\it cfp} refers to the $\langle^{13}$C$|^{14}$N$\rangle$ overlap. %$^{14}$N/$^{13}$C.
The angular distribution obtained from our example is presented in
\fig{example-tr}.
\subsection{Capture}
\label{fresco-capture}
\index{capture!in Fresco}
\index{astrophysical reactions}
{\begin{boxed}
%{\tiny
%\begin{verbatim}
%-------------------------------------------------------------------
%14C(n,g)15C E1 only NAMELIST
% &FRESCO hcm= 0.100 rmatch=100 jtmin=0 jtmax=1.5 absend=-1 thmin=0 thmax=0 iter=1
% elab(1)= 0.005 4.005 nlab=50/
% &PARTITION namep='neutron' massp=1.0087 zp=0 nex=1 namet='14C' masst=14.0032 zt=6 /
% &STATES jp=0.5 ptyp=1 ep=0 cpot=1 jt=0.0 ptyt=1 et=0/
% &PARTITION namep='Gamma' massp=0 zp=0 nex=1 namet='15C' masst=15.0106 zt=6 qval=1.218 /
% &STATES jp=1.0 ptyp=1 ep= 0 cpot=3 jt=0.5 ptyt=1 et=0/
% &partition /
% &Pot kp=1 type= 0 p(1:3)= 14.0000 0.0000 1.3000 /
% &Pot kp=1 type= 1 p(1:3)= 57.0000 1.7 0.7000 /
% &Pot kp=1 type= 3 p(1:3)= 0.0000 1.7 0.5000 /
% &Pot kp=4 type= 0 p(1:3)= 14.0000 0.0000 1.2000 /
% &Pot kp=4 type= 1 p(1:3)= 55.7700 1.2230 0.5000 /
% &Pot kp=4 type= 3 p(1:3)= 5.0000 1.2230 0.5000 /
% &pot /
% &OVERLAP kn1=1 ic1=1 ic2=2 in=2 kind=0 nn=2 l=0 sn=0.5 j=0.5
% kbpot=4 be=1.218 isc=1 /
% &overlap /
% &COUPLING icto=2 icfrom=1 kind=2 ip1=-1 ip2= 1/
% &cfp in=2 ib=1 ia=1 kn=1 a=1.000 /
%-------------------------------------------------------------------
%\end{verbatim}
%}
\centerline{\includegraphics[clip,width=0.8\textwidth]{figures/Example-capture}}
\caption{{\sc Fresco} input for neutron capture by $^{14}$C.}
\label{Example-capture}
\end{boxed}
\begin{figure}[t]
\includegraphics[width=0.75\textwidth]{figures/example-cap}
\caption{Neutron capture cross section for $^{14}$C as a function
of neutron energy and calculated
with the input of \boxref{Example-capture}.} \label{example-cap}
\end{figure}}
Capture reactions are of direct interest in astrophysics.
Although the electromagnetic operator is well understood, coupling effects
may be non-trivial and require focused work. Here we pick a
neutron capture reaction that is completely dominated by $E1$:
$^{14}$C(n,$\gamma)^{15}$C. This reaction was first introduced
in the context of the $r$-process. In our example, the capture is
calculated at 50 different scattering energies, from 5 keV up to 4
MeV. The input is presented in \boxref{Example-capture}
For capture reactions, the first partition is defined in the usual
way, but in the second partition, the projectile should be {\it Gamma}
(with spin {\it jp}=1 and positive parity) and {\it cpot}
should refer to a non-existing potential in order that there be no photon
potential. The $2s_{1/2}$ $^{15}$C overlap
is defined in \&{\it overlap}. Electromagnetic one-photon
couplings are defined through {\it kind}=2. Therein, {\it ip1}
refers to the multipolarity of the transition and {\it ip2}=0,1,2
for including both electric and magnetic transitions, electric only and magnetic only,
respectively.
If {\it ip1} $>$ 0, all multipolarities up to {\it ip1} are included, otherwise only
$|${\it ip1}$|$ is calculated.
%Although it is not needed in the $^{14}$C neutron
%capture, for magnetic transitions you can introduce the projectile
%and target g-factors in this namelist using {\it p1} and {\it p2}.
There are several outputs available specifically for astrophysics.
In \fig{example-cap} we plot the cross section for the
$^{14}$C(n,$\gamma)^{15}$C capture reaction as a function of center-of-mass energy (found in {\sf fort.39}). For charged-particle reactions, astrophysical S-factors are also available (see \tableref{output-files}).
\section{Runtime errors}
\label{fresco-errors}
\index{Fresco!errors}
In a complicated modeling computer program like {\sc Fresco}, accurate results cannot be obtained
if there are obvious numerical errors either in the input, or produced during the calculation. Problems may occur at energies very much below the Coulomb barrier, or at relativistic energies, since {\sc Fresco} should not be used in these cases. The program is written to stop when large numerical inaccuracies occur, but no results can be trusted until they are examined to see that they are not sensitive to further increases
to the maximum radius {\it rmatch}, maximum partial waves {\it jtmax}, maximum non-local range {\it rnl}, and further decreases in the radial step size {\it hcm} and lower radial cutoff parameter {\it cutl}. We give some guidance on these parameters below.
When the program reads the input file, typing errors in variables or their values will cause the program to stop after printing out the complete namelist so you can see which variables have been successfully read in. Alternatively, if the words {\sc namelist} and {\sc cdcc} in the second line of the input are written in lower case (`namelist' or `cdcc'), the compiler's own error handling will be used instead. You can also look at the input echo in {\sf fort.3}, to see up to which line the input has been read successfully.
During the running of {\sc Fresco}, a number of induced or cancellation errors can be detected.
These are:
{\small
\begin{description}
\item[Step size too large:] If $k$ is the asymptotic wave number for a channel, then its product with the step size should be sufficiently small for the Numerov integration method to avoid large errors:
$k*${\it hcm} $ < 0.2$.
\item[Bound-state search failure:] Bound states are found in subroutine {\sf eigcc} by a Newton-Raphson method with at most 40 iterations. Such states can be found at a specific energy by varying some part of the potential: this part cannot be zero.
\item[Insufficient non-local width:] If the non-local coupling form factors are too large when
$|R-R'| \ge$ {\it rnl}, then {\it rnl} should be increased as recommended.
\item[$L$-transfer accuracy loss:] In the transfer Legendre expansion, if $\ell+\ell'$ is too large, there can be cancellation errors between different multipoles $T$. This can be remedied by increasing the input parameter {\it mtmin} to use the slower $m$-dependent method of
\cite{TUM}.
\item[Matching deficiency:] The program will print an error message if the nuclear potentials are not smaller than 0.02 MeV at the outer matching radius, and stop if they are larger than 0.1 MeV. Increase {\it rmatch}, or correct some potential that does not decay sufficiently fast.
\item[Iteration failure in solving coupled equations:] If in some partial wave set, more than {\it iter} iterations still do not appear to converge, then all cross sections will be affected. Either use Pad\'e acceleration, increase {\it iter} or make it negative so that the `best' intermediate value is used, or increase {\it ips} slightly. The detailed progress of the iterations can be seen by setting {\it smats} $\ge 5$.
\item[Accuracy loss in solving coupled equations:] If any of the channels are propagating in
a classically forbidden region, there will be loss of linear independence of the separate solutions
$\{{\sf Y}_{\alpha\beta}(R)\}$, as all solutions will tend to become exponentially increasing.
This is monitored in the subroutine {\sf erwin} during the summations, and will lead to a halt if the cancellation errors are expected to be more than 3\%. In this case, increase the lower cutoff parameters {\it cutl} or {\it cutr},
decrease the matching radius {\it rmatch} if possible, or else use the R-matrix expansion
method to solve the equations.
An extreme error occurs if the simultaneous equations from the matching conditions are singular.
\item[Internal parameter error:] Sometimes, the precalculation of array sizes is inadequate. For advanced users these can be improved by selective specification of the {\sf maxcoup(1:3)} and {\sf expand(1:11)} input arrays.
\end{description}
}
At the end of a {\sc Fresco} calculation, a final `accuracy analysis' is presented. This rechecks that the step size {\it hcm} is small enough, and that {\it rnl} is large enough. Then, using the Coulomb trajectories, it calculates the minimum scattering angles expected to be accurate because of the finite values of {\it rmatch} and {\it jtmax}.
\section{Fitting data: {\sc Sfresco}}
\label{fresco-fitting}
\index{fitting!in Fresco}
\index{SFresco}
A first calculation of cross sections using {\sc Fresco} will rarely be near the experimental data.
Perhaps the reaction model is too simplified, or perhaps the input parameters are not accurate enough.
The optical potential, binding potentials, spectroscopic amplitudes or R-matrix reduced widths could well be adjusted to see if the agreement between theory and experiment can be improved. The program {\sc
Sfresco} searches for a $\chi^2$ minimum when comparing
the outputs of {\sc Fresco} with sets of data, using the {\sc Minuit} search routines.
The inputs for {\sc Sfresco} specify the {\sc Fresco} input and output files,
the number and types of search variables, and the experimental data sets to be compared with.
These experimental data can be ({\it type}=0) an angular distribution for fixed energy,
(1) an excitation and angular cross-section double distributions, or
(2) an excitation cross section for fixed angle.
They could also be (3) an excitation function for the total, reaction, fusion or inelastic cross section,
(4) an excitation phase shift for fixed partial wave,
(5) a desired factor for bound-state search, or
(6) even a specific experimental constraint on some search parameter.
The simplest and most common fitting requirement is to determine an optical potential to fit the observed elastic scattering angular distribution. For example, to find a proton optical potential to fit cross sections for scattering on $^{112}$Cd at 27.90 MeV, we start with the normal {\sc Fresco} input of \boxref{Example-optical-frin}. The cross sections will be calculated at the experimental angles, not those specified here.
In order to vary the real and imaginary potential strengths in this input, as well as the real radius, we have the {\sf search file} for {\sc Sfresco} of \boxref{Example-optical-search}. This search file begins by identifying the previous {\sc Fresco} input and naming the temporary output file, then giving the number of search variables and the number of experimental data sets.
\begin{boxed}
%{\footnotesize
%\begin{verbatim}
%-------------------------------------------------------------------
%p + 112Cd elastic
%NAMELIST
% &FRESCO hcm= 0.100 rmatch= 20.000 jtmin= 0.0 jtmax= 200.0
% thmin= 0.00 thmax=180.00 thinc= 2.00 xstabl= 1
% elab(1)= 27.9 /
% &PARTITION namep='Proton ' massp= 1.0000 zp= 1 nex= 1
% namet='112Cd ' masst=112.0000 zt= 48 qval= 0/
% &STATES jp= 0.5 ptyp= 1 ep= 0.0000 cpot= 1 jt= 0.0 ptyt= 1 et= 0.0000/
% &partition /
% &pot kp= 1 type= 0 p(1:3)= 112.000 0.0000 1.2000 /
% &pot kp= 1 type= 1 p(1:6)= 52.500 1.1700 0.7500 3.5000 1.3200 0.6100 /
% &pot kp= 1 type= 2 p(1:6)= 0.000 0.0000 0.0000 8.5000 1.3200 0.6100 /
% &pot kp= 1 type= 3 p(1:3)= 6.200 1.0100 0.7500 /
% &pot /
% &overlap /
% &coupling /
% -------------------------------------------------------------------
%\end{verbatim}
%}
\centerline{\includegraphics[clip,width=0.42\textheight]{figures/Example-optical-frin}}
\caption{{\sc Fresco} input file {\sf p-cd.frin} for proton scattering on $^{112}$Cd}
\label{Example-optical-frin}
\end{boxed}
\begin{boxed}
%{\footnotesize
%\begin{verbatim}
%-------------------------------------------------------------------
%'p-cd.frin' 'p-cd.frout'
%4 1
% &variable kind=1 name='r0' kp=1 pline=2 col=2 potential=1. step=0.01/
% &variable kind=1 name='V' kp=1 pline=2 col=1 potential=50.0 step=0.1/
% &variable kind=1 name='W' kp=1 pline=2 col=4 potential=5.0 step=0.1/
% &variable kind=1 name='WD' kp=1 pline=3 col=4 potential=10. step=0.1/
% &data iscale=0 idir=1 lab=F abserr=T/
% 22. 0.548 0.044
% 26. 0.475 0.024
% 30. 0.481 0.014
% 38. 0.447 0.009
% 50. 0.144 0.004
% 66. 0.499 0.010
% 70. 0.248 0.005
% 86. 0.463 0.014
% 90. 0.485 0.015
% 106. 0.087 0.003
% 110. 0.135 0.004
% 130. 0.161 0.005
%&
% -------------------------------------------------------------------
%\end{verbatim}
%}
\centerline{\includegraphics[clip,width=0.42\textheight]{figures/Example-optical-search}}
\caption{{\sc Sfresco} input {\sf p-cd.search} for proton scattering on $^{112}$Cd}
\label{Example-optical-search}
\end{boxed}
\begin{figure}
\includegraphics[clip,width=0.37\textheight]{figures/p-cd-fit}
\caption{Initial and fitted proton scattering on $^{112}$Cd at 27.9 MeV.}
\label{example-sfresco-fit}
\end{figure}
There are 3 \&{\it pot} namelist lines in \boxref{Example-optical-frin} for
the nuclear parts of potential {\it kp}=1, so the variables of the interaction potentials ({\it kind}=1)
in the search are identified by the specification of {\it kp} and {\it pline} in the \&{\it variable} namelist
in \boxref{Example-optical-search}, along with {\it col} for the index to the {\it p} array. The {\it potential} value gives the initial value for the search, and {\it step} the initial magnitude for trial changes.
Deformations can as well be searched upon.
There are also parameters {\it valmin} and {\it valmax},\footnote{
Both or none of these must be present.}
to limit the range of that variable. Spectroscopic amplitudes to be varied ({\it kind}=2) are identified by the order {\it nfrac} in which they appear in the {\sc Fresco} input in a \&{\it cfp} namelist, and then by their initial value {\it afrac}. Other variable {\it kind}s are described in the {\sc Fresco} input manual.
Experimental data sets are identified by their {\it type} specifications as listed above, and then by
{\it data\_file} for name of data file with data, which can be `$=$' for this search file,
`$<$' for stdin (the default is `$=$').
Then {\it points} gives the number of data points (default: keep reading as many as possible),
%{\it delta}: if non-zero, construct linear $x$-scale from {\it xmin} in steps of {\it delta},
% (default 0),
{\it lab} is T or F for laboratory angles and cross sections (default false), and
{\it energy} is lab energy for a {\it type}=0 dataset (default: use {\it elab(1)} from \&{\it fresco} namelist).
The {\it abserr} is true for absolute errors (default: false).
Next, {\it idir} is $-1$ for cross-section data given as astrophysical S-factors,
0 for data given in absolute units (the default), and 1 as ratio to Rutherford.
Finally, {\it iscale} is $-1$ for dimensionless data, and
0 absolute data in units of fm$^2$/sr, 1 for b/sr, 2 for mb/sr (the default) and 3 for $\mu$b/sr.
With these two files (Boxes \ref{Example-optical-frin} and \ref{Example-optical-search}),
{\sc Sfresco} is invoked interactively. First the name of the search file in \boxref{Example-optical-search}
is given, then commands as in \tableref{Example-sfresco-commands}.
A minimum set of commands will be
{\sf p-cd.search / min / migrad / end / plot}. The output plot file (default name {\sf search.plot}) also contains the final values of the searched variables. The fitted parameters in this example are
$V=52.53$ MeV, $r0=1.179$ fm, $W=3.46$ MeV and {\em WD} $=7.43$ MeV with $\Chi^2/N = 2.19$.
The initial and final fits to the data are shown in \fig{example-sfresco-fit}.
\begin{table}
\caption{{\sc Sfresco} input commands. Words in {\tt typewriter font} are to be replaced with user values.}
{\footnotesize
\begin{tabular}{l | l} \hline\hline
Command & Operation \\
\hline
Q& query status of search variables
\\ SET {\tt var val}& set variable number {\tt var} to value {\tt val}.
\\ FIX {\tt var}& fix variable number {\tt var} (set {\tt step}=0).
\\ STEP {\tt var step}& unfix variable {\tt var} with step {\tt step}.
\\ SCAN {\tt var val1 val2 step}& scan variable {\tt var}
from value {\tt val1} to {\tt val2} in steps of {\tt step}.
\\ SHOW& list all datasets with current predictions and $\chi$ values.
\\ LINE {\tt plotfile}& write file (default: {\sf search.plot}) %for reading by {\tt xmgr, xvgr},
with theoretical curves only.
\\ READ {\tt file}& read plot output {\tt file} for further searches, if not:
\\ READ {\tt snapfile}& if the name of {\tt snapfile} contains the string `snap',\\
& read last set of snap output {\tt snapfile} from a previous fort.105.
\\ ESCAN {\tt emin emax estep}& scan lab. energy in incident channel
%, to give
%phase shifts in fort.71, fusion \& reaction cross sections in fort.40,
% and S-factors in fort.35 and 75.
%This is to look for/at resonances, or at behavior out of data's energy range,
%not for $\chi^2$ fitting.\\
\\ MIN& call {\sc Minuit} interactively.
\\ MIGRAD & in {\sc Minuit}, perform MIGRAD search.
\\ END& return to {\sc Sfresco} from {\sc Minuit}.
\\ PLOT {\tt plotfile}& write file (default: {\sf search.plot}) %for reading by {\tt xmgr, xvgr},
with data and theory curves.
\\ EX& exit (also at end of input file)\\
\hline\hline
\end{tabular}
}
%\centerline{\includegraphics[width=0.8\textwidth]{figures/Example-sfresco-commands}}
\label{Example-sfresco-commands}
\end{table}
\section{System requirements, compilations and installation}
\label{fresco-install}
\index{Fresco!compilation and installation}
The website {\sf www.fresco.org.uk}
contains a complete distribution set of files for {\sc Fresco} and {\sc Sfresco}, {\sc sumbins}
and {\sc sumxen}. It contains the input and output files for all the examples described here.
The distribution files also contain input and output
files for a range of test cases for your installation, as well as copies of the detailed input manuals for
{\sc Fresco} and {\sc Minuit}.
The distribution set contains a set of precompiled binaries for {\sc Fresco} and {\sc Sfresco}, as well as the source code. To compile the source, you will need a {\sc Fortran} compiler for at least Fortran 90
or 95.
The code is in the directory {\sf fres/source/}, where the script {\sf mk} attempts to find the best compiler for your system, and then compile in a subdirectory named by your system architecture {\it arch} and the compiler chosen using the {\sf makefile}.
You may have to edit the {\sf makefile} to set {\sc fflags} for your compiler, and set {\sc time} and {\sc flush} according to your system libraries. After compilation, {\sf mk install} copies the binaries to a standard {\sf bin}/{\it arch}/ directory for execution in other places.
The website will be regularly updated to include descriptions of any further changes needed or advisable for the programs. All of the information on the website is published under the conditions of the GNU Public License described
at {\sf www.gnu.org/copyleft/gpl.html}.
\begin{thebibliography}{9}
\bibitem{TUM}
T. Tamura, T. Udagawa, K. E. Wood and H. Amakawa, {\em Comput. Phys. Commun.} {\bf 18} (1979) 63.
\end{thebibliography}
%\endinput
%\newpage
%\begin{enumerate}
% \renewcommand{\theenumi}{(\alph{enumi})}
%\item more info on sumbins and sumxen ??
%\item Distribution on CD and website
%\item System requirements, compilation, installation, and test runs
%\item Front-end for two-partition reactions
%\item Front-end for CDCC breakup calculations
%\item Accessing the example calculations for the book topics
%\item Plotting results
%\item Further details
%\end{enumerate}
%{\em to be completed in later Drafts}
\end{document}