next up previous
Next: Appendix I : FRESCO Up: Fresco Previous: 1 Introduction

Subsections

2 Input Cards

2.1 Definition Cards

Card 0

10A8 HEADNG(10)

Heading to describe the nature of this run, for user's information only.


Card 1

10F8.4  HCM,RMATCH,RINTP,HNL,RNL,CENTRE

Wave functions calculated at intervals of HCM up to abs(RMATCH).

If RMATCH < 0, then read Card 1a for coupled Coulomb wave functions.

Non-local kernels Kfi'(Rf,Dfi) calculated at Rf intervals of RINTP, and for a non-local (Dfi) range of RNL centred at CENTRE in steps of HNL. RMATCH and RINTP are rounded to multiples of HCM, and HNL is rounded either to a multiple or a sub-multiple of HCM.

Card 1a

2F8.6, 2F8.2    RASYM,ACCUR,SWITCH,AJSWITCH

Use coupled Coulomb wave functions from CRCWFN out to asymptotic radius RASYM from inner radius abs(RMATCH) for those partitions in which PWF is TRUE.
If RASYM < 0, then determine the outer radius in order that classical Coulomb trajectories reach an angle abs(RASYM) degrees.

ACCUR is an accuracy parameter controlling the piecewise step length. Default is 0.01: smaller values give greater accuracy.

SWITCH is the radius at which to switch from Airy functions to sines and cosines in piecewise method. Default is 1000 fm.

AJSWITCH - default is 0.0
Normally the Coupled Coulomb wfns are matched to zero and the Numerov integration is omitted, when the Coulomb distance of closest approach is more than 4.5 fm outside abs(RMATCH) (or the -CUTR distance if CUTR negative). This is only allowed when $J_{total} \geq $AJSWITCH.


Card 2

2F4.0,      F8.4,  L2, I2,  1X,A1,   I2     4(I4,F4.0)
JTMIN,JTMAX,ABSEND,DRY,CSET,  RELA, NEARFA,(JUMP(i),JBORD(i),i=2,5)

Calculate coupled-channels sets with total J in the interval max(0,JTMIN) < J < JTMAX, stopping sooner if the absorbtion from the elastic channel is less than ABSEND millibarns of three successive J/parity sets. (If ABSEND < 0, this takes the full J interval.)
If JTMIN < 0, then for J < abs(JTMIN) include only the incoming channel in the calculations, ignoring tranfers and excited states. This is needed if the elastic scattering cross sections are to be given correctly.

DRY is normally F (false), but if T (true) the code does a `dry run' to check that all arrays are of sufficient size. All coupled channel sets are generated, but only the elastic channels should be non-zero.

CSET = number of coupled-channels sets to be solved, for each energy on Card 19. CSET = 0, blank or F, for no special limit.
If CSET = 'P', include only positive parity coupled channels sets,
If CSET = 'M' or 'N', include only negative parity coupled channels sets.

RELA = `T' for relativistic kinematics for the incident projectile. Not yet implemented.

NEARFA
= 0 or 1 for the usual cross sections,
= 2 or -2 for printing `far side' cross sections too,
= 3 or -3 for printing far and near side cross sections too.
> 0 for printing far & near-sides for elastic channel only.
< 0 for printing far & near-sides for all channels.

JUMP(i),JBORD(i) : Calculate coupled-channels sets not for every J value, but at intervals of JUMP(i) for J $\geq$ JBORD(i), for i=1,5. (The program sets JUMP(1)=1 & JBORD(1)=JTMIN, to give no J jumping initially). The omitted J values are provided by interpolation on the scattering amplitudes A(m'M':mM; L) prior to calculating cross sections.

Card 3

2I1,F6.4,F8.4,F7.4,I1,3F8.4  KQMAX,PP,THMIN,THMAX,THINC,KOORDS,CUTL,CUTR,CUTC

Give cross sections (and tensor analysing powers up to rank K = KQMAX) for centre-of-mass scattering angle from THMIN to abs(THMAX) in steps of THINC.
Elastic channels normally output the ratio to Rutherford, unless THMAX < 0.
Calculate analysing powers/polarisations for projectile (PP=0 or blank), target (PP=1), ejectile (PP=2) or residual nucleus (PP=3). PP=4 gives projectile (PP=0) analysing powers, along with Kyy results.

KOORDS determines the coordinate systems used for the analysing powers:

KOORDS
= 0 : Madison coordinates (default)
= 1 : Madison + Transverse
= 2 : Madison + Transverse + Recoil
= 3 : Madison + Transverse + Recoil + Hooton-Johnson

CUTL = radial points per L (angular momentum of partial wave) of lower radial cutoff when integrating the radial equations. Default = -1.6
When CUTL>0, use $\ell=J_{total}$ (total angular momentum of CC set),
When CUTL<0, use $\ell=L_{in}$ (orbital angular momentum of incoming partial wave).
Using CUTL<0 gives more accurate analysing powers.

CUTR = lower radial cutoff (fm). Use max(CUTL*$\ell$*HCM,CUTR).
If CUTR < 0, put cutoff at point-Coulomb turning point - |CUTR|.

CUTC = lower radial cutoff (in fm) for off-diagonal couplings.

Card 4

F6.4,I2, I4,  I2,    I2,  A1,  I3, 2I4        I2,    F6.4
IPS, IT0,ITER,IBLOCK,PADE,ISO, NNU,MAXL,MINL, MTMIN, EPC,

2F8.4     2I2,      2e8.1,              4x,i4
ERANGE,DK,INH,plane,smallchan,smallcoup,   initwf

Solve the coupled channels equations by at least IT0 iterations, and up to ITER iterations. Stop sooner if the absolute differences between successive S-matrix elements (scaled by (2J+1)/(2.JTMAX+1)) are less than IPS percent. (Excited state pairs with IGNORE set on Card 7 are not counted against IPS).

Putting IT0=ITER zero solves only the elastic channel (along with the IBLOCK channels: see below). Putting IT0=ITER = 1 or 2 etc. gives 1 or 2-step DWBA.
Normally, a run is terminated if more than ITER steps are required for convergence. Setting ITER < 0 allows continuation even after convergence has failed after abs(ITER) iterations.
Iterations are normally also stopped if the successive differences are smaller than the errors estimated for the numerical integration of the coupled equations. Setting IPS < 0 uses abs(IPS), without this extra check.

IBLOCK is the number of pairs of excitation levels (starting from partition 1, excitation 1) that are coupled exactly by blocking together.
If IBLOCK<1, then read Card 4.5 (see below) for R-matrix solution of the coupled equations.

PADE
= 0 for no Pade acceleration,
= 1 for Pade acceleration by the epsilon algorithm.

ISO (not implemented in this version)

NNU is the number of Gaussian integration points in the angular integration used for the non-local transfer kernels. NNU should be a multiple of 6; NNU = 18 is the minimum, and 24 or 36 give acceptable accuracy for all the reactions tried so far.

MAXL,MINL are the maximum and minimum L values for the non-local kernels. If zero, MAXL has the default value JTMAX+6, and if MINL < 0 it takes the default value |JTMIN|-6.

MTMIN is the lowest L-transfer for calculating transfer form factors using the m-dependent expressions for spherical harmonics. Putting MTMIN = 0 gives default value MTMIN = 6 (use MTMIN < 0 to avoid invoking default, if all transfers are to use this method).

EPC = percentage cutoff accuracy in the NNU angular integration. If zero, the default is (30/NNU)2%.

ERANGE, DK - ignored

INH
= 0 : zero-range transfer forms in intervals of HCM exactly
= 1 : stored in steps of HCM * (proj. core)/(proj. composite mass)
= 2 : stored in steps of HCM * (targ. core)/(targ. composite mass)
So INH=2 corrects for longitudinal recoil during transfers with zero-range projectiles.

PLANE - ignored

SMALLCHAN = fraction of unitarity to define a 'small channel'.
A channel that is 'small' for NSMALL=2 times is dropped permanently.

SMALLCOUP = if all nonelastic channels are weaker than the fraction SMALLCOUP of unitarity, then permanently change from coupled-channels to DWBA.

INITWF = file number from which to read fixed channel wave functions during all iterations. The channels whose wfs are changed must have the same IT index, that counts cards 7 as excited state pairs. This file as the same format as the fort.17 produced when WDISK is nonzero. The INITWF > 0 for formatted wf file, and < 0 for unformatted (the same sign convention as for WDISK).

Card 4.5

If IBLOCK<0, then read:
2i4,           I1,    L1,     I2,  i2,   f6.4, 3f8.4,
NRBASES,NRBMIN,BUTTLE,PRALPHA,PCON,MEIGS,RMATR,EBETA(2),WEAK

NRBASES = target number of radial basis states in each channel. (Use 2*NRBASES for the elastic and first-inelastic channel for more accuracy).

NRBMIN = minimum number of radial basis states.

BUTTLE = 4 for none, 0,2 for complex, 1,3 for real (2,3 without energy shift) Buttle correction (default 0)

PRALPHA = print basis-state eigenvalues to files fort.60,61,62,63

PCON = trace variable for calculation of radial basis states (same meaning as IPC on Cards 13 for single-particle bound states).

MEIGS = maximum number of bound states to find, if ENLAB < 0 on Card 19.

RMATR = R-matrix matching radius (default is RMATCH on Card 1). Warning: RMATR will be changed to an even multiple of HCM.

EBETA(i) = energy $\hbar^2k^2/2m$, where k= f'/f, the logarithmic derivative for all radial basis states at r=RMATR, with k having the same sign as EBETA(i). Use i=1,2 for positive,negative parity coupled channels sets (respectively).

If R-matrix solutions are selected (by IBLOCK<0) then all channels are `blocked' together and solved in a full CRC procedure.

If WEAK>0, then non-elastic columns of the R-matrix are set to zero, when penetrabilities < WEAK.

2.2 Trace Control Variables:

Card 5

14I2
CHANS,LISTCC,TRENEG,CDETR,SMATS,XSTABL,NLPL,WAVES,LAMPL,VEFF,KFUS,WDISK,
               BPM,MEL,CDCC,NFUS
A value of 0 gives no trace, increasing values give progressively more printed output.
Decremented variables are decreased by 1 on each use.


CHANS $\geq$ 1 : Print the sets of coupled partial waves for each J,parity. Decremented.

LISTCC
= 1 : Print coupling coefficients between these channels. Decremented.
= 1,2,... Print progressively more detail of couplings.

TRENEG
$\geq$ 1 : Print all potentials as they are calculated from Cards 10
$\geq$ 3 : Print all potentials as they are calculated from Cards 9

CDETR
$\geq$ 1 : Print information on the solving of the coupled equations. (decremented).

SMATS
$\geq$ 1 : Print absorbtion & reaction cross sections for successive partitions and excitations.
$\geq$ 2 : Print elastic S-matrix elements (Sel). Also `punch' these elastic Sel on output file 7, in format (2F15.10,L,J,JTOT) for Sel complex, L, J and JTOT. See WDISK below for description of these quantum numbers.
$\geq$ 3 : Print all S-matrix elements for the `grazing partial waves' defined by 0.05 < Re(Sel) < 0.95
$\geq$ 4 : Always print all the S-matrix elements.
$\geq$ 5 : Print all S-matrix elements at each iteration of the coupled equations (or, if PADE > 0, the Pade approximant)
$\geq$ 6 : Print all actual S-matrix elements at each iteration (these may be divergent before Pade acceleration).

XSTABL $\neq$ 0 : If XSTABL is non-zero, in file 16 punch output cross sections for all excitation levels in all partitions. A header card in FORMAT(5I6) gives partition IC, level pair IA, number of tensor ranks of analysing powers 1 < KQ1PR < XSTABL, number of angles NANGL, and NEARF. NEARF=1 for total cross section, 2 for far-side component, and 3 for near-side component.
Then follow NANGL print operations in FORMAT(1P,6E12.4), repeating the FORMAT for each operation if KQ1PR is large, of THETA, elastic xs (mb), T10, iT11, T20, T21, T22, iT30, iT31 etc.

NLPL > 0 : print a `contour plot' of the non-local kernels Kfi(Rf,Dfi). This is useful to determine if the parameters on card 1 are adequate. Decremented.

WAVES
$\pm$1 or $\pm$3 : print out wave function solutions of the coupled equations at the end of the iterations. (If WAVES<0 : print out the RATIO of the w/f to its asymptotic form ((G-iF) - S.(G+iF)).i/2)
2 or 3 : print out the source terms at each iteration of the coupled equations.

LAMPL
$\neq$ 0: Print out (on Fortran file 36) the coefficients A(m'M':mM; L) for the Legendre coefficients in the scattering amplitude for the partition number abs(LAMPL), and print out the $f(m'M':mM; \theta)$ for each angle $\theta$.
<0 : only print out on file 37 the amplitudes f, not the A's, for partition abs(LAMPL).
The phase convention here is that there is no Coulomb phase shift for L = 0 in the Coulomb scattering amplitude : factors such as $\exp i(\sigma_L-\sigma_0))$ appear in the A's.

VEFF
$\neq$ 0 : Calculate the `coupled channels effective potential' found be averaging the `trivially equivalent potential' over all the $J,\pi$ sets, with weights of the elastic wave functions times the reaction cross section, in each set.
< 0 : Add this effective potential to optical potential of the elastic channel before printing.
= -2 or +2 : Exclude partial waves with elastic S-matrix element $S_\ell< 0.1$ from the averaging sum.
The results show the real and imaginary parts for successive values of J-L, for the projectile only.

KFUS,NFUS
> 0 : Calculate `core fusion' using the imaginary and scalar parts of potential number KFUS (i.e. cards 10 with TYPE = 1 or 2, and KP = KFUS), also for the first NFUS inelastic chamnels

WDISK
= 1 : Print elastic wave functions on output file 17, FORMATTED
= 2 : Print all wave functions on output file 17, FORMATTED
=-1 : Print elastic wave functions on output file 17, UNFORMATTED
=-2 : Print all wave functions on output file 17, UNFORMATTED

The following data formats are used when WDISK > 0 :
card A: (I4,2F8.4,F8.1,I3)
         NR,H,ENLAB,JTOTAL,PARITY,MP,MT,ZP,ZT :
           number radial points, step size, lab. energy,  J,pi,
           projectile and target masses and charges
card B: (2I4,2F6.1,I4,F6.1,2F15.10,f12.8)
         IT,L,J,JTOT,LIN,JIN,SMAT (complex), ETA
where

IT = index to excited state pair, counts cards 7.
L = partial wave
J = L + projectile spin
JTOT = total spin = Jtotal = J + target spin
LIN = incoming partial wave
JIN = incoming J value.
SMAT = S matrix element for this partial wave.

card C: (6E12.4)   (psi(I),I=1,NR)   wave function
card C is repeated until NR complex values given
NB: the first point psi(1) = 0 always, as at r=0
Cards B & C are repeated for each channel, until IT < 0.
When WDISK < 0, successive records contain the two real values of psi(I), starting IN THIS CASE, from I=2 (i.e. r=h).

BPM
$\geq$ 1 : Calculate fusion cross sections in the Barrier Penetration Model using first the bare potential, and then the bare potential + the `weighted equivalent potential' calculated when VEFF $\neq$ 0.
$\geq$ 2 : Print out L-distributions of the fusion cross section.

MEL - ignored

CDCC
$\neq$ 0: Print out the $f(m'M':mM; \theta)$ for each angle $\theta$ on file 57 for partition PEL, after the following information:

card A: (F10.4,3F8.4)   ENLAB,Bproj,H2SM,e^2    lab energy,projectile binding energy
                                                hbar^2/2.m, e^2
card B: (4f8.4)         massp,masst,massc,massv masses projectile, target, core, valence
card C: (4f8.4)         Zp,Zt,Zc,Zv             charges
card D: (4A8)           namep,namet,namec,namev names
card E: (4f8.1)         Jp,Jt,Jc,Jv             g.s. spins
card F: (4i8)           Pp,Pt,Pc,Pv             g.s. parities
card G: (4I4)           NBINS,NKMAX,NEXB,NNJMAX no. CDCC bins, max NK,
                                                no. excited states, max(2*Jex+1)
card H: (I4,2f8.4)      NANGL,THMIN,THINC       (cm angular range from Card 3)
for each of the NBINS bins:
 card I:(i2,2f4.1,3f8.4,2i4)
         l,j,Emid,kmin,kmax,NK,KN,ISC
            l,j: quantum numbers  (s==Jv)
            Emid:  centre of bin with respect to continuum threshold
            kmin,kmax,NK: Min,max and number of k values in bin integral
            KN:  original KN index for bin state
            ISC:  normalisation used for bin
     for each IK=1,NK
         card J: (10f8.4) delta(IK): nuclear phase shift used in bin integral (radians)

for each excited state pair in the entrance partition: IA=1,NEXB::
    card K: (f4.1,i4,f8.4,i4) Jex,Parity,Eex,IBIN:
            Jex :          spin of this projectile excited state
            Parity:        parity of this projectile state
            Eex:           excitation energy of this state above g.s.
            IBIN:          (first) bin defined for this excited state
    for each IANG=1,NANGL: read complex numbers:
    card L: (6E12.4): ((FAM(MEX,MP,IANG,IA),MEX=1,2*Jex(IA)+1),MP=1,2*Jp+1)
The phase convention again is that there is no Coulomb phase shift for L = 0 in the Coulomb scattering amplitude : factors such as $\exp i(\sigma_L-\sigma_0))$ appear in the A's.

Summary of bin normalisation factors for different ISC values:

ISC
= 2: $\exp(-i\delta(k))$
= 4: $\sin(\delta(k))\exp(-i\delta(k))$
= 12: $k\exp(-i\delta(k))$
= 14: $k\sin(\delta(k))\exp(-i\delta(k))$

2.3 Partitions and Excited States

Card 6

(for each partition IC=1,2,... repeat until a blank card 8) :
A8,      2F8.4,           I4,   L1,1X, A8,      2F8.4,           F8.4
NAME(1,),MASS(1,),ZZ(1,), NEX(),PWF(), NAME(2,),MASS(2,),ZZ(2,), QVAL(IC)

name-----mass-----charge-              name-----mass----charge-
----projectile----------               ---------target--------

Abs(NEX) = number of pairs of excitations of projectile and/or target,

PWF = T, if acceleration of long-range Coulomb couplings to be performed when card 1a present. Default is T.

QVAL = Q value of this partition, relative to the Q-values of the other partitions, for Ep = Et = 0.

If NEX < 0, then cross sections for this partition are not printed.

Card 7

(repeat NEX times before the next Card 6) :
F4.1,I2,   I2,    F8.4,2F4.1, I4,1X,A1
Jp,  COPYp,BANDp ,Ep,  Kp,Tp, KP,PWF,
-------projectile state-----

F4.1,I2,   I2,    F8.4,2F4.1,  2L2,         2I4
Jt,  COPYt,BANDt ,Et,  Kt,Tt,  EXCH,IGNORE, INFAM,OUTFAM
-----target state-----------

This card is repeated NEX times, IA=1,NEX

For both projectile (p) and target (t), the following are read :

J = spin of the state

BAND > 0 for positive parity state, and < 0 for negative parity. The value of BAND defines the rotational band : all states of the same BAND may be coupled by rotational mechanisms.

If COPY is positive, then this level is a copy of a previous level number (COPY < IA) in the same partition.

If COPYp < 0, then this level is an exchange copy of the same IA numbered level in the previous partition abs(COPY). That is, a copy with projectile and target nuclei exchanged.

If COPYt = -IC, then the target state is an exchanged projectile. If that state has spin zero, then only even partial waves are allowed.

COPYs are needed to define identical states of one nucleus in different pairings IA. Spectroscopic amplitudes (see Cards 17) should be defined for only the original `copied' level.

E = excitation energy of state relative to ground state.

K = K of rotation band containing this state. (If zero, use spin of first level in this band.)

T = isospin of this state (if zero, use INT((mass-2*charge)/2)) (The T value is not used by this version of FRESCO).

KP = index of optical potential for this pair of excited states.

PWF= T to use CRCWFN coupled Coulomb wave functions in this partition (see Cards 1,1a)
If zero, use as default KP = IC, the number of this partition.

If EXCH = T calculate and write out all cross sections and amplitudes for $180-\theta$.

If IGNORE = T, ignore the convergence of this state pair (see IPS discussion on Card 4).

If INFAM $\ne$ 0, read in complex amplitudes later to add to calculated scattering amplitudes (see cards A3). (If INFAM < 0, read in only one amplitude (non spin-flip) for each scattering angle.)

If OUTFAM $\ne$ 0, write out complex amplitudes later to add to calculated scattering amplitudes (see cards A3). (If OUTFAM < 0, write out only one amplitude (non spin-flip) for each scattering angle: the average of diagonal amplitudes.)

Card A3

Cards A3 : Extra scattering amplitudes -


If INFAM on a Card 7 is non-zero, then read on file |INFAM|, on a new card for each scattering angle, some complex amplitudes $f(\theta)$, to be added to those calculated by the program before the cross sections are calculated.

If INFAM < 0, then only one f(external) is read, and this is used for all the diagonal f(mM:mM), the off-diagonal values being zero.

If INFAM > 0, then all the f(m'M':mM) are read in.

Card 8

Blank card for end of reading partitions.

2.4 Potentials


Cards 9,10 (to define the potentials to be indexed by KP) :

Each potential is defined by a Card 9, defining the radii to be used and the diagonal Coulomb potential, followed by any number of Cards 10 which cumulatively construct the components of the nuclear potential. The first input on each card is KP, identifying the potential of which this is to be a component.

Card 9

Card 9: I3,I2,3X,2F8.4,    2F8.4
        KP, 0,   A#1,A#2,  R0C,AC  (i.e. would be Card 10, but TYPE=0)
All the radii are multiplied by CC, which is updated by a TYPE=0 Card 9 to CC = A#11/3 + A#21/3 and by TYPE>0 Cards 10, if P0 > 0, to CC = P01/3

R0C = radius (when multiplied by CC) of charge distribution
AC = diffuseness of charge distribution


The values of A#1 and A#2 affect only the conversion from r0 radii to R radii in the definition of potentials in Cards 9 & 10. If however you adopt the convention that A#1 is the target mass number and A#2 the projectile's, then FRESCO can later display the fractional $\beta$ values corresponding to given deformation lengths. (These are for information only, and do not affect the couplings.)

Card 10

Card 10:  I3,I2,  A1,I2,     3F8.4,      3F8.4,       F8.4
          KP,TYPE,IT,SHAPE,  P1,P2,P3,   P4,P5,P6,    P0
                             ---REAL---  -IMAGINARY-  A#
                             V,Vr0,Va,   W,Wr0,Wa,    A#


      

for TYPE
1 SHAPE Central potential, Volume
2 SHAPE Central potential, Derivative (i.e. surface)
3 SHAPE Spin-orbit for projectile
4 SHAPE Spin-orbit for target
5 SHAPE Tr tensor force for projectile
6 SHAPE Tr tensor force for target
7 SHAPE Tensor force between L and combined projectile+target spins(e.g. n-p tensor force)
8 SHAPE Spin.spin force for target & projectile spins
10 SHAPE Deformed projectile (matrix elements from ROTOR)
11 SHAPE Deformed target (matrix elements from ROTOR)
12 SHAPE Projectile coupled by matrix elements read in.
13 SHAPE Target coupled by matrix elements read in.
0 - Coulomb potential : see Card 9.

Where TYPE indicates type of spin tensor or excitation coupling,
IT is `1' or `3' to make this component included only iteratively,
`2' or `3' to NOT subtract this component in KIND=3,4 single-particle couplings,
the SHAPE indicates the radial shape of the form factor,
and P1-P3 & P4-P6 are parameters for the real and imaginary parts (respectively).


If TYPE < 0, add new potential numerically into previous potential, and use abs(TYPE) instead. This saves space, and is necessary if adding together components to be deformed by a following card 10 $\leq$ TYPE $\leq$ 13.

If KP < 0, this will be the last Card 10. Use abs(KP) below, and omit Card 12.


The SHAPEs for volume potentials (TYPE=1, 8 & 15) are,
with R = P2 * CC, RH = (r - R)/P3, and E = exp(-(r-R)/P3),
(replacing (P1,P2,P3) by (P4,P5,P6) throughout for the imaginary parts)
0: Woods-Saxon = - P1 / (1 + 1/E)
1: WS squared = - P1 / (1 + 1/E)2.
2: Gaussian = - P1 * exp(-RH2.)
3: Yukawa = - P1 * E / r
4: Exponential = - P1 * E
5: Reid soft core for T=0, central part
6: Reid soft core for T=1, central part
7: Read Real
8: Read Imaginary
9: Read Complex
-1: Fourier-Bessel = j0(RH) = $\sin$(RH)/RH

For `SHAPE's 7-9, immediately read in a comment card, then (free format) NPOINTS, RSTEP, RFIRST, followed by NPOINTS potential points for the shape in steps of RSTEP starting at r=RFIRST, reading card images all from Input File 4 in free format.
Rescale the real part by P1, the imaginary part by P2, and use radius = P3 * CC for subsequent deformation calculations.
If SHAPE = -7, -8, or -9, rewind file 4 before reading from it.


The SHAPEs for surface potentials (TYPE=2), with R, RH and E as before, are the first derivative forms (normalised to -1 when E=1, if reasonable):

0 : Woods-Saxon = - P1 * 4 * E / (1+E)2
1 : WS squared = - P1 * 8 * E*E / (1+E)3
2 : Gaussian = - P1 * 2 * exp(-RH*RH) * RH
3 : Yukawa = - P1 * E * (1 + r/P3) / (r*r)
4 : Exponential = - P1 * E
5 : Reid soft core for T=0, spin-orbit part x r,
6 : Reid soft core for T=1, spin-orbit part x r,
7,8,9 : Read from cards, as above.


The SHAPEs for spin-orbit potentials (TYPE=3 or 4) are the following surface shapes,
where CONLS = $\hbar^2/(m_\pi^2 c^2)$ = 2.000.

0 : Woods-Saxon = - CONLS * P1 * E / [(1+E)2 * P3 * r]
1 : WS squared = - CONLS * P1 * E*E / (1+E)3 / (2*r*P3)
2 : Gaussian = - CONLS * P1 * exp(-RH*RH) * RH / (2*r*P3)
3 : Yukawa = - CONLS * P1 * E * (1 + r/P3) / (4*r3*P3)
4 : Exponential = - CONLS * P1 * E / (4*r*P3)
5 : Reid soft core for T=0, spin-orbit part
6 : Reid soft core for T=1, spin-orbit part
7,8,9 : Read from cards, as above.

To become interaction potentials, these form factors will be multiplied by $j(j+1) - l(l+1) - s(s+1)= 2{\bf l}.{\bf s}$, where s = spin of projectile or target in each of its ground state and excited levels. There is no option here for state-dependent spin-orbit forces.


The SHAPEs for tensor potentials (TYPE=5, 6 & 7), with R, RH and E as before, are the second derivative forms (normalised to unity when E=1, if reasonable):

0 : Woods-Saxon = P1 * 8 * E*(E-1) / (1+E)3
1 : WS squared = P1 * 4 * E*E / (1+E)3
2 : Gaussian = P1 * 2 * exp(-RH*RH) * RH
3 : Yukawa = P1 * E * (1/P32. + 2/r2.) / r
4 : Exponential = P1 * E
5 : Reid soft core, 12 * tensor part
6 : Vol Gaussian = 12 * P1 * exp(-RH*RH)
7,8,9 : Read from cards, as above.


Deformations

The SHAPEs for deformed potentials (TYPE $\geq$ 10) are constructed either by reading external forms (SHAPE = 7, 8 or 9) for each P(k) non-zero (the input k-form being multiplied by P(k)), or (for SHAPE $\geq$ 10) by deforming the potential on the immediately preceding input card. This preceding card must be of TYPE=0 for Coulomb deformations, or TYPE = 1 or 2 for nuclear deformations: deformation of non-central forces is not yet implemented.


For Nuclear deformations P(k) = DEF(k), the deformation lengths (in fm.). (P0 = DEF(0) for inelastic monopoles.)

For Coulomb deformations P(k) = Mn(Ek), the intrinsic reduced matrix elements in units of e.fmk (P0 gives Mn(0) for inelastic monopoles). These intrinsic matrix elements Mn(Ek) use the reduced matrix elements $\langle I ' \vert\vert Ek \vert\vert I \rangle$ given by the definitions of Alder & Winther (not by those of Brink & Satchler), by assuming a rotational model where K is a good quantum number:

\begin{eqnarray*}
Mn(Ek) = \langle I ' \vert\vert Ek \vert\vert I \rangle
/ (\sqrt {2I+1} \langle IK k0 \vert I ' K \rangle ) \ .
\end{eqnarray*}


Thus, for (possibly hypothetical) $0 \rightarrow k$ transitions, $ Mn(Ek)= \langle k \vert\vert Ek \vert\vert 0 \rangle$.
Mn(E2) is related to the intrinsic quadrupole moment in the rotational model by $Mn(Ek) = Q _ 0 ~ \sqrt {5 / {16 \pi}}$. For a uniformly charged sphere of radius R, to first order in $\beta_k$ we have

\begin{eqnarray*}
Mn(Ek) = {{3 ~ Z ~ \beta _ k ~ R^k} \over {4 \pi}} \ .
\end{eqnarray*}


The Mn(Ek) are thus of the same sign as $\beta$ and Q0 for all rotational bands.


TYPES 12 & 13, for detailed and model-independent specifications, directly use the reduced matrix element

\begin{eqnarray*}
M(Ek) & = &i^{ I-I ' + \vert I-I ' \vert } ~ \langle I ' \ver...
...le \\
& =& \pm \sqrt{ (2I+1) ~ B(Ek, I \rightarrow I ' ) } \ ,
\end{eqnarray*}


where the sign convention has been chosen to make M(Ek) real and symmetric under $I \leftrightarrow I'$ interchange. The diagonal reduced matrix element gives the spectroscopic quadrupole moment

\begin{eqnarray*}
Q_2 = \sqrt {{16 \pi} /5} ~ (2I+1)^{-1/2} ~ \langle II 20 \vert II \rangle ~\langle I \vert\vert E2 \vert\vert I \rangle \ .
\end{eqnarray*}


The intrinsic and reduced matrix elements are therefore connected by

\begin{eqnarray*}
M(Ek) = Mn(Ek) ~ (-1)^{ [I-I ' +\vert I-I ' \vert]/2} ~ \sqrt{2I+1} \langle IK k0 \vert I ' K \rangle
\end{eqnarray*}


in the rotational model where K is a good quantum number, for states of spin I,I', projection K, charge Z, and radius R (DEF(k) = $\beta_k \times R$ for such states).


So Mn(Ek) is the square root of $B(Ek, 0 \rightarrow k)$ with some sign within K=0 bands. Mn(Ek) always has the same sign as $\beta$.

IN OTHER CASES, DO NOT ASSUME THAT ALL TRANSITIONS HAVE THE SAME M(Ek) VALUES. THUS Mn(Ek) ARE THE SAME FOR ALL PURE ROTOR TRANSITIONS, IN CONTRAST TO M(Ek). In order to guide the user here, effective $\beta$ values are calculated from the given Mn(Ek) or M(Ek), for each possible transition. From version FRV onward, Mn(Ek) is used in place of M(Ek) for TYPE 10 & 11 potentials.


Card 11

For TYPEs 12-13, Cards 11 are read to determine the details of the required couplings between each pair of excited states and of the reorientation terms for each individual state.

Card 11      4X,3I4,F8.4       IB,IA,k, STR
to put in a coupling to state IB from state IA, of multipolarity k and strength STR.

For each k used here, P(k) must have been non-zero, and the required coupling has the same form factor but with relative strength STR / P(k). That is, STR is the reduced matrix element <IB | M(Ek) | IA> = M(Ek, IA$\rightarrow$IB) for Coulomb transitions (NOT Mn(Ek)), and is the `reduced deformation length' RDEF(k) for nuclear transitions.

This `reduced deformation length' in the rotational model is

\begin{eqnarray*}
RDEF(k, I \rightarrow I')
&=& DEF(k) (-1)^{ [I-I'+\vert I-I'...
...le IK k0 \vert I'K\rangle\\
&=& M(Ek) * 4\pi / [ 3 Z R^{k-1} ]
\end{eqnarray*}


but may in general be varied independently of DEF(k) and M(Ek). Both these nuclear and Coulomb interactions are thus designed to give channel couplings which are independent of the projection K values read in the Cards 7. This of course requires making RDEF(k) and M(Ek) dependent on the spins of the excited states, not on any simple model.


The SHAPEs for deformed potentials (TYPE=10 to 13) are constructed by:

SHAPE = 0 to 6 : not used (as card gives deformation strengths).
= 7,8,9 : Read from cards, as above for TYPE=1. The multipoles are then multiplied by P(k) (by 1.0 for k=0), and should end up as scaled to replace the form factors calculated below:
= 10 : Coulomb multipoles have the usual simple form for a deformed charged sphere to produce M(Ek) = P(k), varying at large distances as

\begin{eqnarray*}
F(r) = M(Ek) ~ e^2 ~ {\sqrt{4\pi} \over (2k+1)}~ r^{-k-1}
\end{eqnarray*}


where e2=1.440, and where this form factor will later be multiplied by the charge number of the opposing nucleus.
= 10 : Nuclear multipoles are the deformation length DEF(k) = P(k) times the derivative of the undeformed potential times $-1/\sqrt{4\pi}$:

\begin{eqnarray*}
F(r) = - DEF(k) ~ \frac{1}{\sqrt{4\pi}} ~ \frac{dU(r)}{dr}
\end{eqnarray*}


The shape is the same for all multipoles k > 0.
= 11: Nuclear multipoles calculated by numerically deforming the radii of the previous potential, and projecting by Gaussian quadrature onto the required multipoles (those k > 0 for which P(k) is non-zero). For small deformation lengths, these SHAPEs are the same as for SHAPE = 10.
= 12: Same as SHAPE = 11, except that the monopole k = 0 part of the potential is also recalculated by quadrature. A first order correction is made to the radius to improve volume conservation.
= 13: Same as SHAPE = 12, except that no first order correction is made to the radius to improve volume conservation.


Card 12

Blank card for end of reading potential parameters.
            (Unless the last KP on card 10 was negative)

2.5 One- and Two-Particle Form Factors


Read in a card for each desired particle-nucleus form factor, then a blank card (13).

Each form factor is indexed by a number KN, which may refer to a single form, or for two-nucleon form factors will refer to a range of forms KN1 - KN2 (one for each distance and angular momentum between the two nucleons).

Each form factor is the binding of one or two particles to a specified nucleus, where the composite system is another specified nucleus.

To specify the core and composite nuclei, their partition numbers IC1 and IC2 are required (either order), with |IN|=1 for projectile and |IN|=2 for target nuclei.

The mass of the particle is the strict difference of the masses of the core and composite nuclei, except that if IN < 0, then a relativistic correction is made for effect of the relative Q-values on the mass of the composite nucleus, when extracting by differences the mass of the bound fragment.

Card 13

 2I3,     3I2,        I2,  1X,A1,  3I2,      F4.1,I2,F4.1,I2,
 KN1,KN2, IC1,IC2,IN, KIND,   CH1, NN,L,LMAX,SN,IA,JN,IB

   2I3,         F8.4, 2I3,     I3,  I3,F8.4,  I3,F8.4
   KBPOT,KRPOT, BE,   ISC,IPC, NFL, NAM,AMPL, NK,ER
where

KN (=KN1) or the range KN1-KN2 indexes this form factor,

IC1, IC2, IN define the particle and core nuclei as explained above,

KIND is the kind of couplings:
= 0 for (LN,SN) JN couplings   $\leftarrow$ use for typical transfers
= 3 for $\vert(LN,SN)Jn, J_{core}; J_{com}\rangle$ (IA & IB must be given)
= 6 for $\vert L_{nn}, (\ell,S_{12})j_{12}; J_{12}\rangle$ with isospin (.5,.5) T = 9 for $(L_{nn},(\ell,S_{12})j_{12})J_{12}, J_{core}; J_{com}\rangle$ & (.5,.5)T,Tcore;Tcom

where
Jcore = spin of core nucleus (state Ia if given)
Jcom = spin of composite nucleus (state IB if given)


CH1 = single-character identifier to distinguish clusters of nucleons of different structures that are not further described but should not be confused with each other. Use A-M for positive parity clusters, and N-Z for negative parities.

NN = number of nodes (include the origin, but not infinity, so NN > 0)

L = LN = angular momentum of bound cluster relative to the core

LMAX = maximum value of L in states in deformed potential,

SN = intrinsic spin of bound nucleon (one-particle states)
= total angular momentum (L+S) of bound custer (KIND 9)

IA = index (within core partition) of excited state of core, or zero if to be specified later.

JN = vector sum LN + SN

IB = index (within composite partition) of excited state of composite, or zero if to be specified later.

KBPOT = index KP of potential in which to bind this state

KRPOT = index KP' of potential with which to multiply this states' wave function for transfer interactions. If zero, use KBPOT. If the binding potential was adjusted for a specific binding energy, then this adjusted potential (not the original) is used for transfers.

BE = Binding Energy (positive for bound states, negative for continuum bins)


BOUND STATES:

ISC = 0 to vary the binding energy for fixed potential,
> 0 to vary the TYPE = ISC component of the potential KBPOT by a scaling factor to give binding energy BE.
< 0 to vary the TYPE = ISC component of the potential KBPOT, as above, but also to permanently rescale all the varied potential components. This affects all later bound and scattering states using the potential KBPOT.


CONTINUUM BINS:

ISC (default value 2)
= -2: no weighting or normalisation,
= 1,2 : weight wave functions by $\exp(-i\delta(k))$, so they are real, before integrating over bin width.
= 3,4 : weight wave functions by T(k)* (useful for resonances)
= -1,1,3 : normalise wave functions to unity (by usual square norm). (This option is not recommended, for physics reasons!)
$\geq$ 10: use additional k factor in the weighting function, with mod(ISC,10) for above choices. Recommended for low-energy bins.

The values ISC>0 give real-valued bins for single-channel states, but not, in general, for coupled-channels bins. In that case, Fresco stores all bound and continuum states as complex functions. KIND=3 and 4 forward and reverse couplings are both calculated explicitly, but not KIND=7 transfer couplings. For transfers, the reverse couplings must be put in explicitly.
Warning: imaginary parts of bins give imaginary parts of long-range Coulomb couplings that are ignored between abs(RMATCH) and RASYM in the CRCWFN calculations (the values on Card 1a).
In general, ISC=2 is recommended (or ISC=4 for resonances), since then the coupled-channels bins will be nearly real. With ISC=-2 (no phase weighting) there will be different results because of different weighting within the bin. Note that all channels within a bin have the same scalar weighting factor.


IPC = print control for further infomation:


IPC   Print Iterations   Print Final Result  Print W/F

0 no no no
1 no yes yes
2 no yes no
3 yes yes yes
4 yes yes no

For bin states, read this table with `iterations' replaced by `intermediate phase shifts'.

NFL < 0 : to write wave-function u(R)/R and potential overlap V(R)u(R)/R to file number abs(NFL).
> 0 : to read a previously-written wave function from file number NFL.
These wave function files contain a comment card, then (free format) NPOINTS, RSTEP, RFIRST, followed by NPOINTS wf points in steps of RSTEP starting at r=RFIRST, and then NPOINTS for the vertex function (potential*wf). The file numbers NFL should be in the range 20-33 (see appendix IV).

NAM, AMPL = if IA and IB are both non-zero, then there is enough information to set up the spectroscopic amplitude now, to $\sqrt{NAM} \times AMPL$. See Card 17, and discussion after Card 15, for further information about these amplitudes.
Note: If this is a form factor with mixed core levels, then AMPL should be specified here rather than on a Card 17 (as Card 17 requires an IA specification, and such a state would have multiple IA assignments).

If NAM = -1, then use AMPL for the mass of the bound particle, independent of the MASSes on Cards 6. (If NAM $\geq$ 0, then the default particle mass is the difference of the MASSes of the projectiles (IN=1) or targets (IN=2) for partitions IC1 and IC2).

If NAM < -1, then use AMPL instead of ERANGE for the range ER of the energies of the upper and lower boundaries of the continuum bins, and use at least NK=5 x |NAM| integration steps over this range. (These ER and NK override the input values.)

2.5.1 One-particle KINDs


For KIND = 0, (LN,SN) JN couplings.
One form factor with LN,SN, & JN as read in.
LMAX is not used.
IA & IB are used only if NAM & AMPL are non-zero, to specify spectroscopic amplitudes.


For KIND = 3, sum over coupled core and (ls)j particle states :
Coupled form factors (Ln,SN)Jn, Jcore; Jcom (summing over Ln, Jn & Jcore).
NN is the required number of radial nodes for the component wave function with core state IA and partial wave Ln = L.
LMAX is maximum Ln in summation.
SN as read in, the intrinsic spin of the bound particle.
All core states are included that can be coupled to form Jcom, using a deformed binding potential.
Note that such a deformed potential must be TYPE = 11, whether the projectile or target is deformed: not 10 or 12 or 13.
JN is the maximum Jn in the summation (single particle l+s)
IB gives Jcom (fixed) : spin of composite nucleus.
BE is the single-particle binding energy for core state IA.

2.5.2 Two-particle KINDs


For KIND $\geq$ 6, two-particle bound states are constructed out of sums of pairs of previously-defined one-particle states, and input parameters NN through to BE on Card 13 are given new meanings.


NN $\rightarrow$ NPAIRS, the number of pair-products to be summed

L $\rightarrow$ $\ell_{min}$ , minimum orbital angular momentum $\ell$.

LMAX $\rightarrow$ $\ell_{max}$ , maximum orbital angular momentum $\ell$.

SN $\rightarrow$ Smin , minimum sum S12 of the two nucleons' intrinsic spins (Smax = 1.0 always).

IA, IB give core and composite states, as before.

JN $\rightarrow$ J12 , total angular momentum of the two-particle state outside Jcore.

KBPOT $\rightarrow$ T , total Isospin of the 2-nucleon state (0 or 1)
(used to enforce $\ell$ + S12 + T = odd)

KRPOT $\rightarrow$ KNZR, the KN index to a single-particle state u12(r) of KIND 0 or 1, giving the N-N relative motion in the other participating nucleus (usually in the light ion).
If KNZR > 0, then just the overlap u(R) = <u12(r) | U(r,R) > is produced, suitable for zero-range two-nucleon transfer calculations.

BE $\rightarrow$ EPS , the threshold percentage to define components with square norms sufficiently small to be omitted in the final two-nucleon state.

ISC: If ISC $\le$ 0, use Gaussian quadrature grid, in blocks of 6th-order positions, for the N-N distance RMIN to RNN ( Card 1).
If ISC $\geq$ 1, use uniform grid for the N-N distances. Not so accurate.
If ISC $\neq$ 0, print out numerical values of resulting two-nucleon wavefunction U(r,R).

IPC controls the details printed (along with ISC as just above).

IPC $\geq$ 0 : one-line summary of U(r,R) form factor for each r.
$\geq$ 1 : overall norm and rms radius of total NN state.
$\geq$ 2 : contour plot of the L,$\ell$ components included.
$\geq$ 3 : contour plot of the U(r,R) x interaction potential

NFL < 0 : to write two-nucleon wave-function U(r,R) to file number abs(NFL).
> 0 : to read a previously-written wave function from file number NFL. The values of NPAIRS, lmin, lmax, Smin of the present run are ignored. Thus NPAIRS can be set to zero.

NPAIRS must be less than or equal to the compilation parameter MPAIR.


Data cards 14 are now read in, to give details of the single-particle wave functions and their amplitudes in the sum of pairs.

Card 14     3(4I3,                         F8.4)
             (NT1(I),NT2(I),NT3(I),NT4(I), COEF(I)), I=1,NPAIRS

The sum over I of COEF(I) $\vert (l_1,s_1)j_1, (l_2,s_2)j_2 ; J_{12},T \rangle$,
for (l1,s1)j1 given by state u1(r1) of KN1 = NT1(I) > 0,
and (l2,s2)j2 given by state u2(r2) of KN2 = NT2(I) > 0,
is then transformed into the required KIND = 6 format.
The (r1,r2) coordinates become (r,R) coordinates, where r = distance between the two nucleons (angular spin $\ell$), and R = distance from the core to their centre of mass (corresponding angular momentum is Lnn.

If NT3(I) > 0, then the wave functions u1(r1) x u2(r2) are further multiplied by u3(r) (of KN3 = NT3(I)) before coordinate transformations. Only the radial shape of KN3 is used, not any angular momentum numbers.


If NT1(I) = -1, then an external form factor is read in from Fortran file number NT2(I), and processed using the subroutine EXTERN1. At present, this routine is written to read triton wave functions from the Grenoble Faddeev calculations, only reading wave functions, not the potential x wavefunction (so for e.g. stripping, only use prior interactions).

If NT1(I) = -2, then an external form factor is read in from Fortran file number NT2(I), and processed using the subroutine EXTERN2. At present, this routine is written to read 3-body wave functions from HH calculations, reading wave functions, as well as the potential x wavefunction. The 'vrr' file format is assumed, and the breakup (third) vertex function is ignored.


If NT3(I) < 0, then the I'th component of the pair summation is simply the product of cluster wave functions u1(r) u2(R) where u2 may be KIND = 0 or 1.


NT4(I) is not used in this version of FRESCO.


For KIND = 6, construct components of the form $
L_{nn}, ((\ell, (s_1,s_2)S_{12})j_{12}; J_{12},T\rangle
$ for all different Lnn, $\ell$, S12 & j12 values permitted within the limits set by $\ell_{min}$, $\ell_{max}$, Smin, and Smax (J12 and T are fixed) by summing over pairs of single-particle wave functions.


For all KINDs, the printout also lists


DZ = derived charge of the bound particle (always positive).
DM = derived mass of the bound particle
K = the wave number of the bound state asymptotically

NORM=overall square norm of this bound state. The wave functions of the single-particle bound states are always normalised to unity.

RMS= root-mean-square radius of this bound state
D0 = zero-range stripping strength for transfers from this stare
D = asymptotic stripping strength, as used e.g. in sub-Coulomb transfers

2.6 Couplings

Cards 15-17:

Now read in sets of cards for each kind of coupling between partitions and/or between excited states, ending with a card 18 which has ICTO=0. Note that if deformed potentials were given as channel optical potentials then there will already be some couplings between excited states.

Card 15
3I4,               3I2,          2F8.2,  2F4.1     2I4
ICTO,ICFROM,KIND,  IP1,IP2,IP3,  P1,P2,  JMAX,RMX, IP4,IP5

The coupling is from all the states in partition ICFROM to all the states in partition ICTO.

Couplings in the reverse direction are also included unless ICTO < 0, except for KINDs 1 & 2 where finer control is allowed.

Couplings are only active for J < JMAX and Radius < RMX, (if JMAX=0, use JTMAX, and if RMX=0, use RMATCH from card 1).


The couplings can be of 8 KINDs:

= 1 :general spin transfer for projectile/transfer couplings
= 2 : electromagnetic one-photon couplings
= 3 : single-particle excitation of the projectile
= 4 : single-particle excitation of the target
= 5 : zero-range or LEA transfer with strength P1 and finite range radius P2 (in fm.)
= 6 : LEA transfer using D0. & D from bound states
= 7 : finite-range transfer
= 8 : non-orthogonality correction to a KIND 5,6,7 transfer
= 9 : same as KIND=1
=10 : same as KIND=2

After cards 15 for KINDs 2,3,4,5,6,7 (& 8 if IP2>0), spectroscopic amplitudes are read in by means of cards 17. These coupling types are those for which ICTO & ICFROM are different, one being a core partition and the other a composite nucleus. One table is used for all the amplitudes: it is indexed (besides the partition numbers) by IN,IB,IA & KN (see glossary), and stores a real number A. The table is filled cumulatively, starting perhaps with cards 13, so amplitudes need only be entered after the card for the first KIND of coupling in which they appear. (So if all the amplitudes are already entered for a certain KIND of coupling, then the following card 17 will be blank, to indicate no more cards 17 are to be read).


KIND = 1 : General Spin Transfer (with local or non-local external form factors on FILE 4) See Appendix V for more details.

IP1     = 0 : local form factor
1 : non-local (two-dimensional) form factor

IP2     = 0 : read in real values (only if IP3 $\geq$ 0)
1 : read in imaginary values (only if IP3 $\geq$ 0)
2 : read in complex values (only if IP3 $\geq$ 0)

IP3     =3 : read data from JLM folding program jlmP, for projectile couplings only
2 : read data from JLM folding program jlmP, for target couplings only
1 : read data from charge exchange program CHEX2, with appropriate scaling.
0 : no jlmP or CHEX2 scaling factors, only P1, P2 and FSCALE
-1 : write out typical non-local grid (R,R') coordinates
-2 : calculate non-local grid (R,R') coordinates, and call subroutine FFNL to calculate form factor.
-3 : calculate non-local grid (R,R') coordinates, call subroutine FNLSET to calculate an initial form factor, and the call FNLCC for each pair of coupled partial waves, for L-dependent factors.

P1,P2 : scaling for the real and imaginary parts (respectively).

If IP3 $\geq$ 0, read FROM FILE 4 the following cards (until a blank card):

     Card 16.6: I4, 3F8.4,          I4,  2F4.0,   2I4,   A35
                NP ,HNP,RFS,FSCALE, LTR, PTR,TTR, IB,IA, COMMENT
            for NP = number of radial points
                HNP = step size
                RFS = radius of first point
                FSCALE = scaling factor to be applied
                LTR = L-transfer
                PTR = projectile spin transfer
                TTR = target spin transfer
                IB  = excited-state pair fed by coupling
                IA  = excited-state pair feeding the coupling
                COMMENT = comment for display in printout.
       then free format, repeatedly until sufficient data is read.

If IP1 = 0, read local form factor from i=1 to N
1, read successively for j=1,NLO the non-local form factors FNL(i,j), i=1 to NP [NLO = RNL / max(HNL,HCM)]

When IP3=1 for CHEX2 input, the RFS and its F8.4 is omitted from Card 16.6, with default value RFS=HNP.


KIND = 2 : Electromagnetic one-photon couplings (for $E\lambda$ and $M\lambda$ processes).

ICTO is the gamma partition and ICFROM the particle partition. The photon must be on the `projectile' side, and the bound state between target states.

IP1 = $\lambda$ : The multipolarity of the radiation. If $\lambda >$ 0, include all multipoles 1,...,$\lambda$ permitted by parity, whereas if IP1 < 0, include only the multipole $\lambda$ = abs(IP1).

IP2    = 0 : Calculates both electric and magnetic convection couplings
1 : for electric only
2 : for magnetic only

IP3: not implemented yet


KIND = 3 or 4 : Single-particle excitations of the projectile (3) or target (4)

For these KINDs, ICTO is the partition of the nuclei being excited, and ICFROM is used to indicate the core partition if the single particle were removed. No couplings are generated to or from the ICFROM partition, only within ICTO partition.

IP1 = Q : The multipole order of the deforming potential due to the colliding nucleus. If Q > 0, include all multipoles 0,1,...,Q permitted by parity, whereas if IP1 < 0, include only the multipole Q = abs(IP1).

IP2
= 0 : Coulomb & nuclear (complex)
= 1 : nuclear (complex) only
= 2 : Coulomb only

IP3
= 0 or 10 : include all re-orientation terms
= 1 or 11 : no re-orientation terms for Q > 0
= 2 or 12 : ONLY re-orientation terms
= 3 or 13 : include only couplings to and from the ground state, but NOT gs reorientation,
= 4 or 14 : include diagonal couplings, and couplings to and from the ground state
= 5 or 15 : include diagonal couplings, and couplings to and from any bound state
$\geq$ 10 : read cards 16.5, repeating FORMAT(6E12.4) as required, for complex factors
QSCALE(Q), Q=max(0,-IP1),abs(IP1) to scale the folded form factors for multipoles Q.

IP4 = $Q_{\rm max}$, the max deformed core potential multipole
IP5 = $\Lambda_{\rm max}$, the new multipole order for formfactor reduction (see Appendix III).

P1 = FLOAT( potential KP index for fragment - target interaction)
P2 = FLOAT( potential KP index for core - target interaction)
(only SCALAR parts of the potentials P1 and P2 are used).


KIND = 5 or 6 : Zero-range & LEA transfers for projectiles

IP1 and IP2 not used.

For KIND=5,
P1 = D0 - ZR coupling constant
P2 = FNRNG - Effective finite-range parameter for use in LEA (in units of fm.)

For KIND=6, use D0 and FNRNG = $\sqrt{(D/D_0 - 1)/k^2}$. from the projectile bound states. With unbound states, or if IP3 = 1, use D0 from state, but FNRNG = P2 from input card here.

In both KINDs, read in cards 17, noting that for KIND = 5 spectroscopic factors for the projectile are not needed, and are ignored.

Users of these interactions kinds should also pay attention to the PARAMETER INH as described in Appendix I.


KIND = 7 : Finite-range transfers.

IP1 =     0,-2 : POST interaction
1,-1 : PRIOR interaction (N.B. meaning of IP1 thus depends on ICTO & ICFROM!!!)
$\le$ -1 : Use $\theta$ quadrature from $\theta = \pi$ down to $\theta = 0$. (Useful for finite-range knock-on with light projectiles)
$\le$ -3 : `VCORE' interaction: Use ONLY the core-core interaction potential (Useful for finite-range knock-on with light projectiles)

IP2 =     0 : no remnant
1 : full real remnant
-1 : full complex remnant
2 : ``non-orthogonality remnant" - this works by inserting a KIND = 8 coupling card after this coupling, before any subsequent couplings. This is does not affect the one-step amplitudes, and is only useful if another transfer step follows this coupling.

IP3 = KPCORE : the number KP of the potential to use between the two cores, in the remnant part of the interaction potential.

If IP3=0, use as KPCORE the optical potential given for the first pair of excited states in the partition of projectile core. (this uses the observation that optical potentials tend to depend more on the projectile than the target, and must clearly be re-examined if the projectile is heavier than the target.)

P1,P2 : not used.


KIND = 8 : Non-orthogonality supplement appropriate to a previous KIND 5,6 or 7 interaction.

IP1 =     0 : post
1 : prior (N.B. IP1 should be the same as the previous interaction!)

IP2
> 0 : read in spectroscopic factors as cards 17 (Only useful if you did NOT have a previous KIND 5,6,7 interaction, which would have needed the amplitudes then.)
= 0 : no cards 17 to read.

Note that you should either use KIND=7, |IP2|=2, or use KIND=7, |IP2|=1 and a KIND=8. If you have KIND=7, |IP2|=2 and a KIND=8 card, then this is double counting.


Cards 17:
Spectroscopic amplitudes for the overlaps between partitions ICTO & ICFROM already defined by a card 15.

Read until IN = 0 or IN < 0:
 4X,4I4,         F8.4
    IN,IB,IA,KN, A

meaning that the overlap of the composite nucleus in excitation state IB with the core nucleus in excitation state IA is the bound-state form factor KN with amplitude A. The IN=1 specifies projectile overlap, and IN=2 target overlap.

If the form factor KN mixes different IA levels, then the spectroscopic amplitudes should rather be specified with a Card 13.

N.B. The amplitudes A are signed, and are NOT the spectroscopic factors, but will typically be the square roots of these factors. For transfers out of or into closed shells of N antisymetrised nucleons, the spectroscopic factors will usually contain factors of N, so the spectroscopic amplitudes needed by FRESCO will typically need to already contain factors of $\sqrt{N}$.
The sign of A should be consistent with the spin coupling order used in the program, which is

\begin{eqnarray*}
(\ell,s)j, J_{core}; J_{com}
\end{eqnarray*}


for binding a $\ell sj$ nucleon onto a core of spin Jcore to form Jcom.
If IN<0, use abs(IN) in this the last card 17
If IN=0, no more cards 17 to be read.

2.7 Incoming Channel, Energy

Card 18: (has the first 4 columns blank or zero : ie card 15 with ICTO=0)

 3I4,      3I2
 0,PEL,EXL,LAB,LIN,LEX

Incoming plane waves are present in partition PEL with excitation pair EXL. The energies on card 19 are the laboratory energies for partition LAB's nucleus LIN (1 or 2 for projectile or target) in excitation pair LEX.

The defaults for PEL,EXL,LIN & LEX are all 1, and the default for LAB is PEL, so this card can be completely blank usually.


Card 19 :  Projectile Energies
3(F8.4,I8),F8.4) :  (ELAB(i),NLAB(i),i=1,3),ELAB(4)
A series of runs at different (real) energies until ELAB(i) = 0 is found. If NLAB(i) > 1, then the range from ELAB(i) to ELAB(i+1) is covered in NLAB(i) linear intervals.

2.8 Additional Cards on Fortran input file 4:


All these cards involve reading real or complex floating point numbers by repeating

  FORMAT(6E12.4)
Cards A1 : External Potential Form Factors (SHAPEs 7-9 on Cards 10)


Cards A2 : External Coupling Potentials for KIND 1 & 2 couplings (NO < 0 on Cards 16)


next up previous
Next: Appendix I : FRESCO Up: Fresco Previous: 1 Introduction
Ian Thompson 2011-09-08