&OVERLAP namelists for one- and two-particle form factors

Read in a namelist for each desired particle-nucleus form factor, then an empty namelist.

KN1, KN2, IC1, IC2, IN

 
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 $\vert$IN$\vert$=1 for projectile and $\vert$IN$\vert$=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.

KIND

the kind of coupling order (0 to 4 for one-particle states, and 6 to 9 for two-particle states) :

= 0 for (LN,SN) JN couplings   $\leftarrow$ use for typical transfers
= 1 for $\vert L_{n}, (SN,J_{core})S ; J_{com}\rangle$ (IA & IB must be given) *
= 2 for eigenstate in deformed potential (fixed SN, K, Parity) *
= 3 for $\vert(LN,SN)Jn, J_{core}; J_{com}\rangle$ (IA & IB must be given)
= 4 for form of leg of the Dalitz-Thacker Triton
= 5 (not used)
= 6 for $\vert L_{nn}, (\ell,S_{12})j_{12}; J_{12}\rangle$ with isospin $(.5,.5) T$
= 7 for $\vert(L_{nn},\ell)L_t, (S_{12},J_{core})S_t ; J_{com}\rangle$ & $\vert(.5,.5)T,T_{core};T_{com}\rangle$ *
= 8 (not used)
= 9 for $(L_{nn},(\ell,S_{12})j_{12})J_{12}, J_{core}; J_{com}\rangle$ & $(.5,.5)T,T_{core};T_{com}$

where
$J_{core}$ = spin of core nucleus (state Ia if given)
$J_{com}$ = spin of composite nucleus (state IB if given)
$K_{core}$ = projection K of core nucleus (state IA)
$K_{com}$ = projection K of composite nucleus (state IB)
$T_{core}$ = isospin of core nucleus (state IA)
$T_{com}$ = isospin of composite nucleus (state IB)

and * signifies that transfers using these KINDs are not yet implemented.

CH1, NN, L, LMAX, SN, IA, JN, IB

 
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 cluster (KINDs 6 & 9)
= combined cluster and core intrinsic spins (KIND 7)

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

JN = vector sum LN + SN,
but for KIND=1, JN = SN + J$_{core}$ (i.e. S in LS coupling)
and for KIND=7, JN = $L_{nn} + \ell$ (i.e. L in LS coupling)

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

KBPOT, KRPOT, BE, ISC, NK, ER

 
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 in the &Fresco namelist).
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, NFL

 
IPC = print control for further information:

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 line, 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 section 7).

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 discussion for the &cfp namelists, 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 &cfp namelist (as &cfp namelists require 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 in the &partition namelists. (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 leastgr NK=$5\times\vert NAM\vert$ integration steps over this range. (These ER and NK override the input values.)

DM

Set particle mass specifically, rather than using mass differences.

NK, ER, E

Specific values for this continuum bin.

RSMIN, RSMAX

If specified, set the overlap form factors to zero outside the interval [RSMIN,RSMAX]

NLAG, PHASE, AUTOWID

Parameters for PLUTO with bound or resonant states.

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 = 1:

$L_{n},(SN,J_{core})S; J_{com}$ i.e. LS coupling.

Coupled form factors with sum over $L_{n}$ and $S$.

SN as read in, the intrinsic spin of the bound particle.

IA giving $J_{core}$, spin of core nucleus.

IB giving $J_{com}$, spin of composite nucleus.

LMAX is maximum $L_{n}$ in summation.

JN is maximum S in summation.

NN & L restrict the number of radial nodes of one component wave function in the coupled set :

NN gives the number of radial nodes of the last partial wave of angular momentum L$_{n}$ = L input.

For KIND = 2:

eigenstates in a deformed potential, for fixed parity and K projection :

Coupled form factors $(l_{n},SN) JN,K$ , with sum over $L_{n}$ & Jn.

SN, LMAX, NN & L are as for KIND = 1 :

SN as read in, the intrinsic spin of the bound particle.

LMAX is maximum L$_{n}$ in summation.

NN & L restrict the number of radial nodes of one component wave function in the coupled set.

IA is a core state with correct K$_{core}$ projection quantum number.

IB is a composite state with correct K$_{com}$ projection number, so $K = K_{com} - K_{core}$

JN is the maximum Jn in the summation.

For KIND = 3:

sum over coupled core and (ls)j particle states :
Coupled form factors (L$_n$,SN)Jn, J$_{core}$; J$_{com}$ (summing over L$_n$, Jn & J$_{core}$).

NN is the required number of radial nodes for the component wave function with core state IA and partial wave L$_n$ = L.

LMAX is maximum L$_n$ in summation.

SN as read in, the intrinsic spin of the bound particle.

All core states are included that can be coupled to form J$_{com}$, 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 $J_{com}$ (fixed) : spin of composite nucleus.

BE is the single-particle binding energy for core state IA.

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 in the &overlap namelist 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 $\rightarrow$ S$_{min}$ , minimum sum S$_{12}$ of the two nucleons' intrinsic spins (S$_{max}$ = 1.0 always).

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

JN $\rightarrow$ J$_{12}$ , total angular momentum of the two-particle state outside J$_{core}$.

KBPOT $\rightarrow$ T , total Isospin of the 2-nucleon state (0 or 1)s (used to enforce $\ell$ + S$_{12}$ + T = odd)

KRPOT $\rightarrow$ KNZR, the KN index to a single-particle state $u_{12}(r)$ 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 $u(R) = <u_{12}(r) \vert U(r,R) >$ 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.

If ISC $\le$ 0, use Gaussian quadrature grid, in blocks of 6th-order positions, for the N-N distance RMIN to RNN ( &Fresco namelist).
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):
$\geq$ 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 $U(r,R) \times$ 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, $l_{min}$, $l_{max}$, $S_{min}$ of the present run are ignored. Thus NPAIRS can be set to zero.

Namelist &TWONT

This namelist gives the details of the single-particle wave functions and their amplitudes in the sum of pairs.

NT(1:4, :), COEF(:)

 
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 $(l_1,s_1)j_1$ given by state $u_1(r_1)$ of KN1 = NT(1,I) $>$ 0,
    and $(l_2,s_2)j_2$ given by state $u_2(r_2)$ of KN2 = NT(2,I) $>$ 0,
is then transformed into the required KIND = 6 format.

The $(r_1,r_2)$ 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 L$_{nn}$.

If NT(3,I) $>$ 0, then the wave functions $u_1(r_1)\times u_2(r_2)$ are further multiplied by $u_3(r)$ (of KN3 = NT(3,I)) before coordinate transformations. Only the radial shape of KN3 is used, not any angular momentum numbers.

If NT(1,I) = –1, then an external form factor is read in from Fortran file number NT(2,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 $\times$ wavefunction (so for e.g. stripping, only use prior interactions).

If NT(1,I) = –2, then an external form factor is read in from Fortran file number NT(2,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 $\times$ wavefunction. The 'vrr' file format is assumed, and the breakup (third) vertex function is ignored.

If NT(3,I) $<$ 0, then the I'th component of the pair summation is simply the product of cluster wave functions $u_1(r) u_2(R)$ where $u_{2}$ may be KIND = 0 or 1.

NT(4,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 $L_{nn}$, $\ell$, S$_{12}$ & j$_{12}$ values permitted within the limits set by $\ell_{min}$, $\ell_{max}$, S$_{min}$, and S$_{max}$ (J$_{12}$ 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

$D_{0}$ = zero-range stripping strength for transfers from this stare

$D$ = asymptotic stripping strength, as used e.g. in sub-Coulomb transfers