&POT potential namelists

Each potential is started by a &pot namelist with TYPE=0, defining the radii to be used and the diagonal Coulomb potential, followed by any number of further &pot namelists with TYPE $>$

0, which cumulatively construct the components of the nuclear potential. The KP value in each namelist identifies the potential of which this is to be a component. The &pot namelists are repeated until KP $< 0$

indicates the last one to be considered, or an empty namelist is read in.

First kind of &pot namelist

KP, TYPE, p1, p2, p3, p4

with TYPE=0, or

KP, TYPE, AT, AP, RC, AC

with TYPE=0 :
All the radii are multiplied by CC.
CC is updated by a TYPE=0 &pot namelist to CC = AT $^{1/3}$ + AP $^{1/3}$ where AT $\equiv$ p1 and AP $\equiv$ p2,
CC is updated by TYPE $>$

0 namelists, if P0 $>$

0, to CC = P0 $^{1/3}$ .

RC $\equiv$ p3 = surface radius (when multiplied by CC) of charge distribution
AC $\equiv$ p4 = diffuseness of charge distribution

The values of AT and AP affect only the conversion from $r_0$ radii to R radii in the definition of potentials in these namelists. If however you adopt the convention that AT is the target mass number and AP 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.)

Second kind of &pot namelist: TYPE0

KP, TYPE, IT, SHAPE, p1, p2, p3, p4, p5, p6, p0: or
KP, TYPE, IT, SHAPE, p(:): or
KP, TYPE, IT, SHAPE, V,Vr0,Va, W,Wr0,Wa
also: LSHAPE, JL, XLVARY, ALVARY, DATAFILE

An empty namelist ends reading potential parameters, as does a negative KP value in the last namelist.


      


For		TYPE


0		 		Coulomb potential (see above)


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


9		SHAPE		Effective mass: reduction from unity


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.


14		SHAPE		Projectile second-order coupled by matrix elements read in.


15		SHAPE		Target second-order coupled by matrix elements read in.


16		SHAPE		Target & projectile simultaneous second-order coupled by matrix elements read in.


17		 		Target & projectile all-order coupled by matrix elements read in.


20		NUM		Super-soft N-N potl. of de Tourreil & Sprung (SSC(C))


21		NUM		User-supplied N-N potential via subroutine NNPOT.

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,

SHAPE indicates the radial shape of the form factor,

P1-P3 & P4-P6 are parameters for the real and imaginary parts (respectively).

P0 $\equiv$ P7 can give a further parameter.

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

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(-RH $^2$ .)
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 = $j_0$ (RH) = $\sin$ (RH)/RH

For `SHAPE's 7-9, immediately read in free format from from Input File 4, or from datafile if specified: First a comment line,
then (free format) NPOINTS, RSTEP, RFIRST,
followed by NPOINTS potential points for the shape in steps of RSTEP starting at r=RFIRST.
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 $\times r$ ,
6 : Reid soft core for $T$ =1, spin-orbit part $\times r$ ,
7,8,9 : Read from external file, 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*r $^3$ *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 external file, 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/P3 $^2$ . + 2/r $^2$ .) / 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 external file, as above.

If SHAPE $\geq$ 30, then use SHAPE-30, and define a $J$ - or $L$ -dependent potential by a factor defined using
JL, LSHAPE, XLVARY, ALVARY
so that the potential is multiplied by a form factor depending LSHAPE as

=0 : Woods-Saxon = 1 / (1 + 1/E),
=1 : WS squared = 1 / (1 + 1/E) $^2$
=2 : Gaussian = exp(-RH $^2$ ),
where X=L (for JL=`L'), X=J (JL=`J'), RH = (X-XLVARY)/ALVARY, and E=exp(-RH).

If SHAPE $\geq$ 10 and $<$ 19, then use SHAPE-10, and write out the resulting potential to Output file 25 in FORMAT(6E12.4): a form suitable for subsequent reading by a SHAPE $\geq$ 20.

If SHAPE $\geq$ 20 and $<$ 30, then a $J_{T}/\pi$ -dependent potential is defined.
For each subsequent coupled-channels set with total angular momentum $J_{T}$ , a form factor is read into this potential from Input file # SHAPE, reading M = RMATCH/HCM+1 complex numbers in FORMAT(6E12.4) from the file as a random-access, with (M-1)/3+1 lines for each form factor.
For each CC set, form factor number $int(J)+1$ (for SHAPE $\geq$ 24) or 1/2 for +/- parity (for SHAPE $<$ 24) is read.

If SHAPE = 40, then a $\pi$ -dependent potential is defined: use KP = P1 for + parity CC sets, and P2 for – parity.

If SHAPE = 41, then a $L$ -dependent potential is defined: use KP = P( $L$ +1) for L=0–5, and P0 for L $\geq$ 6.

If SHAPE = 42, then a $J$ -dependent potential is defined: use KP = P(Ji+1) for Ji=0–5, and P0 for Ji $\geq$ 6, with Ji = int( $J$ ).
There need only be one &pot namelist for 40 $\leq$ SHAPE $\leq$ 43. The values of TYPE and P2-P7 are ignored. No files are used. Choices are listed in fort.48

Nucleon - nucleon Potentials

The SHAPEs for the nucleon-nucleon SSC (C) potential (TYPE = 8) are taken from the subroutine SSCC in FRESCO. The input NUM is the number of angular momentum components stored, using the order
$^1S_{0}$ , $^3S_{1}$ , $^3S-^3D$ , $^3D_{1}$ , $^1P_{1}$ , $^3P_{0}$ $^3P_{1}$ , $^3P_2$ , $^3P-^3F$ , $^3F_2$ , $^1D_2$ , and $^3D_2$
Note that TYPEs 8 & 9 can only be used in KIND = 1 ( $LS$ -coupled) single-particle states.

The SHAPEs for the user's nucleon-nucleon potential (TYPE = 9) are taken from the subroutine NNPOT, with NUM the same meaning 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 namelist. This preceding namelist 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 $e.fm^k$ (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:

$\displaystyle Mn(Ek) = \langle I ' \vert\vert Ek \vert\vert I \rangle / (\sqrt {2I+1} \langle IK k0 \vert I ' K \rangle ) \ .$

(2)

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

$\displaystyle Mn(Ek) = {{3 ~ Z ~ \beta _ k ~ R^k} \over {4 \pi}} \ .$

(3)

The Mn(Ek) are thus of the same sign as $\beta$ and $Q_{0}$ for all rotational bands.

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

$\displaystyle M(Ek)$	$\textstyle =$	$\displaystyle i^{ I-I ' + \vert I-I ' \vert } ~ \langle I ' \vert\vert Ek \vert\vert I \rangle$	(4)
	$\textstyle =$	$\displaystyle \pm \sqrt{ (2I+1) ~ B(Ek, I \rightarrow I ' ) } \ ,$	(5)

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

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

(6)

The intrinsic and reduced matrix elements are therefore connected by

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

(7)

in the rotational model where $K$

is a good quantum number, for states of spin $I,I'$

, projection $K$

, charge

, 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.

Pairwise couplings

For TYPEs 12–17, namelists &step 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.

&step: IB,I A, k, STR /

to put in a coupling to state IB from state IA, of multipolarity k and strength STR. These are terminated by an empty &step namelist, or by IB $<$

0 indicating the last entry.

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 $<IB \vert M(Ek) \vert IA>$ = 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

$\displaystyle RDEF(k, I \rightarrow I')$	$\textstyle =$	$\displaystyle DEF(k) (-1)^{ [I-I'+\vert I-I'\vert]/2} \sqrt{2I+1} ~ \langle IK k0 \vert I'K\rangle$	(8)
	$\textstyle =$	$\displaystyle M(Ek) * 4\pi / [ 3 Z R^{k-1} ]$	(9)

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 &state namelists. This of course requires making RDEF(k) and M(Ek) dependent on the spins of the excited states, not on any simple model. Note: monopole potentials are included automatically, even if not listed in a &step namelist.

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

SHAPE = 0 to 6 : not used
= 7,8,9 : Read from external file, 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

$\displaystyle F(r) = M(Ek) ~ e^2 ~ {\sqrt{4\pi} \over (2k+1)}~ r^{-k-1}$

(10)

where

, 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 $-1/\sqrt{4\pi}$ :

$\displaystyle F(r) = - DEF(k) ~ \frac{1}{\sqrt{4\pi}} ~ \frac{dU(r)}{dr}$

(11)

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.

For TYPEs 14–17, &step namelists are again read to determine the details of the required couplings between each pair of excited states and of the reorientation terms for each individual state. Only nuclear couplings are yet implemented. Specify values IB,I A, k, STR to put in a coupling to state IB from state IA, of multipolarity $\vert$ k $\vert$ and strength STR. For TYPEs=14–17, the IB and IA can reference any combination of projectile and target states, but k $>$ 0 for projectile and k $<$ 0 for target states. STR refers to the RDEF(k) measure above; the factor $(-1)^{ [I-I'+\vert I-I'\vert]/2}$ must be included.

For TYPE=17, couplings, to all orders in RDEF, are calculated between all pairs of projectile and target states using the matrix eigensolution method of Kermode and Rowley, Phys. Rev. C48 (1993) 2326. The SHAPE and P0-P6 parameters on the TYPE=17 &pot namelist are not used.