New effective interactions for the 0f1p shell

Nuclear Physics North-Holland

A523 (1991) 325-353

NEW EFFECTIVE

INTERACTIONS

W.A. RICHTER

FOR THE Oflp SHELL

and M.G. VAN DER MERWE

Physics Depariment, University of SteNenbosch, Stellenbosch 7600, Republic

of

South Africa

R.E. JULIES Physics Department,

University of the Western Cape, Private Bag Xl 7, Bellville, C. P. 7530, Republic of South Africa B.A. BROWN

National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, Mf 48824, USA Received 11 June (Revised 1 August

1990 1990)

Abstract: New two-body interactions are derived for nuclei in the lower part of the Oflp shell by fitting semi-empirical potential forms and two-body matrix elements to 61 binding and excitation energy data in the mass range 41 to 49. The shell-model calculations assumed a ?a core and valence nucleons distributed over the fuil fp space. Care was taken to exclude intruder states from the selected data set. The r.m.s. deviations between fitted and experimental energies of 176 keV have been achieved by varying only 6 two-body parameters and 4 single-particle energies in the method utilizing a modified surface one-boson exchange potential (MSOBEP) form. in an alternative fit an r.m.s. deviation of 163 keV has been obtained by fitting 12 linear combinations of single-particle energies and two-body matrix elements, while constraining the remaining two-body matrix elements to values of a G-matrix interaction. The results are comparable in quality to recent similar fits in the 1sOd shell. The effects of a mass dependence of the two-body matrix elements are also investigated. An excellent reproduction of ground-state magnetic moments and quadrupole moments is obtained with the new interactions.

1. Introduction The best currently available two-body effective interactions in nuclei are obtained using an unconstrained hamiltonian, in which all two-body matrix elements and single-particle energies are allowed to vary as free parameters in a fit to experimental data. This method will hereafter be referred to as the TBME method or the model-independent (MI) method. In the most comprehensive work employing the TBME method Wildenthal IS’) has been able to reproduce a selection of 447 IsOd shell binding and excitation energies with an r.m.s. deviation between experiment and theory of about 185 keV. The introduction of a mass dependence for the empirical IsOd interaction of Wildenthal was an impo~ant ingredient in its improved reproduction of IsOd-shell observables over previous work 3). However, with such parameterizations very little is generally revealed about the nature or form of the nucleonnucleon interaction. It is also much more difficult to extract an unconstrained 0375Q474/91/$03.50

0

1991 - Elsevier

Science

Publishers

B.V. (North-Holland)

326

W.A. Richter et al f New effective interactions

hamiltonian

from fits in spaces

of two-body of sufficient

matrix elements (195 for the Oflp shell) data on the other.

In ref. ‘) a semi-empirical (OBEP)

larger than the 3sOd shell, due to the large number

interaction

plus core-polarization

was systematically

developed

correction by carrying

based terms

on the one hand,

on one-boson

exchange

and a lack potentials

of the multipole-multipole

out numerous

least-squares

type, fits to the

same basic IsOd-shell data set used by Wildenthal 2”). For the Oflp shell the same interaction form has been adopted. Other refinements to the interaction used in ref. ‘), such as the inclusion of a density dependence, and a mass dependence of the two-body matrix elements, are also employed here. In this work the complete set of basis states which can be generated from the four orbits Of,,, , Of,,, , 1p3j2, lp,,, of the Oflp (or fp) shell is used. By using such a set intra-shell truncation problems are eliminated and the problem of the effective interaction in the given model space can be addressed fully “). In this sense these fp-shell studies follow in the footsteps of pioneering work in the Op shel16), the 1sOd shell ‘) and the Oflp shell ‘). Progress in modern large-basis shell-model calculations have facilitated calculations in the full fp shell for the range of nuclei from mass 41 to 46, and some specific states in nuclei with higher mass. Previous calculations of shell-model wave functions for natural-parity states in the lower part of the fp shell 8-26) have generally been more limited in scope than the present work in the sense that one or more of the following approaches have been adopted: Various truncations of the full fp model space were the calculations concentrated on one mass number or at most employed 8-‘3,17,1QU20), a few nuclei ‘5*18~21~23~25*26), or only the Ca isotopes were considered g~1’~‘2~‘4~15~‘7~23), so that the n-p residual

interaction

did not enter. The calculations

that did not have

usually the Kuo-Brown these restrictions 7,22*24)all used a G-matrix type interaction, renormalized G-matrix I”), generally with some modifications. In the present work effective interactions are systematically derived in the full Oflp space by fitting (i) A semi-empirical interaction to 61 energy data from 12 nuclei in the mass range 41 to 49 (see table l), (ii) Two-body matrix elements

to the same energy data using the so-called

linear

combination method (discussed in sect. 2). Some initial results have been reported in ref. “). In sects. 2 and 3 the basic forms of the model-independent and model-dependent (potential form) interactions, respectively, are introduced, and in sect. 4 the iterative fitting procedure whereby the unknown parameters of the interactions are determined, is discussed. The spin-tensor decomposition method used to analyze the two-body interactions is outlined in sect. 5, and the data selection discussed in sect. 6. The results of the fits with the two interaction forms are presented in sects. 7.1 and 7.2, and the reproduction of energy level spectra is discussed in sect. 7.3. In sect. 7.4 the magnetic dipole and electric quadrupole moments calculated with our two new interactions are compared with experiment and other theoretical values.

327

W.A. Richter et al. / New effective interactions

A comparison

of the various

spin-tensor

with the bare and renormalized

Kuo-Brown

components

of our interactions

interactions

is made

in sect. 7.5. The conclusions

are given in sect. 8.

2. The model-independent

effective

interaction

When fitting two-body matrix elements and the four single-particle energies in the full fp shell to the present energy data set, one is faced with the problem that there are 61 data). elements function

more than three times as many parameters as data (199 parameters versus These parameters are the single-particle energies SPE and two-body matrix TBME in the expression for the eigenenergies corresponding to an eigenJK):

E=(KIHIK)=COBTD(i,K)SPE(i)+CTBTD(i,K)TBME(i), I with the one-body (TBTD) as defined

(I)

transition density (OBTD) and two-body transition density in ref. ‘). The TBME have been assumed to have a simple mass

dependence (V)(A)

= (V)(A = 42)(A/42)-“,

(2)

which reflects the fact that the basis states in a finite model space are mass-dependent. The large number of parameters can be restricted by a method referred to as the linear combination (LC) method ‘,3) (also known as the diagonal correlation matrix method ‘*)). The least-squares fit problem is reformulated in terms of uncorrelated linear combinations of the one- and two-body matrix elements, or “orthogonal” parameters. The eigenvectors of the error matrix given uncorrelated linear combinations of the parameters, and the corresponding eigenvalues give the uncertainties in these linear combinations. The orthogonal parameters can be ordered according to their uncertainty, and the well-determined parameters separated from the poorly determined ones. Some constraints are now added to the least-squares fit so that most of the matrix

elements

are held fixed at values corresponding

to some starting

interaction, e.g. a reasonable G-matrix type interaction, except for those that are well-determined by the data set. In the present case a selected set of 60 (out of a possible 199) matrix elements were varied and the 12 best determined linear combinations determined, while the remaining ones were fixed as described in sect. 7.2.

3. The semi-empirical

effective

interaction

As the starting point for our semi-empirical two-body interaction we have used the parameterization in terms of a sum of one-boson exchange potentials (OBEP) [ref. ‘) 1. The general forms for the central (c), spin-orbit (s) and tensor (t) interactions

WA. Richter et al. / New efective interactions

328

are: V,= 1 PSTV~sTD(R)Y,(r,z/~i), iST

V, = 1 PSTVEsTD(R) Ys(r12/~i)L.

S,

iST

Vt= C PSTVfSTD(R)

yt(T1z/pi)S1z(T).

(3)

iST

PST is the projection operator for the four (ST) channels (00), (Ol), (10) and (II), ViST are strength parameters to be determined and the pi are the interaction ranges corresponding to the various mesons exchanged. The Y(x) are the one-bosonexchange potential (OBEP) type functions Y,(x) = exp (-x)/x, Y,(x) = [I +

(l/x)1 w (-x)/x’,

Y,(x)= [1+(3/x)

+(3/x2)1

exp (-x)/x,

where (02

(4)

x= rrz/pi. S=$(cr1+u2), and S,, is the usual tensor operator * uJ. The ri2 and R are the relative and center-of-mass * r12) -(a, r12=rl-r2,

3(al . r12) coordinates

R=$(r,+r,).

In addition to the central, L * S spin-orbit and tensor parts, there may also be contributions from the so-called antisymmetric spin-orbit (ALS) component, but we neglect this part as in ref. ‘) since the ALS terms are expected to be relatively unimportant. Hosaka, Kubo and Toki 2g)* recently parameterized a bare G-matrix interaction based on the Paris potential in terms of a sum of OBEP terms by treating the ViST as parameters in a fit to the oscillator G-matrix elements. For the central part they used four ranges: pi = 0.20, 0.33, 0.5 and 1.414 fm; for the tensor part two ranges: pi = 0.25 and 1.414 fm; and for the spin-orbit part two ranges: pi = 0.25 and 0.40 fm. Their parameterization has been used as a guide in the IsOd-shell fits of ref. ‘) and in the present case, but we use fewer ranges and modify the interaction strengths (seee table 4). D(R) describes

the density

dependence.

We assumed

a density

dependence

with

the form

with F(R)={l+exp[(R-Ro)/a]}-l.

(9

Note the following corrections in ref?): in eq. (2) the CTshould be replaced by s, in eq. (3) the YLs(x) term should have the factor e-“/x2 rather than e-“/x, and in table 4 the range should be 0.40 l

rather

than 0.50.

329

W.A. Richter et al. / New effective interactions

The values

A, = -1

interactions. Renormalizations model

space

determined

and

Bd = 1 are used

in this work for the density-dependent

of the results due to neglected

are obtained empirically

from schematic

in the energy

The core-polarization

contribution

terms of multipole-multipole

configurations

core-polarization

outside terms

the active

with strengths

fits. to the interaction

operators

V,“( s, T) =

has been parameterized

I). For the central

c P”‘C,( s, T)

component

,

in

we take

(6)

ST

where C,, are strength parameters to be determined. For this work we consider only the monopole contribution to the central component. This term is equivalent to the potential

form discussed

above with p = co and D(R)

= 1.

Our total interaction consists of the sum of the terms V,, V, and V, of eq. (3) plus the monopole core-polarization terms of eq. (6). We have calculated the jj-coupled two-body matrix elements of this interaction in an harmonic-oscillator basis. The two-body matrix elements can in turn be written as a linear combination of the various components of the interaction: TBME (i) = C SP (j) TBMECOMP

(i,j)

,

(7)

where TBMECOMP is a coefficient which represents the known amount of a particular component in the interaction, and the SP(j) are unknown strengths (e.g. the ViST of eq. (3) to be determined from least squares fits of the calculated energies to experimental data. In principle the total interaction may have a large number of parameters if all the OBEP ranges of ref. *“) are considered (24 OBEP terms in total) as well as the various core-polarization correction terms in the different (T, S) channels. However, they are not all independent because of correlations in the data. An essential feature of our method is to achieve as large a reduction in the number of parameters as possible without sacrificing physical content. Therefore the object was to identify those components of the interaction which are most important in determining shell-model eigenvalues. A programme used for the IsOd-shell work ‘) was used to calculate the known parts of the interaction. Then the unknown strength parameters in eq. (3) and in the core-polarization terms were adjusted to give optimal agreement with the experimental energy data. The results of numerous trials showed that one could get results that are as good as any by varying only the central range terms of the OBEP (a medium range of 0.5 fm) and the central monopole core-polarization terms, while fixing the spin-orbit and tensor OBEP terms at the values found in the 1sOd shell fits (see sect. 7.1.1). Because the density dependence of the interaction varies with distance from the centre of the nucleus, it has some features in common with the surface delta

330

interaction,

W.A. Richter e? al. / New elective interactions

which

is evaluated

also occur in the modified interaction

only at the nuclear

surface

surface.

delta interaction,

Since monopole

it is appropriate

as a modzjied surface one-boson-exchange

terms

to refer to our

potential (MSOBEP).

4. The iterative fitting procedure In the general fit method the OBTD and TBTD required the energies have been calculated on a VAX-780 computer,

in the expressions and more recently

for on

a VAX-8530, at Michigan State University using the shell-model code OXBASH ‘O). The CPU time required on the VAX-780 to calculate the eigenenergies for one iteration is about 7 h, and for the transition densities it is about 150 h. The limitation on mass and J, T values is determined by the calculation of the two-body transition densities. The largest (J, T) dimension included was 1484 (the $- ground state in 49Ca). Five iterations were required in order to obtain proper convergence of the interaction. The energies of the selected states were calculated relative to a 40Ca core. A correction for the Coulomb energies of the valence protons was applied to the experimental binding energies relative to 40Ca. The Coulomb energies were estimated from the difference in binding energies between pairs of analog states, one of the states generally being a ground state. We believe that this procedure is justified since the unce~ainties in the Coulomb corrections are small compared to the energy fit deviations (see also ref. “I)). The Coulomb corrections used for the fits are given in table 2. The Coulomb-corrected ground-state energy values in table 1 are very close to a set calculated by Cole (table 3, set I, ref. “>). A modified version 32) of the renormalized G-matrix of Kuo and Brown lo) was employed as the starting interaction for the model-dependent fits. For the modelindependent fits the starting interaction was the original renormalized interaction of Kuo and Brown, In both cases the single-particle energies were taken from ref. “).

5. Spin-tensor

decomposition

of the interactions

The technique of applying a spin-tensor decomposition to the two-body matrix elements of an interaction [see refs. 1,33)*] plays a central role in our analysis of effective interactions. At various stages of our calculations the interactions derived were subjected to a spin-tensor decomposition to monitor the changes in the different interaction components and to compare their values with realistic interactions. Although jj-coupled matrix elements are convenient for shell-model calculations, a transformation to the E&coupling scheme allows more insight into the underlying physics of the two-body interaction. The LS-coupling scheme permits a spin-tensor * The scale of the matrix elements in fig. 2 of ref. 33) should matrix elements in figs. 3 and 4 should be divided by four.

be reduced

by three and the scale of the

1 3 11 9 15 19 13 17

0 4 8 12

%c

Ya

“) An average

7 5 3 11 3 15

‘?a

-38.906 1.157 2.283 3.285

4 4 4 4

TABLE

1

-38.849 1.619 2.552 3.187

-32.104 0.876 1.871 2.601 2.917 3.361 3.616 4.380

-27.861 0.516 0.909 1.805 2.019 2.811

0.820 0.564 1.354 1.166

-19.814 1.781 2.709 3.145

-8.388 3.909 1.892 6.491

E (FPD6)

-38.849 1.544 2.435 3.133

-32.142 1.029 1.813 2.633 3.076 3.172 3.878 4.693

-27.849 0.367 0.917 1.746 2.016 2.776

0.986 0.507 1.497 1.579

-19.837 1.690 2.821 3.162

-8.364 3.951 2.031 6.210

E (FPM13)

0 4 8 12 0 4 3

Ya

@Ca @Ca

by the single-nucleon

-

5 5 5 5 5 5 5

7 5 3 11 9 1 15

45Ca

exD

E

strength.

-79.086

-73.939 3.832

-56.724 1.346 2.574 2.973

-46.320 0.174 1.435 1.562 1.895 2.249 2.877

-48.236 1.083 2.454 4.015 6.509 7.671 8.040

-41.696 0.271 0.350 0.667 0.763 0.947 1.052 2.672 3.567 4.113

spectroscopic

9

8 8

0 0 0 0 0 0 0

2 2 2 2 2 2 2 2 2 2

27-

0 4 8 12 16 20 24

4 12 8 2 6 14 10 18 22 20

2J

44Ti

%c

Nuclide

See text.

-78.997

-73.856 3.659

-56.958 1.576 2.757 3.119

-46.490 0.446 1.522 1.662 1.825 2.914 2.973

-48.142 1.300 2.498 3.716 6.248 7.614 8.313

-42.035 0.651 0.529 1.242 0.807 1.231 1.156 3.017 4.075 4.439

E (FPD6)

-79.998

-73.166 3.609

-56.871 1.505 2.731 3.043

-46.433 0.350 1.441 1.563 1.848 2.313 2.894

-48.029 1.249 2.512 3.639 6.017 1.491 8.027

-41.928 0.375 0.482 0.968 0.688 1.119 1.094 2.783 3.815 4.541

E (FPM13)

binding energies (relative to 4”Ca) and excitation energies for the 61 levels of the final fits. The experimental energies have been Coulomb-corrected as described in the text

value for the two levels at 0.473 and 1.179 MeV, weighted

-32.009 0.980 “) 1.830 2.459 2.987 3.123 3.469 4.361

1 1 1 1 1 1 1 1

-27.769 0.373 0.593 1.678 2.046 2.754

0.611 0.617 1.491 1.511

0 0 0 0

2 14 6 10

%3C

3 3 3 3 3 3

-19.837 1.525 2.752 3.189

2 2 2 2

0 4 8 12

-8.364 3.900 2.100 6.500

E erp

%a

2T

ground-state

1 1 1 1

25

and theoretical

Wa

Nuclide

Experimental

W.A. Richter ef al. / New effective

332

interactions

TABLE 2

Coulomb corrections (A&-) to experimental binding energies (based on Coulomb displacement energies between analogue states} Nucleus

AE, (MeV)

Nucleus

A& (MeV)

42SC

7.214 7.242 7.222

&Ti %k 45Ti

14.811 7.216 14.778

%C

&SC

decomposition of the U-coupled matrix elements (&LSfT~ V~l,ZdL’S’JT} according to the possible combinations of the total spins S and S’ allowed by the tensor rank k:

v=t: v,=c UkXk, k

k

where U and X are irreducible tensors of rank k in space and spin coordinates, respectively. The interaction components are central for k = 0, spin-orbit for k = 1, and tensor for k = 2. The k = 1 terms include both the normal spin-orbit (S = S’ = 1) and antisymmetric spin-orbit (S # S’) terms. A given set of jj-coupled two-body matrix elements can be separated into spin-tensor components according to eq. (8) by doing a jj to LS coupling transformation, followed by a spin-tensor decomposition. 6. Data selection

In the fp shell intruder states, i.e. states with significant admixtures of configurations lying outside the adopted model space, often occur in the low-energy spectra of nuclei. If the mixing with fp-shell states is weak, the properties of the fp-shell states can be described by renormalizing the effective one- and two-body operators. Initially a set of 41 levels considered to be relatively free from admixtures of intruder states was selected, and after some iterations the data set was expanded to include another twenty levels which appeared to have predominantly fp shell-model configurations. With the exception of one case, only the lowest energy state of each spin was used. The 61 energy levels selected for the fits are given in table 1. The weighting of the experimental data used corresponded to adding a theoretical error of 150 keV in quadrature to the experimental error. {The normalized x value for the “best-fit” empirical interaction in the IsOd-shell with the same weighting was about 1.0.) In our model space 4’Ca can have only four single-particle levels, whereas more than 40 levels have been observed below 5 MeV excitation. Single-particle stripping to 41Ca [ref. ‘“)I suggests that the f7/z strength is almost entirely concentrated in the ground state, the p3,2 strength is spread over two states, and the f5,2 and p,,~ strengths

333

W.A. Richrer et al. f New effective interactions

over

several

single-particle

levels.

For

energies,

each

single-particle

weighted

used for the experimental used by Kuo and Brown

value.

The values

in their calculation

Since the single-particle levels in 43Sc, an average

level

the centroid

by single-nucleon

spectroscopic

spectroscopic

of the observed factors,

has been

given in table

1 are the same values

of the effective

fp interaction

strength

is split between

lo).

the first two g-

value of 0.98 MeV for the two levels at 0.47 and 1.18 MeV

[weighted by the single-nucleon spectroscopic data from ref. ‘*)I was used in the fits.

factors

of 0.3 1 and 0.77 respectively;

7. Results 7.1. THE

POTENTIAL

MODEL

FITS

7.1.1. Parameters of the potential. The x-values of the successive iterations in the fp shell are given in table 3. The last letter in the interaction name refers to the parameter set used (i.e. the terms in eqs. (3) and (6) chosen as variables), and the numeral to the iteration number. The central part of our final interaction consists of 4 central density-dependent OBEP terms (for the four (T, S) channels) added onto the standard density-independent one-pion exchange potential [OPEP, assuming a pion-nucleon coupling strength given by fz/4n = 0.081, see ref. ‘“)I plus 4 monopole core-polarization terms. The monopole terms for the (T, S) = (0,O) and TABLE 3 fp-shell Interaction used for OBTD, TBTD (A)

Number of levels used

X

41 41 41 61 61 61

1.774 1.091 1.036 1.304 1.284 1.283

239 138 133 177 176 176

8 12 11 11 10 10

61 61 61

1.299 1.208 1.209

175 163 163

12 12 12

r.m.s. (kev)

Number parameters

of fit varied

Potential model method FPE FPAl FPB2 FPC3 FPC4 FPD5

(B)

Interaction from fit

iterations

Model-independent KBFP FPMIl FPMIZ

FPAl FPB2 FPC3 FPC4 FPD5 FPD6 method FPMll FPMIZ FPM13

The parameter set denoted by A in FPAl consisted of three central range terms, one monopole term, and four single-particle energies. Set B consisted of four central range terms (0.5 fm), four monopole terms and four SPE. C used the same parameters as B except that the T = 0, S = 0 monopole term was fixed at the 1sOd value (SDPOTA interaction, table 4). D used the same parameters as C except that a further monopole term ( T = 1, S = 0) was fixed at the SDPOTA value.

334

W.A. Richter et al. / New effective interaciions

(1,O) channels were fixed in the final fits at the values found in the IsOd-shell fits because they were not well-determined by the data. [See table 4, where the parameters of the FPD6 interaction ref. ‘).I The spin-orbit obtained

are compared and tensor

in the IsOd-shell

with the IsOd-shell

components

SDPOTA

were fixed at values

fits since these components

interaction

of

equal to those

were not reliably

determined

by the data. The final 22-parameter set used, of which only 10 are varied (4 central range terms, 2 monopole terms and 4 single-particle energies), is not unique but is a compromise between goodness of the fit and the number of parameters employed. The error in the range parameter for the (0,O) channel is quite large, but all the other parameters have relatively small errors. The importance of the density dependence is indicated by removing the density dependence of the central range terms in the above fit. For no density dependence in the central parameters (A,= 0 in eq. (5)) a x-value of 1.50 is found, compared to 1.28 (table 3) with a density dependence using Ad = -1.0, Bd = 1.0, &= 1.1, a = 0.6. [For the various iterations in table 3, the same values of these parameters were used as in the 1sOd shell ‘).I The sensitivity of the fits to the parameters of the TABLE 4 Comparison

Component

central

tensor

spin-orbit

of potential-fit parameter values in the fp shell (FPD6 interaction) and the 1sOd shell [SDPOTA interaction from table 13, ref. ‘)] and uncertainties A

S

T

Form

0 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1

DD-M DI-L C DD-M DI-L C DD-M DI-L C DD-M DI-L DD-S DD-L DD-S DD-L DD-S DD-S

Range (fm)

C

0.50 1.414 0.50 1.414 0.50

1.414 0.50 1.414 0.25 1.414 0.25 1.414 0.25 0.25

FPD6 parameter value 0.7454 -10 31.389 -0.967 -560.3 -10.463 0.2430 -562.8 - 10.463 0.701 -1497.1 3.488 -18 400 10.57 6780 -9.28 -31000 -60 600

A

“) 1210 “) 0.075 37.0 a) “) 14.1 “) 0.035 175 aa: “) aa1 “) “)

SDPOTA parameter value 0.7454 -1761 31.389 -1.9624 -462.9 -10.463 0.2430 -493.6 -10.463 0.6562 -454.3 3.488 -18 400 10.57 6780 -9.28 -37 000 -60 600

A

0.21 305 0.04 9.5 0.05 8.0 0.03 50 1900 2.0 2200 1.9 24 000 1400

In the fourth column of the table, C is the monopole core-polarization parameter, DI stands for density-independent, DD for density-dependent, and S, M and L for short, medium and long range, respectively. The central DI-L parameters are the OPEP values from ref. 29). “) These parameters were held fixed in the fit at the SDPOTA values.

335

W.A. Richter et al. / New effective interactions

The in eq. (5), viz. Ad, B,, Ro, and a, were also investigated. changed only by a few per cent for variations in the individual

density dependence x values generally parameters

by as much

as fifty per cent. Hence

the IsOd values

were adopted

for

all these parameters. The mass dependence Amp of the two-body matrix elements in the final fit was also investigated (see eq. (2)). There was no significant dependence of x on p. The actual

value

7.2. THE

used in our fits was 0.35, the same as in the 1sOd shell.

MODEL-INDEPENDENT

FITS

The results of the model-independent linear combination method described

fits are shown in table 3. By means in sect. 2, the 12 best-determined

combinations from a selected set of 60 two-body Excluded as variable parameters were all matrix

of the linear

matrix elements were varied. elements involving the lp,,,

subshell, and the nondiagonal matrix elements involving the Of,,, subshell. The remaining linear combinations, as well as the TBME that are not part of the selected set, were held fixed in each case at values obtained from the renormalized KuoBrown interactions After two iterations an r.m.s. deviation of 163 keV was obtained. This suggests that the linear combination method can be used in conjunction consisting of the more important two-body matrix elements.

7.3. CALCULATED

LEVEL

with a selected

set

SPECTRA

The final interactions, FPD6 in the model-dependent potential method, and FPM13 in the model-independent case, give an excellent reproduction of the energies of the fitted levels. Some representative results are shown in figs. 1 to 5, where the experimental spectra for the low-lying normal parity states fitted are compared with those of our interactions FPD6 and FPM13, and the renormalized Kuo-Brown interaction lo) (KBFP), which was used as the starting interaction in the modelindependent method. Levels connected by dashed lines were those included in the fits. Some additional calculated levels are also shown to indicate where some of the lower-lying levels are predicted to be, All calculated levels up to the highest level shown have generally been included (note the caption to fig. 2). The single-particle energies used for the &to-Brown interaction are based on the experimental 4’Ca spectrum. The values from ref. 18) have been used. It will be noticed that the Kuo-Brown interaction poduces too little binding for the ground states in general, except for 49Ca, where there is overbinding. The shortcomings of the Go-Brown interaction, and some modified versions thereof, have been discussed extensively by Cole 22,23,25*26), ‘t is not the intention to repeat these here an d 1 in any detail.

W.A. Richter et al. / New eflective interucrions

336

1 0 0 I

,

3

I

I

I

,

I

1

Fig. 1. Comparison of theoretical and experimental energy spectra for low-lying natural-parity levels (J indicated for A even and 2f for A odd) in fp-shell nuclei. This diagram is for 43Ca (T=f). The experimental ground-state energy is taken as zero. The ground-state energies shown for the FPD6, FPM13 and KBFP interactions have been calculated relative to ?Za. The spectra are complete up to the highest level shown unless stated otherwise. The experimental levels are from refs. 35S36). Uncertain spin and parity assignments are indicated in brackets. Observed levels of indefinite spin and parity are omitted from the experimental spectrum.

The correspondence between levels calculated with the FPD6 and FPM13 interactions and the experimental levels is generally very good, if probable core-excited intruder states are disregarded. [It has been assumed that the foliowing experimental states have significant admixtures of intruder configurations: The O+ and 2+ states at 1.84 and 2.42 MeV [ref. ‘)] in 42Ca, the lowest $-, z- and second $- states - mainly 5p2h configurations

‘) - in 43Sc, the 0’ and 2+ states at 1.88 and 2.66 MeV [ref. ‘)I

in ‘?Za, and the O+ state at 2.42 MeV in 46Ca. Other cases are discussed in connection with the figures.] In particular the ground-state binding energies are well reproduced. [See also table 1. Excitation energies in table 1 are from refs. 35736).] With a few exceptions, the correct level ordering is reproduced by our two new interactions FPDG and FPMI3. For 43Ca the inversion of the first q- and second $- levels, found for KBFP, is corrected, and the correspondence between experiment and the predicted levels for both our interactions is generally very good. For 44Sc neither the FPD6 nor the FPM13 interactions reproduce precisely the correct ordering, although the energy deviations are not very large, and the calculated

WA.

337

Richter et al. / New effective interactions

KBFP

I

FPt.413 I

I

FPD6

r

FXP

.5.O.5-

--__

.O-

__--

.5.o.5.o.5.o.5.o1

I

,x-D __ \ \--___ , 1

-I

2

,_ ,

I

I

Fig. 2. Energy levels in MS~ (T= 1). Conventions are the same as in fig. 1, except that for FPM13 and FPD6 only one level of each spin is shown above the 8+ state for the sake of clarity.

ground states are too low by 200-300 keV. However, the high-spin states 9+, lO+ and ll+, which are expected to have predominantly fyi2 configurations, are reproduced at about the right energies. For 45Ca the lowest i- state is reproduced too high by about 0.5 MeV for FPD6 (this is in fact the largest deviation in the entire fit), but is correct for FPM13. Otherwise the correspondence between theory and experiment for the fitted levels is excellent

for both

interactions.

Another

conspicuous

difference

between

FPD6

and FPM13 is the second I- state which is predicted about an MeV higher for FPD6. In 48Ca where only the ground and first excited 2+ state were fitted, good agreement is obtained for these levels, and most of the next few levels predicted by FPD6 (up to about 6.5 MeV excitation) appear to have probable experimental counterparts. Evidence is presented in ref. 37) that the lowest excited O+ state at 4.28 MeV is a proton-pairing vibrational state (intruder), and the second excited Of state at 5.46 MeV a neutron-pairing vibrational state, which is reproduced at a similar energy by FPD6. For FPM13 some additional levels (4+, 2+, 3+) are predicted in this energy region. Another obvious difference between the two interactions is the position of the second Of state. A notable discrepancy in the Kuo-Brown spectrum is the low position of the second O+ state. In 49Ca where only the ground state was fitted, the energy of the ground state is well reproduced by FPD6 and the i- first excited state is predicted at approximately

338

W.A. Richter et al. / New e$ective interactions KEFP

FPMI3

FPD6

I

EXP

1

I

5

J5L

j+. \

-AL_, 3 3.

-\ 7

1 j

\ \

k

\

\

5

___---_

-_

\ \\

I

,

I

Fig. 3. Energy

’

7-__

_ I

I

levels in 45Ca (T=$).

KBFP c ’

7 I

1

Conventions

A _-

I

I

I

are the same as in fig. 1.

FPD6

FPM13

8.0-

.-_

!

FXP

2

7.57.0-

(2)

6.5-

-

6.0-

2

'5 4.52x 4.03.5lJJ 3.0-

0

5 gq= -+-/-n-

5.55.0-

AL

-+-a%?-

_-A-._ 4

/ -m2

(4)

-

,j’

0

2.5 2.0

\\ I

Fig. 4. Energy

I

\

---------I

0 I

levels in 48Ca (T = 4). Conventions

I

1

I

I

are the same as in fig. 1.

I I

339

W.A. Richter et al. / New effective interactions .KB

FP

4.5

A f?L

4.0

&

3.5

Yy-

A 7 5

1

0

3

-A J&=

3.0 2.5 '5: 2 2.0

5

5 1

7

lz 1.5 1.0

is

(I)

7

A-

5

3

0.5

7

0.0

,,A----

_-.A----

- _-

/

T0.E i-l.C )Fig. 5. Energy

levels in @Ca (T=;).

Conventions

are the same as in fig. 1

the right excitation, but the experimental spectrum is too uncertain to make any further conclusions. A defect in the FPM13 interaction is apparent in that it predicts the $- state below the ground state. It is also evident that the next five states are also much lower for the FPM13 interaction than Predicted levels for three nuclei not included in with the FPD6, FPM13 and KBFP interactions, figs. 6-8. For 45Sc an excellent reproduction of both FPD6

and FPM13,

and if it is assumed

for FPD6. the fits (45Sc, 45Ti, 47Ca), calculated are compared with experiment in the ground state is obtained with

that the first g$, $- and s- states are

intruders 23,25), reasonable agreement between the predicted and observed spectra is achieved up to about 2 MeV excitation for both our interactions (the first t- state is predicted somewhat high). For higher excitation the experimental spectrum has many uncertainties. However, good agreement is found for the two high-spin states y- and $-_ For 45Ti the low-energy triplet 5-, s-, s- is well reproduced by both the new interactions. The next two states around 1.5 MeV excitation are also correctly predicted, and despite the uncertainties in the experimental spectrum at higher energies the observed high-spin states (y,y, y), with mainly fylz configurations, have theoretical counterparts at about the right energies. In 47Ca the f- ground state and first excited $- state are well reproduced by FPD6 and FPM13. The first i- state is evidently too low in the case of FPM13, a feature

340

W.A. Richter et al. / New effective interactions FPM13

FPD6

I

EXP

1

9.0 8.0 7.0 6.0

zLd& ,

I

I

Fig. 6. Energy

3

3

7

7

I

I

levels in %k

I

(T=f).

/ AKBFP .-

FPH13

.5

EXP

A 7

3 .5

Iz

I

7

$& 7

4 .O

z

I

t

I

5

.O

(17)

_$&

.O

23

I

are the same as in fig. 1.

FPO6

A

.5

I

Conventions

(15)

-j+

-e-

As

2 .5

JZL

2 n5 1 1

I

Fig. 7. Energy

I

,

!

I

levels in 45Ti (T = f). Conventions

I

1

3

are the same as in fig. 1.

I

341

WA. Richter et al. / New effective interactions KBFP

,

10

FPD6

FPM13

I

EXP

!

.o

9.0 8.0

7.0 6.0 T 2

5.0

z

4.0 -

3.0 2.0

1

3

3

1.c

l-

0.c

I -

7 7 !

I

Fig. 8. Energy

I

levels in 47Ca (T=g).

7

7 I

I

Conventions

I

I

I

are the same as in fig. 1.

which has already been noted in the 49Ca spectrum. The density of states up to about 5 MeV excitation predicted by the new interactions is similar to that of the experimental spectrum. In summary, the two new interactions FPD6 and FPM13 generally give a good account of the low-lying spectra of the A = 41-49 nuclei investigated. They are also quite similar to each other in this respect. The most obvious difference between them is the prediction of the position of the first t- state in 49Ca, for which FPD6 gives a much better

7.4. MAGNETIC

result.

DIPOLE

AND

ELECTRIC

QUADRUPOLE

MOMENTS

The calculated ground-state magnetic dipole and electric quadrupole moments for the FPD6 and FPM13 interactions are compared with experimental values in tables 5 and 6, respectively. Some values calculated with the modified Kuo-Brown interaction of McGrory ‘) [KBl, see also refs. 22,23)] are also included. The effect of variations in the effective g-factors and charges on the calculated moments is generally much larger than using different modified versions of the Kuo-Brown interaction 23). The magnetic dipole moments calculated with bare nucleon g-factors in table 5a are reasonably well reproduced by the new interactions except for 45Ti, and are

W.A. Richter et al. / New effective interactions

342

TABLE Sa Comparison of ground-state magnetic dipole moments /_L(in units of 1~~) for the FPD6 and FPM13 interactions with experiment using free-nucleon g-factors. Some values of the KBl interaction of McGrory 7.23) are also included. Experimental values are from ref.“) unless stated otherwise; the experimental error in the last significant digit(sj is indicated in parentheses. Nuclei marked with an asterisk were not included in the fits Nuclide

25

2T

r&p

‘Wa +c 43Ca 4% “SC %a %c* 45Ti*

7

1 1 3 1 2 5 3 1 7 9

-1.594780

47Ca* ‘%a “) two b, ‘)

7 I 7 4 7 7 7 7 3

j~:p (9)

&=(FPM13)

-1.913 5.793 -1.754 4.547 2.218 -1.663 4.753 -0.354 -1.585 -1.451

5.43 (2) -1.3174 (2) “) 4.62 (4) 2.56 (3) -1.3278 (9) ‘) 4.7547 (2) “) ( -)0.095 (2) bj -1.380 (24) “j

Value is avera8e of two experimental measurements. From ref. 38). From ref. 39). g/Q > 0.

(FPD6)

values.

-1.913 5.193 -1.688 4.678 2.641 -1.520 4.714 -0.148 -1.363 -1.331

The error is assigned

&=

(KBl)

-1.913 5.793 -1.67 4.80 2.60 -1.520 4.958 -0.343 -1.38 -1.40

to reflect the deviation

in the

similar to those of KBl. However, for the fp-shell it is apparent that effective g-factors are required. For example, the measured magnetic dipole moments for 4’Ca and 41Sc are significantly smaller than the values based on the free-nucleon dipole moments. The values based on effective g-factors obtained from least-squares fits are shown for FPD6 and FPM13 in table 5b. (The isoscalar (IS} and isovector (IV) spin and orbital g-factors for FPD6 are gs(IS)=f(g$‘+gy) = 1.67, g,(IV)= TABLE 5b As for table 5a, but the effective isoscalar and isovector g-factors used for FPD6 are g,(IS) = 1.67, g,(W) = 5.08, g,(iSj = 0.38, g,(W) =0.33, with g,(IV) = 4.20 and g,(W) =0.49 replaced for FPM13. The KBl values are based on effective g-factors which reproduce the dipole moments of4’Ca and 4’Sc [ref. “)I Nuclide

2.l

2T

Wa 4’sc 4”Ca ‘+3sc @SC 45Ca ‘%c* 45Ti* 4’Ca* 49Ca

7 7 7 7 4 I 7 7 7 3

1 1 3 1 2 5 3 1 7 9

Footnotes

as in table 5a.

CLexp -1.594780 (9) 5.43 (2) -1.3174 (2) “) 4.62 (4) 2.56 (3) -1.3278 (9) h, 4.7567 (2) “) ( - )0.095 (2) hj -1.380

(24) ‘)

EL;: (FPD6) -1.545 5.511 -1.402 4.589 2.559 -1.319 4.715 -0.223 -1.248 -1.234

,u;; (FPMI3) -1.610 5.560 -1.465 4.515 2.716 -1.362 4.586 -0.082 - 1.263 -1.012

IL:: (KBlj - 1.595 5.430

-1.265 4.641 -0.172

343

W.A. Richter et al. / New effective interactions TABLE 6

Comparison of calculated ground-state electric quadrupole moments Q (in units of e. fm’) with experivalues are from ref. 35) ment. The KBl values (modified Kuo-Brown) are from refs. 7,23). Experimental unless stated otherwise; the experimental error in the last significant digit(s) is indicated in parentheses. Nuclei marked with an asterisk were not included in the fits

Q (FPW Nuclide

2J

7 I 7 I

1 1 3 1

4 7 7 I 7 3

2 5 3 1 7 9

-6.2 (12) “) -4.0 (8) “) -26 (6) 10 (5) 4.3 (9) “) -22 (1) (-)1.5 (15) “)

Nuclei marked with an asterisk “) From ref. 41). ‘) From ref. 38). p/Q>O.

e, = 0.64

Q (FPMI3) ep= 1.33 e, = 0.64

-7.3 -15.1 -3.0 -20.0 10.1 3.1 -23.6 -11.5 10.4 -5.6

-1.3 -15.1 -2.5 -20.4 7.2 3.4 -24.0 -11.6 10.0 -5.8

ep = 1.33

Q =P

2T

were not included

Q (-1) ep= 1.2 e, = 0.5

-16 6 2.3 -18.3 -6.2 7.4 -4.1

in the fits

i(g:-g:) = 5.08, g,(IS) =+(gB+g;) = 0.38, g,(IV) =s(gP-g:) = 0.33, assuming orbitindependent g-factors. The optimum isoscalar g-factors for FPM13 are practically the same, but for the isovector terms g,(W) = 4.2 and g,(IV) = 0.49 are used.) A remarkable reproduction of the magnetic dipole moments is achieved for FPD6, with the results for FPM13 only marginally different. The values shown for KBl are based on effective g-factors which reproduce the measured ground-state magnetic dipole moments of 41Ca and 41Sc (assuming I_L~= 2.43pN, CL,= -1.59~~ as in ref. 23), which gives g,(IS) =0.84, g,(IV) =4.03, g,(IS) = 0.5, g,(W) = 0.5). The excellent overall agreement of both our interactions, which also give good results for mass 41, indicate the importance of renormalizing also the orbital magnetic moment operators and not only the spin operators. The optimal

effective

charges

for the quadrupole

moments

calculated

with our

two interactions are very close to those extracted by Dhar and Bhatt 40) (er = 1.33 and e, = 0.64) in projected Hartree-Fock calculations which reproduce B( E2) data. Hence these values are also used for FPD6 and FPM13. Quadrupole moments calculated with these effective charges [table 6; see also Cole “~“)] for FPD6 and FPM13, agree quite well with the experimental values, with the exception of 45Ti. The agreement is generally better than for the KBl values calculated with er, = 1.2 and e, = 0.5. The wave functions based on the final interaction need to be subjected to tests which are more sensitive to specific components of the interaction, e.g. the calculation of Gamow-Teller strengths and electromagnetic transition probabilities. The FPD6

344

W.A. Richter et al. / New effective interactions

interaction

has been applied

in a shell-model

beta deay of 48Ca, and appears and B(GT+)

values

is envisaged

that

comparison

than previous many

more

calculation

to give better

results

calculations

observables

for two-neutrino

double

for the experimental

B(GT-)

42). In the next phase of the work it will be calculated

for the purpose

of

with experiment.

7.5. SPIN-TENSOR

DECOMPOSITION

OF THE

INTERACTIONS

In figs. 9 and 10 the spin-tensor components of the central matrix elements are compared for the bare (triangles) and renormalized (crosses) Kuo-Brown G-matrix interactions, and our FPD6 (solid line) and FPM13 (dashed line) interactions. It is evident that, apart from the (T,S) = (0,O) channel, the FPD6 and FPM13 interactions generally have values fairly close to each other. However, even small differences often account for important differences IsOd-shell 1*33). It will also be noticed

in the level spectra, as was found for the that the effect of renormalizing the bare

-6.+

-I

”

0

I

2

”

”

” I

4

6

”

” I

1

6

10

”

” I

I

12

14

”

” I

I

16

10

I

1

20

J

”

” I

1

22

24

Fig. 9. Matrix elements for the central T=O component. The FPD6 (solid line) and FPM13 (dashed line) interactions are compared with the bare (triangles) and renormalized (crosses) Kuo-Brown Gmatrices. The labels corresponding to the matrix element numbers along the x-axis are the quantum numbers I,I,&LL’SS’T in the U-coupled matrix element (I,I,LSJT) VI IJJ’S’JT).

345

W.A. Richter et al. / New effeciive interactions

0

2

Fig. 10. Matrix

4 elements

6

8

for the central

10

12

T = 1 component.

14

16

Conventions

18

20

22

24

are the same as in fig. 9.

Kuo-Brown G matrix is relatively small in most cases. The FPMU model-independent interaction is very similar to the Kuo-Brown renormalized interaction (KBFP), on which it is partly based, as has been described in sect. 7.2. However, it gives an excellent reproduction of the energy data which is comparable in quality to that of the FPD6 interaction. The fitting of the single-particle energies appears to an important factor in the improvement of the FPM13 interaction over the Kuo-Brown interaction. In figs. 11 and 12 the spin-tensor components for the tensor matrix elements are compared. For a number of matrix elements for T = 0 the FPD6 values are significantly smaller in magnitude than those of FPM13. FPM13 and the renormalized Kuo-Brown interaction are once again very close. In a few cases for T = 0 the bare Kuo-Brown interaction is noticeably more attractive than the renormalized interaction, although there is generally little difference between them. The T = 1 matrix elements are much weaker than for T= 0 and a11 the interactions have similar magnitudes; however, the FPD6 values are opposite in sign. For the T = 0 spin-orbit components in fig. 13 most of the matrix elements are very small except for FPM13. For the T = 1 case in fig. 14 the FPM13 interaction

346

W.A. Richter et al. / New efective

interactions

a

---_ E -_-_ P

+:=-x a

__--__--

__--

a

_---

---___

--‘_xQ

_---

a-_--_

---___ _A--

FCT_

_*--

---___

_--

--*m

---a--

+c

a

W.A. Richter et al. / New effective

347

interactions

0.c I-

-0.

ci-

T

E -1.1I-

0

2

Fig. 12. Matrix

4 elements

6

8

for the tensor

10

12

T= 1 component.

14

16

Conventions

18

20

22

24

are the same as in fig. 9.

follows the trend of the two Kuo-Brown interactions (which have practically identical values), and differs in a number of cases from the FPD6 interaction. The antisymmetric spin-orbit components are shown in fig. 15. The FPD6 and bare Kuo-Brown interactions have no ALS components, whereas the renormalized Kuo-Brown and FPM13 interactions show the same general trend, with some of the Kuo-Brown values being somewhat smaller in magnitude. Finally, the most significant differences between the FPD6 and FPM13 interactions are for the T = 0 central components, the T = 0 tensor and T = 1 spin-orbit components, and the antisymmetric spin-orbit components. The FPM13 interaction is generally

very similar

to the renormalized

Kuo-Brown

interaction

KBFP.

8. Conclusion It was shown that the same semi-empirical or potential interaction form (eqs. (3) and (6)) that was capable of reproducing IsOd-shell data also yields good results in the Oflp shell. Starting from a slightly modified Kuo-Brown G-matrix, and proceeding through successive iterations involving recalculation of the transition

348

W.A. Richter et al. / New eflectioe interactions

I,

1

,

,

,

I

,

,

,

I

1

,

(

(

,

,

,

,

1

_-I 0

Fig. 13. Matrix

2 elements

4

6

for the spin-orbit

8

10

T = 0 component.

12

14

Conventions

16

18

20

are the same as in fig. 9.

densities and the determination of new parameter sets by fitting to a selection of 61 experimental binding and excitation energies, good convergence of the interaction is obtained. The inclusion of a density dependence in the interaction has been shown to be important. However, the mass dependence of the two-body matrix elements does not seem to be particularly important in the mass range investigated. In the final result an excellent fit (r.m.s. deviation 176 keV) to a set of level energies, spanning a wide range of nuclei in the lower part of the fp shell, is obtained with a relatively small number (10) of variable parameters. It was further demonstrated that a fit to the same 61 energy data with an r.m.s. deviation of 163 keV could be obtained by varying 12 linear combinations of single-particle energies and two-body matrix elements, while constraining the remaining matrix elements as described earlier. The reproduction of the 61 levels fitted is of comparable quality as in the case of the potential method, except for the problem of the ;- state in 49Ca, which is predicted far too low. (The two-body matrix elements for both interactions are given in table 7.) The predicted low-energy spectra for some nuclei not included in the fits (45S~, 4sTi, 47Ca) also agree well with experiment for both new interactions. Effective

12 11 1 32 11 1

3331 < 3331

54 11 1 11 11 1 22 11 1 32 11 1

3331 < 3311 3131 < 3131

43 11 1 44 11 1 21 11 1 11 11 1

< 3131 3131 < 3111 1111

3131 33 11 1

34 11 1

3331

11 1

55 11 1

3333

3331 33

33 11 1

3333

w s

2

; g. m 9 6 2 2.

3

% ,

z? i% 5 2

350

WA. Richter et al. / New effective interactions TABLE

Single-particle FPD6

energies

and jj-coupled

and FPM13.

The shell-model

two-body

7

matrix

orbits are labelled

(abJTI VIcdJT) for the interactions

elements

1=

Of,,,,

2 = lp,,,,

3 = Of,,,,

SPE(FPD6):

1

-8.3876

2

-6.4952

3

-1.8966

4

-4.4783

SPE(FPM13):

1

-8.3637

2

-6.3325

3

-2.1539

4

-4.4068

abed

2J

2T

FPD6

1111 1111

1 3

0 0

-0.1774 -0.4993 - 1.0460

-0.7676 -0.4498

FPM13

I

4= lp,,,

obcd

2J

2T

FPD6

FPM13

2122 2131 2131 2131

2 2 3 4

1 0 0 0

-0.3963

-0.4503 0.7350 -0.3280

2131

5

0

-0.4943

0.6136 -0.5183 0.1145

1111

5

1111

7

0 0

-2.4742

1111

0

1

-2.2680

-2.6034 -2.1900

1111

2

1

-0.8875

-0.8690

2131

2

1

0.2368

1111 1111

4 6

1

-0.1436

3 4

1 1

0.3894 0.1166

0.1180 0.1140

3 5 2 4

0.1683 -0.4264

2131 2131

1121 1121 1121 1121

1 0

-0.0868 0.1519

5

1

0.3445

0.0100

-0.7828 -0.6414 -0.4479

-0.83 15 -1.0085 -0.8826 -0.4527

2131

0 1 1

1 3

0

-0.4252 -0.4174

2 3 4 2 3

0 0

1122 1122

2132 2132 2132 2132 2132

1.2486 -0.9173 1.0319 0.2992

1.1230 -0.4080 0.6140 0.4020

1122 1122

0 2

4 3

-0.0787 0.5489 0.2437

-0.0260 0.6190 0.1580

1131

1

1 0

2132 2133

1131 1131

3 5

2133 2133

5 2

0.0469 -0.5164

0.0480 -0.4460

1131 1131 1131

2133 2141 2141

4 3 4

-0.1842 - 1.4047

-0.1560 -1.2950

2141 2141

3 4

-0.5712 0.0538 0.5340 0.4498 0.4602

-0.1080 -0.0780 0.5020 0.4810 0.3190

1A497 0.9221

0.8380 0.5180

0 1

-1.1682 -0.3728

-0.4830

-0.4268 -0.5049 -1.1133

-1.9766

-0.4233 - 1.8940

0 0

-1.3217 -1.3067

-1.0050 -0.9010

2 4 6

1 1 1

1132 1132

1 3

0 0

-0.2805 0.3286 0.5773 0.3309

1132

2

1

0.2669

0.2850

1132 1133

4 1

1 0

0.3050 1.0710

1133 1133 1133 1133 1133 1141 1141

3 5 0 2 4 3 4

0 0 1 1 1 0

0.2639 2.0797 1.2596

1142 1142

1 2

1143 1143 1144

3 2 1

1144 2121 2121 2121 2121 2121 2121 2121 2121 2122

0 2 3 4 5 2 3 4 5 3

1 0 1 0 1 0 1 0 0 0 0 1 1 1 1 0

-0.1165

0.5962 -2.2640 -0.5113 -0.3681 -0.6235 0.2543 -0.5786 0.2794 0.2171 -0.4152 0.3469 -0.7390 0.0028 -0.8232 -0.5474 -2.5224 -0.8402 0.2490 0.0899 0.6558 -0.5368

0.0000 0.4060 0.7160 0.2270 -0.0870

0.5170 0.1700 -2.7880 -0.6380 -0.4000 -0.6400 0.2930 -0.3920 0.2500 0.0330 -0.4700 0.1840 -0.7140 -0.3564 -0.7433 -0.1742 -2.3544 -0.9036 0.2396 0.1891 0.5781 -0.6812

0 1 1 1 0 0 1 1 0 0 1 1 0

2142

2

2142 2143

2 2

2143 2143 2143 2222 2222 2222 2222

3 2 3 1 3 0 2

1 0 0 1 1

2231 2231

1 3

0 0

2231 2232 2232 2232 2233 2233 2233 2233 2241 2242 2242 2243 2243 2244

2 1 3 2 1 3 0 2 3 1 2 3 2 1

1 0 0 1 0 0 1 1 0 0 1 0 1 0

1 0 0 1

0.0390 -0.4050 -0.0140

-0.8564 -0.1236 -0.2959 -1.9515 -0.9190 0.1964

-0.9660 -0.0590 -0.6655 -1.8885 -1.2613 -0.3755

-0.8507 -0.4872 -0.0823

-0.8710 -0.5380 0.0180

-0.0436 -0.2876 0.1103 0.4126 0.0697 -0.9288 -0.1994 -0.5988 - 1.6747 0.6413 -0.0348 -0.2219 1.2251

-0.0730 -0.3400 0.0680 0.0400 -0.1820 -0.7770 -0.1280 -0.4610 -1.5540 0.6010 -0.0840 -0.0470 0.7090

W.A. Richter et al. / New e$ective inieraciions TABLE

7-continued

2T

FPD6

FPM13

a

0

1

-1.1009

-1.4650

1 2 3

0 0 0

-3.7062 -2.7595 -1.0140

3131 3131

4 5

0 0

-4.6645 -2.9498 -1.2631 -2.1877

- 1.9080

3131 3131

6 1

0

-0.0084 -2.4016

-0.1265 -2.2650

2 3 4

0.8104 -0.7480

-0.2890

3131 3131 3131

1 1

0.5599 -0.2038

3131 3131 3132 3132

5 6 1 2

1 1 1

3132 3132

3 4

3132 3132

1 2

3132 3132 3133 3133 3133 3133

3 4 1 3 5 2

3133 3141 3141

4 3 4

3141 3141 3142 3142 3142 3142 3143

3 4 1 2 1 2 2

3143

3

3143 3143 3144

2 3 1

3232 3232 3232

1 2 3

3232

4

3232 3232 3232 3232 3233 3233

1 2 3 4 1 3

abed

2J

2244 3131 3131 3131

Important p = 0.35.

1 0 0 0 0 1 1 1 1 0 0 0 1

0.7636 -1.0100 -1.0597 -0.6618 -0.4647 -0.5580 0.3461 -0.3540 -0.1206 -0.4596 0.1415 -0.8859 -1.1009 0.5823

b

c

d

25

27

3233

2

1

0.0563

3233 3241

4 3

1 0

0.2694 0.1078

-0.1390

3241

4

0

- 1.6022

-1.1880

3241 3241

3 4

1

0.3481

3242 3242 3242 3242

1 2 1 2

3243

2

1 0

3243

3

0

- 1.0726

-0.9971 -0.9530 -0.7050

3243 3243 3244

2 3 1

1 1

0.4984 -0.0930 -0.6792

0.3890 -0.0070

-0.6280 -0.5440

3333 3333 3333 3333 3333

1 3 5 0 2

-0.2212

-0.0120 -0.1960

3333 3341

4 3

1 0

0.1700 0.0607

3341

4

1

0.2675

0.2520

3342 3342

1 2

0

0.0794

3343 3343 3344 3344 4141 4141

3 2 1 0 3 4

1 0

0.3198 -0.5744 -0.4797

-0.1590 0.2560

4141 4141

3 4

4143 4143 4242

3 3 1 2 1 2 2

-0.1061 0.0191 0.0085 0.1523

-0.0350 -0.2950 -0.1320 -0.4930 0.4160 -0.6750 -1.1380 0.63 10 0.4680 -0.4400

1 0

0.3536 -0.653 1

0 1 1

-0.5359 0.5023 -0.3628

0 0 1 1

- 1.4997 -0.5550 0.0212 -0.3242 -0.4497

-0.6590 -0.0790 -0.1500 -0.4270

-0.4906

-0.5380

0.1854 -0.0142

0.3420 -0.0800

-0.2234 -3.0666

-0.1680 -2.1727

4242 4242 4242 4243

0 0

- 1.4447 -0.9873

-1.2448 -0.4640

4243 4244

2 1

0 1

-0.6692

-0.7336 -0.0279 0.2156 0.1290 -0.2570 -0.2920 -0.3270

4343 4343

2 3

4343 4343 4444 4444

2 3 1 0

0 0 1 1 0 0

1 1 1 0 0

0.3586 0.0490 0.5354 -0.3154 -0.7195 -0.4168

351

-0.6740 -0.0120 -0.0930 - 1.4490

note: To allow for a mass-dependence

the TBME

FPD6

1 0

-0.6280 -0.7481

0 1

-0.3393 0.2277 -0.2243

0 0 0 0 1 1

1 0 1 0 0 1 1 0 1 0 0

FPMI3 0.0230 0.1020

0.1400 -0.5310 -0.6200 -0.0710 -0.0580 -0.1590

0.0849

-0.2550 -0.9880

-0.3930

-0.8009 -1.9952 -1.6144 -0.2452

- 1.6899 -0.8837 -0.2180 0.2949 0.2700

-0.3910 -0.1860

-0.3551 -0.7572

-0.0840 -0.3920

- 1.8237 -0.7865

- 1.4840 -0.7460

0.6952 -0.1150

0.0290 -0.2740 -0.4000

-0.4659 0.0655 - 1.9030

0.0780 -2.2910 -1.8970

1 1 0

- 1.8072 0.8237 -0.2571 -0.4075

1 0 0

0.4086 -0.7932 -0.4328

0 1

- 1.6996 -0.2287

1 0 1

0.423 1 -1.1921 -0.1406

must be scaled according

0.1520 -0.6880 -0.4640 0.2410 -0.6860 -0.0810 -1.2340 -0.1350 0.2050 - 1.0740 -0.2490

to eq. (2) with

352

W.A. Richter et al. / New efective interactions

g-factors and charges have been extracted which produce, respectively, outstanding agreement between calculated and measured ground-state magnetic dipole moments, and good agreement in the case of electric quadrupole moments. In the next phase of this work extensive tests of the wave functions yielded by the new interactions are envisaged; this includes the calculation of other observables such as Gamow-Teller strengths and strength distributions, electromagnetic transition probabilities and spectroscopic factors. Work is presently in progress to investigate the application of the semi-empirical MSOBEP in a truncated model space consisting of 0f~,2+Of~;1 (1p3~20f5~21p1~2)1 configurations. Nuclei up to mass number A = 56 and some 300 levels can be included in the fits. If this study proves to be successful it would be a valuable extension of our method to a mass region where calculations in the full fp space are not feasible. Further extensions of this work are being planned. These include: (i) improvements in the shell-model code to allow more nuclei in the fp shell to be considered, and (ii) extensions to further model spaces e.g. for heavy nuclei up to the few particle and hole states around “‘Pb. This last extension may lead to a new universal parameterization of the effective interaction. Simultaneous fits in a multiplicity of model spaces with a common set of parameters are envisaged, with hopefully only a small number of parameter values that are specific to each model space. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)

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