Self-consistent calculation of charge radii of Pb isotopes

Nuclear Physics A551 (1993) North-Holland

Self-consistent

NUCLEAR PHYSICS A

434-450

calculation

of charge radii of Pb isotopes

N. Tajima’

and P. Bonche

Service de Physique Thkorique, CE Saclay, 91191 Gif-sur- Yvette, Cedex, France

H. Flocard Division de Physique 7Morique’,

lnstifut de Physique Nucliaire, 91406 Orsay, France

P.-H. Heenen Service de Physique Nucl&ire The’orique, U.L.B., CP229, 1050 Brussels, Belgium

M.S. Weiss Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA

Received

3 September

1992

Abstract: Charge radii of lead isotopes are calculated with the HF plus BCS method, using Skyrme forces (SkM*, SIII and SGII) for the mean field. When these forces are combined with a seniority pairing force, all of them fail to reproduce the experiment. Neither higher-order corrections, nor ground-state correlations due to the collective modes can resolve the discrepancy. However, by introducing a density-dependent pairing force quenched inside the nucleus, one can explain the odd-even staggering as well as the large kink of charge radii at ‘08Pb when plotted versus A.

1. Introduction The isotope

shift of atomic levels is a demanding

testing ground

where to compare

nuclear theory with experiment I,‘). Isotope shifts which are measured with high precision can be used to obtain the variation of root-mean-square (r.m.s.) nuclear charge radii (r’,) with relatively little ambiguity. Moreover, the advent of laser absorption atomic spectroscopy has made systematic data available 3,4). From a theoretical point of view, lead isotopes are particularly interesting because the doubly magic nucleus “‘Pb is one of the anchor points in the parameterization of effective interactions for mean-field calculations. Moreover, recent measurements have extended to very neutron-deficient lead isotopes 5,6) an already very long sequence [see ref. ‘), p. 5821 which ranges from ‘94Pb to ‘r4Pb, including a number of short-lived specimens far from the P-stability valley. Correspondence to: H. Flocard, Division de Physique Theorique, Institut de Physique 91406 Orsay Cedex, France. ’ Permanent address: Institute of Physics, University ofTokyo, Komaba, Meguroku,Tokyo ’ Unite de recherche des Universites Paris XI et Paris VI, associee au CNRS. 0375-9474/93/$06.00

@ 1993 - Elsevier

Science

Publishers

B.V. All rights reserved

Nucleaire, 153, Japan.

N. Tujima et al. / Pb isotopes

The observed is the presence

charge radii of Pb isotopes of a kink at the shell closure

versus the mass number

even-A

nucleus

two conspicuous

Another

has a charge radius

is about

while for A > 208 the slope is about

model,

value.

features.

One

208Pb when the charge radius is plotted

A. For A < 208, the size of the slope drE/dA

the slope of the liquid-drop size as the liquid-drop

exhibit

435

feature

is the odd-even

larger than the average

half

the same

staggering.

Each

of the radii of adjacent

odd-A isotopes. These features are not specific to the Pb isotopic chain; they have been observed in other regions of the mass table [see ref. ‘) and references therein]. A possible origin of this kink is the change of the average r.m.s. radii of neutron orbitals below and above the major shell gap of N = 126. The attractive protonneutron force transfers this change onto the proton radii. Sagawa et al. ‘) estimated this effect in first-order perturbation theory and interpreted their results in terms of the incompressibility of nuclear matter. As for the odd-even staggering, it seems natural to associate it with pairing correlations, which are diminished in odd-A nuclei because of the blocking effect. As a mechanism in which neutron pairing influences the proton distribution, Zawischa proposed 2p-2n correlations *). In this paper, we first calculate the charge radii of lead isotopes in the framework of the self-consistent mean-field theory and examine its ability to reproduce experiment. We use the Hartree-Fock plus BCS (HFBCS) method 9), whose features essential to the present study are explained in sect. 2. In sect. 3 we present the results of calculations using three effective Skyrme forces, SkM” lo), SIII ‘I) and SC11 I’). It is shown that the SkM” and SGII forces produce very different behaviors for the charge radii although their incompressibilities are almost the same. We also discuss the difference between our self-consistent calculations and the first-order perturbation theory ‘). Compared with experiment, however, none of these interactions reproduces the experiment at the HFBCS level. With the Skyrme SkM” interaction, we show in sect. 4 that corrections to the method for converting measured isotope shifts to r.m.s. radii cannot be expected qualitatively to modify the above negative conclusion. In sect. 5, we analyse the effects of ground-state correlations induced by the isoscalar EO (breathing) mode and by higher-multipolarity modes (quadrupole and octupole). The associat’ed configuration mixing is calculated with the generator-coordinate method. We show that these correlations

cannot

account

for the experimental

data, either.

Given this lack of success, in sect. 6, we question the effective two-body interaction itself and analyse its least determined part, namely the parameterization of the pairing force. For that purpose, we introduce a density-dependent zero-range pairing force instead of the schematic seniority force used in the previous sections. We choose to quench the interaction inside the nucleus, as is usually the case of Landau-Migdal particle-hole effective forces 13). With this choice, the nucleus tends to expand when the pairing correlation is switched on because the interaction is stronger at lower densities, e.g. in the vicinity of the surface. This effect accounts for the magnitude of the kink. The odd-even staggering, too, can be understood by

N. Tajima et al. J Pb isotopes

436

the same mechanism in even-A ones. Finally,

because

in the last section,

pairing

correlations

we summarize

are weaker

in odd-A

nuclei

than

our results and discuss the three Skyrme

forces. 2. Formulation A detailed account of our HFBCS formalism can be found in ref. “). The following presents only those points which are important to this study. For the effective interaction in the mean-field hamiltonian, we use the Skyrme parameterization. For the pairing interaction, we use a seniority force in the calculations shown in the next three sections, and we introduce The seniority force is written as fig=-G,

zero-range

C

forces in the following

at,a?;aTTa,,

sections.

(1)

i>O,j>O

where Uti is the creation for ~=n) in the orbital

operator of a nucleon (a proton for r = p and a neutron i. The time-reversal partner of the orbital i is expressed

as i. For neutrons, we choose the lowest 92 x 2 orbitals as the pairing-active space. For all the calculated isotopes a major shell gap exists just above the 92nd level and the single-particle energy becomes positive from the 93rd level on. As a result, the pairing-active space is well defined and it is not necessary to introduce a smooth cut-off wave function as is done in ref. “). The strength is parameterized as G, = g,/(ll+ N), where N is the number of neutrons. For protons, the pairing never switches on with a reasonable strength of G, for all the cases we calculated. The calculations presented here are done in coordinate space with a mesh size of 1 fm. The nuclei we obtain are almost spherical: their mass quadrupole moments are always less than 3 fm’. In sect. 4, we discuss the accuracy of our results with respect to the mesh size. The pairing-force strength is adjusted to be g, = 12.5 MeV so that the empirical pairing isotopes

gaps are well reproduced. as a function

Fig. 1 shows

of the mass number

the neutron-pairing

A. Empirical

gap of lead

gaps are derived

from the

binding energies 14) of the neighboring five isotopes 15). The calculated values are obtained with seniority force and the SkM” force. For the isotopes A = 194, 202, 214 and 220, calculations have also been done with the SIII force and the SGII force. One sees that the pairing gaps are reproduced fairly well with this average strength. 3. Mean-field effects We have used three kinds of Skyrme forces (SkM”, SIII and SGII) to calculate the proton and neutron r.m.s. radii. The SIII force ‘I), which is one of the most successful forces proposed in the 1970’s, features a good single-particle spectrum

N. Tajima et al. / Pb isotopes

437

Pb

Fig. 1. Neutron-pairing gaps of even-A Pb isotopes. The experimental values are designated by solid circles connected by solid lines. The calculated values with the SkM* force are expressed by open circles connected by dashed lines. The plus and cross symbols for A = 194, 202, 214 and 220 denote the gaps calculated with the SIII and SGII forces, respectively. The seniority pairing force with g, = 12.5 MeV is used for the pairing part of the interaction.

among

other qualities.

It has, however,

a too large nuclear-matter incompressibility to study isotope shifts ‘). In addition The SkM* and SGII forces are both derived

K, = 355 MeV, which may be a major drawback

SIII predicts

too high fission barriers.

from the SkM force r6). The latter force was designed to give correct energies of isoscalar EO and isovector El giant resonances. The SkM” force lo) improves upon the SkM results for the actinide fission barrier height. On the other hand, the SGII 12) force aims at producing correct spin and spin-isospin Landau-Migdal parameters. Both the SkM” and SGII force have reasonable incompressibility moduli K, of 217 and 215 MeV, respectively. Let us make some comments concerning the procedure which determines the numerical values of the parameters of all these forces. The main ingredients which enter their fitting procedure are binding energies and r.m.s. radii of a small number of doubly closed shell nuclei as well as infinite nuclear matter, besides the specific properties

mentioned

above. The symmetry-energy

property

of these forces is mostly

tested near the bottom of the valley of stability. They must therefore be used with care for very neutron-deficient or very neutron-rich nuclei. We will see that the three Skyrme forces studied in this work do not account very well for the symmetry properties. Let us mention that is is also the case for the DlS force 17) to which we compare our results later in this section. Let us define quantities to be used. The mean-square radii are defined as r:(A) = w h ere 7 = p (n) denotes protons (neutrons) and [A) stands for (Ai+:iA)l(AkjA), the HFBCS ground state of A nucleons. The differences, Arz(A) = r:(A) - rf(208), can be deduced with high precision for r =p from the measured isotope shifts. The mean-square radius of the liquid-drop model is given by r”,, =$rzA2’3 with r. = 1.2 fm.

438

N. Tajima et al. / Pb isotopes

In fig. 2, calculated of the mass number

proton

and neutron

mean-square

A. The left, the middle

radii are shown as functions

and the right parts are calculated

with

SkM”, SIII and SGII forces, respectively.

In order to emphasize

the deviation

the smooth

which

liquid-hrop-model

values

trend,

we plot values

are subtracted:

from

Arz(A) - A&(A).

values “). The agreement

corresponding

The figure also includes

with experimental

proton

from

the experimental

r.m.s. radii is excellent

on the

neutron-deficient side for the SkM” or SGII forces. On the neutron-rich side, however, all calculations fail to reproduce the abrupt change observed experi: mentally. We defer further discussions on the value of Ar”, for each nucleus to the last section. Until then, we consider mostly the change of the slope occurring at “‘Pb which we call the kink. Because the calculated Ar”, behaves almost linearly versus A except at A = 208, one can define the magnitude of the kink k, A = 208) using the three nuclei ‘94Pb, 208Pb and The values of k, are given in table 1 together 214Pb. The size of the proton kink increases

(i.e., the change of slope d@dA at 214Pb as: k, =iAr:(214) -$A& 194). with the values of Ar’, for lg4Pb and as one goes from SkM*, to SIII, to

SGII and experiment. The neutron kinks are larger than the proton ones, which is natural since the former induces the latter through the proton-neutron attractive interaction. It should be noted however that calculated neutron kinks themselves are smaller than the experimental proton ones. For this reason it appears difficult to obtain a kink of the magnitude required by experiment only by modifying the

194

210

202

A

218

194

210

202

A

218

194

202

210

218

A

Fig. 2. Proton and neutron mean-square radii of Pb isotopes calculated with the SkM* (left), SIII (middle) and SGII (right) forces. Plotted values are the differences from the values for ‘08Pb. To emphasize the liquid-drop-model values are subtracted. Open the deviation from smooth trends, circles (solid squares) denote the proton (neutron) radii. The experimental values are designated by dots connected by dotted lines. A seniority force with g,= 12.5 MeV is used for the pairing part of the interaction.

N. Tajima et al. / Pb isotopes

439

TABLE 1 Differences

of proton

and neutron mean-square those of *“Pb calculated

Force

SkM* SIII SGII exp.

radii (fm’) of the isotopes with three Skyrme forces

‘94Pb and 214Pb with

k,

k,

-0.7573 -0.8680 -0.7448 -0.6830

0.3545 0.448 1 0.4576 0.6099

0.0050 0.0127 0.0231 0.0529

‘94Pb

‘14Pb

-1.5931 -1.4939 -1.4589

0.8530 0.8335 0.8843

0.0284 0.0322 0.0432

Skyrme-force parameters. This has motivated the analysis of the pairing part of the interaction which we present below in sect. 7. At this point, one can note that the size of the kinks is not determined solely by the incompressibility as it has been suggested in ref. ‘): the SkM” and SGII forces give kinks of different magnitude although they have almost the same incompressibility modulus. This difficulty in reproducing the experimental kink is not specific to Skyrme forces. The HFB calculations with the DlS force 17) [a more elaborate effective force than those of Skyrme with finite ranges and self-consistent pairing) produce a kink k, of 0.012 fm’ [ref. 18)], which is as small as that of SIII. As mentioned earlier, a study of the isotope shift cannot be done independently of a study of the symmetry property of the effective force. Fig. 3 gives the HFBCS binding energies obtained for some lead isotopes with three Skyrme effective interactions: SkM”, SGII and SIII. Instead of the binding energies, we use differences, taking *“Pb as a reference point (AE(A) = E(A) - E(208)) to subtract out systematic numerical inaccuracies linked with the evaluation of the kinetic energy on a mesh of 1 fm [see ref. “)I.

Fig. 3. Errors of calculated masses of Pb isotopes using three types of connected by solid lines), SIII (squares connected by dotted lines) by dashed lines). A seniority force (g, = 12.5 MeV) is used for the masses from the mass of “‘Pb are plotted instead of

Skyrme forces, SkM* (solid circles and SGII (open circles connected pairing interaction. Differences of masses themselves.

440

N. Tajima er al. / Pb isotopes

This figure confirms improved. “‘Pb.

However,

SGII SIII.

feature

appears

it predicts The

SkM*

masses force

properties

to give results

gives very good results

light isotopes than

that the symmetry SIII

of the three

of similar

for heavy lead nuclei.

which gives

disagree results

must

either

of intermediate between

be

side of

On the other hand,

with experiment

can be seen on table 1 where the difference

forces

quality

for

three times more quality.

calculated

The

same

and experi-

mental isotope shifts also favors SIII. For instance, had we chosen lg4Pb instead of 208Pb as a reference point, none of the three forces would have correctly predicted the “‘Pb binding energies, but SIII alone would have given remarkably good agreement for 214Pb both for the binding energy (AE < 100 keV) and for the isotope shift (Ar2 < 0.023 fm’). This comparison between these three forces - with the same pairing interaction - illustrates the importance of the symmetry energy. It appears, that, for SIII, this behavior as a function of (N - 2) is qualitatively correct across the stability valley, though its strength is wrong. Before closing this section, we would like to comment on the difference between a first-order perturbation calculation ‘) and our self-consistent mean-field calculation. The perturbative results with the SGII force as Arz(‘96Pb) = -0.6156 fm’ and Ari(2’4Pb) = 0.5071 fm2. The kink value (where ‘96Pb is used instead of “‘Pb) is 0.033 fm’, which is larger than our result by 43%. The difference in the treatment of pairing does not seem relevant because in ref. ‘) pairing was used only to estimate the distribution of the Cooper pairs. Since the resulting occupation probabilities vf are similar, the treatment of the pairing does not seem to be the source of the difference between our results and theirs. In our opinion it should rather be looked for in the first-order perturbation method used to calculate charge radii. The perturbative treatment estimates the amplitude of the small admixture of the giant monopole

resonance

due to a neutron

(hole)

added

to the 208Pb core. For nuclei

other than those with A = 208 f 1, the response of the core to each neutron (hole) is added assuming linear response. The self-consistent mean-field calculation takes into account this core-polarization effect because a small admixture of the giant monopole

resonance

leads principally

to a small change

of the nuclear

radius.

In

addition to this core-polarization effect, the self-consistent calculation takes into account the change of the single-particle wavefunction occupied by the additional neutron (hole). This effect is a second-order effect and not included in ref. ‘>. It is likely that higher-order effects like this one decrease large responses predicted by lower-order calculations. 4. Corrections 4.1. NUCLEON

SIZE

The experimental isotope shifts measure a difference whereas our calculations determine the proton density.

between charge densities, Since a better calculation

441

N. Tajima et al. / Pb isotopes

based

upon

the charge and

densities

between

the light

densities

with the charge distribution

themselves

the heavy

lead

might

isotopes,

we have

of the proton

This has been done as an illustration

resolve

the different folded

to recalculate

behavior

the HF proton the isotope

only for the SkM” force and results

shifts.

are given

in table 2. In this table we have also reported

calculations

done with a smaller

mesh size,

0.8 fm. As can be seen minor modifications are indeed observed on the resulting isotopes shifts, but no dramatic change, so that the agreement for 194Pb and the discrepancy for 214Pb still holds. The systematic 10% reduction of the isotope shifts observed in the calculation done with a smaller mesh size does not produce any kink and is not significant in that respect. Another small effect can be taken into account, namely a contribution from the charge distribution of the neutron itself. This contribution to the isotope shifts reads as: Ar2=Arfh+----

N-126 82

v>n.

(2)

The first term is the charge contribution of the protons. The second one comes from the neutrons and is proportional to the neutron mean-square charge distribution (r*), which is 0.13 fm’. This last contribution is small and proportional to the neutron number difference with respect to “‘Pb. It changes the average slope but does not give rise to a kink. All the above corrections could not explain experimental data where the HFBCS calculation

4.2. HIGHER

why there is such a kink in the yields rather smooth trends.

MOMENTS

The mean-square

charge radii are not the only nuclear

ingredient

which enter the

experimentally measured isotope shifts. In the analysis of the experiments, they are given in terms of a parameter h proportional to a weighted sum of all the even TABLE 2 Proton and charge r.m.s. radii (fm) and their isotope shifts with the SkM” effective force and for two different values of the mesh size (fm) Proton

density

Charge

density

Mesh size

‘94Pb *“Pb *14Pb d( 194-208) d(214-208)

1.0

0.8

1.0

0.8

5.3666 5.4368 5.4729

5.398 5.461 5.495

5.3742 5.4450 5.4812

5.407 5.472 5.505

-0.759 0.394

-0.684 0.373

-0.766 0.396

-0.707 0.362

442

N. Tajima et al. / Pb isotopes

moments

of the proton

charge

density:

A =Ar2+gAr4+gAr6+.

In the above expansion, it is usually

most of the contribution

approximated

by the simpler

. .. comes from the first term, so that

expression

h = K Ar2 where the parameter

K is given by K=l+This parameter radii

has been

C2Ar4 C3Ar6 -+.... ClAr2+ClAr2

(4)

estimated

I”,‘) utilizing a uniform charge distribution of R = 1.2A1’3 fm and the coefficients Cn tabulated by Seltzer *‘). One obtains h = Ar2( l-0.084+

0.014) = 0.93Ar2 .

(5)

This model of a uniform charge density with such a large radius seems so crude that we have extracted from our calculation the expectation values of y4 to compare with the above number. It turns out that instead of -0.084, we obtain -0.079 for ‘94Pb and -0.085 for *14Pb. Even though both the numerator and the denominator in the first-order correction in K may be overestimated in the model, their ratio agrees quite well with our numbers. In view of that, we do not think that the even smaller correction coming from higher moments will resolve the discrepancy on the kink. If this K-value was responsible for the kink, it should be kept the same for A<208 and double to account for the experimental number for isotope shift of lead nuclei heavier than 2o8Pb. This is more than unlikely in view of the physical content of K. 4.3. PAIRING-INTERACTION

STRENGTH

We have also varied the strength g, of the pairing interaction to sensitivity of the isotope shift upon this component of the force. increasing by 10% the neutron-pairing strength results in negligible proton radius of ‘r4Pb. Modifications of the proton-pairing strength With a proton

pairing

increased

investigate the Decreasing or changes in the was also done.

by 10% over the value used in the same mass region, it does not result in a significant

proton BCS correlation sets in. However, modification of the 2’4Pb proton radius. 5. Correlations 5.1. QUADRUPOLE

and collective

modes

MODE

We now turn to the corrections coming from the admixture of quadrupole correlations into the Hartree-Fock plus BCS ground state. This configuration-mixing problem is treated by a discretized version of the generator-coordinate method (GCM) where the generating variable is chosen to be the quadrupole moment. The

N. Tajima et al. / Pb isotopes

collective generate HFBCS

basis

by means

of constrained

HFBCS

calculations

a set of Slater determinants for quadrupole deformations spherical minimum. For that purpose we use the techniques

ref. “). We then space spanned tion.

is obtained

443

diagonalize

the effective

by these static constrained

The solution

of such

diagonalization

interaction solutions

which

around the described in

SkM* in the non-orthogonal obtained

with the same interac-

in a non-orthogonal

basis

has been

presented elsewhere *‘). The modification of the isotope shift coming from this GCM calculation is very small. Its increase is less than 1% in 194Pb and one order of magnitude smaller in *14Pb.

5.2. MONOPOLE

MODE

A GCM analysis for the isoscalar breathing mode leads to a similar result. The GCM gives rise to a constant increase (about 0.015 fm) of the r.m.s. charge radius for all the isotopes. However, the differences measured from **‘Pb are hardly changed compared with the HFBCS solutions. This result also holds when density-dependent delta forces (see sect. 6) are used instead of seniority forces for pairing. The contributions of the monopole and of the quadrupole modes to the isotopic shifts do not resolve the discrepancy. As in most calculations 3,19,22)and models 23*2‘1), isotope shifts are more or less correctly reproduced below 208Pb and off by a factor of two above 208Pb.

5.3. OCTUPOLE

MODE

All the effects we have been studying so far do not present any dramatic change across the doubly magic ‘*‘Pb nuclei. In contrast, the experimental level scheme throughout the lead isotopic chain shows evidences for a modification of the octupole properties across the magic-shell closure. For nuclei with a mass smaller than 208, the lowest 3- state is similar to that of *08Pb although its collectivity decreases a.s one removes neutrons. For neutron-rich isotopes, this 3- state is still present; however another one, lower in energy, appears already for 2’0Pb at 1.87 MeV. This is a direct consequence of the population of the 2g9,* orbital in “*Pb particle-hole excitation to the j,5,2 level. In 2’4Pb, the splitting subshells is predicted to be 1.8 MeV in our HFBCS calculation

which allows a 3between these two with SkM”.

On the basis of this analysis, we have studied the possibility of corrections to the isotope shift due to octupole mode. For nuclei heavier than 208Pb, the neutron part of the wavefunction may be more sensitive to this mode and, via the proton-neutron force, may generate corrections to the r.m.s. charge radii larger in *14Pb than in 194Pb. With the octupole operator, we have therefore performed HFBCS calculations similar to those presented above for the quadrupole and monopole case. First we have done constrained HFBCS calculations with a value of the quadrupole moment fixed to zero and for different values of the octupole moment. Before doing a full

444

N. Tajima et al. / Pb isotopes

GCM calculation coordinate

with the expectation

variable,

value of the octupole

we have projected

tions 25). The energy

onto parity

of the even-parity

increases

to reach

operator

of 750 fm3 for ‘94Pb and “*Pb,

states

an absolute

states

are more bound

minimum

than the pure

operator

the different

decreases

as generator-

constrained

as the octupole

for an expectation

solumoment

value of the octupole

and of 820 fm3 for 2’4Pb. These correlated

HFBCS

ones by about

1 MeV for the three

isotopes. However, the influence on the isotope shift is negligible, even though it is slightly larger for *14Pb as expected. The complete GCM calculation with the value of the quadrupole moment fixed to zero does not significantly modify our results. In view of the smallness

of these

corrections,

we do not think

that

a more

complete two- or three-dimensional GCM calculations mixing these three moments (monopole, quadrupole and octupole) together would change our conclusion. 6. Effects of density-dependent

pairing forces

In this section, we study the effects of the pairing on the charge radius. In what follows, the calculations are done with SkM”. Instead of the schematic seniority force (l), we use a more realistic zero-range force (delta force 26) for the particle-particle channel of the interaction (i.e. for the BCS calculation). Expecting radius-driving effects, we introduce density dependence into this effective force. Our pairing interaction is written as follows: 1 -U1’U* V’(r,,u,; As density-dependence dependence

r2,(+*)= function

on the nucleon

Vi f(r),

density

rl + r2 %c-r2lf

4

we choose,

parameterized

f(r)=l-@$,

(

-y--

>

.

for the sake of simplicity, by a reference

density

linear pc, (7)

In what follows, we suppress the superscript T as we consider only neutrons. An approximate method is employed to solve the HFBCS equation. When the seniority force is used for pairing, the pairing energy does not depend on the single-particle wavefunctions but only on the occupation probabilities 0:. In this case, the HFBCS equation is separated into the HF equation and the BCS equation: the former determines the single-particle wavefunctions while the latter determines the occupation probabilities “). For a realistic force like eq. (6), however, this method does not give the exact solution. Instead, we calculate the potential-energy surface (PES) versus the r.m.s. mass radius and regard its minimum point as the solution. The states on this PES are constructed by solving the HFBCS equation constrained on the r.m.s. mass radius. The method of solution appropriate for seniority force can be applied with enough accuracy to this radius-constrained equation because the pairing is not likely to influence collective degrees of freedom

N. Tajima et al. / Pb isotopes

other than radius. proportional nucleons

To generate

to r2fcUt(r) on the nucleus.

from escaping.

and begins

the constraint,

445

we exert an isoscalar

A cut-off factorf,,,(

external

r) is necessary

potential to prevent

a cut-off function which decreases from 1 to 0 at r = 9 fm (9.4 fm) for 194Pb (220Pb) over an interval of 1 fm.

To show that

We choose

a force such as eq. (6) will affect the radius,

antisymmetrized

two-body

matrix

5;iw= v, Assuming a constant C,, I&(r, a)/‘, by &A,

Since the absolute the pairing tends

element

of a pair-scattering

C I$;(r, ~1)1” C IGj(r, a2)l’f(r) I UI ‘T1

let us consider

an

process, (8)

dr.

density p0 within a sharp surface and replacing the sum, the average size of the matrix element (8) is estimated to be

value of the above quantity reaches its maximum value at p. = $p,, to shift the nuclear density p. towards fp,. For example, in the

case of a pure delta force (p, = CO), the pairing

compresses

the nucleus

compared

with the HF solution. In the opposite extreme case where pc= po, the pairing interaction is turned off inside the nucleus like a surface force, and the nucleus is expanded by the pairing. Let us emphasize that, in this mechanism, the nucleus swells not only because high-energy orbitals having large r.m.s. radius are occupied through the smearing of the fermi surface, but also because the overall energy gain due to the pairing correlation becomes larger when the density is decreased through expansion. Table 3 gives the parameters of the neutron-pairing force used in the calculations. In this section too, proton pairing is not considered. Parameter set (a) is a pure delta force which tends to compress the nucleus. The seniority force, whose radiuschanging effect should be very small, is put on the next line (b) for the sake of comparison. The parameter sets (c), (d) and (e) have finite pc’s of 0.1603, 0.1382 TABLE 3 Effects of the density dependence of the pairing interaction on the proton and neutron mean-square radii in units of fm2: (a) pure delta force, (b) seniority force, (c), (d) and (e) density-dependent delta force Al; “: (MeV. fm’)

(f:3) (a)

CQ

;:1 (c) exp.

-230

g, = 12.5

(b)

0.1382 0.1603

-1150 -750

0.13

-1450

‘94Pb

‘14Pb

-0.8948 -0.7573 -0.5165 -0.6662

0.3097 0.3545 0.4454 0.3753

-0.4464 -0.6830

0.4761 0.6099

-0.0123 0.0050 0.0150 0.0373 0.0475 0.0529

-1.7440 -1.5931 -1.5137 -1.3571 -1.2825

0.7930 0.8530 0.8668 0.9453 0.9817

0.0076 0.0284 0.0364 0.0606 0.0720

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N. Tajima et al. / Pb isotopes

and 0.13 fme3. Since ip, is always smaller than the usual nucleon density inside the these three pairing interactions are expected to expand the nucleus. The

nucleus,

reference force,

density

while

of 0.1603 fmw3, case (c), is the nuclear

0.1382 fme3, case (d), corresponds

with r,, = 1.2 fm. The average the SkM* force is about

density

matter density

to the liquid-drop

of the interior

of Pb isotopes

0.158 fmp3, which is between

pc’s

of the SkM” model

calculated

density with

of (c) and (d). An even

smaller value, 0.13 fmm3, is used for pc in the parameter set (e) for the sake of comparison. The strength Vi for each pc was determined so that the minimum quasi-particle energies agree well with the experimental pairing gaps. In fig. 4, calculated and experimental Ari are shown. Table 3 includes the values of Ari and Arf, for ‘94Pb and 2’4Pb as well as the sizes of kinks k, and k,. With the seniority force (b), the kink has the correct sign, although its size is much smaller than experiment. With the pure delta force (a), the sign of the kink is reversed. This incorrect sign for the kink k, becomes even larger (-0.0234 fm2) when the strength V,” is increased (by 22%) to -280 MeV. fm3. The change of sign of the kink occurs because pure delta forces compress the nucleus: since the effects of pairing interaction exist only when the pairing is switched on, the closed-shell nucleus is unchanged. Other isotopes are compressed more or less, depending on the size of the pairing correlation. Owing to the continuing growth versus A of the pairing gap in the neutron-deficient side (see fig. l), this effect leads to a kink rather than a bump around

20*Pb. On the neutron-rich

side, the increase

210

i94

of the pairing

gap is not steady

218

A Fig. 4. Proton mean-square radii of Pb isotopes given in the same manner as in fig. 2. The SkM* force is used for the mean-field part of the interaction, while several types of density-dependent pairing forces are employed for the pairing part. Open circles, squares, reversed triangles, solid circles and triangles denote the results calculated with forces (a) to (e) given in table 3, respectively. The experimental values are designated by dots connected by dotted lines.

N. Tajima et al. / Pb isotopes

and the Ar; is not so linear dependence

is introduced,

as in the neutron-deficient

the sign of the kink is reversed

with the experiment.

One observes

larger the magnitude

of the kink.

Regarding comments.

the

447

symmetry

that, the smaller

property

of the force,

side. When again

so that it agrees

the reference we can

As can be seen from fig. 4, the overall behavior

the density

density

make

pc, the

the following

of the isotope

shift does

not improve. This fact is related to the symmetry-energy property of SkM* which has remained almost unchanged. The total binding energies do not vary substantially: less than 0.8 MeV both for ‘94Pb and ‘14Pb from case (a) to (e). One can also explain the odd-even staggering in terms of the density dependence of the pairing force. Odd nuclei have a smaller pairing gap than adjacent even nuclei because of the blocking effect. Weaker pairing leads to a diminution of the expansion due to pairing. Assuming that the mean field does not contribute to the odd-even staggering, we can determine pc from the size of the staggering. We estimate the effects of blocking in the following manner. Blocked states are obtained by solving the BCS equation after removing a given pair of orbitals i and 7 from the pairing-active space. As it is more convenient, instead of considering odd-A nuclei, we adjust the Fermi level such that the expectation value of the number of neutrons becomes even: 2 Cjpi ,. v?, + 1= even. The orbital i does not contribute to the pairing energy. We set ui?= i in the evaluation of the mean-field energy. By minimizing the total energy, we obtain a new solution, whose r.m.s. charge radius, rz(blocked), can be regarded as the average charge radius of the two adjacent odd-A isotopes. In the calculation, blocked orbitals are determined from the experimental spin of the neighboring odd-A nuclei. (3p,,,, 2f,,, and 2g,,, for ‘94Pb, 202Pb and ‘14Pb, respectively. 2g9,2 is assumed for 220Pb.) The differences of t-2 between the ground states of even nuclei and the corresponding blocked states are given in table 4 for five kinds of pairing forces. The experimental values for 194Pb and 2’4Pb are obtained by extrapolation. With a delta force without density dependence (a), even nuclei have smaller r.m.s. charge radii than TABLE

Effects of the

4

density dependence of the pairing interaction on the odd-even proton

mean-square

staggering

of the

radii in units of fm* Arz(g.s.) -Arg(blocked) zropb

(f$)

(4

03

(b) ii;

0.1382 0.1603

(e) exp.

0.13

-230 g, = 12.5 -11.50 -750 -1450

-0.0139 0.0075 0.0204 0.0463 0.0647 -0.045

0.0455 0.03

-0.0066 0.0077 0.0120 0.0285 0.0395 -0.02

0.0121

448

N. Tajima et al. / Pb isotopes

the blocked

states, in contrast

odd-even

staggering

spectrum

is compressed

dependent although

probably

to experiment. because,

and the pairing

forces

[(c) to (e)],

calculated

values

modifications

The seniority

when the nucleus

energy gain increases.

(d) reproduces are

too

force (b) leads to small

expands,

large

which may give a better agreement

the single-particle Among

the experimental for

heavier

isotopes.

with experiment,

the density-

value

for ‘94Pb,

Among

the

one can mention

(i) changing Vg as a function of A, (ii) taking into account the finite-nucleon-number effect with e.g. the variation-after-projection method, (iii) performing self-consistent calculations which consider all the changes of the mean-field due to the blocking, (iv) changing the Skyrme-force parameters (the blocking effect depends strongly on the single-particle

spectrum).

7. Conclusion In the first part of this work, we have shown that none of the currently used effective interactions can reproduce the isotope shifts in the lead region of the mass table. This result holds for Skyrme-type interactions where pairing correlations are usually taken care of by means of a seniority pairing interaction. These results agree with similar calculations by other authors 33’9322).It also holds for the DlS force r7) where pairing is included in a more consistent manner. We have then pointed out the relation between the quality of the calculated isotope shifts versus the symmetryenergy property of the interaction. We have then studied three kinds of possible corrections: (i) using charge density instead of the proton distribution to calculate the r.m.s. radii, (ii) considering the charge distribution of the neutron, (iii) an analysis of coefficients entering the extraction of the isotope shifts from experiment, None of these effects can explain the discrepancy with the experiment, nor do they give measurable contributions compared with the mean-field predictions. Furthermore, no correlations associated to configuration mixing in the groundstate wavefunction provide significant contributions. We have analysed monopole, quadrupole and octupole collective modes in the GCM framework without success. This indicates that a correct estimate of the isotope shifts is unlikely to result from higher-order correlations beyond mean-field calculations. In principle if one uses the 208Pb single-particle states as a basis to make shell-model calculations of any other lead isotopes, one must include core-polarization effects through correlations up to first or even second order. Our completely self-consistent calculations, include core polarization automatically and no further correlations seem to be relevant. The fact that we cannot reproduce the experimental isotope shifts must be therefore related to the effective forces themselves which at present are not tailored to reproduce the symmetry-energy property of nuclei far from the line of stability. Our calculations show that the isotope shift is not simply connected to the incompressibility but seems to provide independent information. Therefore, one

449

N. Tajima et al. / Pb isotopes

can utilize this quantity,

together

mean-field interactions. For this purpose, however, account.

Indeed,

property

of the effective

the isotope

with the symmetry

the effect of pairing

it seems to us from the present nuclear

shift, the observed

interaction

magnitude

property, interaction

to improve

effective

must be taken

into

study that, even if the symmetry

is adjusted

to fit the overall

trend

of

of the kink is too large to be reproduced

only with mean-field effects. With a density-dependent pairing force which almost vanishes inside the nucleus, one can explain a large fraction of the kink as well as the odd-even staggering of the charge radius. Assuming the parameters ,pC= 0.1382 fmm3 and V,” = -1150 MeV*fm3, the odd-even staggering is reproduced for the 194Pb and k, increases by 0.032 fm* compared to the value obtained with the seniority force (SkM” was used for the mean field). With this pairing force, the part of the kink attributable to the mean field is 0.020 fm2, a value which is intermediate between those calculated with the SIII force and the SGII force (with the seniority force). Hence it should be feasible to construct a Skyrme force which reproduces the observed kink when it is combined with an appropriate density-dependent pairing force. Let us also note that in the DlS force, there is no density dependence in the part of the force responsible for pairing correlations. We think that this work has proven the need for a better parameterization of the effective nuclear force to account correctly for the proton-neutron degrees of freedom away from the stability line. We conjecture that, if such a force were to be derived to give correct binding energies, the overall trend of the isotope shift would also be predicted correctly. However, it is also a conclusion of our study that the magnitude of the experimentally observed kink will demand a density-dependent parameterization of the pairing part of the interaction.

One of us (P.B.) wishes to thank J.F. Berger and D. Gogny for fruitful discussions. This work was performed in part under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405 ENG-48 and NSF grant No. PHY90-13248.

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