nuclear-matter symmetry coefficient and nuclear masses

Ž .Nuclear Physics A 668 2000 163–171

www.elsevier.nlrlocaternpe

Nuclear-matter symmetry coefficient and nuclear massesq

J.M. Pearson a, R.C. Nayak a,b

a ( )Departement de Physique, UniÕersite de Montreal, Montreal Quebec , H3C 3J7 Canada´ ´ ´ ´ ´b Department of Physics, G. M. College, Sambalpur, 768004 India

Received 20 August 1999; received in revised form 4 October 1999; accepted 14 October 1999

Abstract

Ž .Within the framework of the ETFSI extended Thomas–Fermi plus Strutinsky integral massformula, a precision fit of nuclear masses with Skyrme forces, subject to the constraint thatneutron matter does not collapse at nuclear or subnuclear densities, is possible if, but only if, thenuclear-matter symmetry coefficient J lies close to 28 MeV. q 2000 Elsevier Science B.V. Allrights reserved.

PACS: 21.10.Dr; 21.60.Jz; 95.30.CqKeywords: Nuclear masses; Nuclear matter; Symmetry coefficient

1. Introduction

We are involved in a program to develop a microscopic theory of nuclear systemsapplicable to the wide variety of phenomena encountered at subnuclear and nucleardensities during and after stellar collapse, and in particular to describe all thesephenomena, as far as possible, in terms of a single, universal, effective interaction. Themain achievement so far has been the development, for the first time, of a mass formula

w xbased entirely on microscopic forces, the ETFSI-1 mass formula 1–5 . The astrophysi-cal interest of such a mass formula lies in the fact that the r-process of nucleosynthesis

Ž .depends crucially on the binding energies among other properties of nuclei that are soneutron-rich that there is no hope of being able to measure them in the laboratory. It isthus of the greatest importance to be able to make reliable extrapolations of massesaway from the known region, relatively close to the stability line, out towards theneutron-drip line.

q Ž .Supported in part by the NSERC Canada .

0375-9474r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9474 99 00431-5

( )J.M. Pearson, R.C. NayakrNuclear Physics A 668 2000 163–171164

w xFull details of this mass formula are to be found in the earlier papers 1–5 , and welimit ourselves here to a reminder of some essential points. The ETFSI method is

Ž .essentially a high-speed approximation to the Hartree–Fock HF method, with aŽ . w xmacroscopic part given by the extended Thomas–Fermi ETF method 6,7 , and shellŽ . w xcorrections calculated by the so-called Strutinsky-integral SI method 1,5 . Pairing is

handled in the BCS approximation with a d-function force. Although this is really amicroscopic-macroscopic mass formula, there is a much greater coherence between the

Ž .two parts than is the case with mass formulas based on the drop let model, since thew xsame Skyrme force underlies both parts. In fact, it has been shown 1,2 that the ETFSI

method is equivalent to the HF method in the sense that when the two methods fit thesame form of Skyrme force to the mass data they give essentially the same extrapolationout to the neutron-drip line. This equivalence to the HF method presumably accounts forthe fact that with just 8 parameters the underlying force of the ETFSI-1 mass formula,

w xSkSC4, fits 1492 mass data for A 0 36 with an rms error of only 0.736 MeV 4 .The chief defect of the original mass formula from the astrophysical point of view

lies in the fact that the force SkSC4 leads to the collapse of neutron matter at aroundnuclear densities. In view of the known stability of neutron stars this collapse is clearlyunphysical, and we must reject the use of the force SkSC4 to calculate the equation of

Ž .state EOS in highly neutron-rich environments, even if it gives an optimal fit tofinite-nucleus masses. In this paper we show how the force can be modified to avoid the

Ž .neutron-matter collapse without compromising the mass fit. We first describe Section 2a change to the form of the density dependence of our force that we make entirely in theinterests of simplification, the overall quality of the fits remaining unchanged. In Section3 we then deal with the neutron-matter condition and make the appropriate modificationof the force, the implications of which are discussed in Section 4.

2. Form of density dependence

The Skyrme forces that we consider in this and our earlier papers have the generalform

12Õ s t 1qx P d r qt 1qx P p d r qh.c.Ž . Ž . Ž . Ž .� 4i j 0 0 s i j 1 1 s i j i j22"

1 1 gq t 1qx P p Pd r p q t 1qx P a r qrŽ . Ž . Ž . Ž .½2 2 s i j i j i j 3 3 s q q2 i j6"

igqbr d r q W s qs Pp =d r p , 2.1Ž . Ž . Ž . Ž .5 i j 0 i j i j i j i j2

"

where P is the two-body spin-exchange operator, and the index q denotes n or p,s

according to whether the term in question relates to neutrons or protons, respectively.Ž . Ž .Also, the t , x and t , x parameters are constrained through the relations1 1 2 2

1t sy t 5q4 x 2.2aŽ . Ž .2 1 13

( )J.M. Pearson, R.C. NayakrNuclear Physics A 668 2000 163–171 165

Table1Parameters of the forces used in this paper

SkSC4 SkSC4o SkSC14 SkSC15

a 1 0 0 0b 0 1 1 1

3Ž .t MeV.fm y1789.42 y1788.76 y1792.47 y1789.8105Ž .t MeV.fm 283.467 283.037 291.334 285.60015Ž .t MeV.fm y283.467 y283.037 y291.334 y285.60023Ž1qg .Ž .t MeV.fm 12782.3 12775.0 12805.7 12783.73

x 0.790000 0.794340 0.364025 0.6212990

x y0.5 y0.5 y0.5 y0.51

x y0.5 y0.5 y0.5 y0.52

x 1.13871 1.18439 0.455431 0.89557335Ž .W MeV.fm 124.877 124.943 125.239 125.1220

g 0.333333 0.333333 0.333333 0.3333333Ž .V MeV.fm y220.0 y220.0 y220.0 y220.0p

and

4q5x1x sy 2.2bŽ .2 5q4 x1

in order for the effective nucleon mass M ) to be equal to the real nucleon mass M, acondition that has been shown to improve the mass fit and the description of fission

Ž w x.barriers see Ref. 3 , besides simplifying enormously the ETF formalism.Ž .As far as the density-dependent term t is concerned, in all previous papers we3

w xfollowed the prescription of Kohler 8 , setting a s 1, b s 0, so that the interaction¨between two protons, for example, depends only on the proton density. On the other

w xhand, most other workers follow the Orsay group 9 and set as0, bs1, no distinctionbeing made between neutrons and protons in the density dependence. The Orsayprescription is certainly much simpler to apply, but the Kohler prescription is more¨

w xcompatible with Brueckner theory 10 . Nevertheless, we show here that the twoŽprescriptions are effectively equivalent for all practical purposes. All necessary formu-

w x .las can be found in Ref. 11 .The first two columns of Table 1 show the parameters of the original force SkSC4

and of a new force SkSC4o, respectively. The only difference in form between the twoforces is that while the former follows the Kohler prescription for the density depen-¨dence, the latter has rather the Orsay form. Both were fitted to the same mass data,

Table2Ž . w xMacroscopic infinite nuclear matter parameters of the forces of Table 1, defined as in Ref. 3


Ž .a MeV y15.86 y15.86 y15.92 y15.88Õy1Ž .k fm 1.335 1.335 1.335 1.335F

Ž .J MeV 27.0 27.0 30.0 28.0Ž .K MeV 234.7 234.7 235.4 234.9Õ


Table3Ž . Ž . Ž .Errors in the data fit of the forces of Table 1. s M ,s S , and s Q denote the rms errors in the fit to then b

absolute masses, the neutron-separation energies, and the beta-decay energies, respectively, while the e

Ž w x.quantities refer to the corresponding mean errors see also Ref. 5 . All quantities are in MeV


Ž .s M 0.736 0.736 0.795 0.741Ž .e M y0.0610 0.0195 y0.00499 0.0325Ž .s S 0.524 0.525 0.517 0.520nŽ .e S 0.0120 0.0118 0.0181 0.0227nŽ .s Q 0.683 0.683 0.686 0.683b

Ž .e Q 0.0465 0.0461 0.0609 0.072b

w xnamely the 1492 measured nuclei with A036 given in the 1988 data complilation 12Ž w xwe did not use the more recent compilation 13 that has become available since SkSC4

.was defined, since we wished to be able to compare the new forces with the old .Likewise both forces were fitted to the same nuclear-matter parameters, a ,k , and JÕ FŽ .Table 2 , and have the same parameter V for the pairing force, defined byp

Õ r sV d r . 2.3Ž . Ž . Ž .pair i j p i j

It will then be seen from Table 3 that the two forces give virtually identical fits to thedata.

We next compare the extrapolations of these two forces to large neutron excesses. InTables 4, 5, and 6 we show the absolute masses M, the neutron-separation energies S ,n

Ž Ž . Ž ..and the beta-decay energies Q always defined as M A,Z yM A,Zq1 , respec-b

tively, for a number of typical nuclei close to the neutron-drip line. The agreementbetween the two forces will be seen to be remarkably close. Even for the pure neutrongas, shown in Fig. 1, a perceptible difference between the two forces emerges only forsupernuclear densities, where the Skyrme description breaks down anyway. Thus,although the two prescriptions are not formally identical, they are essentially equivalentfrom every practical standpoint, and since most existing codes have been written for the

Table4Ž .Masses in MeV of several nuclei close to the neutron-drip line for the forces of Table 1

Z A SkSC4 SkSC4o SkSC14 SkSC15

20 60 14.27 14.29 9.38 12.2233 101 26.30 26.27 20.95 24.0544 136 44.38 44.27 40.58 42.9750 153 51.64 51.53 47.36 49.5958 184 112.05 111.93 107.55 110.1366 202 105.08 104.95 102.82 104.2973 218 105.64 105.50 103.68 104.3382 266 283.35 283.14 281.25 282.8390 274 222.82 222.69 221.99 222.5198 300 300.68 300.52 300.73 300.75


Table5Ž .Neutron-separation energies S in MeV of several nuclei close to the neutron-drip line for the forces of Tablen

1


20 60 2.13 2.12 2.58 2.3233 101 2.12 2.13 2.44 2.4044 136 0.37 0.37 0.54 0.4150 153 0.05 0.05 0.35 0.7858 184 2.97 2.98 3.30 3.1966 202 1.95 1.95 2.02 1.9673 218 0.81 0.80 1.00 1.6782 266 1.80 1.81 1.90 1.7990 274 2.78 2.78 2.84 2.7798 300 2.66 2.66 2.63 2.71

much simpler Orsay prescription we shall henceforth adopt it in preference to the Kohler¨prescription.

3. Neutron-gas properties

Ž .The solid line FP in Fig. 1 shows as a function of density the energy per nucleon ofw xpure neutron matter, as calculated by Friedman and Pandharipande 14 for the realistic

force v q TNI, containing two- and three-nucleon terms. More recent realistic14w xcalculations of neutron matter 15,16 give essentially similar results up to nuclear

densities; higher densities do not concern us here.It will be seen from Fig. 1 that both forces SkSC4 and SkSC4o deviate significantly

from FP at nuclear densities and beyond; more seriously, the implied collapse of neutronmatter at these relatively low densities is incompatible with the stability of neutron stars.ŽAdmittedly, one does not expect Skyrme-type forces to be relevant at supernucleardensities, but the above forces would not even be able to match correctly at nuclear

.densities to a suitable high-density model of neutron matter.

Table6Ž .Beta-decay energies Q in MeV of several nuclei close to the neutron-drip line for the forces of Table 1b


20 60 17.31 17.33 15.96 16.7633 101 20.25 20.25 19.18 19.7344 136 18.89 18.89 18.23 18.6850 153 18.54 18.54 17.40 17.8258 184 15.16 15.15 14.52 14.8466 202 13.78 13.77 13.56 13.6873 218 16.41 16.42 15.40 15.4282 266 14.35 14.32 14.15 14.3790 274 9.58 9.57 9.46 9.5898 300 9.90 9.89 10.52 9.83


Fig. 1. Energy-density curves of neutron matter for the forces of this paper, and for the calculations of Ref.w x14 .

In determining all of the forces described in this paper the nuclear-matter symmetryŽcoefficient J is treated as a fitting parameter actually, we varied it manually in steps of

.1 MeV . Now both forces SkSC4 and SkSC4o give the same value of 27 MeV for J,w xand it has been shown 3 that the form of the neutron-matter energy curve is intimately

related to the value of this coefficient. In fact, we have found that an excellent fit to thew x Ž .neutron-matter curve of FP 14 can be had with Js30 MeV. One such force SkSC6

w xhas already been given by Onsi et al. 17 ; here we present another such force, SkSC14,obtained by minimal modification of SkSC4o, subject to the constraint Js30 MeV, andoptimized with respect to a . On the other hand, we see from Table 3 that the fit to theÕ

Žabsolute masses given by this force is significantly worse than with force SkSC4 or.SkSC4o . This deterioration in the mass fit is reflected in the masses of drip-line nuclei,

Ž .for which shifts of up to several MeV with respect to force SkSC4 or SkSC4o will beŽseen in Table 4. The fact that the force with higher J leads to lower masses at the drip

line for all but the heaviest nuclei is easily understood in terms of the compensationbetween volume-symmetry and surface-symmetry terms that takes place in fitting the

Ž . w x .data: see, for example, Eq. 29 of Ref. 3 .We therefore seek a compromise between these two values of J, and find that the

slight shift from 27 to 28 MeV is enough to stop the collapse of neutron matter. Thiscompromise force, obtained again by minimal modification of SkSC4o, subject this timeto the constraint Js28 MeV, and optimized with respect to a , is labelled SkSC15.Õ

Table 3 shows that the fit to the mass data for this force is much better than withŽ .SkSC14 Js30 MeV , and only insignificantly worse than with the original SkSC4

Ž .Js27 MeV .On the other hand, we note that although force SkSC15 avoids the collapse of

neutron matter, the fit to the FP curve of Fig. 1 is not as good as that of SkSC14.w xHowever, even at subnuclear densities, the neutron-energy curves of Refs. 15,16 do not

w xagree exactly with that of FP 14 , and there is a sufficient margin of uncertainty to makew x w xit difficult to exclude Js28 MeV on this basis. Moreover, both Ref. 15 and Ref. 16

calculate the symmetry energy of nuclear matter as a function of density for various


realistic forces, and we find that their results are compatible with all values of J lying inthe range 27 to 30 MeV.

4. Discussion and conclusions

Our main conclusion is that a precision fit of nuclear masses with Skyrme forces,subject to the constraint that neutron matter does not collapse at nuclear or subnucleardensities, is possible if, but only if, the nuclear-matter symmetry coefficient J lies closeto 28 MeV. An identical conclusion emerges from extensive Skyrme-HF calculationsthat are presently being performed in collaboration with Onsi and Tondeur. On the otherhand, the most refined mass formula to be based on the droplet model, the FRDMŽ . w xfinite-range droplet model leads to Js32.73 MeV 18 . This much higher value of Jis quite incompatible with a Skyrme-force fit of the masses, although it would certainlyensure the stability of neutron matter.

In an attempt to resolve this contradiction we have generalized the Skyrme force ofŽ . Ž .Eq. 2.1 by adding a second density-dependent t term. While the extra degrees of3

freedom might have been expected to enable us to fit the same mass data as before withan arbitrary value of J, what happened in reality was a massive cancellation between thetwo t terms, resulting in severe instability. This artifice thus fails, and we are led back3

to the conclusion that the value of Js28 MeV is quite robust within the framework ofSkyrme forces. However, we are still left with the problem of accounting for thediscrepancy with the much higher FRDM value, and now investigate two possibleexplanations.

Wigner term

One significant difference between the FRDM and ETFSI calculations is that thelatter contains no Wigner term, i.e. no term representing the enhanced binding of nucleiwith N,Z. The importance of such a term is clearly apparent in the ETFSI masses,

Ž w x.which have anomalously large errors for such nuclei see the discussion in Ref. 4 . AllŽ w x.of the many explanations of these anomalies see, for example, Refs. 19–24 involve

going beyond the HF-BCS framework, and for this reason we have made no attempt inthe ETFSI calculations to account for the anomalies, the problematical nuclei beingrelatively few in number and readily identifiable. Nevertheless, the fact that we includedsuch nuclei in the fits described above could conceivably have spuriously affected thevalue of J that we extracted, and so we re-ran the fits excluding all nuclei with N,Z.Our conclusion remains unchanged: Js28 MeV still gives better fits than Js30 MeV.

Malacodermous terms

Turning now to the FRDM as the possible origin of the discrepancy between the twovalues of J, we notice that this model lacks ‘‘malacodermous’’ terms, i.e. higher-ordersurface-symmetry terms that lead to a softening of the nuclear surface for large neutron

w xexcesses 25 . The FRDM fit to masses of nuclei far from beta-stability might thencompensate for this deficiency by finding an abnormally low value for the surface-stiff-


ness coefficient Q. An abnormally high value for J would then result, since the twocoefficients are inversely correlated in fitting to masses of nuclei closer to the line of

Ž w x Ž . Ž . w x.beta-stability see Refs. 26,27 and Eqs. 29 and 30 of Ref. 3 .Since we have found no reason why Skyrme forces should lead to an abnormally low

value of J, we believe that a more detailed study should be made of the extent to whichthe addition of malacodermous terms to the FRDM would lower the value of Jdetermined by the mass fits. At the same time, since Skyrme-type forces are not the lastword in effective forces, there is an obvious need for parallel studies with finite-rangeŽ .Gogny-type forces and also in RMF theory; such mass fits would have to be asextensive as those already performed in the ETFSI framework.

Implications for mass formula

In view of the foregoing we have no option but to regard 28 MeV as the definitivevalue of J for the ETFSI approach to the mass formula. This value leads to a force,SkSC15, that avoids the unphysical collapse of neutron matter that occurred with ouroriginal force SkSC4, and is therefore much better adapted to the calculation of the EOSof stellar nuclear matter. At the same time, the quality of its fit to the mass data is onlyinsignificantly worse than that of SkSC4. Indeed, if we consider the undoubted stabilityof neutron stars as a datum that has to be respected then the new force must be regardedas being in better overall agreement with phenomenology than the original force.

Now although the fits to the mass data given by the two forces are almost identical,we see from Tables 4–6 that their extrapolations to highly neutron-rich nuclei differsomewhat, not only with respect to the absolute masses but also with respect to the Sn

and the Q , the mass-related quantities of primary importance for the r-process. A newb

mass table in which the constraint Js28 MeV is imposed at the outset is thus required,w xand is currently being constructed by Goriely 28 ; this new table will be based not on

w xforce SkSC15, but on a new force fitted to the 1995 mass-data compilation 13 .

Acknowledgements

We acknowledge valuable communications with J.M. Lattimer.

References

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nuclear-matter symmetry coefficient and nuclear masses

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