Nuclear-matter symmetry coefficient and nuclear masses

Nuclear Physics A 668 Ž2000. 163–171 www.elsevier.nlrlocaternpe

Nuclear-matter symmetry coefficient and nuclear masses q J.M. Pearson a , R.C. Nayak a

a,b

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 Within the framework of the ETFSI Žextended Thomas–Fermi plus Strutinsky integral. mass formula, a precision fit of nuclear masses with Skyrme forces, subject to the constraint that neutron matter does not collapse at nuclear or subnuclear densities, is possible if, but only if, the nuclear-matter symmetry coefficient J lies close to 28 MeV. q 2000 Elsevier Science B.V. All rights reserved. PACS: 21.10.Dr; 21.60.Jz; 95.30.Cq

Keywords: Nuclear masses; Nuclear matter; Symmetry coefficient

1. Introduction We are involved in a program to develop a microscopic theory of nuclear systems applicable to the wide variety of phenomena encountered at subnuclear and nuclear densities during and after stellar collapse, and in particular to describe all these phenomena, as far as possible, in terms of a single, universal, effective interaction. The main achievement so far has been the development, for the first time, of a mass formula based entirely on microscopic forces, the ETFSI-1 mass formula w1–5x. The astrophysical interest of such a mass formula lies in the fact that the r-process of nucleosynthesis depends crucially on the binding energies Žamong other properties. of nuclei that are so neutron-rich that there is no hope of being able to measure them in the laboratory. It is thus of the greatest importance to be able to make reliable extrapolations of masses away from the known region, relatively close to the stability line, out towards the neutron-drip line.

q

Supported in part by the NSERC ŽCanada..

0375-9474r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 4 3 1 - 5

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Full details of this mass formula are to be found in the earlier papers w1–5x, and we limit ourselves here to a reminder of some essential points. The ETFSI method is essentially a high-speed approximation to the Hartree–Fock ŽHF. method, with a macroscopic part given by the extended Thomas–Fermi ŽETF. method w6,7x, and shell corrections calculated by the so-called Strutinsky-integral ŽSI. method w1,5x. Pairing is handled in the BCS approximation with a d-function force. Although this is really a microscopic-macroscopic mass formula, there is a much greater coherence between the two parts than is the case with mass formulas based on the dropŽlet. model, since the same Skyrme force underlies both parts. In fact, it has been shown w1,2x that the ETFSI method is equivalent to the HF method in the sense that when the two methods fit the same form of Skyrme force to the mass data they give essentially the same extrapolation out to the neutron-drip line. This equivalence to the HF method presumably accounts for the fact that with just 8 parameters the underlying force of the ETFSI-1 mass formula, SkSC4, fits 1492 mass data for A 0 36 with an rms error of only 0.736 MeV w4x. 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 around nuclear densities. In view of the known stability of neutron stars this collapse is clearly unphysical, and we must reject the use of the force SkSC4 to calculate the equation of state ŽEOS. in highly neutron-rich environments, even if it gives an optimal fit to finite-nucleus masses. In this paper we show how the force can be modified to avoid the neutron-matter collapse without compromising the mass fit. We first describe ŽSection 2. a change to the form of the density dependence of our force that we make entirely in the interests of simplification, the overall quality of the fits remaining unchanged. In Section 3 we then deal with the neutron-matter condition and make the appropriate modification of 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 general form Õi j s t 0 Ž 1 q x 0 Ps . d Ž r i j . qt 1 Ž 1 q x 1 Ps . q t 2 Ž 1 q x 2 Ps . qbr g d Ž r i j . q

5

1 2 "2

 pi2j d Ž ri j . q h.c. 4

1

1 g p i j P d Ž r i j . p i jq t 3 Ž 1 q x 3 Ps . a Ž r q i q r q j . 6 "

½

2

i "2

W0 Ž s i q s j . P p i j = d Ž r i j . p i j ,

Ž 2.1 .

where Ps is the two-body spin-exchange operator, and the index q denotes n or p, according to whether the term in question relates to neutrons or protons, respectively. Also, the Ž t 1 , x 1 . and Ž t 2 , x 2 . parameters are constrained through the relations t 2 s y 13 t 1 Ž 5q4 x 1 .

Ž 2.2a .

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Table 1 Parameters of the forces used in this paper

a b t 0 ŽMeV.fm3 . t1 ŽMeV.fm5 . t 2 ŽMeV.fm5 . t 3 ŽMeV.fm3Ž1q g . . x0 x1 x2 x3 W0 ŽMeV.fm5 . g Vp ŽMeV.fm3 .

SkSC4

SkSC4o

SkSC14

SkSC15

1 0 y1789.42 283.467 y283.467 12782.3 0.790000 y0.5 y0.5 1.13871 124.877 0.333333 y220.0

0 1 y1788.76 283.037 y283.037 12775.0 0.794340 y0.5 y0.5 1.18439 124.943 0.333333 y220.0

0 1 y1792.47 291.334 y291.334 12805.7 0.364025 y0.5 y0.5 0.455431 125.239 0.333333 y220.0

0 1 y1789.81 285.600 y285.600 12783.7 0.621299 y0.5 y0.5 0.895573 125.122 0.333333 y220.0

and x2 s y

4 q 5 x1

Ž 2.2b .

5 q 4 x1

in order for the effective nucleon mass M ) to be equal to the real nucleon mass M, a condition that has been shown to improve the mass fit and the description of fission barriers Žsee Ref. w3x., besides simplifying enormously the ETF formalism. As far as the density-dependent term Ž t 3 . is concerned, in all previous papers we w8x, setting a s 1, b s 0, so that the interaction followed the prescription of Kohler ¨ between two protons, for example, depends only on the proton density. On the other hand, most other workers follow the Orsay group w9x and set a s 0, b s 1, no distinction being made between neutrons and protons in the density dependence. The Orsay prescription is certainly much simpler to apply, but the Kohler prescription is more ¨ compatible with Brueckner theory w10x. Nevertheless, we show here that the two prescriptions are effectively equivalent for all practical purposes. ŽAll necessary formulas can be found in Ref. w11x.. 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 two forces 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,

Table 2 Macroscopic Žinfinite nuclear matter. parameters of the forces of Table 1, defined as in Ref. w3x

a Õ ŽMeV. k F Žfmy1 . J ŽMeV. K Õ ŽMeV.

SkSC4

SkSC4o

SkSC14

SkSC15

y15.86 1.335 27.0 234.7

y15.86 1.335 27.0 234.7

y15.92 1.335 30.0 235.4

y15.88 1.335 28.0 234.9

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Table 3 Errors in the data fit of the forces of Table 1. s Ž M ., s Ž Sn ., and s Ž Qb . denote the rms errors in the fit to the absolute masses, the neutron-separation energies, and the beta-decay energies, respectively, while the e quantities refer to the corresponding mean errors Žsee also Ref. w5x.. All quantities are in MeV

s ŽM. eŽM. s Ž Sn . e Ž Sn . s Ž Qb . e Ž Qb .

SkSC4

SkSC4o

SkSC14

SkSC15

0.736 y0.0610 0.524 0.0120 0.683 0.0465

0.736 0.0195 0.525 0.0118 0.683 0.0461

0.795 y0.00499 0.517 0.0181 0.686 0.0609

0.741 0.0325 0.520 0.0227 0.683 0.072

namely the 1492 measured nuclei with A 0 36 given in the 1988 data complilation w12x Žwe did not use the more recent compilation w13x 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 Õ ,k F , and J ŽTable 2., and have the same parameter Vp for the pairing force, defined by Õpair Ž r i j . s Vp d Ž r i j . .

Ž 2.3 .

It will then be seen from Table 3 that the two forces give virtually identical fits to the data. We next compare the extrapolations of these two forces to large neutron excesses. In Tables 4, 5, and 6 we show the absolute masses M, the neutron-separation energies Sn , and the beta-decay energies Qb Žalways defined as M Ž A,Z . y M Ž A,Z q 1.., respectively, for a number of typical nuclei close to the neutron-drip line. The agreement between the two forces will be seen to be remarkably close. Even for the pure neutron gas, shown in Fig. 1, a perceptible difference between the two forces emerges only for supernuclear densities, where the Skyrme description breaks down anyway. Thus, although the two prescriptions are not formally identical, they are essentially equivalent from every practical standpoint, and since most existing codes have been written for the

Table 4 Masses Žin MeV. of several nuclei close to the neutron-drip line for the forces of Table 1 Z

A

SkSC4

SkSC4o

SkSC14

SkSC15

20 33 44 50 58 66 73 82 90 98

60 101 136 153 184 202 218 266 274 300

14.27 26.30 44.38 51.64 112.05 105.08 105.64 283.35 222.82 300.68

14.29 26.27 44.27 51.53 111.93 104.95 105.50 283.14 222.69 300.52

9.38 20.95 40.58 47.36 107.55 102.82 103.68 281.25 221.99 300.73

12.22 24.05 42.97 49.59 110.13 104.29 104.33 282.83 222.51 300.75

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Table 5 Neutron-separation energies Sn Žin MeV. of several nuclei close to the neutron-drip line for the forces of Table 1 Z

A

20 33 44 50 58 66 73 82 90 98

60 101 136 153 184 202 218 266 274 300

SkSC4

SkSC4o

SkSC14

SkSC15

2.13 2.12 0.37 0.05 2.97 1.95 0.81 1.80 2.78 2.66

2.12 2.13 0.37 0.05 2.98 1.95 0.80 1.81 2.78 2.66

2.58 2.44 0.54 0.35 3.30 2.02 1.00 1.90 2.84 2.63

2.32 2.40 0.41 0.78 3.19 1.96 1.67 1.79 2.77 2.71

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

3. Neutron-gas properties The solid line ŽFP. in Fig. 1 shows as a function of density the energy per nucleon of pure neutron matter, as calculated by Friedman and Pandharipande w14x for the realistic force v14 q TNI, containing two- and three-nucleon terms. More recent realistic calculations of neutron matter w15,16x 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 neutron matter 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 supernuclear densities, but the above forces would not even be able to match correctly at nuclear densities to a suitable high-density model of neutron matter..

Table 6 Beta-decay energies Qb Žin MeV. of several nuclei close to the neutron-drip line for the forces of Table 1 Z

A

20 33 44 50 58 66 73 82 90 98

60 101 136 153 184 202 218 266 274 300

SkSC4

SkSC4o

SkSC14

SkSC15

17.31 20.25 18.89 18.54 15.16 13.78 16.41 14.35 9.58 9.90

17.33 20.25 18.89 18.54 15.15 13.77 16.42 14.32 9.57 9.89

15.96 19.18 18.23 17.40 14.52 13.56 15.40 14.15 9.46 10.52

16.76 19.73 18.68 17.82 14.84 13.68 15.42 14.37 9.58 9.83

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Fig. 1. Energy-density curves of neutron matter for the forces of this paper, and for the calculations of Ref. w14x.

In determining all of the forces described in this paper the nuclear-matter symmetry coefficient J is treated as a fitting parameter Žactually, we varied it manually in steps of 1 MeV.. Now both forces SkSC4 and SkSC4o give the same value of 27 MeV for J, and it has been shown w3x 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 the neutron-matter curve of FP w14x can be had with J s 30 MeV. One such force ŽSkSC6. has already been given by Onsi et al. w17x; here we present another such force, SkSC14, obtained by minimal modification of SkSC4o, subject to the constraint J s 30 MeV, and optimized with respect to a Õ . On the other hand, we see from Table 3 that the fit to the absolute masses given by this force is significantly worse than with force SkSC4 Žor SkSC4o.. This deterioration in the mass fit is reflected in the masses of drip-line nuclei, for which shifts of up to several MeV with respect to force SkSC4 Žor SkSC4o. will be seen in Table 4. ŽThe 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 compensation between volume-symmetry and surface-symmetry terms that takes place in fitting the data: see, for example, Eq. Ž29. of Ref. w3x.. 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. This compromise force, obtained again by minimal modification of SkSC4o, subject this time to the constraint J s 28 MeV, and optimized with respect to a Õ , is labelled SkSC15. Table 3 shows that the fit to the mass data for this force is much better than with SkSC14 Ž J s 30 MeV., and only insignificantly worse than with the original SkSC4 Ž J s 27 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. However, even at subnuclear densities, the neutron-energy curves of Refs. w15,16x do not agree exactly with that of FP w14x, and there is a sufficient margin of uncertainty to make it difficult to exclude J s 28 MeV on this basis. Moreover, both Ref. w15x and Ref. w16x calculate the symmetry energy of nuclear matter as a function of density for various

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realistic forces, and we find that their results are compatible with all values of J lying in the 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 subnuclear densities, is possible if, but only if, the nuclear-matter symmetry coefficient J lies close to 28 MeV. An identical conclusion emerges from extensive Skyrme-HF calculations that are presently being performed in collaboration with Onsi and Tondeur. On the other hand, the most refined mass formula to be based on the droplet model, the FRDM Žfinite-range droplet model. leads to J s 32.73 MeV w18x. This much higher value of J is quite incompatible with a Skyrme-force fit of the masses, although it would certainly ensure the stability of neutron matter. In an attempt to resolve this contradiction we have generalized the Skyrme force of Eq. Ž2.1. by adding a second density-dependent Ž t 3 . term. While the extra degrees of freedom might have been expected to enable us to fit the same mass data as before with an arbitrary value of J, what happened in reality was a massive cancellation between the two t 3 terms, resulting in severe instability. This artifice thus fails, and we are led back to the conclusion that the value of J s 28 MeV is quite robust within the framework of Skyrme forces. However, we are still left with the problem of accounting for the discrepancy with the much higher FRDM value, and now investigate two possible explanations. Wigner term One significant difference between the FRDM and ETFSI calculations is that the latter contains no Wigner term, i.e. no term representing the enhanced binding of nuclei with N , Z. The importance of such a term is clearly apparent in the ETFSI masses, which have anomalously large errors for such nuclei Žsee the discussion in Ref. w4x.. All of the many explanations of these anomalies Žsee, for example, Refs. w19–24x. involve going beyond the HF-BCS framework, and for this reason we have made no attempt in the ETFSI calculations to account for the anomalies, the problematical nuclei being relatively few in number and readily identifiable. Nevertheless, the fact that we included such nuclei in the fits described above could conceivably have spuriously affected the value of J that we extracted, and so we re-ran the fits excluding all nuclei with N , Z. Our conclusion remains unchanged: J s 28 MeV still gives better fits than J s 30 MeV. Malacodermous terms Turning now to the FRDM as the possible origin of the discrepancy between the two values of J, we notice that this model lacks ‘‘malacodermous’’ terms, i.e. higher-order surface-symmetry terms that lead to a softening of the nuclear surface for large neutron excesses w25x. The FRDM fit to masses of nuclei far from beta-stability might then compensate for this deficiency by finding an abnormally low value for the surface-stiff-

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ness coefficient Q. An abnormally high value for J would then result, since the two coefficients are inversely correlated in fitting to masses of nuclei closer to the line of beta-stability Žsee Refs. w26,27x and Eqs. Ž29. and Ž30. of Ref. w3x.. 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 which the addition of malacodermous terms to the FRDM would lower the value of J determined by the mass fits. At the same time, since Skyrme-type forces are not the last word in effective forces, there is an obvious need for parallel studies with finite-range ŽGogny-type. forces and also in RMF theory; such mass fits would have to be as extensive 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 definitive value 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 our original force SkSC4, and is therefore much better adapted to the calculation of the EOS of stellar nuclear matter. At the same time, the quality of its fit to the mass data is only insignificantly worse than that of SkSC4. Indeed, if we consider the undoubted stability of neutron stars as a datum that has to be respected then the new force must be regarded as 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 differ somewhat, not only with respect to the absolute masses but also with respect to the Sn and the Qb , the mass-related quantities of primary importance for the r-process. A new mass table in which the constraint J s 28 MeV is imposed at the outset is thus required, and is currently being constructed by Goriely w28x; this new table will be based not on force SkSC15, but on a new force fitted to the 1995 mass-data compilation w13x.

Acknowledgements We acknowledge valuable communications with J.M. Lattimer.

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