- Email: [email protected]

Nuclear mass formulas for astrophysics J.M. Pearson a,∗ , S. Goriely b a Dépt. de Physique, Université de Montréal, Montréal (Québec), H3C 3J7 Canada b Institut d’Astronomie et d’Astrophysique, CP-226, Université Libre de Bruxelles, 1050 Brussels, Belgium

Received 1 April 2004; received in revised form 28 May 2004; accepted 3 June 2004 Available online 15 June 2004

Abstract We review the Hartree–Fock–Bogolyubov mass models of the Brussels–Montreal group and compare their suitability for astrophysical purposes with three other modern mass formulas: the FRDM of Möller et al., the KUTY model of Koura et al., and the Duflo–Zuker 1995 model. In addition to considering the quality of their respective fits to the data of the atomic mass evaluation of December 2003, we also compare their extrapolations out towards the neutron drip line. The implications for fission barriers and the role of the equation of state of neutron matter are both discussed. © 2004 Elsevier B.V. All rights reserved. PACS: 21.10.Dr; 21.60.Jz; 21.65.+f; 25.85.-w; 95.30.Cq Keywords: Binding energies and masses; Skyrme–Hartree–Fock–Bogoliubov; Neutron matter; Fission barriers

1. Introduction One of several factors limiting our present understanding of the r-process of nucleosynthesis (see, for example, Ref. [1] and references therein) is the fact that its evolution depends on the masses of nuclei that may contain 30 or so neutrons more than the heaviest measured isotope of the same element. A knowledge of the masses of similarly neutronrich nuclei is also required for the determination of the chemical and isotopic composition of the outer crust of neutron stars (see the review [2]). Since there is no prospect of the * Corresponding author.

E-mail address: [email protected] (J.M. Pearson). 0375-9474/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.06.005

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masses of such neutron-rich nuclei being measured in the foreseeable future (see the recent review [3]), progress in both these areas of astrophysics will depend on the development of theoretical methods for making reliable estimates of nuclear masses. Generally speaking, the more fundamental the theory used the greater should be one’s confidence in the nuclear masses thereby calculated, other things being equal. From this perspective it is thus perhaps natural that one should first address the possibility of deriving masses, along with all other nuclear properties, from the “real” basic interactions between nucleons, as determined by the properties of two- and three-nucleon systems. This corresponds, in a nonrelativistic framework, to solving the Schrödinger equation, H Ψ = EΨ,

(1)

where H =−

h¯ 2 2 ∇i + Vij + Vij k , 2M i

i>j

(2)

i>j >k

in which Vij and Vij k are potentials representing the real two-nucleon and three-nucleon interactions, respectively. Such a “fundamentalist” approach is, of course, possible in principle, and in a sense can be regarded as the ultimate objective of nuclear theory. Indeed, an enormous effort has already been expended on this “nuclear many-body problem”, both for finite nuclei and nuclear matter (see below). For finite nuclei the calculations are exceedingly complicated, and it is only in the last few years that it has become possible to calculate ab initio nuclear masses with an accuracy approaching that required in astrophysics, and even then only in the case of the lightest nuclei, A 12 (see Refs. [4,5] for a recent review of the situation). However, such nuclei are involved in big-bang nucleosynthesis and hydrogen burning, and even though the masses of these nuclei can be measured anyway, the calculations still have a vital contribution to make to astrophysics, since they can be used to determine also the rates of electroweak capture rates of these nuclei, quantities that are difficult or impossible to make in the laboratory [6] (see also Ref. [7]). Actually, it is not quite true to say that these ab initio calculations are limited by their complexity to the lightest nuclei, since a considerable simplification arises on jumping to an essentially infinite number of nucleons, on account of the translational invariance that then holds. Homogeneous nuclear systems of this sort, referred to as “infinite nuclear matter” (INM), have long been recognized as a limiting case of ordinary nuclei with an infinite number of nucleons, and with the Coulomb force switched off; without this latter fiction the energy per nucleon would diverge. Although such a system is purely hypothetical, it is of enormous theoretical interest in that being the simplest many-body nuclear system it serves as a test bench for the various ab initio methods that we alluded to above: the first task of nuclear many-body theory is to derive from the basic forces between nucleons the properties of Coulomb-free INM, as inferred from the properties of real nuclei (see below). But there is a second form of INM that has a very real existence, well approximating the core of neutron stars. Ab initio calculations of such “neutron-star matter” play a crucial role in our understanding of neutron stars: see, for example the review [8]. However, while these contributions of fundamental nuclear theory to astrophysics are of great importance, our primary concern in this review is with neither very light nuclei nor neutron stars, but rather with heavy neutron-rich nuclei. In this context it is fairly safe to

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say that there is no chance of the “fundamentalist” approach being of any direct use in the foreseeable future, although we shall see that it can still provide powerful constraints on alternative approaches. Of necessity, all viable approaches to the calculation of the masses of experimentally inaccessible nuclei must have an essentially “semiempirical” character, in the sense that the theoretical description of the nucleus that is adopted always consists of a model with a number of free parameters that are fitted to the measured masses (and maybe to other nuclear data). Thus theory is used simply as a means of extrapolating from the mass data to the unknown nuclides (and to INM). The very first mass model was the 1935 “semiempirical mass formula” of Weizsäcker [9], which in a somewhat updated version (see Section IIIA of Ref. [3]) gives the internal energy (the negative of the binding energy) of a nucleus with Z protons and N neutrons as E = avol A + asf A2/3 +

3e2 2 −1/3 Z A + asym A + ass A2/3 I 2 + δ(N, Z), 5r0

(3)

where A = N + Z, the mass number, and I = (N − Z)/A. The five leading terms correspond to the liquid-drop model, with a sharp radius of R = r0 A1/3 being assumed. The last term represents pairing effects, which were already known in 1935, even though shell effects were not established until the end of the forties. Eq. (3) permits a trivially easy extrapolation from the mass data to the limit of uncharged INM, the energy per nucleon of which becomes e∞ (ρ0 , I ) = avol + asym I 2 .

(4)

Actually, this refers to INM with a neutron/proton ratio of I = (ρn − ρp )/ρ, where ρn and ρp refer to the neutron and proton densities, respectively, and ρ = ρn + ρp . Since, furthermore, the nuclei whose masses are being extrapolated are in their ground states, we must presume the corresponding INM to be at its equilibrium density ρ0 = (3/4π)r0−3 . Thus, while they are characteristic of the liquid-drop model, the three parameters avol , asym , and ρ0 (or r0 ) of the Weizsäcker mass formula (3) can also be interpreted in terms of INM. All the mass models that we consider here can be similarly extrapolated to INM parametrized in terms of these same three parameters; in this paper it is the volume-symmetry coefficient asym (labelled J in the droplet-model literature [10]) that will be of primary interest. The two remaining parameters, i.e., the surface coefficient asf and the surface-symmetry coefficient ass , can be interpreted in terms of “semiinfinite nuclear matter”: see, for example, Refs. [11–13]. It should be realized that all these parameters depend to some extent on the model (see Table II of Ref. [3]): the extrapolation from finite to infinite systems cannot be made in a totally model-independent way. This mass formula works remarkably well, considering its antiquity, and the small number of its adjustable parameters. In particular, it provides qualitative insight into a wide range of mass-related nuclear properties of astrophysical interest: approximate position of the drip lines, demarcation of zones of alpha instability and of beta-delayed nucleon emission, and neutron stars: see Ref. [14] and Appendix C of Ref. [3]. But its failure to include shell effects (see, for example, Fig. 8 of Ref. [3]) and the resulting large rms error of 2.97 MeV with which it fits recent mass data, as opposed to less than 0.7 MeV for modern mass models (see Table 1), means that it is unacceptable for astrophysical applications.

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Table 1 The rms error (σ ) and mean error (¯ , experiment–theory) of fits given by various mass formulas to the data of the 2003 AME [66]. Nfit denotes the number of nuclei to which the corresponding model was originally fitted. For the definitions of the various sub-sets of the 2003 data see Section 3. All errors in MeV

HFB-2 [44] HFB-6 [46] HFB-7 [46] FRDM [17] KUTY [18] DZ [19]

Nfit

2149 nuclei σ ¯

2135 2135 2135 1654 1835 1751

0.659 0.666 0.657 0.656 0.671 0.360

−0.005 0.014 0.026 0.058 0.058 0.009

70 “new” nuclei σ ¯

18 “new” n-rich nuclei σ ¯

0.835 0.816 0.824 0.522 0.709 0.449

0.834 0.698 0.717 0.562 0.873 0.381

0.211 0.244 0.276 −0.015 0.235 0.030

−0.075 0.215 0.229 0.192 0.308 0.061

Reconciling the drop-model (DM) and shell-model (SM) aspects of nuclear structure was one of the major preoccupations of theoreticians during the early fifties, but the two aspects can easily be subsumed within the Hartree–Fock (HF) framework, which is completely microscopic, in the sense that all properties of the nucleus can be derived from internucleonic forces. A major thrust of the present review concerns the success that the nonrelativistic HF method has recently enjoyed as a mass model, but it should not be thought that the ultimate object of an ab initio derivation of nuclear masses has thereby been achieved. On the contrary, the forces of the HF approach are not the “real” basic internucleonic forces but rather “effective” forces with free parameters that are fitted to the mass data. In any case, the HF method is computationally very demanding, and it was only with the new millennium that it became possible to fit the force parameters to essentially all the mass data [15,16]. Thus for many years the scene was dominated by the much simpler hybrid “macroscopic–microscopic” (abbreviated as “mac–mic”, or more euphonically, as “mic– mac”) approach. This approach simply decreed that the two aspects of nuclear structure, DM and SM, must cohabit, with shell-model corrections grafted on to a refined version of Weizsäcker’s liquid drop. Even if some ambiguity arises from the decision to ignore the common origin that the two aspects must have in the HF framework, this approach has been highly successful, especially in its latest manifestation, the “finite-range droplet model” (FRDM) [17], which we shall consider here. Two more “mass formulas” are of astrophysical interest, in the context of long-range extrapolation out towards the neutron drip line. (Our use of the term “mass formula” follows the usual designation of any semiempirical mass model that has been fitted to essentially all the mass data and for which a complete mass table, running from one drip line to the other, has been constructed. Actually, hardly any of the models since the original one of Weizsäcker [9] can be described by simple formulas, and it would be more precise to speak of “semiempirical mass algorithms”, the results being expressed in tabular form.) The first of these, the KUTY method [18], is a variant of the mic–mac approach. The other, DZ [19], lies outside the main stream, as described above: being based on a parametrization of multipole moments of the nuclear Hamiltonian it is more fundamental than the mic–mac approaches, and yet is not strictly microscopic, since no nucleonic interaction appears explicitly. All four of these approaches will be considered in this paper. They are all global models, in the sense that their parameters are fitted to essentially all measured masses, and they are

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intended to be applicable to all bound nuclei in their ground states. We shall not consider local mass formulas, which are simple algebraic formulas expressing the mass of any given unknown nuclide directly in terms of the masses of known neighbouring nuclides. As discussed in Section IIIE of Ref. [3], they cannot be used for long-range extrapolation out towards the neutron drip line, and whereas they could in principle serve to fill in isolated holes in the proton-rich data required for the rp-process, in practice the systematics of the atomic mass evaluations (AME) [20] work better [3]. Nor do we consider the relativistic Hartree method, also known as the relativistic mean-field (RMF) method. While being a most promising approach, the best available meson-parameter set for this method, labelled NL3, was fitted to only ten nuclei, and when used to calculate the masses of the measured even-even nuclides the rms deviation from the experimental values was found to be 2.6 MeV, which is not much better than the Weizsäcker formula (3), and thus unacceptable for astrophysical purposes [21]. Exactly the same situation occurs in the nonrelativistic HF framework with forces that have been fitted to only a few nuclei (see Section 2), and it is reasonable to expect that when fitted to a sufficiently large data base the RMF method will perform at least as well as the nonrelativistic HF mass formulas described in Section 2. Nevertheless there are reasons for supposing that the two approaches will not differ significantly in their extrapolations out to the neutron drip line [22]. The four mass models that we consider for astrophysics are briefly described in Section 2 (several different versions of the HF model will be discussed). In Section 3 we consider the quality of the data fits given by these models, and in Section 4 we compare their respective extrapolations out towards the neutron drip line. We discuss in Section 5 the possibility of extending these mass models to the calculations of fission barriers, since the full elucidation of the r-process requires a knowledge of these for a large number of highly neutron-rich nuclei that are experimentally inaccessible. The emphasis throughout is on the HF models, not because they fit the data better than any other mass formula—so far they do not—but because we believe that insofar as the primary aim is to make reliable predictions of nuclei far from stability these mass models point the way to future progress: with all nuclear properties being grounded in the internucleonic forces the long-term trend in mass models must be towards those that are fully microscopic, with the effective forces related as closely as possible to the real forces. Furthermore, the HF approach is the only one that permits all the other nuclear properties of astrophysical interest such as fission barriers, the equation of state (EOS) of neutron-star matter, level densities, the giant dipole resonance (GDR), and beta-decay strength functions to be calculated in a way that is completely consistent with the mass calculation. Accordingly, in Section 6 we consider in more detail certain aspects of the Skyrme forces that have been used in all the HF mass formulas that are currently available. Our conclusions are summarized in Section 7.

2. The mass models 2.1. Hartree–Fock method This is a variational method, the choice of whose form has been guided by the success of the shell model. That is, the trial wavefunction has the form of a Slater determinant

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Φ = det{φi (xi )}, this being a properly antisymmetrized product of single-particle (s.p.) wavefunctions φi (xi ). Since a model wavefunction of this form can never be identical to the exact wave function Ψ of Eq. (1), whatever the choice of φi (xi ), it follows that the expectation value Φ|H |Φ will always be higher than the exact eigenenergy E of Eq. (1). We shall thus have to replace the exact Hamiltonian H by an effective Hamiltonian if the HF method is to give the exact energy E, and so write in place of Eq. (2) H eff = −

h¯ 2 2 eff ∇i + vij , 2M i

(5)

i>j

in which vijeff is some effective nuclear force that does not have to fit the nucleon-nucleon scattering data, and which will in fact be softer than one that does. One way in which this force can be determined, particularly appropriate to the present context of nuclear masses, is to optimize the fit of the expectation values EHF = Φ|H eff |Φ to all the measured values of E. For a given force the development of the method proceeds by minimizing EHF with respect to arbitrary variations in the unknown s.p. functions φi (xi ), which are then given as eigensolutions to the so-called HF equation, a s.p. Schrödinger equation of the form h¯ 2 2 − ∇ + U φi = i φi , (6) 2M where U is a s.p. field that in general is nonlocal, deformed, and spin-dependent, but ultimately is determined uniquely by the effective force. However, this field depends explicitly on the s.p. functions φi (xi ) themselves, so that Eq. (6) must be solved reiteratively. Once self-consistent solutions φi (xi ) have been determined EHF can be calculated. It should be realized that we are tacitly supposing the shape of the field to be time independent, although vibrational states represent a departure from this condition. Moreover, even if the shape of the field is time independent, if it is not spherically symmetrical it will of necessity rotate, simply by virtue of angular-momentum conservation, giving rise thereby to rotational spectra [23]. The appearance of SM features in such an approach is, of course, guaranteed from the outset. Moreover, provided the effective force, unlike the Coulomb force, is not of infinite range, and has a short-range repulsion, INM will be saturated, i.e., have a finite density and energy per nucleon. Thus both DM and SM aspects emerge automatically and on an equal footing in this picture, so that a much more unified approach to the mass formula is offered than is possible with the hybrid mic–mac methods. A particularly suitable form of effective force is the 10-parameter Skyrme form [24–29], 1 2 pij δ(rij ) + h.c. 2 2h¯ 1 1 + t2 (1 + x2 Pσ ) 2 pij δ(rij )pij + t3 (1 + x3 Pσ )ρ γ δ(rij ) 6 h¯ i + 2 W0 (σ i + σ j )pij × δ(rij )pij , h¯

vij = t0 (1 + x0 Pσ )δ(rij ) + t1 (1 + x1 Pσ )

(7)

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where pij is the momentum conjugate to rij , and Pσ = 12 (1 + σ 1 · σ 2 ) is the two-body spin-exchange operator. All terms here are formally of zero range, although the momentum dependence of the t1 and t2 terms simulates a finite range; the last term is a two-body spin– orbit term. The t3 term is density dependent. With these forces the s.p. equation (6) takes the form

h¯ 2 coul −∇ · ∇ + U (r) + V (r) − iW (r) · ∇ × σ φi,q = i,q φi,q , (8) q q q 2Mq∗ (r) in which i labels all quantum numbers, and q denotes n (neutrons) or p (protons). All the field terms are now local, essentially because one has been able to introduce positiondependent effective masses Mq∗ , one for each of the two types of nucleon. Actually, at any point in the nucleus these two effective masses are determined entirely by the local densities, according to 2ρq h¯ 2 2ρq h¯ 2 h¯ 2 = + 1 − , (9) 2Mq∗ ρ 2Ms∗ ρ 2Mv∗ in which Ms∗ and Mv∗ are the so-called isoscalar and isovector effective masses, respectively, quantities that are determined entirely by the Skyrme-force parameters. The precise expressions for these two quantities, and for all those appearing in Eq. (8), as well as for EHF , can be found in Ref. [30]. With finite-range forces, such as used by the Gogny group [31], the essential nonlocality of the s.p. fields complicates the calculations considerably. It is for this reason that all HF mass formulas constructed so far are based on Skyrme forces: any HF approach to the mass formula makes enormous demands on computer facilities, since every nucleus, which must be assumed a priori to be deformed, has to be calculated many times. Before discussing in more detail these various HF mass formulas we first note a general limitation of the HF method as such. Even when the Slater determinant Φ satisfies the HF equations (6), it can never be identical to the exact nuclear wave function Ψ . Thus we must expect nuclear properties to show features that cannot be accounted for within the HF framework, although it will always be possible to express them in terms of a configuration mixing of Slater determinants; such irreducible deviations from the mean-field picture are referred to as correlations. One such correlation arises when the field U is deformed, since, as noted above, angular-momentum conservation requires that its orientation change with time. Thus, at the very least, if one works with a single Slater determinant a rotational correction of some sort must be made to obtain the energy corresponding to a definite angular momentum of the ground state. Two other highly conspicuous types of correlation that must certainly be taken into account are as follows. Pairing of like nucleons These are the most widespread and conspicuous correlations in nuclear ground states, involving the tendency of like nucleons in time-reversed s.p. states to couple to zero total angular momentum. Their most obvious manifestation lies in the characteristic even–odd effect in binding energies, but they also account for the spherical shape of many open-shell nuclei: a nucleus with even one nucleon outside doubly-closed shells would be deformed in the pure HF picture. The moments of inertia of deformed nuclei can likewise be understood only in terms of pairing forces.

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The simplest way to introduce pairing correlations into the HF framework is as follows. After each HF iteration, in the basis of s.p. states thereby generated, one applies the BCS method (borrowed from the theory of superconductivity) to the pairing interaction, chosen to have the simple δ-function form α ρ δ(r ij ), vpair (r ij ) = Vπq 1 − η (10) ρ0 written here to show a possible density dependence, although for most of our models the pairing force is density independent, i.e., η = 0. This so-called HF–BCS procedure can be cast in variational form [32], but the treatment of pairing is only approximate, in the sense that the pairing matrix elements are treated as constants, and are not subjected to the variation of the φi (xi ). While acceptable for nuclei close to the stability line, this procedure leads to the presence of an unphysical neutron gas outside nuclei that are close to the neutron drip line, essentially because of the continuum of neutron s.p. states: see, for example, Refs. [33,34]. This problem is avoided in the HF–Bogolyubov (HFB) method, which is fully variational, with s.p. and pairing aspects treated simultaneously and on the same footing: see Chapter 7 of Ref. [35], and also Refs. [36,37]. It should be realized that when using a δ-function pairing force both BCS and Bogolyubov calculations will diverge if the space of s.p. states over which such a pairing force is allowed to act is not truncated. However, making such a cutoff is not simply a computational device but rather is a vital part of the physics, pairing being essentially a finite-range phenomenon. To represent such an interaction by a δ-function force is thus legitimate only to the extent that all highlying excitations are suppressed, although how exactly the truncation of the pairing space should be made will depend on the precise nature of the real, finite-range pairing force. Our ignorance on this latter point allows the cutoff to be treated as a free parameter. Wigner effect Even when pairing between like nucleons, i.e., nn and pp pairing (T = 1, |Tz | = 1 in isospin language), is correctly taken into account, HF and other mean-field calculations systematically underbind nuclei with N = Z by about 2 MeV. This effect was taken into account in the first mic–mac mass formula [38], where it was stressed that the effect was highly localized, dying out rapidly as |N − Z| increases from zero. An additional term, having the form (11) EW = VW exp −λ|N − Z|/A , was thus proposed [38], with VW negative and λ 1. Since Wigner’s supermultiplet theory, based on SU(4) spin–isospin symmetry, gives rise to a similar sharp cusp for nuclei with N = Z [39] the term became known as the Wigner term. But the cusp of supermultiplet theory arises from repulsive terms that are proportional to |N − Z|, which become increasingly important as one moves away from the N = Z line, in contrast to the apparent highly localized phenomenon. Fortunately, a more direct description of the observed effect seems to be available in terms of T = 0 neutron–proton pairing, the contribution of which rapidly vanishes as N moves away from Z [40–42]. However, T = 0 pairing is a more complex phenomenon than T = 1, |Tz | = 1 pairing, and no global mass formula constructed so far includes T = 0 pairing explicitly, phenomenological representations such as the one of Eq. (11) having been judged more convenient.

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The Skyrme–Hartree–Fock mass models All the available HF mass formulas have been constructed using a Skyrme force of the form (7), and a pairing force of the form (10) with η = 0 in all but one of the cases discussed here. In the first mass formula [16], HFBCS-1, pairing was treated in the BCS approximation, while the full HFB approach was adopted in all subsequent versions, HFB-1 to HFB-7 [43–46]. Both HFBCS-1 [16] and HFB-1 [43] were fitted to the 1995 mass-data compilation [20], but new data were subsequently made available to us [47], with 382 “new” nuclei (mainly proton-rich). These revealed drastic limitations in both the HFBCS-1 and HFB-1 models, but in a new model [44], HFB-2, considerable improvement was obtained primarily by modifying the prescription for the cutoff of the spectrum of s.p. states over which the pairing force acts, although the use of a generalized Wigner term helped to improve the fit to the lighter nuclei. (Note that because of our purely phenomenological handling of the Wigner term, and also because of the form of the rotational correction, our mass model is not completely microscopic.) We stress that as far as masses are concerned, the choice of pairing-cutoff prescription seems to be more important than the replacement of the HFBCS method by the HFB method (assuming always that the force is refitted to the data). With the pairing cutoff parameter being adjustable, this mass formula had 19 parameters fitted to the mass data. HFB-2 replaces all our earlier mass models, including in particular the various ETFSI models [48–50] based on the “extended Thomas–Fermi plus Strutinsky integral” semi-classical approximation to the HF method. Of the five Skyrme–HFB models published since HFB-2 we mention here only the last two, HFB-6 and HFB-7 [46]. Although these models do not lead to any improvement in the quality of the mass fit (see Table 1), nor to any substantial change in the extrapolations to the neutron-rich region, they are of considerable interest in that they were constrained to take an isoscalar effective mass Ms∗ of 0.8M at the equilibrium density ρ0 of symmetric INM. This is the usually accepted nuclear-matter value (see Section IIIB5e of Ref. [3] for a discussion), but is to be compared with a value of Ms∗ /M much closer to 1 that is needed if one is to reproduce the density of s.p. states close to the Fermi surface of medium and heavy nuclei. (The difference between these two values of Ms∗ can be understood in terms of a particle-vibration coupling [51,52].) In the mass fit of HFB-2 there was no constraint on Ms∗ /M and the value that emerged was close to 1, with a much better s.p. spectra at the Fermi surface than was obtained with HFB-6 and HFB-7. However, the mass fits of HFB-6 and HFB-7 are seen (Table 1) to be as good as that of HFB-2: the mass fit and the s.p. spectra at the Fermi surface are effectively decoupled in HFB-6 and HFB-7 (this was achieved by tuning the pairing cutoff). In an astrophysical context the advantage of models HFB-6 and HFB-7 over HFB-2 is that they are particularly well adapted to following the transition from isolated nuclei to neutron-star matter taking place in stellar collapse, and to the inverse process in the neutron-matter decompression associated with neutron-star mergers. Of course, if one required good s.p. spectra (or if the cutoff prescription turned out to be inconsistent with a more microscopic treatment of pairing), one would have to abandon models HFB-6 and HFB-7, but it would still be possible to obtain good s.p. spectra along with an Ms∗ of around 0.8M by generalizing the Skyrme force (7) to include a t4 term, i.e., a term with simultaneous density and momentum dependence [30,53]. The feature distinguishing HFB-6 from HFB-7 is that the pairing of the latter has a density dependence of the form (10), with η and α taking values suggested by realistic

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INM calculations [54]. This makes very little difference as far as mass fits are concerned, or for any other property that we have so far investigated. We conclude this subsection by stressing the importance of fitting the Skyrme force to essentially all the mass data. Forces that have been fitted to only a few masses usually have very large rms errors when used to calculate the entire data set, e.g., the force SLy4 [29] has an rms error of 4.75 MeV for 570 measured even–even nuclei (we base this result on the table given in Ref. [55].1 2.2. The “finite-range droplet model” (FRDM) mass formula The name of this mass formula, the most sophisticated of the mic–mac family, applies, strictly speaking, only to its macroscopic part, but is used to designate the complete model, which includes shell corrections, BCS pairing corrections, and a Wigner term. Here we shall simply outline the main features of the model; see Refs. [3,17] for more details. The macroscopic term consists basically of the “droplet” model [10], a generalized version of the liquid-drop expression of Eq. (3). In the first place, the liquid drop is both deformable and compressible, responding to the opposing tendencies of the Coulomb force and the surface tension. Also, the surfaces of the neutron and proton distributions are allowed to separate, with the possibility of creating thereby a “neutron skin”. In addition to a considerably refined calculation of the Coulomb energy, the macroscopic term of the FRDM goes beyond the original droplet model in that the simple surface term asf A2/3 in Eq. (3) is replaced by an expression that attempts to take account of the finite range of the nuclear forces. The last major refinement that has been made to the macroscopic term consists of the introduction of the exponential compressibility term, which addresses the tendency of the standard droplet model to overestimate the central density. The shell corrections to be added on to Emac , the macroscopic part of the energy, are determined by using the Strutinsky method. A necessary first step here is the specification of the s.p. field U appearing in a s.p. Schrödinger equation of the form (6), the eigenvalues of which are the s.p. energies i . Then according to the Strutinsky theorem the total nuclear energy can be written as

E = Emac + n i i − n i i , (12) i

i

where the ni are the occupation numbers of the s.p. states. All the shell effects arise in the second term, with the third term being a smoothed form of the second term. The precise way in which this smoothing is made is described in Section 2.10 of Ref. [17], which indicates that some ambiguity may arise near the neutron drip line from failure of the so-called “plateau” condition; there can also be problems associated with a failure to converge with respect to dimensionality [56]. Some effort was made to choose the field U to be compatible with the distribution of nucleons implicit in the macroscopic part of the calculation, 1 Note added in proof: For two more Skyrme–HFB mass models see: M. Samyn, S. Goriely, M. Bender, J.M. Pearson, Phys. Rev. C 70 (2004) 044309; S. Goriely, M. Samyn, J.M. Pearson, M. Onsi, Nucl. Phys. A 750 (2005) 425.

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but a residual ambiguity on this account is inevitable. (All these sources of ambiguity are absent from the HF method.) The pairing model adopted in the FRDM is that of the seniority force, i.e., a pairing force with all matrix elements having the same value, −G, treated in the Lipkin–Nogami variation of the BCS method. The most obvious strategy for fitting such a pairing force to the mass data is to adopt some suitable parametrization of G as a function of N, Z, and the deformation parameters, and to fit this directly to the mass data. However, a more circuitous strategy is adopted in the FRDM, with primacy being given to the pairing gap rather than to the pairing force; a crucial assumption is that the neutron and proton pairing gaps have average N −1/3 and Z −1/3 trends, respectively (see the discussion in Ref. [3]). As for the Wigner term, rather than retain the phenomenologically suggested form (11) for the Wigner term, as in the original mic–mac mass formula [38], the FRDM supposes a form linear in |I |, which is more appropriate to SU(4) symmetry. For a number of other small terms that we have not discussed here Refs. [3,17] should be consulted. In the latter paper it was estimated that of the more than 30 parameters in the FRDM 19 were directly fitted to the masses. The FRDM expresses the binding energy of a nucleus as a function of its deformation, whence the ground-state configuration, and the corresponding mass, are determined by minimization. However, for very large deformations corresponding to the onset of necking, discontinuities arise in some of the small terms on which the model depends for its precision fit to masses, with the result that the model cannot in general be applied to the calculation of fission barriers. Recourse has to be made rather to simpler versions of the model, such as the “finite-range liquid-drop model” (FRLDM): see, for example, Ref. [57]. 2.3. The KUTY mass formula Like the FRDM, the KUTY mass formula [18] has two parts, but the respective parts are not identified with macroscopic and microscopic terms, as such, but rather with general trends on the one hand and fluctuations about these trends on the other hand. In this way, the existence of smoothly varying components in the shell and pairing energies is recognized, but at the price of losing the physical transparency of the macroscopic part of the FRDM. A s.p. field is, of course, an essential feature of the fluctuation part, exactly as in the microscopic part of the FRDM, but there is now no apparent connection with the gross term, so again there seems to be less physics than in the case of the FRDM. Deformation is handled by taking a superposition of translated spherical nuclei. An alternative to the Strutinsky method is used. The quality of the data fit is very similar to that of the FRDM, but more parameters are involved: 34 parameters are fitted directly to the mass data. 2.4. The Duflo–Zuker (DZ) mass formula The approach to the mass-formula problem followed by Duflo and Zuker [19] is more fundamental than the mic–mac methods, and yet is not strictly microscopic, since no nucleonic interaction appears explicitly. Nevertheless, the starting point is the assumption that there exist effective interactions (“pseudopotentials”) smooth enough for HF calculations to be possible. It is then shown that the corresponding Hamiltonian H (the H eff of Eq. (5))

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can be separated into monopole and multipole terms, Hm and HM , respectively. The monopole term is entirely responsible for saturation and s.p. properties, serving in principle as a platform for HF calculations, while the multipole term acts as a residual interaction that permits the method to be pushed beyond pure HF by admitting a very general configuration mixing. It is these monopole and multipole terms, rather than any effective interaction, that are parametrized, the parametrization being formulated through scaling and symmetry arguments in such a way that one takes account of the main features of both saturation and the configuration mixing corresponding to shell-model diagonalizations based on the Kuo–Brown interaction [58–61]. The magic numbers and the regions of strong deformation both arise naturally in this schema, although in earlier versions they were put in by hand [62,63]. (These last two papers nevertheless provide some essential insights into the model; important discussions are also to be found in [64] and [65].) Pairing is taken into account in a very simple way, while the Wigner effect is represented by admitting terms with a T (T + 1) dependence, which gives rise to a cusp of the form |N − Z|. 3. Quality of data fits The mass models whose fits to the data we consider here are HFB-2 [44], HFB-6 [46], HFB-7 [46], FRDM [17], KUTY [18], and DZ [19]. The mass data for which we calculate and show in Table 1 the rms deviations σ and mean deviations ¯ (experiment–calculated) of all these models are those of the latest AME [66], which became available in December 2003. Of these data we exclude all entries that do not satisfy N, Z 8, and also all those for which the stated mass is an estimate based on systematics rather than a measured value (even though we have already remarked that the estimates of earlier AMEs have a rather good track record). We are thus left with 2149 mass data, for which we show σ and ¯ in columns 3 and 4 of Table 1. This data set [66] is so new that none of the mass models considered here was fitted to it, and we thus show (column 2) the number Nfit of masses to which the model in question was originally fitted. The largest data set to which any model was fitted consists of the 2135 masses of the 2001 AME [47], used for the HFB-2, HFB-6, and HFB-7 models. Thus in order to compare the extrapolatory power of the different models we consider the sub-set of 70 “new” nuclei in the 2003 AME [66] that did not appear in the 2001 AME [47] (note that 56 masses quoted as measured in the latter AME, which was provisional and unpublished, had this status removed in the definitive 2003 AME [66]); the corresponding σ and ¯ are shown in columns 5 and 6 of Table 1. In columns 7 and 8 we show the values of these same quantities for the 18 of the 70 “new” nuclei that are neutron rich, and thus of particular astrophysical interest. It will be seen from Table 1 that the DZ model gives a better global agreement with experiment than do any of the other models, although there may be particular regions where other models do better. (Note that we are considering only data included in the 2003 AME [66], which does not give any measured mass for nuclei with A > 265.) The FRDM model likewise is more successful than the HFB models in its predictions for the 70 “new” nuclei; indeed, this model is unique in that its predictions are actually better than the original fit. It should furthermore be realized that both the DZ and FRDM models (along with the KUTY model) would probably fare still better relative to the HFB models if they

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were refitted to the same data as those to which the HFB models were fitted [47]. On the other hand, the predictions of HFB-2, HFB-6, and HFB-7 for these 70 “new” nuclei are tolerably good, and do not present the same crisis as that with which the 2001 AME [47] confronted the HFBCS-1 and HFB-1 models. The DZ model must in fact be performing fairly closely to the limit of precision imposed, according to Bohigas and Leboeuf [67], by considerations of quantum chaos. The precise value of this lower limit is somewhat uncertain, but it appears unlikely that masses could be predicted with rms deviations much smaller than 0.3 MeV. (Note that the argument of Ref. [67] is formulated in terms of a s.p. picture, but it is claimed that the result is applicable in general. We stress also our belief that such a lower limit can only apply to mass predictions, i.e., extrapolations from data fits, since the rms deviations in data fits themselves can be reduced indefinitely simply by increasing the number of parameters in the model, a point that is not discussed in Ref. [67].) However, it must not be concluded that we can eliminate all but the DZ models. We shall see in the next section that the different models give quite different extrapolations into the neutron-rich region, and in the case of the HFB-7 and FRDM models, at least, it is clear that these differences have nothing to do with the quality of the corresponding fits to the presently available data, their rms deviations with respect to the complete data set of the 2003 AME being virtually identical. Furthermore, the DZ model in its present form cannot give ground-state deformations or be used to calculate fission barriers, and is inherently inapplicable to the calculation of other quantities of astrophysical interest such as the EOS of neutron-star matter, the GDR, level densities, and beta-decay strength functions. Thus we believe rather that all three approaches, HFB, DZ, and mic–mac, should be retained, pending further developments, and their implications for the r-process compared. Mutually enhanced magicity It has been known for many years that magic numbers reinforce each other, in the sense that the shell gap for a magic neutron (proton) number is particularly strong when the proton (neutron) number itself is magic (see Ref. [68] for a convenient review). The phenomenon is clearly evident in the sharp peaks displayed by the experimental data in the vicinity of Z = 40 (semi-magic), 82, and 126 in Figs. 1–3,

Fig. 1. The N0 = 50 shell gaps for the mass formulas of this paper.

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Fig. 2. The N0 = 82 shell gaps for the mass formulas of this paper.

Fig. 3. The N0 = 126 shell gaps for the mass formulas of this paper.

respectively, where we plot as a function of Z the neutron-shell gap, defined in terms of the two-neutron separation energy S2n according to Δn (N0 , Z) = S2n (N0 , Z) − S2n (N0 + 2, Z),

(13)

for the magic neutron numbers N0 = 50, 82, and 126, respectively. Equally striking is the failure of all the mass formulas except DZ to reproduce this phenomenon, the DZ model working very well in the neighbourhood of (N = 82, Z = 50) and (N = 126, Z = 82). Doubly-magic nuclei and their immediate neighbours should be the easiest to reproduce within the mean-field framework, and the most likely explanation is that both the HFB and mic–mac models are lacking some correlation that play an essential role in all the openshell nuclei. Since these nuclei are far more numerous and dominate the fit, the failure of the model is compensated by a skewing of the model parameters, a distortion that shows up in those nuclei where the missing correlation is inactive. Charge radii Insofar as a mass model can predict charge radii one has here an independent test of the physical consistency of that model. In Table 2 we show the rms deviations σ and mean deviations ¯ (experiment–calculated) given by the different models with respect to the 523 measured charge radii lised in the compilation [69]. The FRDM radii were

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Table 2 The rms error (σ ) and mean error (¯ , experiment–theory) for the charge radii predicted by various mass formulas. All quantities in fm

HFB-2 [44] HFB-6 [46] HFB-7 [46] FRDM [17]

σ

¯

0.028 0.026 0.026 0.045

0.0140 0.003 −0.003 −0.029

Fig. 4. The N0 = 184 shell gaps for the mass formulas of this paper.

calculated in Ref. [70]; no radii have been published for the other models (see, however, Ref. [71]). 4. Extrapolation towards the neutron drip line As discussed in Ref. [3], different mass models giving comparable fits to the data will give quite different masses when extrapolated out to the neutron drip line. However, it is differential quantities such as the neutron-separation energy Sn and the beta-decay energy Qβ , rather than the absolute masses, that are relevant to the r-process and the composition of the neutron-star crust, and here the differences between the different extrapolations are much less pronounced in general. For example, in the case of all the mass models considered here the neutron drip lines themselves, which are characterized by Sn = 0, more or less coincide, except at the magic neutron numbers, although the neutron drip lines for older drop-model mass formulas can be significantly different [72]. However, even modern mass models, when extrapolated to neutron-rich nuclei, differ considerably in their predicted shell effects, as seen in Figs. 1–4, where we plot the neutronshell gaps, defined in Eq. (13), for the magic neutron numbers N0 = 50, 82, 126, and 184 (there are no data for the last of these). A striking feature for all three of the HFB mass formulas is the quenching of the N = 50 and 82 shell gaps with decreasing Z, i.e., as the neutron drip line is approached. The FRDM model, on the other hand, shows no such quenching. For N0 = 126 and 184 the HFB gaps are more or less constant as a function of Z, while the FRDM gaps are actually enhanced as the neutron drip line is approached.

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This quite different behaviour of the HFB and FRDM shell gaps has important consequences for the r-process (see, for example, Ref. [1]), but there is no experimental evidence at the present time to discriminate in favour of one or the other possibility. Turning to theory for guidance, we know from our experience with HFB-1 and HFB-2 that the treatment of pairing has a strong influence on shell gaps [44], but there is nothing compelling in this respect with any of the presently available models. There is thus an urgent need for a more fundamental theory of pairing, pending the extension of mass measurements into much more neutron-rich regions of the nuclear chart, and in this respect the recent work of Duguet is of considerable interest [73]. We return now to the failure of all the present models except DZ to account for the phenomenon of mutually enhanced magicity, and consider the implications for extrapolation to neutron-rich nuclei. Problems may be expected only in the vicinity of the doubly-magic 78 Ni, 266 Pb, and possibly 176 Sn; the last of these could conceivably be brought inside the drip line by the phenomenon. 5. Fission barriers Any mass model that gives the binding energy of a nucleus as a function of its deformation can be applied in principle to the calculation of its fission barriers. However, at the present time there exist only two published calculations of the fission barriers of all the highly neutron-rich nuclei that are needed for the full elucidation of the r-process. The first of these, due to Howard and Möller [74], is a mic–mac calculation based on an early form of the droplet model, without the refinements of the FRLDM or FRDM. The second, due to Mamdouh et al. [75], is based on the ETFSI approximation to the HF-BCS method, using the SkSC4 force. There are some very striking differences between the predictions made by the two calculations, the most remarkable of which occurs close to the neutron drip line in the vicinity of N = 184 (proton-deficient nuclei): for Z = 84 the Howard–Möller calculation gives 6.7 MeV for the barrier height, while the ETFSI calculation predicts 39.0 MeV. There are several reasons why one can expect this ETFSI value to be too high, the main one being that with our new prescription for the pairing cutoff the N = 184 shell gap is now known to be much smaller (1.2 MeV in the case of HFB-2: see Fig. 4) than that found in the original ETFSI mass table (4.2 MeV) [48]. However, in recent HFB calculations of Samyn [76] using our new forces this barrier height never fell below 28.0 MeV (for force BSk-2, the force of the mass formula HFB-2), while Möller reports in a private communication to us that his new barrier calculations based on the FRLDM [57] lead to a barrier height of this nucleus of about 16.5 MeV (we have explained in Section 2.2 why the FRDM cannot be applied in general to barrier calculations.). There thus remains a very serious contradiction between HF and mic–mac calculations of the barriers of these highly neutron-rich nuclei. 6. Nuclear-matter constraints on Skyrme–HF mass models INM calculations are trivially easy with Skyrme forces, and the results can be expressed in a simple analytic form: see, for example, Refs. [15,24,27,30]. It is therefore natural to ask to what extent the known, or reasonably inferred, properties of INM should constrain

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the Skyrme forces used in HF mass formulas, beyond the constraints imposed by the mass fits themselves. The known properties of INM obviously include those determined by many-body theory from realistic nucleonic interactions, but there are some qualitative, or semi-quantitative, properties that can be inferred without any such calculation. For example, the properties of ordinary nuclei tell us that INM should be stable against spin and joint spin–isospin flip, conditions which require that the Landau parameters G0 and G0 , respectively, should both be > −1 [77–79]. Likewise, simple observation of neutron stars tells us that neutron matter (INM consisting entirely of neutrons) must be stable against a high-density collapse, and also against a spin flip into a ferromagnetic state [80] (either of these two instabilities would favour the beta decay of neutron-star matter into pure neutron matter). The Skyrme forces of all the HF mass formulas published so far [15,16,43–46] satisfy all these conditions. We turn now to the implications of many-body calculations of INM based on realistic nucleonic interactions. One might expect that such calculations could be used to fix once and for all the parameters avol , ρ0 , and asym , among others, the values of which would then be imposed on all mass models. However, these many-body calculations are fraught with ambiguities, and such a procedure would impose unrealistic constraints on the mass models. Nevertheless, realistic INM calculations are valuable for determining the behaviour of mass models in conditions of large charge asymmetry, notably for nuclei close to the neutron drip line. Of especial interest in this respect are calculations on pure neutron matter, which are intrinsically simpler and less ambiguous than the general INM case, for the following reasons: (i) two-nucleon scattering determines the phase parameters better in the T = 1 than in the T = 0 states; (ii) the strongly tensor-coupled 3 S1 and 3 D1 states are absent; (iii) the relatively poorly determined 3-nucleon force is less important in this case (it would not contribute at all to neutron matter if it had zero range). Realistic calculations of neutron matter can thus provide a valuable constraint on Skyrme-force mass models that are intended to be used for highly neutron-rich nuclei, far away from the data, as we now discuss. In Fig. 5 we show the results of the Friedman–Pandharipande (FP) [81] calculation of neutron matter made with realistic two- and three-nucleon forces at zero temperature, with

Fig. 5. The energy per neutron of neutron matter as a function of density for three different calculations (see Section 6 for details).

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the energy per neutron given as a function of density. More recent calculations [82–87] give essentially the same curve over the density range shown; see the review [8]. Fig. 5 shows also the corresponding curve for the force BSk7, the force of the mass formula HFB-7; this curve is indistinguishable from those for mass formulas HFB-2 and HFB-6 over the range shown here (see, however, Fig. 4 of Ref. [46]). It will be seen that the curve for force BSk-7 is definitely softer than the realistic FP curve. In fact, in many of our mass fits there was a tendency for the neutron-matter curve to be still softer, with an optimal mass fit occurring for a value of asym lying between 27 and 28 MeV, for which an unphysical collapse of neutron matter set in at sub-nuclear densities. In the fits HFB-2, HFB-6, and HFB-7 we were able to avoid this contradiction with the known stability of neutron stars by imposing asym = 28.00 MeV, slightly degrading thereby the quality of the mass fit. Nevertheless, it is clear that a still better agreement with the FP neutron-matter curve would be obtained with a somewhat higher value of asym ; preliminary studies show an almost perfect agreement for asym = 30 MeV, although the cost is a further slight deterioration in the quality of the mass fit. Further evidence that the correct value of asym lies significantly higher than 28 MeV comes from measurements of the neutron-skin thickness of finite nuclei, Rnrms − Rprms , where Rnrms is the rms radius of the neutron distribution and Rprms that of the point proton distribution. Taking an experimental value of 0.14 ± 0.04 fm for the case of 208 Pb [88] led Ref. [26] to the value of asym = 29 ± 2 MeV. A more recent measurement [89] of the same quantity gave 0.20 ± 0.04 fm, which we find to be consistent with asym = 32 ± 2 MeV. Both of these experiments involved nucleon–nucleus scattering, and are very difficult, but a newly proposed method based on parity-violating electron–nucleus scattering is promising [90]. However, whatever the outcome of these new measurements of neutron-skin thickness, we find that imposing the constraint asym = 32 MeV leads to mass fits that are unacceptably poor. Moreover, the corresponding neutron-matter curve, labelled J32 in Fig. 5, is definitely stiffer than the FP curve. Imposing asym = 30 MeV would thus seem to be a reasonable compromise, and we are currently investigating the implications of such a constraint; earlier studies on these lines [91] suggested that there will be a minimal effect on the Sn and Qβ , and thus on the r-process. On the other hand, shifting asym from 27 to 30 MeV is found to have a drastic effect on the composition of the inner crust of neutron stars [92], at least within the framework of Skyrme-force models. In any case, the need to impose a value of asym , rather than letting it emerge freely from the mass fits, might be regarded as a limitation of the Skyrme-force model. Even more disconcerting is the possibility that an optimal mass fit might be incompatible with the stability of neutron stars, a datum as well established as any nuclear mass. It would be interesting to see whether generalizing the Skyrme force to include a t4 term [30] could, with the extra degrees of freedom, resolve this dilemma.

7. Conclusions In the foregoing we have briefly described the various currently available nuclear “mass formulas” that can respond to the astrophysical need for estimates of the binding energies

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of nuclei that are so neutron rich that there is no prospect of being able to measure them in the laboratory. All the different approaches that we have described are semiempirical, in the sense that the mass model adopted always contains a number of free parameters that are fitted to the available data. An obviously necessary (but far from sufficient) condition for a reliable extrapolation from the known to the unknown regions of the nuclear chart is that the data themselves be well fitted. We see from Table 1 that the rms deviation σ of each of the mass formulas with respect to the 2149 measured masses of nuclei with N and Z 8 given in the most recent data compilation [66] is always smaller than 0.7 MeV. The DZ model [19], which attempts to incorporate the main features of shell-model diagonalizations, gives by far the lowest σ , while the two mic–mac models, FRDM [17] and KUTY [18], perform comparably to the three HFB models. Far more striking than the slight differences in the quality of the data fits given by the different mass formulas is the way in which they diverge when extrapolated to the neutron drip line; the problem is particularly acute when one considers the possibility of a quenching of the neutron shell gaps as the neutron drip line is approached, a matter of crucial astrophysical importance. It is noteworthy that even two mass formulas with almost identical rms deviations will give quite different predictions for highly neutron-rich nuclei, e.g., the FRDM and any of the HFB models. Thus the most urgent need, aside from new data, is not so much to improve still further the quality of the data fits as to assess which, if any, of the models considered here is to be trusted. In this respect the fact that the FRDM, alone of all the models, actually performs better on the new data that became available in the 2003 AME [66] than in the original fit might be regarded as significant, although there is no guarantee that this performance will be maintained in the future as new data come in. On the other hand, as far as long-range extrapolations are concerned, one might be tempted to suppose that the more microscopic a model is the more reliable its extrapolations will be. However, while this criterion would certainly favour our HFB models, it must be remembered that the effective forces used, with the conventional Skyrme form (7) in the HF channel and the δ-function form (10) in the pairing channel, have only a tenuous connection with the real forces and are far from unique. A salutary experience in this respect was provided by the new data that forced a change in the pairing-cutoff prescription that had been adopted in HFB-1. This led in turn to a radical revision in the predictions for highly neutron-rich nuclei, and no matter how confident one may feel in the HFB models described here (HFB-2, HFB-6, and HFB-7) there is no guarantee that new data acquired in the future will not lead to equally drastic modifications in the extrapolations. The present situation is seen to be far from satisfactory, and it is important that the HFB approach be explored much more systematically. In the first place, the method used so far in all the HFB mass models needs to be improved, both with respect to the rotational correction and to the explicit inclusion of vibrational effects. As for the interaction, the range of possibilities should be restricted by an examination of the constraints imposed by fundamental theory: we have already discussed the constraint imposed on the HF channel by neutron–gas calculations (Section 6), and we believe that the pairing channel should likewise be handled in a more fundamental way. At the same time, in order to have a clearer idea of the remaining level of ambiguity in the extrapolations to the highly neutronrich regions of the nuclear chart, one should take different forms of force in an attempt to

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exploit all the possibilities that are consistent with the fundamental constraints that have been imposed. For example, one could generalize the conventional Skyrme form (7) by including a t4 term. But in widening thereby the range of theoretical possibilities it is very much to be hoped that new mass measurements will at the same time help to discriminate between competing models. It is impossible to exaggerate the importance of new data; while measurements in the vicinity of closed shells on the neutron-rich side would be especially useful, even new data on the proton-rich side could be useful, as proved to be the case with the mainly proton-rich data of the 2001 AME [47] (see Section 2). In any case, for applications to the r-process at the present time one should really compare the implications of all the mass formulas described here, bearing in mind the possibility that none of them is correct. At this point one might be tempted to use the measured abundances of r-process nuclides to discriminate between the different mass formulas, but one would first have to make sure that there was no residual ambiguity associated with the astrophysical conditions under which nucleosynthesis occurs. We have also discussed (Section 5) the extension of mass models to the calculation of the fission barriers of highly neutron-rich nuclei. Some very serious contradictions between mic–mac and HF calculations have been noted.

Acknowledgements We wish to thank P. Möller, C. Thibault, and A. Zuker for valuable communications. An extensive collaboration with M. Samyn is gratefully acknowledged. We are deeply indebted to F. Tondeur for his pioneer efforts over many years. J.M.P. acknowledges financial support from NSERC (Canada). S.G. is a FNRS Associate.

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