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Nuclear surface properties in relativistic e?ective ?eld theory

arXiv:nucl-th/9810026v1 9 Oct 1998

M. Del Estal, M. Centelles, X. Vi? as n Departament d’Estructura i Constituents de la Mat`ria, Facultat de F? e ?sica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain

Abstract We perform Hartree calculations of symmetric and asymmetric semi-in?nite nuclear matter in the framework of relativistic models based on e?ective hadronic ?eld theories as recently proposed in the literature. In addition to the conventional cubic and quartic scalar self-interactions, the extended models incorporate a quartic vector selfinteraction, scalar-vector non-linearities and tensor couplings of the vector mesons. We investigate the implications of these terms on nuclear surface properties such as the surface energy coe?cient, surface thickness, surface sti?ness coe?cient, neutron skin thickness and the spin–orbit force.

PACS: 21.60.-n, 21.30.-x, 21.10.Dr, 21.65.+f Keywords: Nuclear surface properties; spin–orbit potential; semi-in?nite nuclear matter; non-linear self-interactions; Quantum Hadrodynamics; e?ective ?eld theory.

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1

Introduction

Quantum hadrodynamics (QHD) and the relativistic treatment of nuclear systems has

been a subject of growing interest during recent years [1,2,3,4,5]. The σ?ω model of Walecka [1] and its non-linear extensions with cubic and quartic self-interactions of the scalar-meson ?eld [6] have been widely used to this end. This model contains Dirac nucleons together with neutral scalar and vector mesons as well as isovector-vector ρ mesons. At the mean ?eld (Hartree) level, it already includes the spin–orbit force, the ?nite range and the density dependence which are essential ingredients of the nuclear interaction. This simple model has become very popular in relativistic calculations and describes successfully many properties of the atomic nucleus. From a theoretical point of view, the non-linear σ ? ω model with cubic and quartic scalar self-interactions was classed within renormalizable ?eld theories which can be characterized by a ?nite number of coupling constants. However, very recently, generalizations of this model that include other non-linear interactions among the meson ?elds and tensor couplings have been presented on the basis of e?ective ?eld theories by Serot et al. [5,7,8,9]. The e?ective theory contains many couplings of non-renormalizable form that are consistent with the underlying symmetries of QCD. Consequently, one must ?nd some suitable expansion parameters and develop a systematic truncation scheme. For this purpose the concept of naturalness has been employed: it means that the unknown couplings of the theory should all be of the order of unity when written in appropriate dimensionless form using naive dimensional analysis [5,7,8,9]. Then, one can estimate the contributions coming from di?erent terms by counting powers in the expansion parameters and truncating the Lagrangian at a given level of accuracy. One important fact is the observation that at normal nuclear densities the scalar and vector meson ?elds, denoted by Φ and W , are small as compared with the nucleon mass M and that they change slowly in ?nite nuclei. This implies that the ratios Φ/M, W/M, |?Φ|/M 2 and |?W |/M 2 are useful expansion parameters when the e?ective ?eld theory is applied to the nuclear many-body problem. From this viewpoint, if all the terms involving scalar and meson self-interactions are retained in the Lagrangian up to fourth order, one 2

recovers the well-known non-linear σ?ω model plus some additional terms [5,7,9]. For the truncation to be consistent, the corresponding coupling constants should exhibit naturalness and cannot be arbitrarily dropped out without an additional symmetry argument. The e?ective Lagrangian truncated at fourth order contains thirteen free parameters that have been ?tted to reproduce twenty-nine ?nite nuclei observables [5,9]. Remarkably, the ?tted parameters turn out to be natural and the results are not dominated by the last terms retained. This evidence con?rms the utility of the principles of naive dimensional analysis and naturalness and shows that truncating the e?ective Lagrangian at the ?rst lower orders is justi?ed. The term with a vector-meson quartic self-interaction has been considered previously in relativistic mean ?eld (RMF) calculations from a phenomenological point of view. Bodmer [10] considered this coupling to avoid the negative coe?cient of the quartic scalar selfinteraction that appears in many non-linear σ?ω parametrizations that correctly describe the atomic nucleus [11]. In some special situations this negative term can lead to a pathological behaviour of the scalar potential. On the other hand, the equation of state is softened at moderate high densities when the vector non-linearity is taken into account. The quartic vector self-interaction has also been phenomenologically used by Gmuca [12,13] in a nonlinear σ?ω model for parametrizing Dirac–Brueckner–Hartree–Fock calculations of nuclear matter. The same idea was developed by Toki et al. and applied to study ?nite nuclei [14] and neutron stars [15]. Recently, the properties of high-density nuclear and neutron matter have been analyzed in the RMF approach taking into account scalar and vector non-linearities [8]. The tensor couplings of the vector ω and ρ mesons to the nucleon were investigated by Reinhard et al. [3,16] as an extension of the RMF model, and more recently by Furnstahl et al. [9,17] from the point of view of relativistic e?ective ?eld theory. In these works it was shown that the tensor coupling of the ω meson has an important bearing on the nuclear spin–orbit splitting. The surface properties of nuclei play a crucial role in certain situations. This is the case, for instance, of saddle-point con?gurations in nuclear ?ssion or fragment distributions

3

in heavy-ion collisions. Within a context related to the liquid droplet model (LDM) and the leptodermous expansion [18], the surface properties can be extracted from semi-in?nite nuclear matter calculations either quantally or semiclassically (though the total curvature energy coe?cient can only be computed semiclassically [19]). In the non-relativistic case most of the calculations of the surface properties have been carried out using Skyrme forces, quantally [20] or semiclassically with the help of the extended Thomas–Fermi (ETF) method [20,21]. In the relativistic case the nuclear surface has been analyzed within the σ?ω model since a long time ago. The calculations have been performed semiclassically using the relativistic Thomas–Fermi (TF) method or its extensions (RETF), for symmetric [1,6,22,23,24] and asymmetric [25,26] matter, and also in the quantal Hartree approach [27,28,29]. In the framework of the relativistic model and e?ective ?eld theory, the main purpose of the present work is to carefully analyze the in?uence on surface properties of the quartic vector non-linearity, of the newly proposed scalar-vector self-interactions and of the tensor coupling. We shall investigate quantities such as the surface energy coe?cient, surface thickness, spin–orbit strength, surface sti?ness coe?cient and neutron skin thickness obtained from Hartree calculations of symmetric and asymmetric semi-in?nite nuclear matter. The paper is organized as follows. Section 2 is devoted to the basic theory. The results on the surface properties of symmetric matter are discussed in Section 3. Section 4 addresses the case of asymmetric systems. The summary and conclusions are given in the last section.

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2

Mean ?eld equations for symmetric semi-in?nite nuclear matter

Following Ref. [7], to derive the mean ?eld equations one starts from an energy functional

containing Dirac baryons and classical scalar and vector mesons. The energy functional can be obtained from the e?ective Lagrangian in the Hartree approach using many-body techniques [5,9]. However, this energy functional can also be considered as an expansion in Φ/M, W/M, |?Φ|/M 2 and |?W |/M 2 of a general energy density functional that contains all the correlation e?ects. The theoretical basis of this functional lies on the extension of the Hohenberg–Kohn theorem [30] to QHD [31]. Using the Kohn–Sham scheme [32] with the mean ?elds playing the role of Kohn–Sham potentials, one ?nds similar mean ?eld equations to those obtained from the Lagrangian [31], but including e?ects beyond the Hartree approach through the non-linear couplings [4,7,8,9]. A semi-in?nite system of uncharged nucleons corresponds to a one-dimensional geometry where half the space is ?lled with nuclear matter at saturation and the other half is empty, so that a surface develops around the interface. The ?elds and densities change only along the direction perpendicular to the medium. Specifying the energy density functional considered in Refs. [5] and [9] to symmetric semi-in?nite nuclear matter with the surface normal pointing into the z direction one has E(z) = ?? (z) ?iα·? + β[M ? Φ(z)] + W (z) ? α ifv βα·?W (z) ?α (z) 2M m2 2 s Φ (z) 2 gs

α

+

1 Φ(z) 1 κ3 Φ(z) κ4 Φ2 (z) 1 + α1 (?Φ(z))2 + + + 2 2gs M 2 3! M 4! M 2 Φ(z) ζ0 1 4 1 W (z) 1 + α2 (?W (z))2 ? 2 2 2gv M 4! gv Φ(z) η2 Φ2 (z) 1 1 + η1 + 2 M 2 M2 m2 2 v W (z) , 2 gv

? ?

(2.1)

where the index α runs over all occupied states of the positive energy spectrum, Φ ≡ gs φ0 and W ≡ gv V0 (notation as in Ref. [1]). Except for the terms with α1 and α2 , the functional (2.1) is of fourth order in the expansion. We retain the ?fth-order terms α1 and α2 because 5

in Refs. [5] and [9] they have been estimated to be numerically of the same magnitude as the quartic scalar term in the nuclear surface energy. The mean ?eld equations are obtained by minimizing with respect to ?? , Φ and W : α ?iα·? + β[M ? Φ(z)] + W (z) ? ifv βα·?W (z) ?α (z) = εα ?α (z) , 2M (2.2)

2 ? ?Φ(z) + m2 Φ(z) = gs ρs (z) ? s 2 gs 2M

m2 2 κ3 κ4 Φ(z) s Φ (z) + M 2 3! M Φ(z) M m2 2 v W (z) 2 gv

+

η1 + η2

+

2 α1 α2 gs (?W (z))2 , (2.3) [(?Φ(z))2 + 2Φ(z)?Φ(z)] + 2 2M 2M gv

2 ??W (z) + m2 W (z) = gv ρ(z) + v

fv η2 Φ(z) ρT (z) ? η1 + 2 2 M

Φ(z) 2 mv W (z) M

?

1 α2 ζ0 W 3 (z) + [?Φ(z) · ?W (z) + Φ(z)?W (z)] . (2.4) 3! M

The baryon, scalar and tensor densities are respectively ρ(z) =

α

?? (z)?α (z) , α ?? (z)β?α (z) , α i ?· ?? (z)βα?α (z) . α M

(2.5) (2.6) (2.7)

ρs (z) =

α

ρT (z) =

α

The expression of the four-component spinors ?α (z) in the semi-in?nite medium was given by Hofer and Stocker in Ref. [27]. In a semi-in?nite nuclear matter calculation the sum over the single-particle states is replaced by an integration over momenta: ?→ 2 ? (2π)3 dk ,

λ

(2.8)

α

where ? stands for the volume of the box, the factor 2 takes into account the isospin degree of freedom and λ describes the spin orientation of the nucleons. Introducing the Fermi 6

2 2 2 2 2 2 momentum kF, the integration domain is restricted to kx + ky + kz = k⊥ + kz ≤ kF , with

kz ≥ 0 if the bulk nuclear matter is located at z = ?∞. Following the method outlined in Ref. [27] one ?nds two sets (λ = ±1) of ?rst-order di?erential equations for the orbital part of the upper and lower components of the Dirac spinors: dGa (z) fv dW (z) Ga (z) = [εa ? W (z) + M ? (z)] Fa (z) , ? λk⊥ + dz 2M dz ? fv dW (z) dFa (z) Fa (z) = [εa ? W (z) ? M ? (z)] Ga (z) , ? λk⊥ + dz 2M dz (2.9)

(2.10)

where a = (kz , k⊥ , λ) and M ? (z) = M ? Φ(z) is the Dirac e?ective mass of the nucleons. From the asymptotic behaviour at z = ?∞ (bulk nuclear matter), the condition on the energy eigenvalues is εa =

? 2 2 ? k⊥ + kz + M∞ 2 + W∞ , with M∞ and W∞ being the nuclear

matter values of M ? and W . The densities for each spin orientation λ = ±1 read √2 2 kF ?kz 2 kF λ dkz dk⊥ k⊥ |Ga (z)|2 + |Fa (z)|2 , ρ (z) = π2 0 0 √2 2 kF ?kz 2 kF λ ρs (z) = dkz dk⊥ k⊥ |Ga (z)|2 ? |Fa (z)|2 , 2 0 π 0 √2 2 kF ?kz 2 d 2 kF Fa (z)Ga (z) , dkz dk⊥ k⊥ ρλ (z) = T 2 0 π dz M 0 and the total densities are given by ρ(z) =

λ

(2.11) (2.12) (2.13)

ρλ (z) ,

ρs (z) =

λ

ρλ (z) , s

ρT (z) =

λ

ρλ (z) . T

(2.14)

Using the equations of motion the energy density of the semi-in?nite nuclear matter system can be written as follows: √2 2 kF kF ?kz 2 dkz dk⊥ k⊥ E(z) = π2 λ 0 0

2 2 ? k⊥ + kz + M∞ 2 + W∞

|Ga (z)|2 + |Fa (z)|2 m2 2 s Φ (z) 2 gs

1 fv Φ(z) 1 + Φ(z)ρs (z) ? W (z) ρ(z) + ρT (z) ? 2 2 2 4M + ? Φ(z) Φ(z) η1 + η2 4M M ζ0 1 4 m2 2 v W (z) + W (z) 2 2 gv 4! gv

κ3 κ4 Φ(z) + 3 6 M

α1 Φ(z) α2 Φ(z) (?Φ(z))2 + 2 (?W (z))2 . 2 M 4gs 4gv M 7

(2.15)

Finally, the surface energy coe?cient Es is obtained from the expression [18]

2 Es = 4πr0 ∞ ?∞

dz [E(z) ? (av + M)ρ(z)] ,

(2.16)

where av is the energy per particle in bulk nuclear matter and r0 is the nuclear radius constant: r0 = (3/4πρ0)1/3 , with ρ0 the nuclear matter density. Another important quantity in the study of the nuclear surface structure is the spin–orbit interaction. By elimination of the lower spinor in terms of the upper spinor, one obtains a Schr¨dinger-type equation with a term Vso (z) that has the structure of the single-particle o spin–orbit potential for the non-relativistic case [27,29]. In our present model the orbital part of the spin–orbit potential reads as Vso (z) = 1 1 2M εa ? W (z) + M ? (z) fv dW (z) dW (z) dΦ(z) + . + dz dz M dz (2.17)

In the non-relativistic limit, by means of a Foldy–Wouthuysen reduction, Eq. (2.17) becomes

FW Vso (z) =

1 dW (z) dΦ(z) (1 + 2fv ) + 2 4M dz dz

(2.18)

and the nucleons are then moving in a central potential of the form Vc (z) = W (z) ? Φ(z) . (2.19)

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3

Surface properties in the symmetric case

Although previous works [23,24,25,26,27,28,29] have thoroughly investigated the prop-

erties of the nuclear surface in the standard non-linear σ ?ω model, for which κ3 and κ4 are the only non-linearities in (2.1), here we wish to enlarge this study by including the additional non-linear and tensor couplings considered in Refs. [5,7,9]. In concrete, we want to study the role of the quartic vector self-interaction ζ0 that has been used after the work of Bodmer [10], the role of the terms with η1 and η2 that couple the scalar and vector ?elds, of the terms with α1 and α2 that imply the gradients of the ?elds, and of the tensor coupling fv of the vector ω meson to the nucleon. While ζ0 , η1 and η2 can be classi?ed as volume contributions, the couplings α1 , α2 and fv are genuine surface terms. On the basis of the concept of naturalness there is no reason to omit any of these terms in the energy density functional (2.1), unless there exists a symmetry principle to forbid it. However, to clarify the impact on the surface properties of the aforementioned couplings we will analyze each one separately, as it has been similarly done in Ref. [7]. In this section we shall study symmetric systems, while in Section 4 we shall address the case of asymmetric matter.

3.1

E?ect of the quartic vector self-interaction

2 2 In the conventional non-linear σ ?ω model the value of the coe?cients gs /m2 , gv /m2 , s v

κ3 and κ4 can be univocally obtained by imposing that for nuclear matter at saturation the

? density ρ0 , energy per particle av , e?ective mass M∞ /M and incompressibility modulus K

take given values. When the vector-meson quartic self-interaction is switched on, the Dirac equation for the baryons and the Klein–Gordon equation for the vector ?eld in in?nite nuclear matter become (throughout this subsection we set η1 = η2 = α1 = α2 = fv = 0): av =

2 ? kF + M∞ 2 + W∞ ? M ,

(3.1) (3.2)

1 2 3 m2 W∞ = gv ρ0 ? ζ0 W∞ . v 6

3 The saturation density ρ0 and the Fermi momentum kF are related as usual by ρ0 = 2kF /3π 2. ? Specifying ρ0 , av and M∞ /M, from the above equations one extracts the coupling constant

9

gv as a function of ζ0 (the nucleon and ω masses take their empirical values: M = 939 MeV

2 and mv = 783 MeV). The steps to calculate gs /m2 , κ3 and κ4 are then the same as when s

ζ0 = 0, see e.g. Refs. [10] and [11], but now these coe?cients become functions of ζ0 . The reader will ?nd a detailed study of the implications of the vector non-linearity ζ0 in nuclear matter in Refs. [7] and [10]. The assumption of naturalness requires that the couplings gs /4π, gv /4π, κ3 , κ4 and ζ0 should all be roughly of the order of unity. Figure 1 illustrates the variation of these couplings as a function of the non-dimensional parameter η0 = m2 v 2 gv 6m2 v ζ0 ρ2 0 (3.3)

used by Bodmer1 in Ref. [10]. With ms = 490 MeV, the ?gure presents the results for four sets of nuclear matter properties: ρ0 = 0.152 fm?3 (kF = 1.31 fm?1 ), av = ?16.42 MeV,

? ? K = 200 and 350 MeV, and M∞ /M = 0.6 and 0.7. When K = 200 MeV and M∞ /M = 0.6

the equilibrium properties of the interaction and the scalar mass ms are very close to those of the non-linear parametrization NL1 [33]. We realize that gs and gv are only weakly a?ected by η0 (the change is not appreciable in the scale of Figure 1). However, η0 has a direct e?ect on the scalar-meson quartic selfinteraction κ4 , moving it from a negative value when the quartic vector term is absent (η0 = ∞) to a desirable positive value when η0 ? 2. The change of sign of κ4 takes place

? at larger values of η0 for larger K and M∞ /M; in fact, κ4 is already positive at η0 = ∞ if ? K = 350 MeV and M∞ /M = 0.7. Both non-linearities of the scalar ?eld κ3 and κ4 remain

in the natural zone for 2 ≤ η0 ≤ ∞ approximately, excepting the case of K = 350 MeV

? and M∞ /M = 0.7, but they start to depart appreciably from their natural values when

η0 ≤ 2. These trends fairly agree with the assumption of naturalness: η0 ≥ 2 corresponds to the region where ζ0 can be considered as natural (see the ?gure), and for this range of η0 values also the rest of coupling constants are natural. The behaviour gleaned from Figure 1 is rather independent of the saturation density ρ0 and energy av , and e.g. we have found similar trends with the speci?c nuclear matter properties used by Bodmer in Ref. [10].

1

Notice that in Ref. [10] the parameter η0 was called z.

10

Next we turn our attention to the surface properties. In Figures 2 and 3 we have plotted, respectively, the surface energy coe?cient Es and the surface thickness t of the semi-in?nite density pro?le (the 90%–10% fall-o? distance) against the η0 parameter. The selected values of η0 are those employed in Table 1 of Ref. [10]. With ρ0 = 0.152 fm?3 and av = ?16.42 MeV, we have performed the calculations for a few incompressibilities (K = 125, 200 and

? 350 MeV) and e?ective masses (M∞ /M = 0.6 and 0.7). Furthermore, we have considered

two values of the mass of the scalar meson (ms = 490 and 525 MeV). This quantity governs the range of the attractive interaction and determines the surface fall-o?: a larger ms results in a steeper surface and a reduction of Es and t. From Figures 2 and 3 we realize that the quartic vector self-interaction scarcely alters the values of the surface energy coe?cient and of the surface thickness if ζ0 remains in the natural domain, i.e., if 2 ≤ η0 ≤ ∞. At K = 350 MeV and M ? /M ≥ 0.6, Es is raised by decreasing η0 . This tendency may be inverted at smaller values of the incompressibility, depending also on the value of the nucleon e?ective mass, but the global trends are practically independent of the mass of the scalar meson. The thickness t exhibits a more monotonous behaviour: in all cases it stays almost equal to its value at η0 = ∞ and goes down slightly for η0 ≤ 2. Noticeable departures of the surface energy coe?cient and thickness from the η0 = ∞ values can be found only if η0 is decreased beyond the natural limit, a situation where the interaction is mainly ruled by the vector-meson quartic self-interaction. Even though the impact of the vector-meson quartic self-interaction is small, it can help to ?nd parameter sets for which both the surface energy coe?cient and the surface thickness lie in the empirical region. This fact is illustrated in Figure 4, where Es and t are drawn

? versus M∞ /M for η0 = ∞, 2 and 0.5 and for ms = 450, 500 and 550 MeV, using the nuclear

matter conditions of Ref. [10]: ρ0 = 0.1484 fm?3 , av = ?15.75 MeV and K = 200 MeV. (We also performed the calculations for η0 = 5 and η0 = 1 to con?rm the trends we discuss below.) The horizontal dashed lines in Figure 4 serve to indicate the empirical region for the surface energy and thickness. If at η0 = ∞ we concentrate, for instance, on the parametrizations with ms = 450 MeV

? we see that the one with M∞ /M = 0.7 yields, simultaneously, Es and t within the empirical

11

region. The agreement between the calculated values with ms = 450 MeV and the empirical region improves for η0 = 2, where both Es and t are acceptable for practically all the values of the e?ective mass considered. For η0 ≥ 2 the dependence of Es and t upon the nucleon e?ective mass is similar to that found in the usual σ?ω model without a quartic vector self-interaction [23,28]. The general tendencies start to change when η0 is lowered and leaves the natural region. As η0 becomes

? smaller Figure 4 shows that the slope of the curves of Es and t as a function of M∞ /M

changes, and that the curves for the di?erent ms come closer together. To get more insight about the in?uence of the vector-meson quartic self-interaction, we display in Figure 5 the pro?les of the baryon density ρ(z) and of the surface tension density (Swiatecki integrand) σ(z) = E(z) ? (av + M)ρ(z), Eq. (2.16). In turn, we have represented in Figure 6 the orbital part of the spin–orbit potential Vso (z), Eq. (2.17), at the Fermi surface (i.e., evaluated at k = kF ) and the central mean ?eld Vc (z) de?ned in Eq. (2.19). The properties of the interactions used in these ?gures are ρ0 = 0.152 fm?3 , av = ?16.42

? MeV, K = 200 MeV, M∞ /M = 0.6 and 0.7, and ms = 490 MeV. Results are shown for

η0 = ∞, 2 and 0.5. The corresponding values of the surface energy coe?cient and surface thickness can be read from Figures 2 and 3. It can be seen that the local quantities depicted in Figures 5 and 6 oscillate as functions of z (Friedel oscillations). Both the surface tension density and the single-particle spin– orbit potential are con?ned to the surface and average to zero inwards as the bulk matter is approached. The local pro?les for η0 = 2, which somehow marks the limit of naturalness as we have commented, are almost equal to those obtained in the absence of the quartic vector self-interaction. In agreement with Figures 2 and 3, only when η0 is decreased to non-natural

? values one can notice some changes in the pro?les, which are more visible for M∞ /M = 0.6 ? than for M∞ /M = 0.7. Decreasing η0 makes the surface steeper and the thickness t smaller,

produces an enhancement in the surface region of the density ρ(z) and of the mean ?eld Vc (z), and builds up Friedel oscillations in σ(z) and in the spin–orbit potential Vso (z). It is well known that the experimental spin–orbit splittings require within narrow bounds

? a Dirac e?ective mass M∞ /M around 0.6 in the conventional relativistic model [11,7]. Figure

12

6 shows that introducing a quartic vector self-interaction makes the spin–orbit well deeper.

? However, it is only a minor e?ect: with M∞ /M = 0.7 it is not possible to reproduce the spin– ? orbit interaction of the case M∞ /M = 0.6 at η0 = ∞, not even if one sets η0 = 0.5 (which

in addition brings about an unreallistically small t, see Figure 3). A similar conclusion was drawn in Ref. [10] from an analysis in nuclear matter. We also have computed the

FW non-relativistic limit Vso (z) of the spin–orbit potential given by the expression (2.18). FW In agreement with Ref. [29] we have found that while Vso (z) qualitatively reproduces the

behaviour of Vso (z), it strongly underestimates the quantitative depth of the fully relativistic

? ? spin–orbit strength (by ? 20% for M∞ /M = 0.6 and by ? 15% for M∞ /M = 0.7).

3.2

In?uence of the volume cubic and quartic scalar-vector interactions

The next bulk terms in the energy density (2.1) that contain non-linear meson interac-

tions are ? η2 Φ 1Φ η1 + 2M 2M m2 2 v W . 2 gv (3.4)

In analogy to Figure 1 for η0 , in Figure 7 we study the change of the couplings gs /4π, gv /4π, κ3 and κ4 with the parameters η1 and η2 (introducing each one separately) for some speci?c equilibrium properties. With ρ0 = 0.152 fm?3 , av = ?16.42 and ms = 490 MeV, in part (a)

? of Figure 7 it is K = 200 MeV and M∞ /M = 0.6, while in part (b) it is K = 350 MeV and ? M∞ /M = 0.7. ? For K = 200 MeV and M∞ /M = 0.6 both η1 and η2 have a considerable e?ect on the

scalar non-linearities κ3 and κ4 . The coupling κ4 changes sign for η2 ≈ 1, but it remains negative in the interval of η1 values used. To keep all the coupling constants within natural values we see that the range for η1 and η2 (when introduced separately) is restricted to run roughly from ?0.5 to 2.5. Consider now di?erent values of the incompressibility and e?ective mass, as in part (b) of Figure 7. The dependence of the couplings on η2 is not

? signi?cantly altered. However, increasing either K or M∞ /M results in a smoother slope of

κ3 with η1 , while it makes κ4 grow steadily with η1 and become positive at some value of

? this parameter. We have checked for K = 200 MeV and M∞ /M = 0.55, and for K = 125

13

? MeV and M∞ /M = 0.6, that κ4 remains negative at all values of η1 and that κ3 grows with

η1 faster than in part (a) of Figure 7. The in?uence of the non-linear interactions η1 and η2 on the surface properties is analyzed in Figure 8. With ζ0 = α1 = α2 = fv = 0, in this ?gure we have computed Es and t taking into account the terms (3.4). The saturation conditions of the interaction and the scalar mass are the same as in part (a) of Figure 7. The results are displayed in the plane η1 –η2 in the form of contour plots of constant Es (solid lines) and of constant t (dashed lines). The range of variation of η1 and η2 lies in the region imposed by naturalness, and yields values of Es and t within reasonable limits. As it can be inferred from the nearly vertical lines in the η1 –η2 plane, the surface energy coe?cient and thickness depend mostly on η1 and are rather independent of η2 . The consequence of increasing η1 is a reduction of the values of Es and t. The lines of constant t turn out to be, roughly speaking, parallel to the lines of constant Es . This means that from the interplay of the parameters η1 and η2 it is not possible to change the value of t relative to that of Es (for example, we see in Figure 8 that t ? 2.2 fm if Es = 18 MeV). We have calculated the spin–orbit potential Vso (z) at the Fermi surface for several values of η1 and η2 . We have found that these couplings have a marginal e?ect on the spin–orbit strength, as it happened to be the case with the other bulk non-linearity η0 . To get some information about the incidence on the surface energy and thickness of all the volume non-linear meson interactions together, we have repeated the calculations in the η1 –η2 plane setting η0 = 2 for the quartic vector self-interaction. One ?nds similar features to those of Figure 8. The e?ect of η0 = 2 is just shifting Es and t towards smaller values as compared with the case η0 = ∞ (ζ0 = 0), which is in accordance with what was found in

? Figures 2 and 3 at K = 200 MeV and M∞ /M = 0.6.

3.3

In?uence of the non-linear terms with gradients

Now we discuss the non-linear interactions 1 Φ α1 α2 (?Φ)2 ? 2 (?W )2 2 2 M gs gv 14 (3.5)

that vanish in in?nite nuclear matter. We recall that these terms are actually of order 5 in the expansion of the e?ective Lagrangian but, following Refs. [5] and [9], we include them because they can be relevant in the surface due to their gradient structure. Using the same saturation properties and scalar mass of Figure 8, in Figure 9 we have calculated Es and t for several values of α1 and α2 with ζ0 = η1 = η2 = fv = 0. One observes that the curves of constant Es are projected onto the plane α1 –α2 as almost parallel straight lines (at least in the analyzed region, corresponding to natural values of α1 and α2 ). The same happens to the curves of constant t. But in contrast with the situation found in the plane η1 –η2 (Figure 8), the slope of the lines of constant t is di?erent from that of the lines of constant Es . This means that by varying α1 and α2 one can achieve some modi?cation on the surface thickness while keeping the same surface energy. For example, if we consider the contour line of Es = 18 MeV we ?nd that for α2 = 2.0 it is t ? 2.05 fm, whereas for α2 = ?1.5 it is t ? 2.25 fm. From Figure 9 we also see that increasing α1 at constant α2 brings about larger values of Es and t, and that the opposite happens if one increases α2 at constant α1 .

? We have repeated the calculations of Figure 9 (K = 200 MeV, M∞ /M = 0.6) for ? K = 350 MeV and for M∞ /M = 0.7, to verify to which extent the behaviour in the α1 –α2

plane is a?ected by the incompressibility and e?ective mass of the interaction. Certainly, the contour lines of Es and t are shifted with respect to Figure 9, but the trends with α1 and α2 turn out to be qualitatively the same. The range of variation of the surface energy

? and thickness in the α1 –α2 region we are considering is shorter when M∞ /M = 0.7, while

it is more or less the same when K = 350 MeV. To assess the importance of the bulk non-linear meson interactions on our study on α1 and α2 , we have performed calculations as in Figure 9 but setting η0 = 2 with η1 = η2 = 0, and setting η1 = 1 with η0 = ∞ and η2 = 0 (as indicated, the e?ect of η2 is much smaller than that of η1 ). The results show a completely similar behaviour to Figure 9. Even the slope of the contour lines of Es and t in the α1 –α2 plane changes only slightly. Comparing with Figure 9, when η0 = 2 one ?nds that Es is shifted by approximately ?1 MeV, and that when η1 = 1 then Es is shifted by around ?3 MeV. The shifts of the surface thickness t are 15

less regular and their magnitude depends on the value of α1 and α2 . In order to investigate the impact of the gradient interactions α1 and α2 on the spin– orbit potential, in Figure 10 we have plotted Vso (z) at the Fermi surface for a few selected values of α1 and α2 . The nuclear matter properties and the scalar mass are the same as in Figure 6, where we studied the dependence of Vso (z) on η0 . One can see that the meson interaction with α1 = 1 and α2 = 0 reduces the strength of Vso (z) and shifts the position of the minimum slightly to the exterior. On the contrary, the interaction with α1 = 0 and α2 = 1 makes the potential well deeper. The combined e?ect is probed in the case α1 = α2 = 1. Since in the relativistic model the spin–orbit force is strongly correlated with

? the Dirac e?ective mass, we compare in Figure 10 the situation at M∞ /M = 0.6 and at ? ? M∞ /M = 0.7. We realize that the incidence of α1 on Vso (z) is weaker for M∞ /M = 0.7. ? The small perturbations arising from the gradient interactions when M∞ /M = 0.7 are not ? su?cient to produce a spin–orbit strength equivalent to that of the case M∞ /M = 0.6.

3.4

Role of the tensor coupling of the omega meson

To conclude this section we investigate the in?uence of the ω tensor coupling ?? (z) ? α ifv βα·?W (z) ?α (z) 2M (3.6)

α

which adds some momentum and spin dependence to the interaction. The natural combination for this coupling is fv /4. Well known from one-boson-exchange potentials (where fv above is commonly written as fv /gv ), the tensor coupling was included in the ?ts to nuclear properties of Refs. [3,16] (conventional QHD) and [5,9] (e?ective ?eld theory), and in the study of the nuclear spin– orbit force in chiral e?ective ?eld theories carried out in Ref. [17]. These works noticed the existence of a trade-o? between the size of the ω tensor coupling and the size of the scalar ?eld. In other words, the tensor coupling breaks the tight connection existing in relativistic models between the value of the nucleon e?ective mass at saturation and the empirical

? spin–orbit splitting in ?nite nuclei (which constrains M∞ /M to lie between 0.58 and 0.64

[7]). Including a tensor coupling the authors of Refs. [5,9,17] were able to obtain natural 16

parameter sets that provide excellent ?ts to nuclear properties and spin–orbit splittings

? with an equilibrium e?ective mass remarkably higher (M∞ /M ? 0.7) than in models that

ignore such coupling. We want to analyze the nature of this e?ect in the simpler but more transparent framework of semi-in?nite nuclear matter. In Figure 11 we have drawn the surface energy coe?cient and the surface thickness as functions of fv in the range [?0.6, 0.9] for two values of the e?ective mass and of the incompressibility, having set ms = 490 MeV, ρ0 = 0.152 fm?3 and av = ?16.42 MeV. To exemplify the incidence of fv on the spin–orbit potential, Figure 12 displays Vso (z) at the Fermi surface for a few of the cases of Figure 11. We also performed the calculations for ms = 525 MeV: Es and t are shifted downwards with respect to Figure 11 and Vso (z) is deeper than in Figure 12, but the global trends with fv are the same. Figure 11 shows the strong reduction of Es and t as fv increases (the slope of the

? ? curves is milder for M∞ /M = 0.7 than for M∞ /M = 0.6). Figure 12 reveals that this fact

is associated with a deeper and wider spin–orbit potential. This agrees with the results of Hofer and Stocker [27] who showed in the standard RMF model that the spin–orbit coupling reduces the surface energy and thickness. At variance with the individual values of Es and t, the ratio Es /t stays to a certain extent constant with fv . Figure 12 evinces the sensitivity of Vso (z) to fv . The lower the nucleon e?ective mass is,

? the larger the e?ect. For M∞ /M = 0.7 we realize that with positive values of fv (? 0.3 in

the present case) one can get a spin–orbit strength comparable, or even stronger, to that

? of the case M∞ /M = 0.6 and fv = 0, something that could not be achieved with natural

values of the couplings studied in the previous sections. Since our parametrization with

? M∞ /M = 0.7 and K = 200 MeV at fv = 0 already has reasonable surface energy and

thickness (Es = 16.6 MeV and t = 1.97 fm), increasing fv results in smaller values of Es and t. This should be compensated with the other couplings (especially α1 and α2 ) that modify the spin–orbit strength to a lesser degree than fv , or the starting point should have other values of the incompressibility K and the scalar mass ms . The spin–orbit e?ect has to do with the explicit dependence of the nucleon orbital wave functions on the spin orientation λ. As described in Ref. [27] nucleons with λ = +1 feel

17

an attractive spin–orbit potential and are pushed to the exterior of the surface, whereas the spin–orbit force is repulsive for nucleons with λ = ?1 which are pushed to the interior. As a consequence of this a depletion of particles with λ = ?1 occurs at the surface. This

? behaviour is contrasted in Figure 13 for fv = 0 and fv = 0.6 in the case M∞ /M = 0.7.

The ?gure depicts the pro?les of the total baryon and tensor densities as well as those of their spin components ρλ (z) and ρλ (z) for λ = ±1, Eqs. (2.11)–(2.14). When the spin–orbit T strength is large, attraction dominates over repulsion and more particles accumulate at the surface than particles are removed from it. Then the total baryon density is enhanced at the surface region and it falls down more steeply.

18

4

Surface properties in the asymmetric case

We brie?y recall some basic de?nitions concerning nuclear surface symmetry properties

(further details on the relativistic treatment of asymmetric in?nite and semi-in?nite nuclear matter can be found in Refs. [25,26,29]). For a bulk neutron excess δ0 = (ρn0 ?ρp0 )/(ρn0 +ρp0 ) (i.e., the asymptotic asymmetry far from the surface), a surface energy coe?cient can be computed as

2 Es (δ0 ) = 4πr0 ∞ ?∞

dz [E(z) ? (av (δ0 ) + M) ρ(z)] ,

(4.1)

where E(z) is the total energy density of the system of neutrons and protons, av (δ0 ) denotes the energy per particle in nuclear matter of asymmetry δ0 , and ρ(z) = ρn (z) + ρp (z) with ρn and ρp referring to the neutron and proton densities, respectively. According to the liquid droplet model (LDM) [18], for small values of the neutron excess Es (δ0 ) can be expanded as follows: Es (δ0 ) = Es + 9J 2 2Es L 2 δ0 + · · · . + 4Q K (4.2)

In this equation J stands for the bulk symmetry energy coe?cient, L reads for the LDM coe?cient that expresses the density dependence of the symmetry energy, and Q is the socalled surface sti?ness coe?cient that measures the resistance of the system against pulling the neutron and proton surfaces apart. All of these macroscopic coe?cients are familiar from semi-empirical LDM mass formulae. Another quantity of interest is the neutron skin thickness Θ, namely the separation between the neutron and proton surface locations:

∞

Θ=

?∞

dz [ρn (z)/ρn0 ? ρp (z)/ρp0 ] .

(4.3)

In ?nite nuclei Θ would correspond to the di?erence between the equivalent sharp radii of the neutron and proton distributions. In the small asymmetry limit the LDM predicts a linear behaviour of Θ with δ0 : Θ= 3r0 J δ0 . 2 Q (4.4)

For calculations of ?nite nuclei of small overall asymmetry I = (N ? Z)/A, the LDM

19

expansion of the energy can be written as E = av + JI 2 A + Es ? 9J 2 2Es L 2 2/3 I A + aC Z 2 A?1/3 + · · · , ? 4Q K (4.5)

where aC is the Coulomb energy coe?cient. Notice that I = δ0 in ?nite nuclei. To describe asymmetric matter in the relativistic approach we need to generalize the energy density (2.1) by including the isovector ρ meson. In terms of the mean ?eld R = gρ b0 , with b0 the time-like neutral component of the ρ-meson ?eld, the additional contributions to Eq. (2.1) read ?? (z) ? α 1 ifρ τ3 βα·?R(z) ?α (z) + R(z) [ρp (z) ? ρn (z)] 4M 2 m2 2 ρ R (z) . 2 gρ (4.6)

α

1 1 Φ(z) ? 2 (?R(z))2 ? 1 + ηρ 2gρ 2 M The symmetry energy coe?cient turns out to be

2 3 2 gρ kF kF 1 + J= . 2 2 m2 1 + η (1 ? M ? /M) ? 2 )1/2 12π ρ 6 (kF + M∞ ρ ∞

(4.7)

In the conventional model one has fρ = ηρ = 0. The isovector tensor coupling fρ was included in the calculations of Refs. [3,5,9,16]. The new non-linear coupling ηρ between the ρ- and σ-meson ?elds is of order 3 in the expansion and it has been introduced in Refs. [5,9]. We will not consider higher-order non-linear couplings involving the ρ meson since the expectation value of the ρ ?eld is typically an order of magnitude smaller than that of the ω ?eld [5,9]. For example, in calculations of the high-density nuclear equation of state, M¨ ller u and Serot [8] found the e?ects of a quartic ρ meson coupling (R4 ) to be only appreciable in stars made of pure neutron matter. On the other hand, in analogy to the couplings α1 and

2 α2 for the σ and ω ?elds, we also tested a surface contribution ?α3 Φ (?R)2 /(2gρ M) and

found that the impact it has on the properties we will study in this section is absolutely negligible. As we have seen, the quantity that governs the surface properties in the regime of low asymmetries is the surface sti?ness Q. Table 1 analyzes the e?ect on Q and L of the couplings discussed in the preceding sections and of the fρ and ηρ parameters. On the basis of Eq. (4.4), we have extracted Q from a linear regression in δ0 to ?t our results for Θ up 20

to δ0 = 0.1. We have set the equilibrium properties to ρ0 = 0.152 fm?3 , av = ?16.42 MeV,

? K = 200 MeV, M∞ /M = 0.6 and J = 30 MeV, and have used a scalar mass ms = 490 MeV

and a ρ-meson mass mρ = 763 MeV. Though here we are interested in tendencies rather than in absolute values, for comparison we mention that NL1 has (units in MeV) J = 43.5, L = 140 and Q = 27 [29], the sophisticated droplet-model mass formula FRDM [34] implies J = 33, L = 0 and Q = 29, and the ETFSI-1 mass formula [35] based on microscopic forces predicts J = 27, L = ?9 and Q = 112. Table 1 shows that the in?uence on the surface sti?ness of the volume self-interactions η0 , η1 and η2 is not very large for natural values of these couplings. In the present case Q is slightly increased by decreasing η0 (i.e., by increasing the quartic vector coupling ζ0 ). For η1 = 1 we ?nd a non-negligible increase of Q, which signals a larger rigidity of the nuclear system against the separation of the neutron and proton surfaces. The e?ect of η2 is again moderate as compared to that of η1 . Q is augmented by a positive ηρ coupling, while a negative ηρ induces a lower value of Q. Some visible changes in Q take place when the α1 and α2 gradient interactions are taken into account. Due to the opposite behaviour of Q with α1 and α2 , the tendencies compensate in a case like α1 = α2 = 1, but the net e?ect is reinforced e.g. if α1 = ?α2 = 1. As one could expect the isoscalar tensor coupling fv has a notable e?ect on Q, even for the relatively small value fv = 0.3 that we have used in Table 1. On the contrary, Q is virtually insensitive to the isovector tensor coupling fρ . The reason is that the derivative of the R(z) ?eld is much smaller than that of the W (z) ?eld. In the least-square ?ts to ground-state properties of Refs. [3,16] nothing was gained by the ρ tensor coupling. In any case, the best ?ts of Refs. [5,9] have fρ ≈ 4. From Table 1 we recognize that the main changes in the coe?cient L arise from the ηρ coupling. As a rule of thumb, increasing values of Q are associated with decreasing values of L for the bulk couplings η0 , η1 , η2 and ηρ . Since L is a bulk quantity, it is not modi?ed by the surface interactions. Figures 14 and 15 illustrate the dependence on asymmetry of the neutron skin thickness, surface energy and surface thickness for some of the cases considered in Table 1. The ?gures

21

extend up to δ0 = 0.3, which widely covers the range relevant for laboratory nuclei (δ0 ≤ 0.2). As the system becomes neutron rich we can appreciate how a neutron skin develops and Θ grows from its vanishing value at δ0 = 0. For small asymmetries the growth is linear in δ0 , as predicted by the LDM. The surface energy coe?cient grows quadratically with increasing neutron excess and the LDM equation (4.2) is clearly a good approximation. In general, the interactions having thicker neutron skins (smaller values of Q) also have larger surface energies. The parameter ηρ can be used for the ?ne tuning of the symmetry properties of the interaction without spoiling the predictions for symmetric systems. If in the conventional ansatz gρ is ?xed by the value of the symmetry energy J, in the extended model J depends on a combination of gρ and ηρ , Eq. (4.7). Therefore, ηρ provides in practice a mechanism that can help to simultaneously adjust Q (to get the required neutron skin Θ) and J (to keep the ?t to the masses) preserving the symmetric surface properties.

22

5

Summary

Within relativistic mean ?eld theory, we have investigated the in?uence on nuclear

surface properties of the non-linear meson interactions and tensor couplings recently considered in the literature. These interactions, beyond standard QHD, are based on e?ective ?eld theories. The e?ective ?eld theory approach allows one to expand the non-renormalizable couplings, which are consistent with the underlying QCD symmetries, using naive dimensional analysis and the naturalness assumption [5,7,8,9]. The quartic vector self-interaction ζ0 makes it possible to obtain a desirable positive value of the coupling constant κ4 of the quartic scalar self-interaction, for realistic nuclear matter properties and within the bounds of naturalness. This ζ0 coupling has only a slight impact on the surface properties. Nevertheless, it helps to ?nd parametrizations where both the surface energy coe?cient Es ant the surface thickness t lie in the empirical region. The ζ0 vector non-linearity makes the spin–orbit potential well deeper, although the e?ect is almost negligible. Concerning the volume non-linear couplings η1 and η2 , they also allow one to obtain positive values of κ4 in the region of naturalness, depending somewhat on the saturation properties (incompressibility and e?ective mass). The surface properties are not much a?ected by these bulk terms either, and it turns out that η2 has a marginal e?ect as compared to that of η1 . The equilibrium properties do not depend on the couplings α1 and α2 that involve the gradients of the ?elds. Thus, these couplings serve to improve the quality of the surface properties without changing the bulk matter. In the conventional σ ? ω model the only parameter not ?xed by the saturation conditions is the mass of the scalar meson. In the α1 –α2 plane the lines of constant Es have a di?erent slope than those of constant t. It is then possible to keep a ?xed value of Es and to modify the value of t by choosing α1 and α2 appropriately. The range of variation of Es and t with α1 and α2 is wider than with the volume couplings. This justi?es including these gradient terms in the energy functional in spite of being of order 5 in the expansion. The α1 and α2 surface meson interactions also in?uence the spin–orbit potential, but the e?ect is not extremely signi?cant. The e?ective model is augmented with a tensor coupling of the ω meson to the nucleon. 23

An outstanding feature is the drastic consequences it has for the spin–orbit force. We have emphasized how inclusion of fv permits to obtain a spin–orbit strentgh similar to that of

? M∞ /M ? 0.6 with larger values of the equilibrium nucleon e?ective mass, contrary to the

phenomenology known from models without such a coupling. We have discussed the implications of the extra couplings of the extended model on various surface symmetry properties. We have restricted ourselves to the regime of low asymmetries, where the liquid droplet model can be applied and the surface sti?ness coe?cient Q is the key quantity. In particular we have pointed out the role that the non-linearity ηρ of the isovector ρ-meson ?eld may play in the details of the symmetry properties.

Acknowledgements

The authors would like to acknowledge support from the DGICYT (Spain) under grant PB95-1249 and from the DGR (Catalonia) under grant GR94-1022. M. Del Estal acknowledges in addition ?nancial support from the CIRIT (Catalonia).

24

References

[1] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [2] L.S. Celenza and C.M. Shakin, Relativistic nuclear physics: theories of structure and scattering (World Scienti?c, Singapore, 1986). [3] P.-G. Reinhard, Rep. Prog. Phys. 52 (1989) 439. [4] B.D. Serot, Rep. Prog. Phys. 55 (1992) 1855. [5] B.D. Serot and J.D. Walecka, Int. J. of Mod. Phys. E 6 (1997) 515. [6] J. Boguta and A.R. Bodmer, Nucl. Phys. A 292 (1977) 413. [7] R.J. Furnstahl, B.D. Serot and H.B. Tang, Nucl. Phys. A 598 (1996) 539. [8] H. M¨ ller and B.D. Serot, Nucl. Phys. A 606 (1996) 508. u [9] R.J. Furnstahl, B.D. Serot and H.B. Tang, Nucl. Phys. A 615 (1997) 441. [10] A.R. Bodmer, Nucl. Phys. A 526 (1991) 703. [11] A.R. Bodmer and C.E. Price, Nucl. Phys. A 505 (1989) 123, and references therein. [12] S. Gmuca, Z. Phys. A 342 (1992) 387. [13] S. Gmuca, Nucl. Phys. A 547 (1992) 447. [14] Y. Sugahara and H. Toki, Nucl. Phys. A 579 (1994) 557. [15] K. Sumiyoshi, H. Kuwabara and H. Toki, Nucl. Phys. A 581 (1995) 725. [16] M. Rufa, P.-G. Reinhard, J.A. Maruhn, W. Greiner and M.R. Strayer, Phys. Rev. C 38 (1988) 390. [17] R.J. Furnstahl, J.J. Rusnak and B.D. Serot, Nucl. Phys. A 632 (1998) 607. [18] W.D. Myers and W.J. Swiatecki, Ann. Phys. (N.Y.) 55 (1969) 395; Ann. Phys. (N.Y.) 84 (1974) 186; W.D. Myers, Droplet model of atomic nuclei (Plenum, New York, 1977). [19] M. Centelles, X. Vi? as and P. Schuck, Phys. Rev. C 53 (1996) 1018. n [20] M. Brack, C. Guet and H.-B. H? akansson, Phys. Rep. 123 (1985) 275; J. Treiner and H. Krivine, Ann. Phys. (N.Y.) 170 (1986) 406; K. Kolehmainen, M. Prakash, J.M. Lattimer and J. Treiner, Nucl. Phys. A 439 (1985) 537. [21] W. Stocker, J. Bartel, J.R. Nix and A.J. Sierk, Nucl. Phys. A 489 (1988) 252. 25

[22] W. Stocker and M.M. Sharma, Z. Phys. A 339 (1991) 147; M.M. Sharma, S.A. Moszkowski and P. Ring, Phys. Rev. C 44 (1991) 2493. [23] M. Centelles, X. Vi? as, M. Barranco and P. Schuck, Ann. Phys. (N.Y.) 221 (1993) 165; n M. Centelles and X. Vi? as, Nucl. Phys. A 563 (1993) 173; M. Del Estal, M. Centelles n and X. Vi? as, Phys. Rev. C 56 (1997) 1774. n [24] C. Speicher, R.M. Dreizler and E. Engel, Nucl. Phys. A 562 (1993) 569. [25] D. Von-Ei?, J.M. Pearson, W. Stocker and M.K. Weigel, Phys. Lett. B 324 (1994) 279. [26] M. Centelles, M. Del Estal and X. Vi? as, Nucl. Phys. A 635 (1998) 193. n [27] D. Hofer and W. Stocker, Nucl. Phys. A 492 (1989) 637. [28] D. Von-Ei?, W. Stocker and M.K. Weigel, Phys. Rev. C 50 (1994) 1436. [29] D. Von-Ei?, H. Freyer, W. Stocker and M.K. Weigel, Phys. Lett. B 344 (1995) 11. [30] P. Hohenberg and W. Kohn, Phys. Rev. B 136 (1964) 864. [31] C. Speicher, R.M. Dreizler and E. Engel, Ann. Phys. (N.Y.) 213 (1992) 312. [32] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1133. [33] P.-G. Reinhard, M. Rufa, J. Maruhn, W. Greiner and J. Friedrich, Z. Phys. A 323 (1986) 13. [34] P. M¨ller, J.R. Nix, W.D. Myers and W.J. Swiatecki, At. Data Nucl. Data Tables 59 o (1995) 185. [35] Y. Aboussir, J.M. Pearson, A.K. Dutta and F. Tondeur, Nucl. Phys. A 549 (1992) 155.

26

Table captions

Table 1. The surface sti?ness coe?cient Q and the coe?cient L for several values of the couplings analyzed in the text. We have set ρ0 = 0.152 fm?3 , av = ?16.42 MeV,

? K = 200 MeV, M∞ /M = 0.6, J = 30 MeV and ms = 490 MeV.

27

Table 1

η0 ∞ 5 2 ∞

η1 0

η2 0

ηρ 0

α1 0

α2 0

fv 0

fρ 0

Q (MeV) 21 21.5 22

L (MeV) 96 95 90 91 93 89 87 119 96 96 96 96 96 96 96

1 0 1

0 1 1 0

0

0

0

0

0

24.5 22 25.5

∞ ∞

0

1 ?1

0

0

0

0

23 18

0

0

0

1 0 1

0 1 1

0

0

17 25 19 16

1 ?1 ∞ ∞ 0 0 0 0 0 0.3 ?0.3 0 0 0 0 0 0 5 0

24 19 21

28

Figure captions

Figure 1. The couplings gs /4π, gv /4π, κ3 , κ4 and ζ0 versus the parameter η0 de?ned in Eq. (3.3). The naturalness assumption requires all these couplings to be of order unity. We have taken ρ0 = 0.152 fm?3 (kF = 1.31 fm?1 ), av = ?16.42 MeV and ms = 490 MeV.

? Figure 2. Surface energy coe?cient Es for several values of the parameter η0 , K, M∞ /M

and ms , with ρ0 = 0.152 fm?3 and av = ?16.42 MeV. Figure 3. Surface thickness t of the baryon density pro?le for several values of the param? eter η0 , K, M∞ /M and ms , with ρ0 = 0.152 fm?3 and av = ?16.42 MeV.

Figure 4. Surface energy coe?cient Es and surface thickness t for several values of the

? parameter η0 , M∞ /M and ms , with ρ0 = 0.1484 fm?3 , av = ?15.75 MeV and K = 200

MeV (Ref. [10]). Figure 5. Baryon density ρ(z) and surface tension density σ(z) = E(z) ? (av + M)ρ(z) of semi-in?nite nuclear matter for some values of the parameter η0 . It is ρ0 = 0.152 fm?3 , av = ?16.42 MeV, K = 200 MeV and ms = 490 MeV. Figure 6. Orbital part of the spin–orbit potential Vso (z) at the Fermi surface and central mean ?eld Vc (z), Eqs. (2.17) and (2.19) respectively, for some values of the parameter η0 . It is ρ0 = 0.152 fm?3 , av = ?16.42 MeV, K = 200 MeV and ms = 490 MeV. Figure 7. The couplings gs /4π, gv /4π, κ3 and κ4 against the parameters η1 (left) and η2 (right). With ρ0 = 0.152 fm?3 , av = ?16.42 MeV and ms = 490 MeV, results are

? shown for K = 200 MeV and M∞ /M = 0.6 in part (a), and for K = 350 MeV and ? M∞ /M = 0.7 in part (b).

Figure 8. Level curves in the plane η1 –η2 of the surface energy coe?cient Es (in MeV, solid lines) and of the surface thickness t (in fm, dashed lines), with ζ0 = α1 = α2 = fv = 0. The point η1 = η2 = 0 is marked by a cross. It is ρ0 = 0.152 fm?3 , av = ?16.42 MeV,

? K = 200 MeV, M∞ /M = 0.6 and ms = 490 MeV.

29

Figure 9. Same as Figure 8 in the plane α1 –α2 , with ζ0 = η1 = η2 = fv = 0. The point α1 = α2 = 0 is marked by a cross. Figure 10. Orbital part of the spin–orbit potential Vso (z) at the Fermi surface for some values of the couplings α1 and α2 . The equilibrium properties of nuclear matter and the scalar mass are the same of Figure 6. Figure 11. Surface energy coe?cient Es and surface thickness t as functions of the strength fv of the ω-meson tensor coupling, with ζ0 = η1 = η2 = α1 = α2 = 0. We have set ρ0 = 0.152 fm?3 , av = ?16.42 MeV and ms = 490 MeV. Figure 12. Orbital part of the spin–orbit potential Vso (z) at the Fermi surface for some values of the tensor coupling fv . The equilibrium properties of nuclear matter and the scalar mass are the same of Figures 6 and 10. Figure 13. Total baryon density ρ(z), total tensor density ρT (z) and their components ρλ (z) and ρλ (z) for the two spin orientations λ = ±1. They have been calculated for T

? fv = 0 and fv = 0.6, with M∞ /M = 0.7, K = 200 MeV and ms = 490 MeV.

Figure 14. Neutron skin thickness Θ as a function of the bulk neutron excess δ0 . The solid line is the result of the conventional model (ζ0 = η1 = η2 = ηρ = α1 = α2 = fv = fρ = 0). The other lines di?er from the latter in the indicated parameter. Figure 15. Same as Figure 14 for the surface energy coe?cient Es and the surface thickness

2 t as functions of the bulk neutron excess squared δ0 .

30

50 40 30 20 10

Coupling Constants K = 200 MeV M*/M = 0.6 ∞

gs/4π

κ3 κ4

K = 200 MeV M*/M = 0.7 ∞

g v/4π

ζ0

0 -10 -20

K = 350 MeV 40 M*/M = 0.6 ∞ K = 350 MeV M*/M = 0.7 ∞

30 20 10 0 -10 -20

∞

5

2 1 0.5

η0

5

2 1 0.5

20

K = 125 MeV M*/M = 0.7 ∞ ms= 490 MeV ms= 525 MeV

16

12 20

16

12

Es[MeV]

K = 200 MeV M*/M = 0.6 ∞

K = 200 MeV M*/M = 0.7 ∞

35 30 25 20 15

∞

K = 350 MeV M*/M = 0.6 ∞

K = 350 MeV M*/M = 0.7 ∞

5

2 1 0.5

η0

5

2 1 0.5

2.4

ms= 490 MeV ms= 525 MeV

K = 125 MeV M*/M = 0.7 ∞

2.2 2.0 1.8 1.6

2.4 K = 200 MeV M*/M = 0.6 ∞ 2.2 2.0 1.8

t [fm]

K = 200 MeV M*/M = 0.7 ∞

1.6 2.4 K = 350 MeV M*/M = 0.6 ∞ 2.2 2.0 1.8 1.6

∞

K = 350 MeV M*/M = 0.7 ∞

5

2 1 0.5

η0

5

2 1 0.5

24 20 16 12 8 20

Es[MeV]

η0= 0.5

η0= 0.5

3.0 2.5 2.0 1.5 1.0 2.5

t [fm]

η0= 2

η0= 2

16 12 8

η0= ∞ η0= ∞

2.0 1.5 1.0 2.5 2.0 1.5

ms= 450

20 16 12

, 500

, 550 0.60

MeV

0.65 1.0 0.70

8 0.55

0.60

0.65

0.70 M*/M ∞

0.16

ρ

M*/M = 0.6 ∞

σ

ρ σ

M*/M = 0.7 ∞

0.8 0.6 0.4 0.2 0.0 -0.2

σ [MeV·fm-3]

0.12

ρ [fm -3]

η0= ∞ η0= 2 η0= 0.5

0.08

0.04

0.00 -8

-6

-4

-2

0

2

-8 z [fm]

-6

-4

-2

0

2

4

0

M*/M = 0.6 ∞ M*/M = 0.7 ∞

0 -20

Vc [MeV]

η0= ∞

-40

η0= 2 η0= 0.5

Vso

Vso

-2

-60

Vc

Vc

-4

-80 -8

-6

-4

-2

0

2

-8 z [fm]

-6

-4

-2

0

2

4

Vso [MeV·fm]

30

Coupling Constants

K = 200 MeV

η0= ∞, η2= 0

M*/M = 0.6 ∞

η0= ∞, η1= 0

20 10 0 -10 -20 -30 -2

gs/4π gv/4π

(a)

0 2

η1

κ4

κ3

4

6

-2

0

2

η2

4

6

8

30

Coupling Constants

K = 350 MeV

η0= ∞, η2= 0

M*/M = 0.7 ∞

η0= ∞, η1= 0

20 10 0 -10 -20 -30 -2 0 2

η1

gs/4π gv/4π

κ4 κ3

(b)

-2 0 2

η2

4

6

4

6

8

2.0 1.5 1.0 0.5 η2 0.0

2.2

2.1 2.0 1.9

-0.5 -1.0 -1.5

-2.0 -0.5 0.0 0.5 1.0 1.5 η1

2.0

M*/M = 0.6 ∞ 1.5 K = 200 MeV

1 .8 2 .0 2.2

2.4 2 .6

1.0 0.5 α2 0.0 -0.5 -1.0

-1.5 -1.5 -1.0 -0.5 0.0

0.5

1.0

2 .8

1.5

3.0

2.0

α1

M*/M = 0.6 ∞

M*/M = 0.7 ∞

0

VSO [MeV·fm]

-2

α1=0, α2=0 α1=1, α2=0 α1=0, α2=1 α1=1, α2=1

-4

-8

-6

-4

-2

0

2

-8 z [fm]

-6

-4

-2

0

2

4

25

2.8 2.4

20 2.0 15

K = 200 MeV

1.6

M*/M = 0.6 ∞

Es[MeV]

10

M*/M = 0.7 ∞

1.2 2.4 2.0

20

15 1.6 10 -0.6 1.2 0.9

-0.3

0.0

0.3

0.6

-0.6 fv

-0.3

0.0

0.3

0.6

t [fm]

K = 350 MeV

0

VSO [MeV·fm]

M*/M = 0.6 ∞

M*/M = 0.7 ∞

-2

fv= 0 fv= -0.3

-4

fv= 0.3 fv= 0.6

-6 -8

-6

-4

-2

0

2

-8 z [fm]

-6

-4

-2

0

2

4

0.16 0.12

ρ [fm -3] ρ(z)

λ= + 1 λ= - 1 ρ(z)

total 0.08

f v= 0

0.04 10xρT(z) 0.00 -0.04 -8 10xρT(z)

fv= 0.6

-6

-4

-2

0

2

-8 z [fm]

-6

-4

-2

0

2

4

1.0

η0= 2

0.8 0.6 0.4 0.2

η1= 1 ηρ= 1 α1= 1

Θ [fm]

fv = 0.3

0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30

δ0

30

Es [MeV]

η0= 2 η1= 1 ηρ= 1

25

20

15

α1= 1

fv = 0.3

3.3

t [fm]

2.8 2.3 1.8 0.00

0.02

0.04

δ0

2

0.06

0.08

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