Surface Incompressibility from Semiclassical Relativistic Mean Field Calculations

By using the scaling method and the Thomas-Fermi and Extended Thomas-Fermi approaches to Relativistic Mean Field Theory the surface contribution to the leptodermous expansion of the finite nuclei incompressibility has been self-consistently computed. The validity of the simplest expansion, which contains volume, volume-symmetry, surface and Coulomb terms, is examined by comparing it with self-consistent results of the finite nuclei incompressibility for some currently used non-linear sigma-omega parameter sets. A numerical estimate of higher-order contributions to the leptodermous expansion, namely the curvature and surface-symmetry terms, is made.

The curvature of the nuclear matter equation of state (EOS), i.e., the nuclear matter incompressibility K ∞ is a key quantity in nuclear physics because it is related to many properties of nuclei (such as radii, masses and giant resonances), heavy-ion collisions, neutron stars and supernova collapses. One important source of information on K ∞ is provided by the study of the isoscalar giant monopole resonance (GMR) (breathing mode) in finite nuclei.
The nuclear matter incompressibilty K ∞ is not a directly measurable quantity, what is measured is, actually, the energy E M of the GMR of finite nuclei. It is convenient to write this energy in terms of the incompressibility K A for a finite nucleus of mass number A as: where < r 2 > is the rms matter radius and M the nucleon mass. The finite nucleus incompressibilty K A can be parametrized by means of a leptodermous expansion [2] which is similar to the liquid drop mass formula: where I = (N − Z)/A is the neutron excess. Eq. (2) suggests that it is possible to fit the coefficients of the expansion to the experimental data in a model independent way.
Although some effort along these lines has been made in the past [9], the fact that a fit of the parameters of Eq. (2) to experimental data does not lead to a unique determination of the parameters is well established [6,10,11]. Rather, the nuclear matter incompressibility has to be determined from effective forces which reproduce, in a microscopic calculation, the experimental values of the GMR excitation energy in heavy nuclei [6].
It is also possible to fit K A calculated microscopically within the scaling model for a given effective interaction to the leptodermous expansion Eq. (2). This has been done, for example, in the non-relativistic frame using Skyrme forces [12]. In this case the coefficients entering Eq. (2) can be expressed through infinite and semi-infinite nuclear matter properties calculated with the Hartree-Fock approximation for each considered interaction. In particular, the volume-symmetry (K vs ) and Coulomb (K coul ) coefficients depend on some parameters of the liquid droplet model [13] computed only using nuclear matter properties [2]. The surface coefficient K sf , also derived in [2], can be written as [14]: The surface tension σ is calculated in symmetric semi-infinite nuclear matter and is defined as: where ρ is the density profile whose central value is given by ρ c = ρ(−∞), H is the energy density and e ∞ is the energy per particle in nuclear matter at density ρ c . In Eq. (3) dots indicate the derivatives with respect to the central density and all the quantities are evaluated at a central density equal to the nuclear matter saturation density ρ 0 , which is related to the radius constant r 0 through 4πr 3 0 ρ 0 /3 = 1. The key quantity entering Eq. (3) isσ which is the second derivative of σ(ρ c ) with respect to ρ c calculated at ρ c = ρ 0 . The determination ofσ also requires knowledge of how the density profile ρ is modified during compression [15]. In the study of the breathing mode a scaling transformation of the densities is assumed. Actually, the coefficients entering the parametrization (2) can be derived under this hypothesis [2]. The scaling transformation means that the density changes according to the transformation r → λr and consequently ρ λ (r) = λ 3 ρ(λr).
Thus, in the scaling approach: To obtain the surface incompressibility coefficient K sf for a given effective interaction, it is necessary, first of all, to calculate the scaled surface tension σ λ by replacing the densities by the scaled densities given by Eq. (5) in Eq. (4). In the non-relativistic frame this can be easily done within the Hartree-Fock scheme using zero-range Skyrme forces and a simple analytical expression for σ λ is obtained [12,14].
The self-consistent calculation of K sf within the RMF approximation using the σ − ω model is more involved due to the problem of the change in the meson fields induced by the scaled nuclear densities [16]. To our knowledge, only approximate calculations of K sf have been developed in the past for the relativistic model. This is the case of the Relativistic Thomas-Fermi (RTF) calculations of Refs. [16,17] where a local density approximation of the meson fields was used. Another approach is related with the study of nuclei under an external pressure. Starting from a schematic energy density functional and adding a density-dependent constraint which simulates the pressure, analytical expressions for the surface tension σ as a function of the bulk density ρ c can be derived for a wide class of compression modes, in particular, for the scaling mode [15]. This way one obtains the following formula forσ in the scaling mode: where α is the surface diffuseness parameter of a symmetric Fermi density. This pocket formula has been employed to estimate K sf in the RMF model for several non-linear σ − ω parameter sets [19]. A symmetric Fermi function that reproduces in the best way the density profile obtained from a Hartree calculation of semi-infinite nuclear matter has been used in ref. [19] to determine the α parameter of Eq. (7).
Very recently, the scaling method applied to the RMF theory in the RTF and Relativistic Extended Thomas-Fermi (RETF) approaches has been used to self-consistently obtain the excitation energy of the GMR of finite nuclei [20,21]. Our aim in the present paper is, firstly to obtain the surface coefficient K sf self-consistently in the RTF and RETF approaches developed in [20,21] for some linear and non-linear σ −ω parameter sets. On the other hand, we want to check whether the leptodermous expansion of the finite nucleus incompressibility Eq. (2) can reproduce the corresponding fully self-consistent value obtained in the RETF approach [21] with some selected non-linear σ − ω parameter sets.
The key point of our semiclassical approach is that the local Fermi momentum k F and the effective mass m * scale as [20,21]: wherem * is still a function of λ. With the help of Eq. (8), the nuclear part of the energy and the scalar density includingh 2 corrections, which are functionals of k F and m * , scale as: AgainẼ andρ s are functions of the collective coordinate λ because of their dependence oñ m * . Thus the scaled surface tension can be written as [20][21][22]: where all densities and fields depend on the variable λz. With the help of the Klein-Gordon equations for the scaled vector and scalar fields derived from (10), the scaled surface tension can be recast as: Using the explicit RTF or RETF expressions for the nuclear part of the energy and for the scalar density [20][21][22][23] together with the Klein-Gordon equations for V λ , φ λ , ∂V λ ∂λ and ∂φ λ ∂λ derived from Eq. (10), after some algebra the first and second derivatives of the scaled surface tension σ λ with respect to λ at λ = 1 read (see Refs. [20,21] for more details): and The first derivative of e ∞ (λ 3 ρ 0 ) at λ = 1 is just three times the pressure calculated at saturation density and thus it vanishes, while the second derivative gives K ∞ ρ [21,26]. On the other hand, since in the self-consistent RTF and RETF calculations the inputs for computing Eqs. (12)-(13) are quantities obtained from the solution of the variational equations associated with the surface tension (10) at λ = 1, the so-called "sigma dot" theorem is rigorously fulfilled [27]. The method therefore allowsσ and consequently K sf to be computed on top of a self-consistent RTF or RETF calculation of the surface tension in symmetric semi-infinite nuclear matter. This is similar to what happens in the non-relativistic frame with Skyrme forces [14], although in the relativistic case additional Klein-Gordon equations for ∂V λ ∂λ and ∂φ λ ∂λ at λ = 1 have to be solved. Now we shall discuss the results obtained from the self-consistent RTF and RETF methods in the scaling approximation. Table 1 collects K ∞ ,σ and K sf for the non-linear NL-Z2 [28], NL1 [29],NL3 [30], NL-RA1 [31], NL-SH [32] and NL2 [33] and the linear HS [34] and L1 [33] parameter sets. One observes that in both the RTF and RETF calculationsσ and K sf decrease (become more negative) with increasing bulk incompressibility K ∞ . The RTF and RETF values ofσ and K sf for a given parameter set are, in general, rather different from one another, which means that the precise value of these quantities is model dependent.
This is known to happen also with other quantities related with the nuclear surface. For example, such is the case of the surface energy coefficient of the leptodermous expansion of the binding energy of a nucleus, which is calculated as 4πr 2 0 σ. The quality of the RTF and RETF approximations for semi-infinite nuclear matter and finite nuclei with respect to the RMF Hartree approach, and its dependence on the effective interaction, was investigated in Refs. [22,24,25] Table 1.
Another different approach to computing K sf was proposed in Refs. [16,17]. It is based on the scaling method together with a local density approximation for the meson fields within the RTF approach. In Ref. [ [16].
It should also be pointed out that in our self-consistent semiclassical calculations we find that the ratio between the surface and bulk incompressibilities increases with K ∞ (in agreement with the results of [17]). In the RETF case this ratio is close to one, as happens for the non-relativistic Skyrme forces [5], provided that the bulk incompressibility K ∞ of the interaction is not excessively high. In the RTF case the ratio between the surface and bulk incompressibilities increases much faster with K ∞ than in the RETF calculations, and it considerably differs from unity for parametrizations with either a very low or a very high bulk incompressibility. In Figure 1 we plot −K sf as a function of K ∞ for the parameter sets considered in Table 1. As in the non-relativistic case [2], The surface incompressibility coefficient is both large and negative, thus its contribution considerably reduces the finite nucleus incompressibility K A with respect to the nuclear matter limit K ∞ . This result, although obtained in the scaling model, illustrates the physical effect that the compression of the surface provides a considerable reduction of K A , which is also found in more fundamental RPA calculations [6]. In Ref. [21] we have self-consistently computed the finite nucleus incompressibility K A using the RETF approach and the scaling method which we have employed in the present work to obtain K sf . Thus we can now precisely check the ability of the leptodermous expansion Eq. (2) in reproducing the full calculation of K A carried out in Ref. [21] for various finite nuclei.
The coefficients K vs and K coul entering Eq. (2) are computed using nuclear matter properties only. Explicit expressions for these coefficients in the non-linear σ − ω model are reported in [18]. In our analysis we will use the NL1, NL3 and NL-SH parameter sets for which the numerical values of these coefficients are given in [19]. The surface incompressibility coefficient is the self-consistent value taken from Table 1. Table 2  Now we would like to analyze whether the addition of some higher order terms in the leptodermous expansion Eq. (2) improves the agreement with the K A results calculated selfconsistently. In particular, we will focus our attention on the curvature K cv A −1/3 and the surface-symmetry K ss I 2 A −1/3 terms. Although these terms could be derived by enlarging the leptodermous expansion of Blaizot [2], as has been done in the non-relativistic case [12], it becomes more complicated in the relativistic case. Thus, for a fast estimate of the curvature and surface-symmetry terms, we perform a numerical fit. To do this, we follow the same strategy as in Ref. [12]. First we consider symmetric nuclei with the Coulomb force switched off. In this case the leptodermous expansion Eq.(2) (adding the curvature term) reduces to: In Figure 2 we plot [K A − K ∞ ]A 1/3 versus A −1/3 for the three parameters sets used in this analysis. Here K ∞ is the nuclear matter incompressibility given in Table 1 and K A are the self-consistent incompressibilities calculated for A ranging from 300 up to 300000. In the linear fit of these curves the y−axis intercept gives K sf of the corresponding force, while the slope gives K cv . The surface terms obtained in this way are −246.1, −328.4 and −435.8 MeV for the NL1, NL3 and NL-SH parameter sets, which are very close to the corresponding self-consistent values (see Table 1). The estimates of the curvature term in the leptodermous expansion of the finite nucleus incompressibility obtained with NL1, NL3 and NL-SH are −317.2, −229.8 and −185.6 MeV respectively.
To obtain the surface-symmetry contribution, we have found it convenient to parametrize the difference between the self-consistent incompressibilities K A of a given nucleus with neutron excess I and the corresponding symmetric nucleus as: where again uncharged nuclei have been considered. For each parameter set and according to (15), if [K A,I − K A,I=0 ]I −2 is plotted versus A −1/3 a unique curve should be found which is independent of the value of I. However, one obtains a family of almost parallel lines whose slope is K ss . The splitting of these lines gives us information on the higher order symmetry contributions missed in the parametrization (15). Thus we will estimate the surface-symmetry term from a linear fit of the curve corresponding to I = 0.1, which roughly corresponds to an average asymmetry along the periodic table. This curve is plotted in Figure 3 for surface-symmetry contributions, we fit the self-consistent results for the finite nuclei considered in Table 3 to a leptodermous expansion including curvature and surface-symmetry terms. The volume, surface and volume symmetry coefficients are taken from self-consistent infinite and semi-infinite nuclear matter calculations. The results of this calculation show that the difference of the curvature contribution obtained from the fit in the asymptotic region and from finite nuclei is always less than 10%, whereas the difference in the surfacesymmetry contribution lies below 3%.
We have applied the scaling method in the Thomas-Fermi and Extended Thomas-Fermi approximations to the Relativistic Mean Field Theory to self-consistently calculate the surface coefficient K sf of the leptodermous expansion of the finite nucleus incompressibility derived within the Blaizot model. The ratio between the surface and bulk incompressibilities obtained in our semiclassical calculation increases with the nuclear matter incompressibility, more strongly in the RTF than in the RETF case. In the RETF calculations this ratio is close to one, as in the case of non-relativistic Skyrme forces, for the non-linear parameter sets which have a nuclear matter incompressibility not larger than roughly 300 MeV.
For the analyzed σ − ω parameter sets, the leptodermous expansion Eq. (2) is not able to reproduce very precisely the finite nuclei incompressibilities obtained self-consistently.
In particular the macroscopic contribution of the Coulomb force can differ from the selfconsistent contribution up to 6 MeV. We have numerically estimated higher order contributions to the leptodermous expansion, namely curvature and surface-symmetry terms, in the asymptotic region (i.e., for very large uncharged systems). We have found that the finite nuclei incompressibilities are reasonably well reproduced by an extended leptodermous expansion which includes curvature and surface-symmetry contributions.