BOLOGNA SECTION. Paolo Finelli University of Bologna

MB31 BOLOGNA SECTION Paolo Finelli University of Bologna paolo.finelli@unibo.it INFN-MB31 Collaboration Meeting in Trento - December 16/17, 011

latest comment of the MB31 referees...a lot of experience. The project is interesting perhaps a little too broad. Each group makes its own contribution. It would be nice to have more relation between the different groups...

Outline of the talk Previous works: my skills, what I can do and topics on which we can start a collaboration (see latest work with Lecce, Pavia and Orsay) Future works: what I plan to do (a chance for future collaborations?) Recent calculations: pairing from two- and threebody forces in finite nuclei/ ground-state properties from mean-field approaches

PREVIOUS WORKS parity violating electron scattering neutron weak charge >> proton weak charge is small, best observed by parity violation ) ( ) ( ) ˆ( 5 r A r V r V! + = ) ( ) ( / / / 3 r r r Z r d r V! = " # [ ] ) ( ) ( ) 4sin (1 ) ( r N r Z G r A N P W F!! " # # = ) ( Q F d d d d P! Mott =! " "! = ) ( ) ( 4 1 ) ( 0 3 r qr j r d Q F P P " #! = ) ( ) ( 4 1 ) ( 0 3 r qr j r d Q F N N " # Electron - Nucleus Potential electromagnetic axial Neutron form factor Parity Violating Asymmetry A(r) 1 4sin 1 <<! W " Proton form factor! " # $ % & ' ' = ( ) * +, -. + ( ) * +, -. ( ) * +, -. ' ( ) * +, -. = ) ( ) ( 4sin 1 Q F Q F Q G d d d d d d d d A P N W F L R L R / 01 from Urciuoli, PREX workshop@trento 009

PREVIOUS WORKS parity violating electron scattering 08 Pb experimental values (E=1 GeV and Θ=5 o ) (Michaels, PAVI workshop 011, submitted to PRL?) e 05 Theoretical calculations for tin isotopes A l 1e 05 Sn 116 118 10 1 14 850MeV 0 10 F [q]/n 10 7 Sn 116 118 10 1 14 10 1 0 1 3 q[fm 1 ]

PREVIOUS WORKS Relativistic mean field models GROUND STATE PROPERTIES (RHB in open shell nuclei) Meson-exchange non linear density dependent B exp -B th (MeV) 6 4 0 - -4 RHB/DD-ME -6 0 50 100 150 00 50 A FIG.. Absolute deviations of the binding energies calculated with the DD-ME interaction from the experimental values [17]. Point coupling linear density dependent -phenomenological -chiral dynamics inspired

PREVIOUS WORKS Relativistic mean field models excited STATE PROPERTIES (qrpa in open shell nuclei) Meson-exchange non linear density dependent R (10 3 fm 4 ) 0 15 10 (a) ISGM 08 Pb 13.9 MeV R (e fm ) 30 0 (b) IVGD 13.5 MeV 10 5 0 0 10 0 E (MeV) 0 0 10 0 E (MeV) Point coupling linear density dependent -phenomenological -chiral dynamics inspired

PREVIOUS WORKS!-hypernuclei properties Fig. 3. Binding energies of the Λ in different (s, p,...) orbitals of six hypernuclei (cf. Tables and 3), calculated with the FKVW density functional using the three parameter sets for the ΛN couplings (cf. Table 1). Results are plotted as functions of the mass number and compared with experimental energies [1]. Also shown is a Woods Saxon fit [15] DDRMF: C. M. Keil, F. Hofmann and H. Lenske, Phys. Rev. C 61 (000) 064309. (dashed curves) to guide the eye. QMC: P. A. M. Guichon, A. W. Thomas and K. Tsushima, Nucl. Phys. A 814 (008) 66. hypernuclei vanishing spin-orbit splittings RMF + tensor: J. Mares and B. K. Jennings, Phys. Rev. C 49 (1994) 47 Table 4 16 40 89 139 08 P -shell spin orbit splittings #$ Λ (p) for six hypernuclei ( 13 Λ C, Λ O, Λ Ca, Λ Y, Λ La, Λ Pb). Experimental values [44], or empirical estimates [1,47,48], are shown in comparison with our theoretical predictions (FKVW), using a! DDRMF broad range of ζ parameters (see Eq. (1)), and other relativistic calculations with (RMFI [11]) or without (RMFII [14]) QMC tensor coupling. 1.5 All energies are given in kev. The asterisk means that a local fit has been necessary. E RMF + tensor Exp.Experimental data p3/ s 16 O Λ " p1/ 13 C Λ!E (p) (MeV) Nucleus 40 Ca Λ 89 Y Λ 139 La Λ 1 0.5 0 RMFI [11] RMFII [14] [kev] FKVW (0.4! ζ! 0.66) 15 ± 54 ± 36 [44] 160!! 510 310 1100 300!! 600 [47] 800!! 00 [1] 10!! 490 70 1400 140!! 40 10 1400 90 [48] 40!! 180 110 700 0!! 80 50 300 0!! 70 50 300!-hypernuclei properties Pb 08 Λ # &$% $%&'()* " #$%!"#!!"#$%&'$()*+,*)* -0.5 " " spin orbit correction and the Pauli-blocking effect at the quark level. Without these corrections GΛ S,V the resulting energy levels ζshow = anstrong overbinding (cf. Table 4 in Ref. [1]). A very recent 16 G -1 O improvement [13] solved the overbinding problem, introducing the scalar polarizability of the 0.1 Λ0 0. 0.05 S,V 0.15 nucleon in a self-consistent way instead of the-/3 Pauli blocking correction. In Tables and 3 we ratio between nucleon and A have included the latest update ofstrengths these calculations.! coupling (0.3-0.5 test from QCD Fig. 3 provides a further of sum therules) sensitivity of calculated single-λ energies with respect -453 <4in this case 0.5 Λ(0) (0) 678%'$,,=>$+*)&?&@)&; to a variation of the ratio ζ = GS,V /GS,V between contact terms representing the in-medium 13 4*%)%$$9-:;7 $*(=-&$+A condensate background fields for the hyperon and the nucleons. For the six hypernuclei listed in -./0

future WORKS particle-vibration coupling Particle-vibration coupling is still an open issue: so far no theoretical approach is self-consistent k k + k E. Litvinova et al. Corrections are always on top of the mean-field calculation

future WORKS particle-vibration coupling: opm? In this approach E xc is expressed in terms of the KS orbitals and eigenvalues rather than density itself KS potentials Density Total energy Kinetic term External potential Hartree potential Exchange correlation term

future WORKS particle-vibration coupling: opm? External potential Hartree potential Exchange correlation term Explicit dependence on KS orbitals and energies

future WORKS correlations in mean field models Correlations between different observables for several relativistic and non-relativistic functionals Correlations between nuclear matter and finite nuclei observables

future WORKS NON-RELATIVISTIC ENERGY DENSITY FUNCTIONAL FROM CHIRAL DYNAMICS Energy per particle Effective mass Gradient function Spin-orbit function

latest WORKS applications OF IN-MEDIUM CHIRAL DYNAMICS TO NUCLEAR STRUCTURE: a realistic pairing interaction (NUCL-TH/1111.4946, submitted to prc) With D. Vretenar and T. Niksic (University of Zagreb); Many thanks to N. Kaiser, W. Weise and J. Holt (Technical University of Munich) for numerical results.

INTRODUCTION Keep things as simple as possible: find a fast and computationally efficient way to include a realistic pairing in HFB calculations (of course, using explicit formulas is not an option). realistic? -body plus 3-body (with some approximations) Solve the gap equation for symmetric nuclear matter. Use a parametrized pairing force that, on one hand, reproduces the realistic pairing gap in symmetric nuclear matter and, on the other hand, is easy to use for finite systems. Try to include all the physical informations. Not only the pairing gap at some Fermi momentum, but all k F -dependence: fix parameters in order to reproduce Δ(k F ). Test the realistic potential in finite nuclei: pairing gaps in isotopic and isotonic chains.

the gap equation V: pairing potential Δ: pairing field (k, k F )= 1 4π 0 p V (p, k) (p, k F ) [E(p, kf ) E(k F,k F )] + (p, k F ) dp ε: quasiparticle energy ε F : Fermi energy E(p, k F ) E(k F,k F )= p k F M (k F ) V(p,k) = V B (p,k) + V 3B (k F,p,k) Two-body: N3LO by Machleidt & Entem [Phys. Rep. 503 (011) 1]. Three-body: effective two-body density-dependent interaction developed by Holt, Kaiser and Weise [PRC 81 (010) 0400]. See also Hebeler et al. [PRC 8 (010) 014314 and nucl-th/1104.955]. For the energies we could use the bare mass (M N ) but it would be better to use an effective mass (M*) [Fritsch, Kaiser and Weise, NPA 750 (005) 59 or Holt, Kaiser and Weise, nucl-th/1107.5966].

-body interaction (n3lo) Phase shifts of np scattering as calculated from NN potentials at different orders of ChPT LO NLO NLO N3LO (black dots are experimental data) [See also Epelbaum, Hammer and Meissner, RMP 81 (009) 1773]

results FROM n3lo (with bare mass) 4 ~0.3 MeV Charge Dependence [Hebeler, PLB 648 (007) 176] 1 S 0 superfluid pairing gap F (MeV) 3 1 Neutron matter Nuclear matter 0 0 0.5 1 1.5 k F (fm -1 ) Fermi momentum With N3LO interaction, pairing gaps in good agreement with known calculations (several realistic interactions: AV18, CD-Bonn, Nijmegen,...). See Hebeler, Schwenk and Friman, PLB 648 (007) 176 for recent V lowk calculations.

effective mass m* Holt, Kaiser and Weise, nucl-th/1107.5966 1 0.9 Effective effective nucleon mass M*/M M * (ρ) / M 0.8 0.7 Substantial agreement with non relativistic Skyrme phenomenology at saturation density (0.7 < M*/M < 1). 0.6 0 0.05 0.1 0.15 0. ρ [fm -3 ] The two-body interaction includes long-range one- and two-pion exchange contributions and a set of contact terms contributing up to fourth power in momenta. In addition they add the leading order chiral 3N interaction with its parameters c E, c D and c 1,3,4.

results FROM n3lo (with M*) 1 S 0 superfluid pairing gap F (MeV) 4 3 1 Neutron matter Nuclear matter Pairing gaps are changed: a generalized reduction in magnitude and a small shift of the maxima towards smaller momenta. Similar results are well-known in literature [with Gogny interaction, EPJA 5 (005); more details about medium corrections in Cao, Lombardo and Schuck, PRC 74 (006) 64301]. 0 0 0.5 1 1.5 k F (fm -1 ) Fermi momentum Effective masses from Holt et al., 1107.5966

3-body interaction c i known from πn scattering c D and c E fixed from binding energies of light nuclei ( 3 He, 4 He) and/or nd scattering length

3-body interaction 3 body body density dependent Holt et al., [PRC 81 (010) 0400] In-medium NN interaction generated by the two-pion exchange component (c 1,c 3,c 4 ) of the chiral three-nucleon interaction. 1 3 In-medium NN interaction generated by the onepion exchange (c D ) and short-range component (c E ) of the chiral three-nucleon interaction. 4 5 6 in-medium nucleon propagator

3-body interaction Holt et al., [PRC 81 (010) 0400] In-medium NN interaction generated by the two-pion exchange component (c 1,c 3,c 4 ) of the chiral three-nucleon interaction. In-medium NN interaction generated by the onepion exchange (c D ) and short-range component (c E ) of the chiral three-nucleon interaction. 1 3 4 5 6 The attraction is reduced by the action of the three-body forces

results FROM 3-BODY 1 S 0 superfluid pairing gap F (MeV) 4 3.5 3.5 1.5 1 Nuclear matter N3LO 0.5 N3LO+3-body 0 0 0.5 1 1.5 k F (fm -1 ) Fermi momentum Pairing gaps are changed: a small reduction in magnitude and a relevant shift of the maximum towards smaller momenta. Known calculations for pairing gaps with threebody forces included in the nuclear potential are: Hebeler and Schwenk, PRC 8 (010) 014314; Zuo, Lombardo, Schulze and Shen, PRC 66 (00) 037303. Effective masses from Holt et al., 1107.5966

neutron matter with b and 3b preliminary! 3.5 Margueron Cao Fabrocini AFDMC neutron matter F (MeV) 1.5 1 0.5 0 0 0. 0.4 0.6 0.8 1 k F (fm -1 )

separable pairing by Y. tian In nuclear matter the pairing interaction has a separable form in momentum space k V 1 S 0 k = Gp(k)p(k ) Gaussian ansatz p(k) =e a k The two parameters G and a can be adjusted to reproduce the density dependence of the gap at the Fermi surface. The ansatz can be tested over well known calculations (like Gogny). Energies (pairing and binding) and gaps are reproduced with very high accuracy. [Y. Tian et al., PLB 676 (009) 44; Veselý, Dobaczewski, Michel and Toivanen, JoP Conf. Ser., 67 (011) 0107]

finite nuclei calculations h µ h + µ Δ: pairing field U V k = E k U V In the hamiltonian h we have V ph : particle-hole potential : chemical potential Three possible choices for V ph : FKVW: Density-dependent point coupling model constrained by in-medium chiral interactions, see Finelli et al. [NPA 770 (006) 1] for ground-state calculations. Applications also for excited-state properties and, in particular, hypernuclei in the last years. PC-F1: Non-linear point coupling model, see Burvenich et al. [PRC 65 (00) 044308]; see also Madland et al. [PRC 46 (199) 1757] for the first application to finite nuclei. PC-D1: Density-dependent point coupling model, see Niksic et al. [PRC 78 (008) 034318] and, for density-dependent theories, Typel et al. [NPA 656 (1999) 331].

Theory pairing gaps in finite nuclei = State-dependent single-particle states k kv k k v k Occupation factors Alternative choice k = kv k u k k v ku k Δ k are the diagonal matrix elements of the pairing part of the RHB single-nucleon hamiltonian in the canonical basis (the basis that diagonalizes the 1-particle density matrix) [Lalazissis, Vretenar and Ring, PRC 57 (1997) 94; Bender, Rutz, Reinhard and Maruhn, EPJA 8 (000) 59] (5) (N 0 ) = 1 8 [E(N 0 + ) 4E(N 0 + 1)+ 6E(N 0 ) 4E(N 0 1) + E(N 0 )] [Bender, Rutz, Reinhard and Maruhn, EPJA 8 (000) 59] Empirical Estimates Five-points formula Alternative approach Theoretical gaps are provided by Δ LCS (Lowest Canonical State) which denotes the diagonal pairing matrix element Δ i corresponding to the canonical single-particle state Φ i whose quasi-particle energy is the lowest. Experimental gaps extracted from binding energies through three-point mass differences centered on odd-mass nuclei. [Hebeler, Duguet, Lesinski and Schwenk, PRC 80 (009) 4431]

(Neutron pairing gap) results (isotopes) Z = 8 Z = 50 Z = 8 N3LO N3LO and 3-body Exp. data (5 pts.) (Total binding energy deviation) < n > [MeV] B exp - B th [MeV] 1.5 1 0.5 5 0-5 8 40 50 50 8 30 40 50 60 80 N 16 100 10 140 With N3LO pairing gaps in reasonable agreement with exp. data. With the inclusion of three-body forces pairing gaps are reduced by 30/40% [See also PRC 80 (009) 04431]. In some cases the pairing gaps collapse because of the wrong mean field single particle states. Unrealistic shell quenching FKVW interaction [NPA 770 (006) 1] Binding energies in very good agreement.

results (isotones) N = 8 N = 50 N = 8 With N3LO pairing gaps in reasonable agreement with exp. data. (Total binding energy deviation) (Proton pairing gap) < p > [MeV] B exp - B th [MeV] 1.5 1 0.5 5 0-5 0 8 8 50 58 0 30 FKVW interaction [NPA 770 (006) 1]? 30 40 50 Z Spurious shell closure [see Geng et al., CPL 3 (006) 1139] N3LO N3LO and 3-body Exp. data (5pts.) 60 80 With the inclusion of three-body forces pairing gaps are reduced by 30/40 % [See also PRC 80 (009) 04431]. In some cases the pairing gaps collapse because of the wrong mean field single particle states. Binding energies in very good agreement.

(Neutron pairing gap) results (different p-h interactions) (Total binding energy deviation) < n > [MeV] 1.5 1 0.5 Z = 8 Z = 50 Z = 8 FKVW PCD1 PCF1 Exp. data (5 pts.) In some cases the pairing gaps collapse because of the wrong mean field single particle states. B exp - B th [MeV] 5 0-5?? 8 40 50 50 8 16 30 40 50 60 80 N 100 10 140 Of course, different p-h interactions play a role, because of the single-particle states. Binding energies still in very good agreement for every p-h interaction.

CONCLUSIONS Infinite systems: with the introduction of 3-body forces the pairing gap is reduced in magnitude and shifted towards smaller momenta. Δ Finite nuclei: with the introduction of 3-body forces pairing gaps in finite nuclei are strongly reduced respect to the -body case. [See K. Hebeler et al., PRC 80 (009) 04431 and 1104.955, T. Duguet et al., nucl-th/1004.358v1 and references therein]. Additional effects have to be included in order Δ? kf to reproduce empirical data. [Particle vibration Coupling?, Vigezzi et al., nucl-th/1001.1057]. N,Z

latest WORKS RELATIVISTIC AND NON-RELATIVISTIC MODELS GROUND-STATE PROPERTIES IN COLLABORATION WITH PAVIA, MEUCCI, PACATI E GIUSTI LECCE, granada AND ORSAY, CO E DE DONNO LALLENA E ANGUIANO GRASSO submitted to phys. rev. c

Motivation We study the predictions of three mean-field theoretical approaches in the description of the ground state properties of some spherical nuclei far from the stability line. We compare binding energies, single particle spectra, density distributions, charge and neutron radii obtained with different approaches. The agreement between the results obtained with the three different approaches indicates that these results are more related to the basic hypotheses of the mean-field approach rather than to its implementation in actual calculations.

models Skyrme Gogny Text RMF (meson-exchange) Density-dependent coupling

E/A [MeV] E/A [MeV] 100.0 80.0 60.0 40.0 0.0 PNM CBF AFDMC D1M SLy5 DDME 0.0 0.0 0.1 0. 0.3 0.4 0.0 10.0 0.0-10.0 SNM -0.0 0.0 0.1 0. 0.3 0.4 infinite systems (a) (b) e(ρ, δ) =e(ρ, 0) + e sym (ρ) δ + O(δ 4 ) e(ρ, 0) = a V + 1 K V ɛ +... K V =9ρ e(ρ, 0) 0 ρ ρ=ρ0 e sym (ρ) =a sym + L ɛ +... e sym (ρ) L =3ρ 0 ρ ρ=ρ0 E/A [MeV] 100.0 80.0 60.0 40.0 0.0 e sym 0.0 0.0 0.1 0. 0.3 0.4 [fm -3 ] (c) exp AFDMC CBF D1M SLy5 DDME ρ 0 0.16 ± 0.01 0.16 0.16 0.16 0.16 0.15 e(ρ 0, 0) -16.0 ± 0.1-16.00-16.00-16.01-15.98-16.13 K V 0 ± 30 76 69 17 8 78 a sym 30-35 31.3 33.94 9.45 3.66 33.0 L 88 ± 5 60.10 58.08 5.41 48.38 54.74 nuclear matter properties at saturation density ρ for various ca

binding energies 10.0 O Ca Ni Sn 9.0 E/A [MeV] 8.0 7.0 6.0 exp D1M SLy5 DDME 5.0 8 48 60 56 78 114 13 16 4 40 5 48 68 100 116 A

energies (I) 0.0 O Ca Ni Sn h [MeV] -10.0-0.0-30.0-40.0 exp D1M SLy5 (a) DDME 8 48 60 56 78 114 13 16 4 40 5 48 68 100 116 Single particle proton levels below the Fermi surface 10.0 O Ca Ni Sn p [MeV] 0.0-10.0-0.0-30.0 (b) 8 48 60 56 78 114 13 16 4 40 5 48 68 100 116 Single particle proton levels above the Fermi surface 13.0 11.0 O Ca Ni Sn p [MeV] 9.0 7.0 5.0 3.0 (c) 8 48 60 56 78 114 13 16 4 40 5 48 68 100 116 A Proton energy gap ε p -ε h

energies (II) 50.0 O Ca Ni Sn p [MeV] 40.0 30.0 0.0 D1M SLy5 DDME Differences between the s.p. energy of the least bound proton level and that of the 1s 1/ level, calculated with the three different MF models. 10.0 8 48 60 56 78 114 13 16 4 40 5 48 68 100 116 A

proton distributions Proton distributions for the various calcium isotopes Wave functions of the proton s 1/ levels for the calcium isotopes. 0.15 0.15 0.8 0.8 p [fm -3 ] 0.1 0.09 0.06 40 Ca D1M SLy5 DDME 0.1 0.09 0.06 48 Ca [fm -3/ ] 0.6 0.4 0. 40 Ca D1M SLy5 DDME(u) DDME(l) 0.6 0.4 0. 48 Ca 0.03 0.03 0.0 0.0 0.00 0 4 6 8 10 0.00 0 4 6 8 10-0. 0 4 6 8 10-0. 0 4 6 8 10 0.15 0.15 0.8 0.8 0.1 5 Ca 0.1 60 Ca 0.6 0.6 p [fm -3 ] 0.09 0.06 0.09 0.06 [fm -3/ ] 0.4 0. 5 Ca 0.4 0. 60 Ca 0.03 0.03 0.0 0.0 0.00 0 4 6 8 10 r [fm] 0.00 0 4 6 8 10 r [fm] -0. 0 4 6 8 10 r [fm] -0. 0 4 6 8 10 r [fm]

neutron and matter distributions Neutron distributions for the various calcium isotopes Matter distributions for the various calcium isotopes 0.15 0.15 0.5 0.5 n [fm -3 ] 0.1 0.09 0.06 40 Ca D1M SLy5 DDME 0.1 0.09 0.06 48 Ca m [fm -3 ] 0.0 0.15 0.10 40 Ca D1M SLy5 DDME 0.0 0.15 0.10 48 Ca 0.03 0.03 0.05 0.05 0.00 0 4 6 8 10 0.00 0 4 6 8 10 0.00 0 4 6 8 10 0.00 0 4 6 8 10 0.15 0.15 0.5 0.5 0.1 5 Ca 0.1 60 Ca 0.0 5 Ca 0.0 60 Ca n [fm -3 ] 0.09 0.06 0.09 0.06 m [fm -3 ] 0.15 0.10 0.15 0.10 0.03 0.03 0.05 0.05 0.00 0 4 6 8 10 r [fm] 0.00 0 4 6 8 10 r [fm] 0.00 0 4 6 8 10 r [fm] 0.00 0 4 6 8 10 r [fm]

radii Rch [fm] Rn [fm] 6.0 5.0 4.0 3.0.0 6.0 5.0 4.0 3.0.0 O Ca Ni Sn D1M SLy5 DDME rms charge radii exp (a) 8 48 60 56 78 114 13 16 4 40 5 48 68 100 116 O Ca Ni Sn (b) 8 48 60 56 78 114 13 16 4 40 5 48 68 100 116 A neutron radii Rn-Rp (fm) 0.8 0.6 0.4 0. 0.0-0. -0.4 0.8 0.6 0.4 0. 0.0-0. -0.4 0.8 0.6 0.4 0. 0.0-0. O Ca Ni Sn D1M SLy5 DDME -0.4-0. -0.1 0.0 0.1 0. 0.3 0.4 0.5 (N-Z)/A Neutron skins calculated with the three different approaches as a function of the relative neutron excess. The full lines show a linear fit do the data, the dashed lines present the predictions by Angeli et al.

conclusions (i) 1. The properties of infinite nuclear matter at saturation densities are very similar for the three approaches. The only exception is L. Above the saturation point, the behaviors of the EOS are remarkably different, especially in the case of PNM.. In our calculations all the 16 nuclei investigated are bound. This MF prediction could be in contrast with the experimental evidence (the neutron drip line for the oxygen isotopes starts with the 6 O nucleus, therefore 8 O is experimentally an unstable system) 3. The proton s.p. energies around the Fermi surface have similar values for all the three calculations. 4. In each isotopes chain, the energy available to arrange the proton s.p. levels decreases with increasing neutron number. As a consequence the density of states increases. 5. The study of the density distributions indicates a good agreement at the nuclear sur- face for all the three types of calculations. In the nuclear interior we have observed very different behaviors when the proton and neutron densities are separately considered. These differences in the nuclear center are much smaller when the total matter is considered.

conclusions (ii) 6. The large differences of the proton distributions in the nuclear interior are due to the s proton waves. 7. The values of the charge rms radii are very similar in all the three calculations and agree very well with the available experimental data and with their empirical extrapolations. Our results show a small increase of these radii with the neutron numbers. 8. Neutron skins are larger in light than in heavy nuclei. This is not a trivial geometrical effect. The most important message emerging from our investigation is the convergence of the results of the 3 MF models for all the nuclei investigated. This indicates the importance of producing and investigating exotic nuclei. The comparison between the observed properties and the MF predictions is going to confirm, or not, the MF model itself, and not a specific implementation of it.