Global investigation of odd-even mass differences and radii with isospin dependent pairing interactions
aa r X i v : . [ nu c l - t h ] N ov Global investigation of odd-even mass differences and radii with isospin dependentpairing interactions
C. A. Bertulani, Hongliang Liu, and H. Sagawa Department of Physics, Texas A&M University-Commerce, Commerce, Texas 75429, USA Center for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu, 965-8580 Fukushima, Japan (Dated: November 11, 2018)The neutron and proton odd-even mass differences are systematically studied with Hartree-Fock+BCS (HFBCS) calculations with Skyrme interactions and an isospin dependent contact pair-ing interaction. The strength of pairing interactions is determined to reproduce empirical odd-evenmass differences in a wide region of mass table. By using the optimal parameter, we perform globalHF+BCS calculations of nuclei and compare with experimental data. The importance of isospindependence of the pairing interaction is singled out for odd-even mass differences in medium andheavy isotopes. The proton and neutron radii are studied systematically by using the same model.
PACS numbers: 21.30.Fe, 21.60.-nKeywords: effective pairing interaction, isospin dependence, finite nuclei
I. INTRODUCTION
Microscopic theories for calculating nuclear massesand/or binding energies (see, e.g., [1–3]), have been re-vived and further elaborated with the advance of com-putational resources. These advances are now sufficientto perform global studies based on, e.g., self-consistentmean field theory, sometimes also denoted by densityfunctional theory (DFT) [4, 5]. One particular aspectof the nuclear binding problem is a phenomenon of odd-even staggering (OES) of the binding energy. Numerousmicroscopic calculations have been published that treatindividual isotope chains. However, it might be necessaryto examine the whole body of OES data to draw generalconclusions [6].Theoretically, OES values are often inferred from theaverage HFBCS or Hartree-Fock-Bogoliubov gaps [7–9],rather than directly calculated from the experimentalbinding energy differences between even and odd nuclei.It should be mentioned that the average HFB gaps aresometimes substantially different from the odd-even massdifferences calculated from experimental binding ener-gies. In this work, we compare directly the calculatedOES with the ones extracted from experiment. Oneshould say that there are also several prescriptions to ob-tain the OES from experiments, such as 3-point, 4-point,and 5-point formulas [6]. We adopt the 3-point formula∆ (3) centered at an odd nucleus, i.e., odd-N nucleus forneutron gap and odd-Z nucleus for proton gap [2]:∆ (3) ( N, Z ) ≡ π A +1 h B( N − , Z ) − N, Z ) (1)+ B( N + 1 , Z ) i , where B ( N, Z ) is the binding energy of (
N, Z ) nucleusand π A = ( − ) A is the number parity with A = N + Z .For even nuclei, the OES is known to be sensitive notonly to the pairing gap, but also to mean field effects,i.e., shell effects and deformations [6, 7]. Therefore, thecomparison of a theoretical pairing gap with OES should be done with some discretion. One advantage of ∆ (3) o ( N = odd in Eq. (1)) is the suppression of the contri-butions from the mean field to the gap energy. Anotheradvantage of ∆ (3) o ( N, Z ) is that it can be applied to moreexperimental mass data than the higher order OES for-mulas. At a shell closure, the OES (Eq. (1)) does not goto zero as expected, but it increases substantially. Thislarge gap is an artifact due to the shell effect, which istotally independent of the pairing gap itself.Recently, an effective isospin dependent pairing inter-action was proposed from the study of nuclear matterpairing gaps calculated by realistic nucleon-nucleon in-teractions. In Ref. [8], the density − dependent pairinginteraction was defined as V pair (1 ,
2) = V g τ [ ρ, βτ z ] δ ( r − r ) , (2)where ρ = ρ n + ρ p is the nuclear density and β is theasymmetry parameter β = ( ρ n − ρ p ) /ρ . The isovectordependence is introduced through the density-dependentterm g τ . The function g τ is determined by the pairinggaps in nuclear matter and its functional form is givenbyg τ [ ρ, βτ z ] = 1 − f s ( βτ z ) η s (cid:18) ρρ (cid:19) α s − f n ( βτ z ) η n (cid:18) ρρ (cid:19) α n , (3)where ρ =0.16 fm − is the saturation density of symmet-ric nuclear matter. We choose f s ( βτ z ) = 1 − f n ( βτ z ) andf n ( βτ z ) = βτ z = [ ρ n ( r ) − ρ p ( r )] τ z /ρ ( r ). The parametersfor g τ are obtained from the fit to the pairing gaps insymmetric and neutron matter obtained by the micro-scopic nucleon-nucleon interaction.In the literature and in many mean field codes publiclyavailable such as the original EV8 code [10], a pure con-tact interaction is used without an isospin dependence.In our notation, this amounts replacing the isospin de-pendent function g τ in Eq. (2) by the isoscalar functiong s = 1 − η s (cid:18) ρρ (cid:19) α s . (4) TABLE I: Parameters for the density-dependent function g τ defined in Eqs. (2) and (3) for the IS+IV interaction (first row)and g s in Eq. (4) for the IS interaction. The parameters for g τ are obtained from the fit to the pairing gaps in symmetric andneutron matter obtained with the microscopic nucleon-nucleon interaction. The paring strength V is adjusted to give the bestfit to odd-even staggering of nuclear masses. The parameters for g s correspond to a surface peaked pairing interaction with noisospin dependence. interaction V (MeVfm ) ρ (fm − ) η s α s η n α n g τ (isotopes) 1040 0.16 0.677 0.365 0.931 0.378g τ (isotones) 1120 0.16 0.677 0.365 0.931 0.378g s (isotopes) 1300 0.16 1. 1. — —g s (isotones) 1500 0.16 1. 1. — —
700 800 900 1000 1100 1200 1300 1400 1500 1600 1700Pairing strength (MeVfm )0.00.51.0 σ ( M e V ) IS isotopesIS isotonesIS+IV isotopesIS+IV isotones
FIG. 1: (Color online) The mean square deviation σ of OESbetween experimental data and the HF+BCS calculations.The filled circles and squares correspond to the results withIS pairing for neutron and proton gaps, respectively, while thefilled diamonds and triangles are those of IS+IV pairing forneutron and proton gaps. Experimental data are taken fromRef. [17]. See the text for details. The EV8 code has been modified, using the filling ap-proximation, to account for mass calculations for odd-Nand odd-Z nuclei and is publicly available as the EV8oddcode [11]. It has also been modified to include isospin de-pendent paring, by means of Eq. (2). The parametersof the isoscalar interaction were adjusted with EV8 to abest global fit of nuclear masses [12]. They correspondto a surface peaked pairing interaction (Eq. (4) with η s not too far from the unity).A recent publication has explored the isospin depen-dence of the pairing force for the OES effect for a fewselected isotopic and isotonic chains [13]. Here we havemade a more ambitious study by extending the calcula-tion to the whole nuclear chart. We have also explored several other observables such as neutron and protonradii systematically which may allow for more solid con-clusions on isospin dependent pairing interactions.This paper is organized as follows. In section II we dis-cuss our numerical calculation strategy. Our results arepresented in section III for the energies, separation en-ergies, OES energies, and nuclear radii. Our conclusionsare presented in section IV. II. CALCULATION STRATEGY
The HF+BCS calculations are performed by usingSLy4 Skyrme interaction which was found to be the mostaccurate interaction for studying OES for a few selected( N = 50 ,
82) isotonic and (Sn and Pb)isotopic chains [13].Our iteration procedure used in connection to EV8oddachieves an accuracy of about 100 keV, or less, with 500Hartree-Fock iterations for each nuclear state. Our cal-culations were performed with the now decommissionedXT4 Jaguar supercomputer at ORNL, as part of theUNEDF-SciDAC-2 collaboration [14].The HF+BCS calculations were first performed foreven-even nuclei. The variables in the theory are the or-bital wave functions φ i and the BCS amplitudes v i and u i = p − v i . By solving the BCS equations for theamplitudes, one obtains the pairing energy from E pair = X i = j V ij u i v i u j v j + X i V ii v i (5)where V ij are the matrix elements of the pairing interac-tion, Eq. (2), namely V ij = V Z d r | φ i ( r ) | | φ j ( r ) | g τ [ ρ ( r ) , β ( r ) τ z ] , where ρ ( r ) = P i v i | φ i ( r ) | .After determining the single-particle energies of even-even nuclei, the odd-A nuclei are calculated with the so-called filling approximation for the odd particle startingfrom the HF+BCS solutions of neighboring even-even nu-clei: ones selects a pair of i and e i orbitals to be blocked, FIG. 2: (Color online) Binding energy differences between experimental data and calculations using the HF+BCS model withthe IS and IS+IV pairing interactions. The left panels show the differences for even Z isotopes varying neutron numbersincluding both odd and even numbers. The right panel show those for even N isotones varying proton numbers including bothodd and even numbers. The thin lines show the closed shells at N ( Z ) = 20, 28, 40, 50, 64, 82 and 126. Experimental data aretaken from Ref. [17]. See the text for details. and changes the BCS parameters v i and v e i for theseorbitals. The change is to set v i = v e i = 1 / V depends on the energy window adopted forBCS calculations. The odd nucleus is treated in the fill-ing approximation, by blocking one of the orbitals. Theblocking candidates are chosen within an energy windowof 10 MeV around the Fermi energy. This energy win-dow is rather small, but it is the maximum allowed bythe program EV8odd. It is shown that the BCS modelused in the EV8odd code gives almost equivalent results to the HF+Bogoliubov model with a larger energy win-dow, except for unstable nuclei very close to the neutrondrip line [12]. The pairing strengths V for IS and IS+IVpairing interactions are adjusted to give the best fit toodd-even staggering of nuclear masses in a wide region ofthe mass table. III. NUMERICAL RESULTSA. Global data on odd-even staggering
The results for the mean square deviation of our globalmass table calculations are shown in Fig. 1. Table I givesthe values V in Eq. (2) and the parameters for g τ andg s is Eqs. (3) and (4) used in the present work. Optimalpairing strength values were found to be different for iso-tones (varying Z , constant N ) and for isotopes (varying N , constant Z ).Figure 1 shows the mean square deviation σ of OES be-tween experimental data and the HF+BCS calculations.The mean square deviation σ is defined as σ = vuut N i X i =1 (cid:12)(cid:12)(cid:12) ∆ (3) i ( HF + BCS ) − ∆ (3) i ( exp ) (cid:12)(cid:12)(cid:12) /N i (6)where N i is the number of data points. For the IS interac-tion, the results for neutrons show a shallow minimum at V ∼ (1100 − · fm . For protons, the minimum FIG. 3: (Color online) The same as Fig. 2 but for neutron and proton separation energies. See the caption to Fig. 2 and thetext for details. becomes at around V ∼ · fm . This differencemakes it difficult to determine a unique pairing strengthcommon for both neutrons and protons. The results ofIS+IV pairing show a minimum at V ∼ · fm for both neutron and proton OES which makes it easierto determine the value for the pairing strength. Adoptedvalues for the following calculations are listed in Table I.The systematic study of HF+BCS calculations are per-formed for various isotopes and isotones for all availabledata sets with ( Z = 8 , · · · , N = 8 , · · · , δE = | E exp − E cal | (7)are shown for both IS and IS+IV pairing interactionsin Fig. 2. The left panels show the values δE varyingneutron numbers (including both odd and even numbers)for each even Z. The right panels show the values δE varying proton numbers (including both odd and evennumbers) for each even N. With IS pairing, we can see arather large deviation for Z = 50 isotopes in the upperright panel. This difference disappears in the case ofIS+IV pairing shown in the lower right panel. On theother hand, for the N = 82 nuclei, the IS+IV interactiondoes not work that well. As far as the binding energiesare concerned, the best results with IS+IV interaction areobtained for nuclei with N = 60 −
78 and N = 86 − S n are reasonable for medium and heavy massnuclei with N = 60 − N = 82. For heavy nuclei with N = 126, the calculatedresults are poorer than in other mass regions. For protonseparation energy S p , the HF+BCS also gives reasonableresults, except in the Z = 50 and 82 mass regions.In order to see the different outcomes between IS andIS+IV pairing interactions, the HF+BCS model calcula-tions are shown together with empirical data in Fig. 4.In most of cases, the difference between the two pairinginteractions are small. However, we can see a clear im-provement of the agreement of S p with empirical data of N = 136 isotones with IS+IV pairing in Fig. 5.The differences of neutron OES ∆ n and proton OES∆ p between HF+BCS and empirical data are shown inFig. 6 for both IS and IS+IV pairing, respectively, inthe upper and lower panels. The agreement betweenHF+BCS and the empirical data are good in the overallmass region except for masses with Z = 50 and at a smallmass region A <
60. To clarify the difference between ISand IS+IV pairing, the OES differences ∆ n are shown inthe upper panel of Fig. 7 for Z = 52, 78 and 92 isotopes.The HF+BCS results are compared with the experimen-tal data and also the phenomenological parameterizationbased on liquid drop model,¯∆ = c/A α (8)with c = 4 . .
31) MeV for neutrons (protons) and α = 0 .
31 which gives the rms residual of 0.25 MeV [12].We can see clearly a better agreement of IS+IV resultswith empirical data for all isotopes. In the lower panelof Fig. 7, the OES differences ∆ n are shown for N = 76,102 and 112 isotones. The results with IS+IV pairingcertainly improve systematically the agreement with em-
50 60 70N51015 S n ( M e V )
90 100 110 120N 120 130 140N expISIS+IV
Z=46 Z=72 Z=90
30 40 50Z051015 S p ( M e V )
60 70 80Z 90 Z expISIS+IV
N=50 N=90 N=136
FIG. 4: (Color online) Neutron separation energies S n of three isotope chains with Z = 46, 72 and 90 calculated by IS andIS+IV interactions in HF+BCS model (upper panels). The lower panels show the proton separation energies S p for N = 50,90 and 136 isotones. Experimental data are taken from Ref. [17]. See the text for details. pirical data, especially for N = 102 isotones. The largeincrease of the HF+BCS model results at Z = 81 is anartifact due to the shell closure at Z = 82. It is interest-ing to notice that the liquid drop formula gives smoothmass number dependence which reflects well that of veryheavy isotones with N = 112.The average gaps ∆ (3) are tabulated for high and lowisospins in Table II. Each isotope (isotone) in Fig. 7is divided into two subsets of almost equal numbers ofnuclei by a cut at some value of I = ( N − Z ) /A . Both theaverage proton and neutron ∆ (3) show smaller values forhigher isospin so that the pairing interaction is weaker forneutron-rich nuclei. The IS+IV interaction reproducesproperly the difference of the neutron ∆ (3) between highand low isospin nuclei. For proton ∆ (3) also, the IS+IVpairing gives a good account of the isospin effect than theIS pairing. B. Nuclear radii
Nuclear radii provide basic and important informationfor various aspect of nuclear structure problems. Theproton radii, or equivalently the charge radii with thecorrection of finite proton size, can be determined accu-rately by electron scattering and muon scattering exper-iments. However it is difficult to determine the neutron radii of finite nuclei with the same accuracy level as thatof the proton radii while there were several experimentalattempts to determine the difference of the neutron toproton radius [19–21]. It should be noticed that the dif-ference of the neutron and proton radii, δr np = r n − r p , iscalled the neutron skin. It is thought that δr np can giveimportant constrains on the effective interactions used innuclear structure study [22].The neutron and proton radii of various isotopes andisotones are calculated by using the HF+BCS modelwith the two pairing interactions, IS and IS+IV. Theresults of neutron radii are shown in the left panel ofFig. 8. Since we do not find any appreciable differencesbetween the two pairing interactions in the results, theresults of IS+IV interactions will be mainly discussedhereafter. The results obtained by a simple empiricalformula r n = r N / with r = 1 .
139 fm [18] are alsoplotted in the figure. In general, the simple formula for r n agrees well with the HF+BCS results. It is noticedthat the HF+BCS model gives larger neutron radii fornuclei with N <
40 than the simple formula but smallerfor nuclei with
N > N = 20,28, 40, 50, 82 and 126 isotones are shown as a func-tion of proton number Z in the right panel. The sim-ple Z / dependence is also plotted to follow the formula r p = 1 . / . The simple formula in general gives agood account of the HF+BCS data and could be a good TABLE II: Average ∆ (3) for low isospin and high isospin nuclei and its difference. See the text for details.Data set Low isospin High isospin DifferenceNeutrons Z = 52 Exp 1.36 1.08 -0.28IS 1.52 1.41 -0.11IS+IV 1.40 1.19 -0.21 Z = 78 Exp 1.13 0.99 -0.14IS 0.96 1.16 0.20IS+IV 0.87 0.91 0.04 Z = 92 Exp 0.77 0.56 -0.21IS 0.90 0.80 -0.10IS+IV 0.70 0.55 -0.15Protons N = 76 Exp 1.19 0.93 -0.26IS 1.13 0.87 -0.26IS+IV 1.13 0.98 -0.15 N = 102 Exp 0.96 0.63 -0.33IS 0.79 0.39 -0.40IS+IV 0.92 0.59 -0.33 Z = 112 Exp 0.87 0.66 -0.21IS 0.58 0.61 0.03IS+IV 0.67 0.70 0.03
86 88 90 92 94 96Z2468 S p ( M e V ) expISIS+IVN=136 FIG. 5: (Color online) Proton separation energies S p of N = 136 isotones calculated by IS and IS+IV interactionsin HF+BCS model. Experimental data are taken from Ref.[17]. See the text for details. starting point for describing the isospin dependence ofnuclear charge radii. However we can see some deviationbetween the HF+BCS and the simple formula especiallyheavy N = 50 and N = 82 isotones.The neutron skin r n − r p calculated by HF+BCS modelwith the two pairing interactions are shown in Fig. 9.The neutron skin becomes as large as 0.4 fm near theneutron drip line with Z <
28. On the other hand, theneutron skin is at most 0.25 fm in neutron-rich nucleiwith
Z >
50. For proton-rich nuclei, the proton skinbecomes 0.1 fm with
Z <
56 and smaller than 0.05 fm in heavier isotopes, larger than Z = 56. The results ofIS and IS+IV pairings are shown in the left panel andright panel, respectively. In general, the two pairing in-teractions give almost the same results as shown in Fig.9. However, it is noticed that the IS+IV pairing givessomewhat smaller neutron skins than the IS pairing invery neutron-rich nuclei such as Sn,
Ba and
Po.The calculated values are compared with empirical dataof Sn isotopes obtained from studies of spin-dipole reso-nances [19] and antiprotonic atoms [20] in the left panelFig. 10. The calculated values show reasonable agree-ment with the empirical data within the experimentalerror. The isospin dependence of neutron skin is shownin the right panel of Fig. 10 together with empiricalvalues obtained by antiprotonic atom experiments in awide range of nuclei from Ca to
U. The slope of ex-perimental data as a function of the isospin parameter I = ( N − Z ) /A is reproduced well by our calculations.The neutron skin of Pb has been discussed inten-sively in relation with neutron matter properties. Thesystematic studies of scattering data yield the empiricalvalue r n − r p = 0 . ± .
02 fm which is close to anotherempirical value r n − r p = 0 . ± .
02 fm from the studyof antiprotonic-atom systems. The model independentdetermination of parity violation experiment at JeffersonLaboratory [22] has been proposed and performed re-cently to obtain the neutron skin of
Pb. However thestatistics was poor and needs improvement by more dataaccumulation. Our calculated value r n − r p = 0 .
157 fmis close to the experimental values by the two systematicstudies.
FIG. 6: (Color online) The same as in Fig. 2 but for OES for neutrons and protons. See the caption to Fig. 2 and the text fordetails.
IV. SUMMARY AND CONCLUSIONS
In summary, we studied the binding energies, separa-tion energies and OES by using HF+BCS model withSLy4 interactions together with the isospin dependencepairing (IS+IV pairing) and isoscalar (IS pairing) in-teractions. The calculations are performed with theEV8odd code for even-even nuclei and also even-oddnuclei using the filling approximation. For the neu-tron pairing gaps, the IS+IV pairing strength decreasesgradually as a function of the asymmetry parameter( ρ n ( r ) − ρ p ( r )) /ρ ( r ). On the other hand, the pairingstrength for protons increases for larger values of theasymmetry parameter because of the isospin factor inEq. (3). The empirical isotope dependence of the neu-tron OES, ∆ (3) n , is well reproduced by the present calcula-tions with the isospin dependent pairing compared withthe IS pairing. We can also obtain a good agreementbetween the experimental proton OES and the calcula-tions with the isospin dependent pairing for N = 50 and N = 82 isotones.The neutron and proton radii were also studied by us-ing the same HF+BCS model with the two pairing inter-actions. The two pairing interactions give essentially thesame results for the radii except for a few very neutron-rich nuclei. We found systematically large neutron skins in very neutron-rich nuclei with | r n − r p | ∼ . N − Z )dependence of the data.We tested the IS+IV pairing for the Skyrme interac-tion SLy4 and found the results reproduce well the sys-tematical experimental data. Thus, we confirm a clearmanifestation of the isospin dependence of the pairinginteraction in the OES in comparison with the experi-mental data both for protons and neutrons. Acknowledgments
This work was partially supported by the U.S. DOEgrants DE-FG02-08ER41533 and DE-FC02-07ER41457(UNEDF, SciDAC-2), the Research Corporation, and theJUSTIPEN/DOE grant DEFG02- 06ER41407, and bythe Japanese Ministry of Education, Culture, Sports, Sci-ence and Technology by Grant-in-Aid for Scientific Re-search under the Program number C(2) 20540277. Com-putations were carried out on the XT4 Jaguar supercom-puter at the Oak Ridge National Laboratory. [1] A. Bohr, B. R. Mottelson, and D. Pines, Phys. Rev. ,936 (1958). [2] A. Bohr and B. R. Mottelson, “Nuclear Structure” (Ben-jamin, New York, 1969) Vol. I.
50 60 70 80N0.00.51.01.52.0 90 100 110 120N 130 140 150N expISIS+IV4.66/A
Z=52 Z=78 Z=92
48 52 56 60 64Z0.00.51.01.52.0 64 68 72 76 80 84Z 72 76 80 84 88Z expISIS+IV4.31/A
N=76 N=102 N=112
FIG. 7: (Color online) The neutron and proton OES, ∆ n and ∆ p , calculated with the HF+BCS model with IS and IS+IVinteractions. Experimental data are taken from Ref. [17]. See the text for details.[3] D. M. Brink and R. Broglia, “Nuclear Superfluidity, Pair-ing in Finite Systems”, Cambridge Monographs on Par-ticle Physics, Nuclear Physics and Cosmology, vol 24(2005).[4] M. Bender, P.-H.Heenen and P.-G. Reinhard, Rev. Mod.Phys. , 121 (2003).[5] M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz and P.Borycki, Int. J. Mass Spectrum, , 243 (2006).[6] W. Satula, J. Dobaczewski and W. Nazarewicz, Phys.Rev. Lett. , 3599 (1998).[7] T. Duguet, P. Bonche, P.-H. Heenen, and J. Meyer, Phys.Rev. C , 014311 (2001).[8] J. Margueron, H. Sagawa and K. Hagino, Phys. Rev. C , 064316 (2007).[9] M. Yamagami and Y. R. Shimizu, Phys. Rev. C ,064319 (2008).[10] P. Bonche, H. Flocard and P. H. Heenen, Comput. Phys.Commun. , 49 (2005).[11] G. Bertsch and C.A. Bertulani, unpublished. EV8oddcode available upon request. [12] G. F. Bertsch, C. A. Bertulani, W. Nazarewicz, N.Schunck and M. V. Stoitsov, Phys. Rev. C , 034306(2009).[13] C.A. Bertulani, Hongfeng Lu, H. Sagawa, Phys. Rev. C , 027303 (2009).[14] “The Universal Nuclear Energy Density Functional”, aSciDAC project. URL: http://unedf.org[15] T. Duguet, P. Bonche, P. H. Heenen and J. Meyer, Phys.Rev. C , 014310 (2001);[16] J. Margueron, M. Grasso, G. Col`o, S. Goriely and H.Sagawa, J. of Phys. G , 125103 (2009).[17] G. Audi, A. H. Wapstra and C. Thibault, Nucl. Phys. A729 , 337 (2003).[18] Jie Meng et al., Prog. Part. Nucl. Phys. , 470 (2006).[19] A. Krasznahorkay et al, Phys. Rev. Lett. , 3216 (1999).[20] A. Trzcinska et al., Phys. Rev. Lett. , 082501 (2001).[21] J. Jastrzebski et al, Int. J. Mod. Phys. E , 34 (2004).[22] C. J. Horowitz and J. Piekarewicz, Phys. Rev. C ,062802, (2001). r n (f m ) OCaNiZrSnPb1.139 × N r p (f m ) N=20N=28N=40N=50N=82N=1261.264 × Z FIG. 8: (Color online) Neutron and proton radii of various isotopes and isotones calculated by means of the HF+BCS modelwith IS+IV interaction. The solid lines are empirical fits used in Ref. [18]. See the text for details.FIG. 9: (Color online) Neutron skin neutron for isotopes and isotones calculated by means of the HF+BCS model with IS andIS+IV interactions. The thin lines indicate the closed shells with N (or Z)=20, 28, 40, 50 ,64, 82 and 126. See the text fordetails.
100 110 120 130A-0.100.10.20.3 r n -r p (f m ) exp 1exp 2IS+IV Sn r n -r p (f m ) expIS+IV Ca Ni Fe Ni Fe Co Fe Cd Sn Zr Ni Sn Te Te Sn Ca Zr Cd Te Te Sn Te Bi Pb Th U FIG. 10: (Color online)
Left - Neutron skin for Sn isotopes obtained with the HF+BCS model with IS+IV interaction.Experimental data are taken from Ref. [19, 20].
Right - Neutron skin as a function of isospin parameter I = ( N − Z ) /A/A