Shell model study for neutron-rich sd-shell nuclei
Kazunari Kaneko, Yang Sun, Takahiro Mizusaki, Munetake Hasegawa
aa r X i v : . [ nu c l - t h ] J a n Shell model study for neutron-rich sd -shell nuclei Kazunari Kaneko , Yang Sun , , Takahiro Mizusaki , Munetake Hasegawa Department of Physics, Kyushu Sangyo University, Fukuoka 813-8503, Japan Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China Institute of Natural Sciences, Senshu University, Tokyo 101-8425, Japan (Dated: October 21, 2018)The microscopic structure of neutron-rich sd -shell nuclei is investigated by using the spherical shell-modelin the sd - p f valence space with the extended pairing plus quadrupole-quadrupole forces accompanied by themonopole interaction (EPQQM). The calculation reproduces systematically the known energy levels for even-even and odd-mass nuclei including the recent data for S, S and Ar. In particular, the erosion of the N = Si can be explained. Our EPQQM results are compared with other shell model calculationswith the SDPF-NR and SDPF-U effective interactions.
PACS numbers: 21.10.Dr, 21.60.Cs, 21.10.Re
I. INTRODUCTION
The current experimental and theoretical investigation fo-cuses on the study of evolution of nuclear structure aroundthe shell gap at N =
28. The N =
28 shell closure is a tradi-tional one in the nuclear single-particle spectrum driven by thespin-orbit interaction. For example, the Ca nucleus has beenknown to exhibit double-magicity as a result of the neutronsubshell gap separating the f / -orbit and the f / p -shells at N =
28. However, recent theoretical studies and experimentaldata have questioned the persistence of the traditional magicnumbers, and revealed that the N =
28 shell gap is erodedwhen moving away from the stability line. The experimentaldata for the S and Si isotones indicate a clear breaking ofthe N =
28 magicity because of the observed small 2 + en-ergy [1–3]. Recently, a direct evidence of collapse of the N =
28 shell closure due to the level inversion between the lowest7/2 − and 3/2 − levels has been observed in S [4]. In addition,toward N =
28, the degeneracy between the lowest 1 / + and3 / + states in the odd-proton P, Cl, and K isotopes suggests[5] that the shell gap between the d / and s / proton orbitsalmost collapses at N =
28. The reduction of the N =
28 neu-tron gap combined with this degeneracy is regarded as the pos-sible origin of erosion of the N =
28 magicity in the neutron-rich sd -shell nuclei.To gain the insight for the evolution of shell structure inthe neutron-rich sd -shell nuclei and to understand the detailsof collapse of the N =
28 magicity, shell-model calculationsin the full sd - p f space are desirable. However, conventionalshell-model calculations in the full sd - p f space are not pos-sible at present because of the huge dimension in the con-figuration space. To study the neutron-rich sd -shell nuclei,therefore, one needs to incorporate truncations in the valencespace. One choice is such that valence protons are restrictedto the sd shell and neutrons to the sd - p f shell with twelve sd frozen neutrons. Such an approach was originally introducedwith the effective interaction SDPF-NR [6–10]. The nucleiin this mass region are known to have a variety of shapes(e.g. shape coexistence discussed in Refs. [2, 4]), and the nu-cleus Si has been the subject of debate on its magic nature[3, 11]. The SDPF-NR interaction was applied to the neutron- rich sd -shell nuclei, and one found that it describes well theexcitation energies for very neutron-rich isotopes with Z > Z =
14) isotopes where the calculated ener-gies for − Si were too high. To improve the results, the p f -shell effective interaction in SDPF-NR was renormalizedto compensate for the absence of 2p-2h excitations from thecore [12]. With the reduction of pairing in the renormaliza-tion for Z ≤
14, the 2 + excitation energies in silicon isotopesagreed nicely with experiment. Thus, two SDPF-U interac-tions, one for Z >
14 and the other for Z ≤
14, are respec-tively needed for the description of collectivity at N =
28 inthe isotopic chains of sulfur ( Z =
16) and silicon ( Z = N =
28 [13]. From the above discussion, onesees that a consistent shell-model treatment for collectivity atthe N =
28 shell closure in different isotopic chains has notbeen successful so far.The main purpose of the present article is to construct aunified effective interaction for the neutron-rich sd -shell nu-clei, which can be consistently applicable to both Z > Z ≤
14 isotopic chains, and to understand the shell evo-lution in this exotic mass region. It is well known that re-alistic effective interactions used in the low-energy nuclearstructure study are dominated by the pairing plus quadrupole-quadrupole ( P + QQ ) forces with inclusion of the monopoleterm [14]. As documented in the literature, the extended P + QQ model combined with the monopole interaction workswell for a wide range of nuclei [15, 16]. This effective inter-action is called hereafter EPQQM to distinguish it from shellmodels with other effective interactions. The EPQQM modelhas demonstrated its capability of describing the microscopicstructure in different N ≈ Z nuclei, as for instance, in the f p -shell region [15] and the f pg -shell region [16]. Recently, ithas been shown that the EPQQM model is also applicable tothe neutron-rich Cr isotopes [17].The monopole interaction plays an important role in ourdiscussion. The monopole shifts in the spherical shell modelhave been introduced to account for the non-conventionalshell evolution in neutron-rich nuclei. Connection betweenthe monopole shifts and the tensor force [18] has been studiedwithin the self-consistent mean-field model using the Gognyforce [19]. It has recently been clarified that the possible phys-ical origin of these interactions is attributed to the central andtensor forces [20, 21]. For example, when one goes fromthe Ca down to the Si isotopes a significant reduction of the N =
28 shell gap can be found. By removing protons from the d / orbit in nuclei between Ca and Si, the strong attractiveproton-neutron monopole force between the p d / and n f / orbits is no longer present [12, 22]. This makes the neutron f / less bound, and thus reduces the size of the N =
28 gapin nuclei between Ca and Si.The paper is arranged as follows. In Sec. II, we outline ourmodel. In Section III, we perform the numerical calculationsand discuss the results for the neutron-rich nuclei in the sd - p f shell region. Finally, conclusions are drawn in Section IV. II. THE MODEL
We start with the following form of Hamiltonian, whichconsists of pairing and quadrupole-quadrupole terms with themonopole interaction H = H sp + H P + H P + H QQ + H m = (cid:229) a e a c † a c a − (cid:229) J = , g J (cid:229) M k P † JM k P JM k − c / b (cid:229) M : Q †2 M Q M : + (cid:229) a ≤ b (cid:229) T k T m ( ab ) (cid:229) JMK A † JMT K ( ab ) A JMT K ( ab ) , (1)where b in the third term is the length parameter of harmonicoscillator. We take the J = J = QQ ) forces in theparticle-hole channel [15, 16]. The monopole interaction isdenoted by H m , where the global monopole force is neglected + + + - - - - - - - - - - - - - - - - K - - - - Ar S Si Ca Ca E x c it a ti on e n e r gy ( M e V ) calexp FIG. 1: (Color online) Comparison of experimental and calculatedenergy levels for the odd-mass nuclei near the neutron or proton shellclosure in the sd region.
10 12 14 16 18 20 2201234567
EPQQM SDPF-U SDPF-NR EXP E ( + ) ( M e V ) ZN=28
FIG. 2: (Color online) Experimental and calculated first excited 2 + energies E ( + ) for the neutron-rich N =
28 isotones. because it does not affect the excitation energies of the low-lying states. This isospin-invariant Hamiltonian (1) is diago-nalized in a chosen model space based on the spherical basis.We employ the shell-model code ANTOINE [23] for the nu-merical calculation. In the present work, we consider an Ocore and employ the sd - p f model space comprising the 0 d / ,1 s / , 0 d / active proton orbitals and the 0 d / , 1 s / , 0 d / ,0 f / , 1 p / , 0 f / , 1 p / active neutron orbitals with twelve sd frozen neutrons. We employ the same single-particle en-ergies as those of Ref. [12]: e d / = . e s / = . e d / = . e f / = . e p / = . e p / = .
15, and e f / = .
18 (all in MeV). We adopt the following interac-tion strengths for the EPQQM forces: g = . , g = . c = .
474 (all in MeV). Finally, for the monopole terms,the strengths are chosen to be (all in MeV): k T = ( f / , d / ) = − . , k T = ( f / , d / ) = − . , k T = ( d / , p / ) = . , k T = ( d / , p / ) = − . , k T = ( d / , p / ) = . , k T = ( s / , p / ) = − . , k T = ( d / , d / ) = − . , k T = ( f / , f / ) = − . , k T = ( f / , p / ) = − . , k T = ( f / , p / ) = − . , k T = ( f / , f / ) = − . , k T = ( p / , p / ) = − . . (2)The rest of the monopole terms are neglected in the presentcalculations. The above EPQQM force strengths are deter-mined so as to reproduce the experimental energy levels forodd-mass nuclei that have one particle or one hole on topof the neutron and proton shell closures. The results arecompared with experimental data [24] in Fig. 1. Here theroot mean square (RMS) deviations between the experimen-tal and theoretical excitation energies are 0.24, 0.14, and 0.20(in MeV) for the 3 / − , 1 / − , and 5 / − states, respectively,which means that the agreement is good. We emphasize thatin order to describe the data correctly, a very important part ofthe interaction is those monopole forces. The attractive T = p d / - n f / and p d / - n f / have re-cently been proposed to be attributed to the monopole effectof the central and tensor forces in the nucleon-nucleon inter- neutron sdpf-shells N=28 E SP E ( M e V ) Z f5/2p1/2p3/2 f7/2d3/2s1/2d5/2 FIG. 3: (Color online) Effective neutron single-particle energies at N =
28 from Z = Z =
14 16 18 20-3-2-10123 D ( / - - / - ) ( M e V ) (cid:1)(cid:0)(cid:2)(cid:3)(cid:4)(cid:5) N=27 Z
FIG. 4: (Color online) Energy splitting between the lowest 7 / − and3 / − states in neutron-rich N =
27 isotones [4, 25, 27]. action [20, 21], where the central term produces a global con-tribution and the tensor term generates local variations. Theyare responsible for the collapse of the N =
28 shell closure inthe neutron-rich sd -shell nuclei, which gives rise to the drasticchange in nuclear shape of the N =
28 isotones. In fact, it hasbeen reported that the tensor force produces the pronouncedoblate minimum in the potential energy surface for Si [22].Along the K isotopic chains, the repulsive p d / - n p / andattractive p s / - n p / monopole interactions lead to an inver-sion of the expected ordering in the lowest 3 / + and 1 / + levels in K. In addition, these interactions reproduce the evo-lution of the lowest 3 / − state in Si. The monopole interac-tions between p d / and the p f -shells act so as to reproducethe low-lying states in Ca.
III. NUMERICAL RESULTS AND DISCUSSIONSA. The shell erosion in N = We discuss changes in the shell structure when movingaway from the valley of stability. Large energy gaps at the tra- ditional magic numbers in stable nuclei may be washed out inthe neutron- or proton-rich regions. The fundamental questionof how the nuclear shell structure evolves with proton or neu-tron excess is one of the main motivations to study nuclei farfrom stability. The study of shapes in such exotic nuclei servesas a sensitive test for the predictive power of nuclear models.Figure 2 shows the systematics of the first excited 2 + statesfor the N =
28 isotones, which are compared with three shellmodel calculations, respectively with the EPQQM, SDPF-U,and SDPF-NR effective interactions. As one can see, all thecalculations are capable of producing the drastic drop in en-ergy when removing protons from the double magic nucleus Ca.To understand the erosion of the N =
28 shell gap, the neu-tron effective single-particle energies (ESPE) as a function ofproton number are shown in Fig. 3. The n f / and n p / orbitals at Z =
20 becomes degenerate at Z =
16. This de-generacy is due to the attractive proton-neutron monopole in-teraction between the n f / and p d / orbitals. As seen inFig. 4, the inversion of the lowest 7 / − and 3 / − states in S( Z =
16 and N =
27) just reflects the degeneracy of the two or-bitals, n f / and n p / , at Z =
16 in Fig. 3. On the other hand,the neutron ESPE in the SDPF-U interaction does not showsuch a strong reduction of the N =
28 shell gap from Z =
20 to Z =
16. At Z =
16 the gap still equals about 3.5 MeV, and theinversion between the 7 / − and 3 / − states is ascribed to thecombined effect of the gap reduction due to the monopole in-teractions and the increase of the multipole correlations. The N =
28 shell closure is eroded in Ar and S, after the re-moval of only two and four protons, respectively. This rapiddisappearance of rigidity in the N =
28 isotones has been as-cribed to a reduction of the neutron shell gap at N =
28 com-bined with that of the proton subshell gap at Z =
16, leading toincreased probability of quadrupole excitations within the f p and sd shells for neutrons and protons, respectively. In fact,the occurrence of the low-lying isomer at 320.5 keV in Shas been interpreted in the shell-model framework as result-ing from the inversion between ( n / − ) − and ( n / − ) + configurations [25, 26]. We have thus understood that the in-version has the origin of vanishing N =
28 magicity at Z = Z =
20 are shown as a func-tion of neutron number. It is seen that the shell gap betweenthe p d / and p s / proton orbitals in the double closed-shellnucleus Ca ( Z = N =
20) decreases with increasing neutronnumber. The p d / orbit is almost degenerate with the p s / orbit at N =
28, and lies just below the p s / orbit at N = p d / and p s / orbits thus occurs above N =
32. The effect of adding neutrons to the f / orbital isprimarily to reduce the proton s / - d / gap. The relevant or-bitals, p s / and p d / , are known to become degenerate at N =
28. This degeneracy enhances the quadrupole correlationenergy of the configurations with open neutron orbitals. TheESPE from N =
34 to N =
40 for the p d / are different fromthose of the SDPF-U calculations. It would be attributed tothe monopole effect between the p d / and the n f / . Fig-ure 6 shows the energy splitting between the lowest 3 / + and1 / + states as a function of mass number for the neutron-richisotopes of phosphorus, chlorine, and potassium. The calcu-lations are in a good agreement with available experimentaldata [3, 24, 28] for the phosphorus and chlorine isotopes, butnot for the potassium isotopes. The energy splitting between3 / + and 1 / + for K isotopes in Fig. 6 deviates from theexperimental data. This would be related with the disagree-ment of the first excited 2 + enegies in Fig. 2. The variationtrend reflects the reduction of the p d / - p s / splitting in Fig.5 as neutron number increases. The inversion of the p d / - p s / orbits at N =
28 corresponds to the observation that theground-state of K is actually the 1 / + state, not the expected3 / + from the hole state ( p d / ) − . The near degeneracy of p d / and p s / is again attributed to the attractive proton-neutron monopole interaction between the n f / and p d / .The erosion of the N =
28 shell gap is enhanced at Z =
16 bythe degeneracy of the p d / and p s / proton orbits. In Fig.6, we also show the energy splitting between the lowest 3 / + and 1 / + states for the − Cl nuclei [28].
B. Comparison with other shell model calculations
A consistent description of collectivity at N =
28 for dif-ferent isotopic chains of the neutron-rich sd -shell nuclei hasbeen a challenge for shell model calculations. The early cal-culations with the SDPF-NR interaction described the levelscheme and transitions for these isotopic chains. However,it was found that the calculated results deviate from the ex-perimental data. For the silicon isotopes the calculated en-ergy levels are too high. To improve the agreement, Nowackiand Poves have proposed two effective interactions [12], one(SDPF-U) for Z ≤
14 and the other (SDPF-U1) for Z > + energies as a function of neu-tron number for the Ca, Ar, S, and Si isotopes. The EPQQMcalculations agree well with the known experimental data, ex-cept for Si and S. The SDPF-NR calculations also agreewith the data for the Ca and Ar isotopes, but cannot reproducethose of the Si isotopes. The calculated first excited 2 + stateslie higher than experiment. On the other hand, the SDPF-U(SDPF-U1) interaction fails to reproduce the known experi-mental data for Z >
14 ( Z ≤ N = + energy level of Si with N =
20. This suggests that it would be necessary to include theneutron excitations from the sd -shell to the p f -shell across the N =
20 energy gap. The disappearance of the N =
20 magicstructure between the sd and p f shells has already been estab-lished for the neutron-rich nuclei with N =
20 isotones.
18 20 22 24 26 28 30 32 34 36 38 40 42-8-6-4-202468 proton sd-shells
Z=20 E SP E ( M e V ) N d3/2s1/2d5/2 FIG. 5: (Color online) Effective proton single-particle energies at Z =
20 from N =
20 to N = C. Level scheme and B ( E ) values In this subsection, we present the theoretical level schemesand electromagnetic E + energy level for S repro-duces very well the recently observed data [13]. Very recently,energy spectrum for Ar has been experimentally obtained[33, 34]. In S, the calculated E ( + ) / E ( + ) ratio is about2.5, which qualitatively suggests a transitional or g -soft naturefor this nucleus. In S, the E ( + ) / E ( + ) ratio is 3.0, whichis quite close to the rigid rotor value. The situation signifi-cantly differs in S where both deformed and spherical con-figurations are predicted to coexist and mix weakly with each
34 36 38 40 42 44 46 48 50-4-3-2-10123
P_cal Cl_cal K_cal P_exp Cl_exp K_exp D ( / + - / + ) ( M e V ) A FIG. 6: (Color online) Energy splitting between the lowest 3 / + and1 / + states in neutron-rich P, Cl, and K isotopes.
18 20 22 24 26 28 30 32 34 36 38 40N Ca EPQQM SDPF-U SDPF-U1 SDPF-NR EXP S Si
18 20 22 24 26 28 30 32 34 36 380123456 E ( + ) ( M e V ) N Ar FIG. 7: (Color online) Comparison between experimental and calculated first excited 2 + energies E ( + ) with several effective interactions,EPQQM, SDPF-U, SDPF-U1, and SDPF-NR, for Si ( Z = ) , S ( Z = ) , Ar ( Z = ) , and Ca ( Z = ) isotopes. + + S + + + + + + + + + + + + + + + + + + + S + + + S S S S E x c it a ti on e n e r gy ( M e V ) calexp FIG. 8: (Color online)Experimental and calculated energy levels foreven-even Sulfur isotpes [2, 24, 29, 30]. other, which is supported by the observation of the low-lying0 + state at 1.326 MeV [30]. In the shell model calculation,this state lies below the 2 + state. The presence of a low-lying0 + state is considered as a signature of a spherical-deformedshape-coexistence and the present data support the weakeningof the N =
28 shell gap. Figure 9 shows the energy levels forthe even-even argon isotopes. One finds that the calculatedenergy levels can reproduce the experimental data [24, 33].The first excited 2 + energy E ( + ) for Ar isotopes decreaseswith increasing neutron number from Ar to Ar, but witha sudden increase for the N =
28 isotope. This suggests thepersistence of the N =
28 gap in Ar nuclei. In Figs. 8 and9, we note that the RMS deviations for the 2 + , 2 + , and 4 + states are respectively 0.22, 0.71, and 0.92 (in MeV), and the agreement becomes worse with increasing excitation energy.The disagreement may require the further improvement of the EPQQM interaction.To demonstrate the EPQQM model calculation for odd-mass nuclei in this mass region, we show in Fig. 10 energylevels for some odd-mass S and Ar isotopes. In the compari-son, the experimental energy levels for S and Ar are takenfrom the recent experiments [4, 34]. From the calculation, theground state of S is 3 / − , not the expected 7 / − , suggestingthat this nucleus is deformed. The occurrence of a low-lyingisomer at 320.5 keV in S is interpreted in the shell-modelframework as resulting from the inversion between the natu-ral ( n f / ) − and the intruder ( n p / ) + configurations. Boththe low B ( E
2; 7 / − → / − ) value and the absence of cal- + + + + + + + + + + + + + + + + + + + + + Ar + + + Ar Ar Ar Ar Ar E x c it a ti on e n e r gy ( M e V ) calexp FIG. 9: (Color online)Experimental and calculated energy levels foreven-even Argon isotopes [24, 31–33]. - - - - - - - - - - - - - Ar Ar - - - - - - - - Ar - - - - S E x c it a ti on e n e r gy ( M e V ) calexp FIG. 10: (Color online)Comparison of calculated energy levels withthe recent experimental data for S and Ar nuclei [4, 24, 25, 34–37]. culated deformed structure built on the 7 / − isomer suggesta coexistence of different shapes in the low-lying structure ofthe nucleus S. In Ar, the ground state 3 / − and the ex-cited state 1 / − can be interpreted as a coupling of a p / and p / neutron to the 0 + ground state of Ar. The 5 / − and7 / − levels result from the coupling of a p / neutron to the2 + state of Ar.In Table I, we compare B ( E ) values between the theoret-ical calculations and experiments for the S, Ar, and Si iso-topes. The experimental B ( E ) values are taken from Refs.[1, 29, 38, 39]. In our B ( E ) calculations, we take the ef-fective charges e p for protons and e n for neutrons as follows; e p = . e and e n = . e for S isotopes, e p = . e and e n = . e for Ar isotopes, and e p = . e and e n = . e forSi isotopes. On the other hand, in the SDPF-NR and SDPF-Ucalculations the effective charges e p = . e and e n = . e suggested in Ref. [12] are used for S and Si isotopes, andthe effective charges e p = . e and e n = . e are adoptedfor Ar isotopes. The B ( E ) values were previously calculatedand discussed with the SDPF-NR interaction by Retamosa etal. [6]. As one can see, our EPQQM calculation is in a goodagreement with the experimental B ( E ) values for the − S,but larger than the experimental value for S. On the otherhand, both the SDPF-U and SDPF-NR calculations also re-produce the data B ( E ) except for , S. In Table I, we alsocompare the calculated B ( E ) values with data for the Ar iso-topes. The experimental B ( E ) values are taken from Refs.[24, 29, 38, 40]. One observes that the experimental B ( E ) in-creases monotonically from Ar to the midshell nucleus Ar,and then decreases towards Ar. We note that all the theoret-ical calculations agree fairly well for − Ar, but for Ar thecalculated B ( E ) values are all too large [32, 41]. All the cur-rent shell model calculations with different interactions yielda similar value, which may indicate that this is a robust result.Very recently, however, a new experimental value 114 e fm has been reported [42], and is in a good agreement with allthe shell-model predictions. In Table I, this new experimentalvalue is included for comparison. It is apparent from the re-duction in excitation energy of the 2 + states and the enhance-ment in B ( E ) that collective features are gradually developedin neutron-rich isotopes between N =
20 and N =
28. How-ever, a different approach, the self-consistent mean-field cal-
TABLE I: Calculated B ( E + → + ) values for the even-even S,Ar, and Si isotopes. B ( E ) [ e fm ]Exp. EPQQM SDPF-U SDPF-NR S 21 22 22 21 S 47 38 33 34 S 67 63 58 60 S 79 72 75 73 S 63 98 73 73 Ar 26 35 34 34 Ar 66 59 49 50 Ar 86 59 70 72 Ar 69 47 71 77 Ar 114 108 105 106 Si 39 33 34 30 Si 38 40 39 37 Si - 48 54 46 Si - 64 86 53TABLE II: Calculated spectroscopic quadrupole moments for thefirst excited 2 + states in the N =
28 isotones. Q [ e fm ]Exp. EPQQM SDPF-U SDPF-NR Mg - -23 -20 -18 Si - 16 20 15 S - -3 -16 -15 Ar -8 -16 -3.7 -2.5 Ar - 21 21 21 Ca - 5 3 3 culation by Werner et al. [43], yields a much smaller B ( E ) ,in a disagreement with the new experimental data. We com-pare B ( E ) values between theoretical calculations and exper-iment for the Si isotopes. The agreements with the availabledata are satisfactory. The B ( E ) values increase with increas-ing neutron number N . The calculated B ( E ) values are quitesimilar for the EPQQM and SDPF-U interactions. For SDPF-NR, however, the predicted values are smaller for the heavierisotopes , Si.Finally in Table II, we present calculated spectroscopicquadrupole moments for the N =
28 isotones. For all the ef-fective interactions, small, positive Q ’s are found for Ca,indicating a spherical character of this nucleus. For the Snucleus, its small 2 + energy, large B ( E ) value, and the pres-ence of a second excited 0 + isomer at low excitation suggest amixed ground state configuration with spherical and deformedshapes. Recent experiments on the odd mass sulfur isotopesaround S seem to indicate that S is a deformed nucleuswith strong shape coexistence whereas S can be consideredas a well-deformed system. The enhancement in B ( E ) of Ar coincides with a shape change from a small prolate de-formation in Ar to a large deformation in Ar. The pre-dicted spectroscopic quadrupole moment Q s ( + ) in Ar issomewhat larger than the experimental value, but with a cor-rect sign. In Si, the yrast sequence of 0 + , 2 + , 4 + does notfollow the rotational J ( J + ) law. However, its quadrupolemoments are consistent with a deformed oblate structure. Theground state of Mg and that of Si are predicted to beprolately and oblately deformed, respectively. All the shellmodel calculations indicate consistently a rapid variation inquadrupole moment for the N =
28 isotones, suggesting adrastic shape change in this mass region.
IV. CONCLUSIONS
We have investigated the microscopic structure of theneutron-rich sd -shell nuclei using the spherical shell modelwith a new effective interaction in the sd - p f valence space.The interaction is of the extended pairing plus quadrupole-quadrupole type forces with inclusion of the monopole inter-action. The calculation has reproduced reasonably well theknown energy levels for even-even and odd-mass nuclei in-cluding the recent data for S, S and Ar. A special atten-tion has been paid to the N =
28 shell-gap erosion. It has been shown that the attractive T = p d / - n f / and p d / - n f / are very important for the variation ofthe shell gap at N =
28. This shell evolution is consideredas the monopole effect of the central and tensor forces in thenucleon-nucleon interaction. The strong attractive monopoleinteraction p d / - n f / causes the inversion of the 3 / + and1 / + states in odd-proton nucleus K. However, the inver-sion is recovered in the neighboring nucleus K, which is dueto the repulsive monopole interaction p d / - n p / . We havestudied the shape change in the neutron-rich N =
28 isotones,and predicted the drastic shape change when protons are re-moved away from the Z =
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