Proton dominance in the 2^+_2 -> 0^+_1 transition of N = Z\pm 2 nuclei around Si-28
aa r X i v : . [ nu c l - t h ] A p r KUNS-2334
Proton dominance in the +2 → +1 transition of N = Z ± nuclei around Si Yoshiko Kanada-En’yo
Department of Physics, Kyoto University, Kyoto 606-8502, Japan E Si are investigated in relation with intrinsic deformations based on a methodof antisymmetrized molecular dynamics. By comparing E Si and S, transition matrix amplitudes M p and M n for protons and neutrons are discussed inmirror analysis. Particular attention is paid to the M n /M p ratio in the transition from the 2 +2 stateto the 0 +1 state. The M n /M p ratio in Mg and Si is also investigated. It is found that the protondominance in the transition 2 +2 → +1 in Si and Si originates in the oblate trend of the Z = 14proton structure. I. INTRODUCTION
Nuclear systems often show intrinsic quadrupole de-formations. In many nuclei except for shell-closed nu-clei, normal deformations are found in the ground bands.Shell effects are essential for the deformations as de-scribed by the Nilsson model, and therefore, a varietyof shapes appear depending on the mass number par-ticular in mid-shell nuclei. In Z = N nuclei in the sd -shell region, a nuclear shape rapidly changes as a func-tion of Z = N reflecting the proton-neutron coherentshell effects in deformed systems. For instance, the pro-late ground band in Ne and the oblate one in Si areknown. These shapes are easily understood from the shelleffects in the Nilsson model, where the Z = N = 10 and Z = N = 14 shell gaps appear in the prolate and theoblate deformations, respectively.For Z = N nuclei, deformation phenomena are not sosimple as in Z = N nuclei. If the shell effect for pro-ton orbits and that for neutron ones compete with eachother, the proton shape can be affected by the neutronstructure, or decoupling between proton and neutron de-formations may occur. The latter case might be possi-ble in light-mass nuclei and it may be observed in thequadrupole transition properties such as the ratio of theneutron transition matrix amplitude to the proton one(so-called M n /M p ratio). Such the decoupling betweenproton and neutron shapes has been suggested, for in-stance, in C for which the enhanced M n /M p ratio, i.e.,the neutron dominance was observed in the ground-bandtransition [1]. The neutron dominance was described bythe oblate proton and prolate neutron shapes [2, 3]. Oneof the reasons for the characteristic structure of C isthat a Z = 6 system favors an oblate proton deforma-tion while a N = 10 nucleus has the prolate trend ofthe neutron deformation because of each shell effect ofthe proton and neutron parts. To clarify the oblate ten-dency of the proton structure in C, a possible K = 2side band and its transition properties should be exper-imentally observed. In general, such a nucleus havingoblate proton and prolate neutron structures may showan isovector triaxiality. If the case, in addition to the K = 0 band, a K = 2 side band is constructed withthe rotation around the symmetric axis of the prolateneutron part. Since the proton contribution should be dominant while the neutron contribution is minor for thisrotation, the transition from the 2 +2 state in the K = 2band to the ground state may show the proton dominanceresulting in a small M n /M p ratio. Although the K = 2side band was theoretically suggested, the 2 +2 state in C has not been experimentally confirmed yet, unfortu-nately. C is another candidate for the nuclei with theisovector triaxiality[2, 4], but the transition strength fromthe 2 +2 state to the 0 +1 state has not yet been measured.A similar situation of the isovector triaxiality is ex-pected in Z = 14 nuclei, i.e., Si isotopes because theoblate proton structure is favored due to the Z = 14shell effect. Let us consider deformations of Si and Si. From the negative sign of the Q -moments for the 2 +1 states in the Z = N = 12 and Z = N = 16 systems, theprolate trend of neutron structure is expected in N = 12and N = 16 systems. Considering the prolate tendencyof the neutron part, the isovector triaxiality in Si and Si would be possible and it might lead to the protondominance in the 2 +2 → +1 transition resulting in a small M n /M p ratio. For Si and Si, the M n /M p ratios forthe 2 +2 → +1 transition as well as the 2 +1 → +1 was exper-imentally determined by mirror analysis[5, 6] and inelas-tic scattering data[7]. The reported values of the M n /M p ratio in the 2 +2 → +1 are 0 . ± .
07 and 0 . ± .
03 for Si and Si, respectively, and they indicate the protondominance. The data of neutron and proton transitionmatrices, M n and M p , in this mass region are qualita-tively reproduced by shell model calculations[8] and theywere discussed in relation with effective charges comingfrom core polarization [8, 9]. The proton dominance inthe 2 +2 → +1 of Si was discussed also in relation withthe γ softness in the calucaltions based on the quadrupolecollective Hamiltonian[10].Our aim is to understand the properties of proton andneutron transition matrices from the viewpoint of defor-mations of proton and neutron parts. In particular, theproton dominance in the 2 +2 → +1 is a focusing featurewhich might be interpreted in connection to decouplingof proton and neutron shapes. For this aim, we applya theoretical approach of antisymmetrized molecular dy-namics(AMD). The AMD has been successfully appliedfor study of structures of p -shell and sd -shell nuclei[11–13]. The present method of AMD calculations is the sameas those used for investigation of deformation phenomenain Si, Ne, O, and C[14] and those in C isotopes[2].Applying the AMD method, we investigate the transi-tion properties of Si. Assuming the mirror symmetry,the calculated proton and neutron transition matrix am-plitudes are compared with the data evaluated by the ob-served E Si and S, and theyare discussed in connection with proton and neutron de-formations. Transition properties in Mg and Si arealso discussed. The proton dominance in the transition,2 +2 → +1 , for C and C is also shown as well as thatfor Si isotopes.The paper is organized as follows. In the next section,the formulation of the present calculation is explained. InIII, The results for Si and S and those for Mg and Si are shown and the proton dominance of 2 +2 → +1 in Si and Si is discussed. The proton dominance in Cand C is shown in IV. In V, a summary is given.
II. FORMULATION
Here we briefly explain the formulation of the presentcalculations. Details of the formulation of AMD methodsfor nuclear structure study are explained in Refs. [11–13].The method of the present calculations is basically sameas that in Refs. [11, 14].An AMD wave function Φ
AMD for a system with themass number A is given by a single Slater determinantof Gaussian wave packets as,Φ AMD ( Z ) = 1 √ A ! A{ ϕ , ϕ , ..., ϕ A } , (1)where the i -th single-particle wave function is written as, ϕ i = φ X i χ i τ i , (2) φ X i ( r j ) ∝ exp (cid:8) − ν ( r j − X i √ ν ) (cid:9) , (3) χ i = ( 12 + ξ i ) χ ↑ + ( 12 − ξ i ) χ ↓ . (4)Here the isospin function τ i is fixed to be up(proton) ordown(neutron). The orientation of intrinsic spin ξ i is alsofixed to be 1 / − / ν = 0 .
15 fm − which is the optimumvalue for Si used in Ref. [14]. The spatial part, φ X i ,is written by a Gaussian wave packet localized at thecertain position X i in the phase space. Then, the AMDwave function is expressed by { X i } which indicate theGaussian centers for all the single-particle wave functionsand are treated as the independent complex variationalparameters.We perform energy variation for a parity-eigen state, P ± Φ AMD ≡ Φ ± AMD , projected from an AMD wave func-tion by means of the frictional cooling method[11]. Weconsider the AMD wave function obtained by the en-ergy variation as an intrinsic state, and total-angular-momentum projection( P JMK ) is performed after the vari- ation to calculate such observables as energies and tran-sition strengths. The K -mixing is incorporated inthe total-angular-momentum projection for non-zero J states.As shown later, two local minimum solutions are ob-tained in the energy variation for the Z = 14 systems.The two minima almost degenerate to each other andmay correspond to shape coexistence phenomena as al-ready discussed in the N = 14 systems[14]. Compar-ing the calculated structure properties such as transitionstrengths with the experimental data, we assign one oftwo minima to the intrinsic state of the ground band.By using the obtained wave functions, we calculatethe transition matrix amplitudes M p and M n for pro-ton and neutron, respectively, and also the E M p = h f || P ( t z = 1 / rY µ || i i , (5) M n = h f || P ( t z = − / rY µ || i i , (6) B ( E
2) = e J i + 1 M p . (7)(8)Here P ( t z = ± /
2) are the isospin projection operatorsfor protons and neutrons.
III. RESULTSA. Effective interaction
The effective nuclear interactions adopted in thepresent work consist of the central force, the spin-orbitforce and Coulomb force. We adopt MV1 force [15] asthe central force. The MV1 force contains a zero-rangethree-body force in addition to the two-body interaction.For interaction parameters, we use the same parameterset as that in Ref. [14]. Namely, the MV1 force (case1) with the parameters b = 0, h = 0, and m = 0 .
62 isused. For the spin-orbit force, the two-range Gaussianform of the G3RS force [16] is adopted. The strengths ofthe spin-orbit force are u I = − u II = 2800 MeV. Thesestrengths were adjusted to reproduce energy levels of Siin Ref. [14]. B. Si and S After the energy variation for parity projected AMDwave functions, two local minimum solutions (A) and(B) are obtained for Si. The state (A) shows a smallerdeformation as ( β p , γ p ) = (0 . , . π ) and ( β n , γ n ) =(0 . , . π ) and the state (B) has a lager deformationas ( β p , γ p ) = (0 . , . π ) and ( β n , γ n ) = (0 . , . π ).Here the definition of the quadrupole deformation param-eters ( β p , γ p ) for proton density distribution and ( β n , γ n )for neutron one is given in Ref. [4].After the angular momentum projection, the energylevels are obtained from the states (A) and (B) (Fig. 1).The bands constructed from two intrinsic states, (A) and(B), almost degenerate. As shown later, the band (A)shows good agreements with experimental data such asthe Q moment and E J ∼ E K π = 2 + side bandby the K π = 0 + ground band was identified[17]. In thepresent calculation, no K π = 2 + side band member isobtained from the band (A) because of the prolate in-trinsic shape of the state (A). Owing to the Z = 14shell gap in the oblate deformation, it is naturally ex-pected that Si may be soft agaist γ deformation to-ward the oblate region. Therefore, we construct an-other intrinsic state (C) by the alternative energy varia-tion where we vary the single-particle neutron wave func-tions but freeze the proton configuration so that it hasthe same proton structure as that of the oblate groundstate of Si. It corresponds to the energy variationwith the constraint of the oblate proton structure. Thusobtained state (C) has an triaxial neutron structure of( β n , γ n ) = (0 . , . π ) with the oblate proton deforma-tion of ( β p , γ p ) = (0 . , . π ). The single-particle en-ergy levels in the intrinsic wave functions for the states(A) and (C) are shown in Fig. 3. The derivation ofthe single-particle energies of AMD wave functions isdescribed, for example, in Refs. [13, 18]. In the state(C), the single-particle energy spectra are consistent withthose for oblate systems where the N = 14 shell gap isclearly seen, while the N = 14 shell gap vanishes in thestate (A).After the angular momentum projection, the K π = 0 + and the K π = 2 + side bands are generated from theintrinsic state (C). We should stress that the absoluteenergy of the 0 +1 projected from the state (C) is rela-tively lower than that from the state (A) even thoughthe intrinsic energy of the state (C) is higher than (A)before the projection. It is because the triaxial stategains more energy than the prolate state in the angularmomentum projection. Finally we superpose the state(A) and (C), and obtain the energy levels for the groundand side bands. The energy levels calculated by the su-perposition (A+C) correspond well to the experimentallevels for the K π = 0 + and the K π = 2 + bands exceptfor a slightly higher excitation energy for the K π = 2 + band. Considering that the K π = 2 + band members are constructed mainly from the | K | = 2 components of theintrinsic state (C) and the ground band can be describedby the K = 0 states of (C), these bands are considered tobe the K π = 0 + and the K π = 2 + side bands projectedfrom the triaxial intrinsic state (C). We here note that thewave functions of the K π = 0 + components of the state(C) and (A) have large overlap with each other. There-fore, an alternative interpretation is possible. Namely,the ground band is regarded as the rotational band ofthe prolate state (A) and the K = 2 side band is the γ vibration band on the top of the prolate state, which istaken into account by the superposition of (A) and (C).We calculate the quadrupole transition propertiesby the superposition (A+C) and discuss the neutronand proton contributions based on the mirror analy-sis. We first show the calculated values of B ( E M p , and electricquadrupole moments Q in Si in comparison with theexperimental data in Table I. The theoretical values arein good agreement with the experimental data except forthe inter-band transition 4 +2 → +1 .We next discuss the M n /M p ratio for the 2 +1 → +1 and 2 +2 → +1 transitions. We assume here the mirrorsymmetry that the neutron transition matrix amplitudes M n in Si is consistent with the proton transition matrixamplitudes M p in S. Then, the experimental M n valuesfor Si are evaluated by B ( E
2) in S. As shown in TableII, the experimental M n /M p ratio for 2 +1 → +1 in Siis close to a unit indicating that the proton and neutronparts equarly contribute to the ground-band transition.What is striking is that the M n /M p ratio for 2 +2 → +1 is 0.5 in the experimental data. This indicates the dom-inant proton matrix amplitude, which is about twice ofthe neutron one in the transition from the side band tothe ground state. The present calculation reproduces thistrend of the quenched M n /M p ratio, though they muchunderestimate the M n value for 2 +2 → +1 in Si. Theorigin of the dominant proton contribution and the minorneutron one in this transition, 2 +2 → +1 , is understoodby the difference between proton and neutron structures.As mentioned before, the side band members are con-structed mainly from the | K | = 2 components of theintrinsic state (C), while the ground band is also approx-imately written by the | K | = 0 states projected from thestate (C). That is, the 2 +2 state can be interpreted asthe triaxial side band. As shown in Fig. 2, the state (C)shows the larger triaxiality of the proton shape than theneutron part, and therefore, proton contribution shouldbe dominant but neutron one is minor in the 2 +2 → +1 .We should comment again that the K = 0 componentsof the triaxial state (C) have large overlaps with those ofthe prolate state (A). In fact, the overlap between the 0 + states projected from (A) and (C) is 55%. This is becausethe deformation parameter γ is not a coordinate but justan expectation value, and two states with different γ val-ues are not orthogonal to each other. Even if a rotationalband is constructed by the angular-momentum projec-tion from an intrinsic state, the intrinsic shape is notobservable but an interpretation which is useful to inter-prete the microscopic wave functions projected from theintrinsic state as the rotational band members. There-fore, strictly speaking, it is not easy to clearly distin-guish between rotational K = 2 modes of a static triaxialstate and γ -vibrational modes in such the system. Then,we can consider the alternative interpretation for the 2 +2 state as the γ vibration on the prolate K = 0 groundband. Again, the proton dominance can be easily un-derstood by the γ vibration of the proton structure. Inany cases, we can conclude that the proton dominanceof the transition, 2 +2 → +1 , in Si originates in the γ softness of the proton part in the Z = 14 system, andthe difference between proton and neutron structures isessential. E x c it a ti on e n e r gy ( M e V ) Si A+BBA A+CC exp.
024 02 354 0 02 2 3454 4 024 2345 024 2345 02 3453454 024 2345
FIG. 1: Energy levels of Si for positive parity states. Thecalculated levels are obtained by the projection from the state(A), the state (B), the superposition (A+B), the state (C),and the superposition (A+C). The energies are measured fromthe 0 +1 calculated with (A+C). The experimental levels arethe positive parity bands observed by the gamma-ray mea-surements in Ref. [17].
0 0.4 0 0.4 0 0.4 cos β γ s i n β γ cos β γ cos β γ s i n β γ s i n β γ β β β Si(C) Si(A) Si(B) γ γ γ FIG. 2: Deformation parameters β and γ of the intrinsic wavefunctions Si(A), Si(B), and Si(C). The filled trianglesindicate β p and γ p for the proton part and the open circlesare β n and γ n for the neutron part C. Mg and Si In this subsection, we discuss the proton dominancein the 2 +2 → +1 of Si in relation to the γ deformationof the proton structure, in a similar way to the mirroranalysis of Si and S. Before discussing the M n /M p ratio of Si based on the mirror analysis, we first inves-tigate the structure of the ground and excited states of -60-50-40-30-20-10 0 proton neutron -60-50-40-30-20-10 0 proton neutron
Si(A) Si(C) e n e r gy ( M e V ) e n e r gy ( M e V ) FIG. 3: Single-particle energies in the intrinsic wave functions Si(A) and Si(C).TABLE I: E Si. The experimental data are takenfrom [19] a and [17] b exp. cal. B ( E M p B ( E M p Inband ( K = 0 +1 )2 +1 → +1 ± a
14 5.6 12.54 +1 → +1 +1 . − . b
14 8.7 15.5Inband ( K = 2 +1 )3 +1 → +2 +16 − b
15 8.5 15.44 +2 → +2 +3 . − . b
13 7.1 144 +2 → +1 +1 → +1 +2 . − . b
15 7.6 14.55 +1 → + +1 . − . b +2 → +1 ± . a +2 → +1 +11 − b
14 9.1 15.93 +1 → +1 +2 . − . b
11 2.8 8.84 +2 → +1 +4 . − . b
18 0.02 0.84 +1 → +1 Q µ Q µ +1 − ± − . the mirror nucleus Mg for which the existing data isricher than for Si.We apply the AMD method to Mg. After energyvariation for parity projected AMD wave functions, twolocal minimum solutions (A) and (B) are obtained in Mg, similarly to the case of Si. The state (A)shows the triaxiality shape ( β p , γ p ) = (0 . , . π ) and( β n , γ n ) = (0 . , . π ) with a smaller neutron defor-mation, while the state (B) has a prolate deformation( β p , γ p ) = (0 . , . π ) and ( β n , γ n ) = (0 . , . π ) witha larger neutron deformation as seen in Fig. 4. The en-ergy of the intrinsic state (A) degenerates with the state(B) within 0.1 MeV before the angular momentum pro-jection. After the angular-momentum projection, the 0 + energy projected from the state (A) is 1.1 MeV higher TABLE II: The calculated and theoretical B ( E
2) values andthe proton transition matrix for the transitions 2 +2 → +1 and2 +1 → +1 in Si and S. The experimental values of the M n /M p ratio are evaluated based on the mirror analysis inwhich M n of Si is assumed to be equal to M p of S. Theexperimental B ( E
2) values are taken from Ref. [19] and c thoseevaluated from the life times and branching ratios.exp. cal. B ( E M p Mn/Mp B ( E M p Mn/Mp +1 → +130 Si 8.5 ± S 11 c
18 7.6 14.52 +2 → +130 Si 1.7 ± S 0.4 c than that obtained from the state (B). In spite of theslightly higher energy of the state (A) than that of thestate (B), we tentatively assign the states projected from(A) to the experimental K π = 0 + ground band and itsside band K π = 2 +1 , because the observed level struc-ture and E K π = 0 +1 and K π = 2 +1 band members can be reproduced by the resultscalculated with the state (A). The state (B) is inconsis-tent with the experimental fact that the K π = 2 +1 sideband exists by the ground band, and therefore, it wouldcorrespond to an excited K π = 0 + band though it iseventually the lowest in the present calculations. In fact,the state (A) can be the lowest if we tune the interactionparameters, for instance, with m = 0 . b = h = 0 . u I = − u II = 3200 MeV. Hereafter, we concentrateonly on the K π = 0 + band and its K π = 2 + side bandconstructed from the state (A), and omit the mixing ef-fect of the state (B).To see the γ softness of the N = 14 neutron struc-ture in Mg, we construct an intrinsic state (C) by thealternative energy variation that we vary the centers ofthe single-particle Gaussian wave functions only for pro-tons but freeze the neutron configuration so that it hasthe same neutron structure as that of the oblate groundstate of Si. It means the energy variation with the con-straint of the oblate neutron structure. Thus obtainedstate (C) of Mg has an triaxial proton structure of( β p , γ p ) = (0 . , . π ) with the almost oblate neutrondeformation ( β n , γ n ) = (0 . , . π ). The state (C) is2 MeV higher than the state (A) before and after theangular momentum. In Fig. 5, we show the energy lev-els obtained by the superposition (A+C). The calculatedenergy levels are in reasonable agreements with the ex-perimental data except for underestimation of the levelspacing in the ground band. In case of Mg, the mixingof the state (C) gives minor contributions and the energylevels in the (A+C) calculations are qualitatively same asthose obtained by the single intrinsic state (A) without mixing of the state (C). It indicates that the K π = 2 +1 band can be interpreted as the side band of the K π = 0 +1 ground band constructed from the triaxial shape of thestate (A).Let us show the quadrupole transition properties calcu-lated by the superposition (A+C) and discuss the protonand neutron matrix amplitudes based on the mirror anal-ysis. We first show the calculated values of the B ( E Q moments of Mg in Table III. The present re-sults are in reasonable agreement with the experimentaldata. Next we discuss the M n /M p ratio for the transi-tions 2 +1 → +1 and 2 +2 → +1 in Si. Here we assume themirror symmetry that M n and M p of Si equal to M p and M n of Mg, respectively. The experimental valuesof M p are obtained from the B ( E
2) values of Si and Mg. The experimental M n /M p ratio for the ground-band transition 2 +1 → +1 is 0.94 which is close to one,while that for the 2 +2 → +1 is 0.5. The quenched M n /M p for the transition 2 +2 → +1 indicates the proton domi-nance. The present calculations reproduce this featureof the proton dominance.Here we remind the reader that the 2 +2 state of Mg isconstructed from the triaxial neutron shape of the state(A). In other words, the 2 +2 state of the mirror nucleus Si is given by the rotation of the triaxial proton shape.Consequencely, the proton contribution is dominant inthe excitation from the 0 +1 state to the 2 +2 state resultingin the small value of M n /M p for the transition 2 +2 → +1 in Si.
TABLE III: E Mg. The experimental data are takenfrom [20] a and [21] b exp. cal. B ( E M p B ( E M p Inband ( K = 0 +1 )2 +1 → +1 ± a +2 → +1 ± a
24 16.1 25.7Inband ( K = 2 +1 )3 +2 → +2 +7 . − . b
17 22.5 26.84 +4 → +2 +6 . − . b
15 9.5 19.8Inter-band2 +2 → +1 ± a Q µ Q µ +1 − . ± − . IV. M n /M p RATIOS IN p -SHELL NUCLEI In the previous section, we discuss the the proton dom-inance in the transition 2 +2 → +1 between the side bandand the ground band of Si isotopes with N = Z ±
2. The
TABLE IV: The calculated and theoretical B ( E
2) values andthe proton transition matrix for the transitions 2 +2 → +1 and2 +1 → +1 in Si and Mg. The experimental values of the M n /M p ratio are evaluated based on the mirror analysis inwhich M n of Si is assumed to be equal to M p of Mg. Theexperimental B ( E
2) values are taken from Ref. [20]exp. cal. B ( E M p Mn/Mp B ( E M p Mn/Mp +1 → +126 Si 15.4 ± Mg 13.4 ± +2 → +126 Si 1.6 ± Mg 0.35 ±
0 0.4 0 0.4 0 0.4 cos β γ s i n β γ cos β γ cos β γ s i n β γ s i n β γ β β β Mg(C) Mg(A) Mg(B) γ γ γ FIG. 4: Deformation parameters β and γ of the intrinsic wavefunctions Mg(A), Mg(B), and Mg(C). The filled trian-gles indicate β p and γ p for the proton part and the open circlesare β n and γ n for the neutron part proton dominance can be described by the triaxial shapeof the proton structure. This feature originates in theoblate trend of the Z = 14 proton configuration.Let us consider the proton dominance in p -shell nu-clei. In C isotopes, Z = 6 proton configuration has theoblate trend as is known in C. In fact, the oblate protonand prolate neutron shapes are theoretically suggested in C and C in the AMD calculations[2]. In these nuclei,the proton dominance in the transition 2 +2 → +1 is ex-pected as well as Si and Si. In this section, we referto the intrinsic structures of C and C investigated inRef. [2], and discuss the proton dominance in C isotopesin comparison with Si isotopes. We use the AMD wavefunctions calculated in the previous work[2].We show the deformation parameters β and γ for pro-ton and neutron density in the intrinsic states of C and C in Fig. 6. The experimental and theoretical valuesof the M n /M p ratios are written in Fig. 7, in which theexperimental energy levels are drawn. In both nuclei, C and C, the proton structure shows the oblate de-formation in spite of the prolate neutron structure. Itmeans that the Z = 6 proton structure is not so muchaffected by the neutron structure but it keeps the oblatetendency. As a result, the decoupling between proton andneutron shapes is more remarkable in C isotopes thanin Si isotopes. This decoupling is one of the reasons forthe enhanced M n /M p ratio of the ground-band transition K=0 +1 K=2 +1 E x c it a ti on e n e r gy ( M e V ) Mg exp.A+C FIG. 5: Energy levels of Mg. The calculated levels areobtained by the superposition of the states projected from thestates (A) and (C). The experimental levels are the positiveparity bands observed by the gamma-ray measurements inRef. [21].
0 0.6 0 0.6 cos β γ s i n β γ cos β γ s i n β γ γ β β C C γ FIG. 6: Deformation parameters for the intrinsic wave func-tions of C and C. The filled triangles indicate β p and γ p for the proton part and the open circles are β n and γ n for theneutron part +1 → +1 in C as discussed in Ref. [2]. For the tran-sition 2 +2 → +1 in C and C, the calculated M n /M p ratio is quenched and it indicates the proton dominance.The suggested proton dominance originates in the oblateproton structure in the Z = 6 systems. This is a goodanalogy to the proton dominance in Si and Si, whicharises from the oblate trend of the Z = 14 proton struc-ture.Experimentally, the M n /M p ratio for 2 +2 → +1 is un-known. Inelastic scatterings of C and C might be agood probe to evaluate the M n /M p ratio.Note that the shape has large quantum fluctuation insuch light systems and it is not observable. Nevertheless,it is helpful to interpret the angular-momentum projectedstates in terms of rotation of the intrinsic deformation toget semi-classical picture of transition properties. Ourargument is that the M n /M p ratios in C isotopes can bequalitatively understood by the oblate proton and prolateneutron shapes. V. SUMMARY E Si were investigated in relation withthe intrinsic deformation. In the calculation of the AMDmethod, the experimental B ( E
2) values in the K π = 0 +1 Si Si E n e r gy ( M e V ) C C (6.9/5.3) 2+ (2+) 2(1.1/4.6) E n e r gy ( M e V ) (15/13)18/15 2+ 2+0+ 3/6.6(0.6/6.6)2+ 2+0+(17/12)18/197.2/7.8 (13/3.1)−/3.5 (0.8/4.9)2.8/6.0(1.5/5.6) FIG. 7: The experimental energy levels of the 0 +1 , 2 +1 , and2 +2 states and M n /M p ratios for the 2 +1 → +1 and 2 +2 → +1 of Si, Si, C and C[19, 20, 22, 23]. The experimentalvalues of the neutron matrix amplitude( M n ) are deduced fromthe corresponding B ( E
2) values of the mirror nucleus. Thevalues in the parentheses are the present calculations. and K π = 2 +1 bands are reproduced by the calculation.Based on mirror analysis, the transition matrix ampli-tudes M p and M n were discussed. Particular attentionis paid to the M n /M p ratio in the transition from the2 +2 state to the 0 +1 state, whose quenching is experimen-tally known. The M n /M p ratio in Mg and Si wasalso investigated. We have shown that the K π = 2 +1 band can be interpreted as the triaxial side band, andthe proton contribution is dominant in the transition 2 +2 → +1 . The quenched M n /M p , i.e., the proton dom-inance in 2 +2 → +1 in Si and Si originates in theoblate trend of the Z = 14 proton structure. The protondominance in 2 +2 → +1 is suggested also in C and C,where the oblate proton structure is favored.We should comment that the shape has large quantumfluctuation in light-mass systems and it is not observ-able. Strictly speaking, the macroscopic picture may betoo simple for such systems, and therefore, it is not easyto clearly distinguish between two collective pictures,the rotational mode of a static triaxial state and the γ -vibrational mode. Nevertheless, it is helpful to interpretthe angular-momentum projected states in terms of ro-tation of the intrinsic deformation to get semi-classicalpicture of transition properties. Our argument is thatthe quesched M n /M p ratios in 2 +2 → +1 of these nucleican be qualitatively understood by the oblate trend ofproton shapes in the prolate neutron structures. Acknowledgments
The computational calculations of this work were per-formed by using the supercomputers at YITP and donein Supercomputer Projects of High Energy AcceleratorResearch Organization (KEK). This work was supportedby Grant-in-Aid for Scientific Research from Japan So-ciety for the Promotion of Science (JSPS). It was alsosupported by the Grant-in-Aid for the Global COE Pro-gram ”The Next Generation of Physics, Spun from Uni-versality and Emergence” from the Ministry of Educa-tion, Culture, Sports, Science and Technology (MEXT)of Japan. [1] Z. Elekes et al. , Phys. Lett.
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