Shape fluctuations in the ground and excited 0+ states of 30Mg and 32Mg
Nobuo Hinohara, Koichi Sato, Kenichi Yoshida, Takashi Nakatsukasa, Masayuki Matsuo, Kenichi Matsuyanagi
aa r X i v : . [ nu c l - t h ] S e p Shape fluctuations in the ground and excited + statesof Mg and Mg Nobuo Hinohara, Koichi Sato, Kenichi Yoshida,
2, 1
TakashiNakatsukasa, Masayuki Matsuo, and Kenichi Matsuyanagi
1, 3 Theoretical Nuclear Physics Laboratory, RIKEN Nishina Center, Wako 351-0198, Japan Department of Physics, Faculty of Science, Niigata University, Niigata 950-2181, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: September 24, 2018)Large-amplitude collective dynamics of shape phase transition in the low-lying states of − Mgis investigated by solving the five-dimensional (5D) quadrupole collective Schr¨odinger equation. Thecollective masses and potentials of the 5D collective Hamiltonian are microscopically derived with useof the constrained Hartree-Fock-Bogoliubov plus local quasiparticle RPA method. Good agreementwith the recent experimental data is obtained for the excited 0 + states as well as the groundbands. For Mg, the shape coexistence picture that the deformed excited 0 + state coexists withthe spherical ground state approximately holds. On the other hand, large-amplitude quadrupole-shape fluctuations dominate in both the ground and the excited 0 + states in Mg, so that theinterpretation of ‘coexisting spherical excited 0 + state’ based on the naive inversion picture of thespherical and deformed configurations does not hold. PACS numbers: 21.60.Ev, 21.10.Re, 21.60.Jz, 27.30.+t
Nuclei exhibit a variety of shapes in their ground andexcited states. A remarkable feature of the quantumphase transition of a finite system is that the order pa-rameters (shape deformation parameters) always fluctu-ate and vary with the particle number. Especially, thelarge-amplitude shape fluctuations play a crucial role intransitional (critical) regions. Spectroscopic studies oflow-lying excited states in transitional nuclei are of greatinterest to observe such unique features of the finite quan-tum systems.Low-lying states of neutron-rich nuclei around N =20 attract a great interest, as the spherical configura-tions associated with the magic number disappear inthe ground states. In neutron-rich Mg isotopes, the in-crease of the excitation energy ratio E (4 +1 ) /E (2 +1 ) [1–3] and the enhancement of B ( E
2; 2 +1 → +1 ) from Mgto Mg [4–6] indicate a kind of quantum phase tran-sition from spherical to deformed shapes taking placearound Mg. These experiments stimulate microscopicinvestigations on quadrupole collective dynamics uniqueto this region of the nuclear chart with various theo-retical approaches; the shell model [7–10], the Hartree-Fock-Bogoliubov (HFB) method [11, 12], the parity-projected HF [13], the quasiparticle RPA (QRPA) [14,15], the angular-momentum projected generator coordi-nate method (GCM) with [16] and without [17, 18] re-striction to the axial symmetry, and the antisymmetrizedmolecular dynamics [19].Quite recently, excited 0 + states were found in Mg[20, 21] and Mg [22] at 1.789 MeV and 1.058 MeV, re-spectively. For Mg, the excited 0 +2 state is interpretedas a prolately deformed state which coexists with thespherical ground state. For Mg, from the observed pop-ulation of the excited 0 +2 state in the ( t, p ) reaction on Mg, it is suggested [22] that the 0 +2 state is a spheri-cal state coexisting with the deformed ground state and that their relative energies are inverted at N = 20. How-ever, available shell-model and GCM calculations consid-erably overestimate its excitation energy (1 . − . +2 states. For understanding shapedynamics in low-lying collective excited states of Mg iso-topes near N = 20, it is certainly desirable to develop atheory capable of describing various situations in a uni-fied manner, including, at least, 1) an ideal shape co-existence limit where the wave function of an individualquantum state is well localized in the deformation spaceand 2) a transitional situation where the large-amplitudeshape fluctuations dominate.In this article, we microscopically derive the five-dimensional (5D) quadrupole collective Hamiltonian us-ing the constrained Hartree-Fock-Bogoliubov (CHFB)plus local QRPA (LQRPA) method [23]. The 5Dcollective Hamiltonian takes into account all the fivequadrupole degrees of freedom: the axial and triaxialquadrupole deformations ( β, γ ) and the three Euler an-gles. This approach is suitable for our purpose of de-scribing a variety of quadrupole collective phenomena ina unified way. Another advantage is that the time-oddmean-field contributions are taken into account in eval-uating the vibrational and rotational inertial functions.In spite of their importance for correctly describing col-lective excited states, the time-odd contributions are ig-nored in the widely used Inglis-Belyaev cranking formulafor inertial functions. The CHFB + LQRPA method hasbeen successfully applied to various large-amplitude col-lective dynamics including the oblate-prolate shape coex-istence phenomena in Se and Kr isotopes [23, 24], the γ -soft dynamics in sd -shell nuclei [25], and the shape phasetransition in neutron-rich Cr isotopes [26]. A preliminaryversion of this work was reported in Ref. [27].The 5D quadrupole collective Hamiltonian is writtenas H coll = T vib + T rot + V ( β, γ ) , (1) T vib = 12 D ββ ( β, γ ) ˙ β + D βγ ( β, γ ) ˙ β ˙ γ + 12 D γγ ( β, γ ) ˙ γ , (2) T rot = 12 X k =1 J k ( β, γ ) ω k , (3)where T vib and T rot are the vibrational and rotationalkinetic energies, respectively, and V is the collective po-tential. The vibrational collective masses, D ββ , D βγ , and D γγ , are the inertial functions for the ( β, γ ) coordinates.The rotational moments of inertia J k associated with thethree components of the rotational angular velocities ω k are defined with respect to the principal axes. In theCHFB + LQRPA method, the collective potential is cal-culated with the CHFB equation with four constraints onthe two quadrupole operators and the proton and neu-tron numbers. The inertial functions in the collectiveHamiltonian are determined from the LQRPA normalmodes locally defined for each CHFB state in the ( β, γ )plane. The equations to find the local normal modesare similar to the well-known QRPA equations, but theequations are solved on top of the non-equilibrium CHFBstates. Two LQRPA solutions representing quadrupoleshape motion are selected for the calculation of the vibra-tional inertial functions. After quantizing the collectiveHamiltonian (1), we solve the 5D collective Schr¨odingerequation and obtain collective wave functionsΨ αIM ( β, γ, Ω) = X K =even Φ αIK ( β, γ ) h Ω | IM K i , (4)where Φ αIK ( β, γ ) are the vibrational wave functions and h Ω | IM K i are the rotational wave functions defined interms of D functions D IMK (Ω). We then evaluate E sd and pf shells) are taken into accountfor both neutrons and protons. To determine the param-eters in the P+Q Hamiltonian, we first perform Skyrme-HFB calculations with the SkM* functional and the sur-face pairing functional using the HFBTHO code [28].The pairing strength ( V = −
374 MeV fm − , with a cut-off quasiparticle energy of 60 MeV) is fixed so as to repro-duce the experimental neutron gap of Ne (1.26 MeV).We then determine the parameters for each nucleus inthe following way. The single particle energies are deter-mined by means of the constrained Skyrme-HFB calcu-lation at the spherical shape. The resulting single parti-cle energies (in the canonical basis) are then scaled withthe effective mass of the SkM* functional m ∗ /m = 0 .
79, since the P+Q model is designed to be used for single-particle states whose effective mass is equal to the barenucleon mass. In Mg, the N = 20 shell gap be-tween d / and f / is 3.7 MeV for the SkM* functional,and it becomes 2.9 MeV after the effective mass scal-ing. This value is appreciably smaller than the stan-dard modified oscillator value 4.5 MeV [29]. This spac-ing almost stays constant for − Mg. The strengths ofthe monopole-pairing interaction are determined to re-produce the pairing gaps obtained in the Skyrme-HFBcalculations at the spherical shape. The strength of thequadrupole particle-hole interaction is determined to re-produce the magnitude of the axial quadrupole defor-mation β of the Skyrme-HFB minimum. The strengthsof the quadrupole-pairing interaction are determined soas to fulfill the self-consistency condition [30]. We usethe quadrupole polarization charge δe pol = 0 . E β - γ mesh points in the region 0 < β < β max and0 ◦ < γ < ◦ , with β max = 0 . Mg and 0.6 for , , Mg.Our theoretical framework is quite general and itcan be used in conjunction with various Skyrmeforces/modern density functionals going beyond the P+Qmodel. Then the effects of weakly bound neutrons andcoupling to the continuum on the properties of the low-lying collective excitations, discussed in Refs. [14, 15],can be taken into account, for example, by solving theCHFB + LQRPA equations in the 3D coordinate meshrepresentation. However, it requires a large-scale calcu-lation with modern parallel processors and it remains asa challenging future subject. A step toward this goal hasrecently been carried out for axially symmetric cases [26].Figure 1 shows the collective potentials V ( β, γ ) for − Mg. It is clearly seen that prolate deformationgrows with increase of the neutron number. The col-lective potential for Mg is very soft with respect to β .It has a minimum at β = 0 .
11 and a local minimum at β = 0 .
33. The barrier height between the two minima isonly 0.24 MeV (measured from the lower minimum). In Mg, in addition to the prolate minimum at β = 0 . N = 20spherical shell gap) appears. The barrier height betweenthe two minima is 1.0 MeV (measured from the lowerminimum). The spherical local minimum disappears in Mg and Mg, and the prolate minima become soft inthe direction of triaxial deformation γ . In Mg, the po-tential minimum is located at γ = 10 ◦ .In Fig. 2, calculated excitation energies and E +1 and 4 +1 states and the remarkable increase of B ( E
2; 2 +1 → +1 )from Mg to Mg are well described in this calcula-tion. The calculated ratio of the excitation energies, E (4 +1 ) /E (2 +1 ), increases as 2.37, 2.82, 3.26, and 3.26,while the ratio of the transition strengths, B ( E
2; 4 +1 → +1 ) /B ( E
2; 2 +1 → +1 ), decreases as 2.03, 1.76, 1.43, and FIG. 1: (Color online) Collective potentials for − Mg. The HFB equilibrium points are indicated by red circles. E xc i t a t i on ene r g y ( M e V ) (a) EXPCHB+LQRPA 0 50 100 150 200 30 32 34 36 B ( E ; + - > + ) ( e f m ) Mass number (b)
FIG. 2: (Color online) Comparison of calculated excita-tion energies of the 2 +1 and 4 +1 states (upper panel) and B ( E
2; 2 +1 → ) values (lower panel) in − Mg with ex-perimental data [1–6]. Mg to Mg. Thus, the propertiesof the 2 +1 and 4 +1 states gradually change from vibrationalto rotational with increasing neutron number.Let us next discuss the properties of the 0 +2 states andthe 2 + and 4 + states connected to the 0 +2 states withstrong E +2 statesare 1.353 and 0.986 MeV for Mg and Mg, respectively,in fair agreement with the experimental data [21, 22].In particular, the very low excitation energy of the 0 +2 state in Mg is well reproduced. In our calculation,more than 90% (80%) of the collective wave functionsfor the yrast (excited) band members are composed ofthe K = 0 component. Therefore we denote the groundband by ‘the K = 0 band,’ and the excited band by ‘the K = 0 band.’ The 2 + and 4 + states belonging to the K = 0 band appear as the second 2 + and 4 + states in , Mg, while they appear as the third 2 + and 4 + statesin , Mg. Accordingly, we use 2 +2 , and 4 +2 , , to collec- E xc i t a t i on ene r g y ( M e V ) + + + (a) CHB+LQRPAEXProtor 0.1 1 10 100 30 32 34 36 B ( E ) r a t i o Mass number (b)
FIG. 3: (Color online) Excitation energies of the ex-cited 0 +2 , 2 +2 , and 4 +2 , states (upper panel) and the ra-tio B ( E
2; 0 +2 → +1 ) /B ( E
2; 0 +1 → +2 , ) of the inter-band E K = 0 and K = 0 bands (lower panel). Experimental data are taken fromRefs. [21, 22]. See texts for details. tively indicate the second or the third 2 + and 4 + states.The calculated ratios of the excitation energies relative tothe excited 0 +2 state, [ E (4 +2 , ) − E (0 +2 )] / [ E (2 +2 , ) − E (0 +2 )],are 3.18, 2.87, 3.25, and 3.00, for Mg, Mg, Mg,and Mg, respectively. In the upper panel of Fig. 3 wealso plot the rotor-model prediction for the excitationenergies of the 4 + states estimated from the 0 + − + spacings in the K = 0 bands. The deviation from therotor-model prediction is largest in Mg indicating im-portance of shape-fluctuation effects. Although the cal-culated excitation spectrum of the K = 0 band in Mglooks rotational, we find a significant deviation from therotor-model prediction in the E E B ( E
2; 4 +2 , → +2 , ) /B ( E
2; 2 +2 , → +2 ), are 1.05, 1.54,1.47, and 1.51, for − Mg, respectively. The deviationfrom the rotor-model value (1.43) is largest in Mg. Thesignificant deviation from the simple rotor-model pat-
FIG. 4: (Color online) Vibrational wave functions squared P K | Φ αIK ( β, γ ) | of the 0 +1 , +1 , +2 and 2 +2 , states in − Mg. Contour lines are drawn at every eighth part ofthe maximum value. tern of the K = 0 bands in Mg and Mg, noticedabove, can be seen more drastically in the inter-band E B ( E
2; 0 +2 → +1 ) /B ( E
2; 0 +1 → +2 , ) ofthe inter-band transition strengths between the K = 0 and K = 0 bands. If the K = 0 and K = 0 bandsare composed of only the K = 0 component and the in-trinsic structures in the ( β, γ ) plane are the same withinthe band members, this ratio should be one. These ra-tios for Mg and Mg are close to one, indicating thatthe change of the intrinsic structure between the 0 + and2 + states is small. In contrast, the ratios for Mg and Mg are larger than 10, indicating a remarkable changein the shape-fluctuation properties between the 0 + and2 + states belonging to the K = 0 and K = 0 bands.Figure 4 shows the vibrational wave functions squared P K | Φ αIK ( β, γ ) | . Let us first examine the characterchange of the ground state from Mg to Mg. In Mg,the vibrational wave function of the ground 0 +1 state isdistributed around the spherical shape. In Mg, it is re-markably extended to the prolately deformed region. In Mg, it is distributed around the prolate shape. Fromthe behavior of the vibrational wave functions, one canconclude that shape fluctuation in the ground 0 +1 state islargest in Mg. To understand the microscopic mecha-nism of this change from Mg to Mg, it is necessary to take into account not only the properties of the collectivepotential in the β direction but also its curvature in the γ direction and the collective kinetic energy (collectivemasses). This point will be discussed in our forthcomingfull-length paper. As suggested from the behavior of theinter-band B ( E
2) ratio, the vibrational wave functionsof the 2 +1 state are noticeably different from those of the0 +1 state in Mg and Mg, while they are similar in thecase of Mg. Next, let us examine the vibrational wavefunctions of the 0 +2 and 2 +2 , states in − Mg. It is im-mediately seen that they exhibit one node in the β direc-tion. This is their common feature. In Mg and Mg,one bump is seen in the spherical to weakly-deformedregion, while the other bump is located in the prolatelydeformed region around β = 0 . − .
4. In Mg, the nodeis located near the peak of the vibrational wave functionof the 0 +1 state, suggesting that they have β -vibrationalproperties. | Φ α I K ( β , γ ) | (a) 0 Mg Mg Mg00.5 0 0.1 0.2 0.3 0.4 0.5 0.6 | Φ α I K ( β , γ ) | β ( γ =0.5 o ) (c) 0 P ( β ) (b) 0 P ( β ) β (d) 0 FIG. 5: (Color online) (a) Vibrational wave functionssquared, | Φ α,I =0 ,K =0 ( β, γ = 0 . ◦ ) | , of the 0 +1 states in − Mg. Their values along the γ = 0 . ◦ line are plottedas functions of β . (b) Probability densities integrated over γ , P ( β ) ≡ R dγ | Φ α,I =0 ,K =0 ( β, γ ) | | G ( β, γ ) | / , of the 0 +1 statesin − Mg, plotted as functions of β . (c) Same as (a) but forthe 0 +2 states. (d) Same as (b) but for the 0 +2 states. To further reveal the nature of the ground and excited0 + states, it is important to examine not only their vibra-tional wave functions but also their probability densitydistributions. Since the 5D collective space is a curvedspace, the normalization condition for the vibrationalwave functions is given by Z X K | Φ αIK ( β, γ ) | | G ( β, γ ) | / dβdγ = 1 (5)with the volume element | G ( β, γ ) | / dβdγ = 2 β p W ( β, γ ) R ( β, γ ) sin 3 γdβdγ, (6) W ( β, γ ) = { D ββ ( β, γ ) D γγ ( β, γ ) − [ D βγ ( β, γ )] } β − , (7) R ( β, γ ) = D ( β, γ ) D ( β, γ ) D ( β, γ ) , (8)where D k =1 , , are the rotational masses defined through J k = 4 β D k sin ( γ − πk/ β, γ ) is givenby P K | Φ αIK ( β, γ ) | | G ( β, γ ) | / . Due to the β factor inthe volume element, the spherical peak of the vibrationalwave function disappears in the probability density dis-tribution. Accordingly, it will give us a picture quitedifferent from that of the wave function. Needless to say,it is important to examine both aspects to understandthe nature of individual quantum states.In Fig. 5, we display the probability density integratedover γ , P ( β ) ≡ R dγ | Φ α,I =0 ,K =0 ( β, γ ) | | G ( β, γ ) | / ,of finding a shape with a specific value of β , to-gether with the vibrational wave functions squared | Φ α,I =0 ,K =0 ( β, γ ) | for the ground and excited 0 + states( α = 1 and 2). Let us first look at the upper panelsfor the ground states. We note that, as expected, thespherical peak of the vibrational wave function for Mgin Fig. 5(a) corresponds to the peak at β ≃ .
15 of theprobability density in Fig. 5(b). In Fig. 5(b), the peakposition moves toward a larger value of β in going from Mg to Mg. The distribution for Mg is much broaderthan those for Mg and Mg.Next, let us look at the lower panels in Fig. 5 for theexcited states. In Fig. 5(c). the vibrational wave func-tions for Mg and Mg exhibit the maximum peak atthe spherical shape. However, these peaks become smalland are shifted to the region with β ≃ . β ≃ . Mg and Mg, respectively, in Fig. 5(d). On theother hand, the second peaks at β ≃ . β ≈ . Mg and Mg, respectively, seen in Fig. 5(c) becomethe prominent peaks in Fig. 5(d). In Mg, the bump at β ≃ . β ≃ .
3. In this sense, we can regard the 0 +2 state of Mg as a prolately deformed state. In the case of Mg,the probability density exhibits a very broad distribution extending from the spherical to deformed regions up to β = 0 . β ≃ . β ≃ .
3. The position of the node coincides with thepeak of the probability density distribution of the the0 +1 state, as expected from the orthogonality condition.The range of the shape fluctuation of the 0 +2 state in β direction is almost the same as that of the 0 +1 state.Thus, the result of our calculation yields a physical pic-ture for the 0 +2 state in Mg that is quite different fromthe ‘spherical excited 0 + state’ interpretation based onthe inversion picture of the spherical and deformed con-figurations. In Mg, the peak is shifted to the regionwith a larger value of β and the tail toward the sphericalshape almost disappears.In summary, we have investigated the large-amplitudecollective dynamics in the low-lying states of − Mg bysolving the 5D quadrupole collective Schr¨odinger equa-tion. The collective masses and potentials of the 5Dcollective Hamiltonian are microscopically derived withuse of the CHFB + LQRPA method. Good agreementwith the recent experimental data is obtained for theexcited 0 + states as well as the ground bands. For Mg, the shape coexistence picture that the deformedexcited 0 + state coexists with the spherical groundstate approximately holds. On the other hand, large-amplitude quadrupole-shape fluctuations dominate inboth the ground and the excited 0 + states in Mg, sothat the interpretation of ‘deformed ground and sphericalexcited 0 + states’ based on the simple inversion picture ofthe spherical and deformed configurations does not hold.To test these theoretical predictions, experimental searchfor the distorted rotational bands built on the excited 0 +2 states in Mg and Mg is strongly desired.One of the authors (N. H.) is supported by the SpecialPostdoctoral Research Program of RIKEN. The numeri-cal calculations were performed on the RIKEN IntegratedCluster of Clusters (RICC). This work is supported byKAKENHI (Nos. 21340073, 20105003, 23540234, and23740223). [1] A. N. Deacon et al. , Phys. Rev. C , 034305 (2010).[2] S. Takeuchi et al. , Phys. Rev. C , 054319 (2009).[3] K. Yoneda et al. , Phys. Lett. B , 233 (2001).[4] O. Niedermaier et al. , Phys. Rev. Lett. , 172501 (2005).[5] T. Motobayashi et al. , Phys. Lett. B , 9 (1995).[6] H. Iwasaki et al. , Phys. Lett. B , 227 (2001).[7] E. K. Warburton et al. , Phys. Rev. C , 1147 (1990).[8] Y. Utsuno et al. , Phys. Rev. C , 054315 (1999).[9] E. Caurier et al. , Nucl. Phys. A , 374 (2001).[10] T. Otsuka, Eur. Phys. J. A , 69 (2003).[11] J. Terasaki et al. , Nucl. Phys. A , 706 (1997).[12] P.-G. Reinhard et al. , Phys. Rev. C , 014316 (1999).[13] H. Ohta et al. , Eur. Phys. J. A , s1.549 (2005).[14] M. Yamagami et al. , Phys. Rev. C , 034301 (2004).[15] K. Yoshida et al. , Phys. Rev. C , 044312 (2008). [16] R. Rodr´ıguez-Guzm´an et al. , Nucl. Phys. A , 201(2002).[17] J. M. Yao et al. , Phys. Rev. C , 014308 (2011).[18] J. M. Yao et al. , Int. J. Mod. Phys. E , 482 (2011).[19] M. Kimura et al. , Prog. Theor. Phys. , 33 (2002).[20] H. Mach et al. , Eur. Phys. J. A , 105 (2005).[21] W. Schwerdtfeger et al. , Phys. Rev. Lett. , 012501(2009).[22] K. Wimmer et al. , Phys. Rev. Lett. , 252501 (2010).[23] N. Hinohara et al. , Phys. Rev. C , 064313 (2010).[24] K. Sato et al. , Nucl. Phys. A , 53 (2011).[25] N. Hinohara et al. , Phys. Rev. C , 014321 (2011).[26] K. Yoshida et al. , Phys. Rev. C , 061302 (2011).[27] N. Hinohara et al. , AIP Conf. Proc. , 200 (2011),arXiv:1101.2256. [28] M. Stoitsov et al. , Comp. Phys. Comm. , 43 (2005).[29] T. Bengtsson et al. , Nucl. Phys. A , 14 (1985). [30] H. Sakamoto et al. , Phys. Lett. B245