Shape Coexistence and Band Termination in Doubly Magic Nucleus 40 Ca

Commun. Theor. Phys. (Beijing, China) 43 (2005) pp. 509 514 c International Academic Publishers Vol. 43, No. 3, March 15, 2005 Shape Coexistence and Band Termination in Doubly Magic Nucleus 40 Ca DONG Bao-Guo, GUO Hong-Chao, and SHI Yi-Jin Department of Nuclear Physics, China Institute of Atomic Energy, P.O. Box 275(18), Beijing 102413, China (Received July 6, 2004) Abstract Shape coexistence and band structure near yrast line of the Z = N doubly magic nucleus 40 Ca have been investigated by the configuration-dependent cranked Nilsson Strutinsky approach. The observed normal deformed and superdeformed bands are explained and the terminating states are confirmed by the calculations. The transition quadrupole moment Q t of the calculated superdeformed band is in good agreement with the observed one at high spin. There is shape coexistence within the same configuration. Possible normal deformed and superdeformed bands with rotation around the intermediate axis in several interesting configurations of 40 Ca are discussed. Possible favored superdeformed band terminations in 38 Ca and 38 Ar are predicted. The experimental results in 38 Ar are discussed simply. PACS numbers: 21.10.Hw, 21.10.Re, 21.60.Ev, 27.40.+z Key words: shape coexistence, configuration-dependent cranked Nilsson Strutinsky approach, superdeformed band termination, transition quadrupole moment 1 Introduction The features of band structures at high spin and smooth band terminations of nuclei near or far away from the β-stability line in the A 40 mass region, especially for the highly deformed or superdeformed (SD) bands of the Z = N nuclei and their terminating features, are of considerable interest. In this mass region, protons and neutrons occupy the same or similar orbitals, and this would make the specific subtle effects significantly, such as spherical, normal deformed, and SD shape coexistence effects at higher spin, new mode of collective motion. Those effects would be unobservable in other mass region by now. In 40 Ca the valence space is composed of the excited particles and the holes in the 40 Ca core. It is a good chance to study the contribution of the configurations with different f 7/2 orbitals occupied by particles or holes to collectivity, the collective motions of nucleons in the mean field, the deformation effects, and spin contribution of the configurations in the smooth band terminations. Recently the experimental results of the doubly magic nucleus 40 Ca were reported [1] and the four positive parity bands, including one SD band with a transition quadrupole moment Q t = 1.18eb at low spin and 1.81eb at high spin, [2] were observed up to spin 16 +, 16 +, 15 +, and 13 +, respectively. The structures of 40 Ca were studied experimentally for several times from different points of view. [3,4] Early spherical shell model [5] calculations for the low-lying levels and recently fixed configuration deformed Hartree Fock [6] calculations for the SD band were performed for 40 Ca. The cranked relativistic mean-field (CRMF) calculations have been performed in Ref. [1]. In addition to 40 Ca of A 40 mass region, a highly deformed or SD band up to spin I = 16 + of terminating state in 36 Ar [7,8] and several normal deformed rotational bands up to the termination [9] in the A 45 50 mass region, were observed. 2 Results and Discussions The configuration-dependent cranked Nilsson Strutinsky (CNS) approach [10,11] was used to study the band structures of 40 Ca. The Nilsson parameters of Ref. [10] were adopted with pairing correlations neglected. Considerable effort was recently focused on even-even and/or odd-a nuclei, such as 48 Cr, [12] 47,48,49 Cr, [9] and 36 Ar, [7] to compare with the CNS model, the spherical shell model, and the cranked Hartree Fock Bogoliubov method in the description of experimental data. The CNS calculations are thus expected to provide a reliable description of quadrupole properties. [12] The good description of the observed bands at high spin and terminating states in 46 V shows that the CNS approach works well in this mass region. [13] In order to understand the observed bands in 40 Ca, the CNS calculations were performed for the four combinations of parity (π = ±) and signature (α = 0, 1). For (π, α) = (+, 0), (+, 1) the configurations of interest near yrast lines and their possible maximum spins are [1,1] and I max = 10, 9, 8; [2,2] and I max = 16, 15, 13, 12, respectively. Here a shorthand notation used later to label the configurations (relatively to the 40 Ca core) is: [p 1, n 1 ] π(f 7/2 ) p1 ν(f 7/2 ) n1, where p 1 (n 1 ) is the number of protons (neutrons) in the f 7/2 orbitals. To illustrate, the detailed structures of the [2,2] terminating states are π[(d 3/2 s 1/2 ) 2 2,1,0 (f 7/2) 2 6] 8,7,6 ν[(d 3/2 s 1/2 ) 2 2,1,0 (f 7/2) 2 6] 8,7,6, where the subscripts are the possible spin contributions from the occupied valence orbitals forming the configurations at the oblate (γ = 60 ) axis. Similarly, for (π, α) = (, 0), (, 1) the configurations and the possible maximum spins are [1,0] and I max =5, 4; [2,1] and I max = 12, 11, respectively. The [0,1] and [1,2] configurations can be formed by exchanging the above proton and neutron configurations. The project supported by National Natural Science Foundation of China under Grant No. 10275093 E-mail: dongbg@iris.ciae.ac.cn

510 DONG Bao-Guo, GUO Hong-Chao, and SHI Yi-Jin Vol. 43 2.1 Normal Deformed Bands I = 13 and 12. It can be seen from Fig. 1 that the calculated bands are in good agreement with the experimental bands in energies, the curvature and the terminating states, especially when the spin is near terminating states, i.e. the spin region of interest. However, in the low spin range the calculated results are higher than the observed ones about 2 MeV if the bands are renormalized at spin I = 12 or 13, and it comes from the effect of pairing neglected. The good agreement between calculated and observed bands at high spin demonstrates that the CNS model works well in the A 40 mass region. Fig. 1 Comparison of experimental and theoretical E E RLD energies as a function of spin for the positive parity in 40 Ca. The energies are relative to a rigid rotation reference ( 2 /2J rig)i(i + 1). The open symbols are the theoretical data and solid symbols the experimental data. Terminating states are indicated by large open circles. γ > 0 (γ < 0) means the local minimum is located in the γ > 0 (γ < 0) region. 3L(H) means the low (high) part of observed band 3 since band 3 is labeled a band as in Ref. [1] but the states with different signature. A comparison of experimental and theoretical E E RLD energies as a function of spin for the smooth terminating bands in 40 Ca with the positive parity π and two signatures α is shown in Fig. 1, where E RLD ( 2 /2J rig )I(I +1) 0.007(158/A) 5/3 I(I +1) is the energy of a rigid rotor reference with A being the mass number. These E E RLD plots could provide considerably more detailed information about individual and relative properties of the rotational bands. Based on the calculated results, the configurations [2,2]γ < 0, [2,2]γ > 0 0 with α = 0 and [2,2]γ < 0 with α = 1 are suggested to be the observed bands 2, 3L and 4 (labeled in Ref. [1]), respectively. The configuration of band 3H (including two states) would be [2,2]γ > 0 or [2,2], where means that the last two neutrons occupy the α = 1/2 (d 3/2 s 1/2 ) orbitals. The calculated shape trajectories as a function of spin are shown in Fig. 2 for the bands assigned to the observed bands in 40 Ca. Shown in Figs. 1 and 2, the observed bands 2, 3L, and 3H are with the terminating spins of 16, 12, and 15, respectively. Band 4 is predicted to terminate favorably at spin I = 15. Band 3 is composed of two parts which belong to different signatures (π, α) = (+, 0), (+, 1) but they are connected by dipole transition between spin Fig. 2 Calculated shape trajectories as a function of spin in the (ε 2, γ) plane for the positive parity bands in 40 Ca (corresponding to Fig. 1). The step in spin is 2. 2.2 Shape Coexistence There are two minima located at different deformations in the potential energy surface (PES) of the [2,2] configuration for 40 Ca. The PES s with spin I = 8 and 12 are shown in Fig. 3. Two bands coming from different minima are very different with each other in collectivity in the [2,2] configuration. The minima located around γ = 60 axis corresponds to the band which has more single particle like behavior. The minima located around (ε 2, γ) = (0.35, 60 ) at low spin corresponds to the other band, which has more collective nature with γ deformation increasing gradually as spin increasing. Then the two minima merge into one when spin reaches I = 14 and the corresponding band terminates at spin I = 16 and ε 2 = 0.14, see Fig. 2. The configuration of band 3 from I = 8 to 12 is the same as band 2 and forms shape coexistence together with states of band 2 within the configuration in the CNS calculations. The I = 12 state of band 3 is terminating state at about ε 2 = 0.32. This could explain the behavior of the observed bands 2 and 3 qualitatively. The positive

No. 3 Shape Coexistence and Band Termination in Doubly Magic Nucleus 40 Ca 511 γ deformation band, the [2,2] configuration, is about 1.3 MeV higher than negative band in energy at low spin, thus there is only one band observed by experiment. When spin is over 6, the positive γ deformation band is similar to or lower than the negative γ band in energy, so the two bands exhibit coexisting collective triaxial and near-oblate shapes at higher spin and could be observed at the same time. The calculations indicate that the observed bands 2, 3 and 4 are highly-normal deformed bands indeed. Fig. 3 Calculated potential energy surfaces for the [2,2] configuration assigned to the observed band 2 and 3L terminating at spin I = 16 and 12 in 40 Ca. The contour line separation is 0.2 MeV. For the [2,2] γ < 0 configuration assigned to the observed band 4 with parity and signature (π, α) = (+, 1), the proton configuration is the same as band 2 but the neutron configuration is that where two f 7/2 neutrons occupy the two lowest negative signature orbitals rather than the lowest negative and positive signature orbitals in the case of band 2. Therefore, the two bands have very similar behavior in the shape coexistence and the tendency with spin increasing. This configuration reaches terminating state at spin I = 15 and ε 2 = 0.12. Other features are similar to the [2,2] configuration with (π, α) = (+, 0). The calculated two minima in the PES with I = 11 suggests a oblate-collective prolate shape coexistence, which are related to two states. One is a collective state and another is a terminating state, i.e. π[(d 3/2 s 1/2 ) 2 0 (f 7/2) 2 6] 6 ν[(d 3/2 s 1/2 ) 2 0 (f 7/2) 2 3.5+1.5] 5 at ε 2 = 0.31, where the contribution to the total spin could be determined by the single-particle energies versus aligned spin diagram. [9] 2.3 Transition Intensity The relative intensity of the γ-ray transitions of interest, which has simple physical picture, could be understood qualitatively by the shape coexistence and the PES of the [2,2] configuration with α = 0, see Fig. 3. For example, there is only one minimum in the PES at spin I = 14 corresponding to a single state for band 2. At spin I = 12, there are two minima in the PES, one is shallow and the other is deep, determining two states. The I = 12 state of band 2 corresponding to the very shallow one is a collective state with a deformation of (ε 2, γ) = (0.23, 22 ) and is high in energy, see Fig. 3(b). The I = 12 state of band 3 corresponding to the deep minimum is a terminating state with configuration [2,2], i.e. π[(d 3/2 s 1/2 ) 2 0 (f 7/2) 2 6] 6 ν[(d 3/2 s 1/2 ) 2 0 (f 7/2) 2 6] 6 at ε 2 = 0.31 and is low in energy. The state of the shallow one is not easy to be populated compared with one of the deep one. So the relative intensity of the γ-ray transition from the I = 14 state to the I = 12 shallow state of band 2 is much weaker than that to the I = 12 deep one of band 3. Why the relative intensity of the γ-ray transition between I = 12 of band 3 to 10 of band 2 is much stronger than that between I = 12 to 10 within band 3 is a complicated issue and there are many competitions among the states which cannot be understood simply in the CNS model, see Fig. 1 in Ref. [1]. 2.4 Superdeformed Bands The [4,4] configuration is assigned to the observed band 1 of 40 Ca by our calculations and this band is a SD band, see Figs. 2 and 4. This is consistent with the results of shell model and CRMF calculations. The subshell, such as 1f 7/2 or (1f 5/2 p 3/2 p 1/2 ), used here in the CNS model cannot be divided into different single orbital or sub-subshell, such as divided into the 1f 5/2 orbitals and keep the occupied particles on in the whole spin region since the orbitals in the subshell mix seriously due to rotation and deformation. If terminating state can be reached the possible maximum spin would be 20, 24, or 28 including particles excited from the d 5/2 orbitals, i.e. π[(d 5/2 d 3/2 s 1/2 ) 4 2,6 (f 7/2) 4 8] 10,14 ν[(d 5/2 d 3/2 s 1/2 ) 4 2,6 (f 7/2) 4 8] 10,14 at ε 2 = 0.40 and the γ = 60 axis. The SD band is a very γ soft band, see Figs. 2 and 5. In the PES there are very stable two minima located at (ε 2, γ) (0.53, 21 ) and (0.54, 20 ) in the spin region I = 0 22. The γ 21 minimum becomes more and more shallow with spin increasing and disappears when I > 22. There is a very flat region near the γ = 60 axis at ε 2 0.45 and I = 24. For example, the energy of the mesh point with (ε 2, γ) = (0.44, 60 ) in the PES is 45.78 MeV which gradually decreases to 45.67 MeV at the mesh point (ε 2, γ) = (0.48, 36 ). The difference in energy is only 0.11 MeV and it would be possible to form a minimum or terminating state within the accuracy of the CNS calculations. However, there is no really minimum found at the γ = 60 axis, see Fig. 5. There should be two SD bands built in the two minima but only one SD band and a single level were reported in Ref. [1]. Comparing the two minima and results of CRMF calculations, the observed band 1 is built in the γ = 20 minimum in the PES with configuration [4,4]. The γ 21 minimum would be related to the single I = 14 SD level of 18 725 kev (transition of 3 454 kev) but it needs more levels and further experiment to confirm, see Fig. 4. No

512 DONG Bao-Guo, GUO Hong-Chao, and SHI Yi-Jin Vol. 43 terminating state was found at I = 20 even up to 30 in our calculations. two neutrons of the [4,4] configuration in 40 Ca occupy the [321]3/2 orbitals which have strong driving effect towards to negative γ deformation, see Fig. 6 and Ref. [13]. The behavior of the [202]5/2 and [321]3/2 orbitals as a function of γ deformation is very similar at ε 2 = 0.40, where the SD band terminates in 36 Ar and at 0.53, where the minimum of the SD band is located in the PES in 40 Ca. The last four particles occupied the [321]3/2 orbitals drive the nucleus toward large negative γ deformation and lead to no terminating state in the calculations for this configuration in 40 Ca. Fig. 4 Comparison of experimental and theoretical E E RLD energies as a function of spin for the superdeformed bands built in the minima shown in Fig. 2 for 40 Ca. The energies are relative to a rigid rotation reference ( 2 /2J rig)i(i + 1). The open symbols indicate the theoretical data and solid symbols the experimental data. Fig. 5 Calculated potential energy surfaces for the [4,4] configuration assigned to the observed band 1 in 40 Ca. The contour line separation is 0.2 MeV. As a contrast, the SD or highly deformed band with configuration [2,2] in 36 Ar [7] reached the terminating state at spin I = 16. The last two protons and two neutrons of this [2,2] configuration occupy the [202]5/2 orbitals which have strong decreasing effect on total energy for large values of both positive and negative γ deformation but not for prolate deformation (γ 0 ). Besides the same occupation as the above [2,2] in 36 Ar, the adding two protons and Fig. 6 (a) Single-proton energies plotted as a function of the γ deformation. The + and symbols are for signature α = +1/2 and 1/2, respectively. The open circles with + and symbols indicate the two [321]3/2 orbitals occupied by the last two protons in the [4,4] configuration in 40 Ca. The solid circles indicate the orbitals occupied by the proton configurations 4 for 40 Ca and 2 for 36 Ar. The labelling of the orbitals with the γ deformation at a high rotational frequency is only for the convenience of discussion. Note the [440]1/2 orbital is unoccupied for the proton configuration 4. (b) Total energies as a function of the γ deformation for 40 Ca and 36 Ar. The dotted line is not real minimum in the PES of 36 Ar and just for comparison with 40 Ca at the same ε 2 without occupied the [321]3/2 orbitals, which show strong driving effect towards to negative γ deformation on the nuclear shape.

No. 3 Shape Coexistence and Band Termination in Doubly Magic Nucleus 40 Ca 513 2.5 Transition Quadrupole Moment For the 8-particle 8-hole (8p 8h) excitation configuration in 40 Ca, the γ deformation is about 10 and the transition quadrupole moment Q t is 2.0eb in the CRMF calculations without pairing as in Ref. [1]. For the [4,4] configuration, the γ = 20 states give Q t 2.0eb in the spin region I = 0 30 and the γ = 21 states give Q t 1.3eb using the Q t expansion in Ref. [9] when we do not distinguish the high-j and low-j orbitals in the CNS calculations. The CNS results are in good agreement with the experimental data of Q t = 1.81eb [2] at spin I 12 for band 1 that confirms the configuration of higher spin part of this SD band is pure [4,4]. In the spin I 10 region the experimental data of Q t = 1.18eb is different from the CNS calculations for the γ = 20 states of the [4,4] configuration. This discrepancy indicates that the configuration of band 1 is not pure [4,4] at low spin and some mixture of different configurations mainly from the [2,2] configuration having less quadrupole collectivity and probably decreasing Q t at low spin. For the [2,2] configuration, the γ deformation is about 50 and the states give Q t 1.08eb, but the γ deformation is about 40 and Q t 0.44eb for band 2. 2.6 Superdeformed Band Terminations The SD or highly deformed band terminations have been studied by the CNS calculations for A 40 nuclei. Some possible SD band terminations were predicted, such as the [2,4] configuration in 38 Ar and [4,2] in 38 Ca, in which the last two neutrons or two protons occupied the [321]3/2 orbitals comparing with that of 36 Ar. The terminating behaviors of SD bands show the gradually changing tendency from 36 Ar to 40 Ca. The mirror nuclei 38 Ar and 38 Ca have very similar behavior in the SD bands and termination. Both configuration [2,4] in 38 Ar and [4,2] in 38 Ca are very γ soft SD bands, in the spin region I = 12 18 the minima form a long flat valley, especially. The deformation is about ε 2 = 0.46 0.53 and γ = 23 22 at spin I = 10 for these two bands. Both bands terminate at I = 20 and about ε 2 0.40. The tendency of these bands in energy with spin increasing favors the band termination, so SD bands terminating would be smooth. The experimental results in 38 Ar were reported recently and the four observed bands labeled 1, 2, 3, and 4 (including two bands) were observed up to spin 16 +, 14 +, 10 +, 14, and 13, respectively. [14] These observed bands can be well explained by the [1,3], [2,2], [0,2], and [1,2] configurations in the CNS calculations. These calculated configurations terminate at spin 16 +, 14 +, 10 +, 16, and 11 with quadrupole deformations ε 2 = 0.20 0.36 so they are highly-normal deformed bands. For example, the deformation is (ε 2, γ) = (0.38, 39 ) at I = 2 + and (0.32, 60 ) at I = 14 + for the [2,2] configuration. So no SD band in 38 Ar has been observed experimentally. The contribution of valence nucleons to total spin of these SD and normal deformed bands in 36,38 Ar and 38,40 Ca is interesting. A very large spin contribution comes from those nucleons occupying the 1f 7/2 orbitals. It is about 3 or 5 times that in the (d 5/2 d 3/2 s 1/2 ) orbitals at total spin I = 6 or 8. The contribution of the rest or the core is less than 1 and almost zero in general. 2.7 Nuclear Rotation Around Intermediate Axis Comparing the rotational behavior of the calculated rotational bands in 40 Ca with that in 48 Cr [12] of the middle moment of inertia, which is forbidden in classical mechanics rotation, a conclusion of the rotation around the intermediate axis can be drawn for 40 Ca. The results of the CNS calculations, such as the triaxial shape at spin above 8 + (e.g. the 12 + state has γ = 15 ) corresponding to rotation around the intermediate axis in 48 Cr, the B(E2) values, and the spectroscopic quadrupole moments, were confirmed by the spherical shell model calculations. [12] The calculated configurations clearly showing rotation around the intermediate axis are [2,2] ( γ = 60 40 and I = 0 10, 1 11 with (π, α) = (+, 0), (+, 1)) and [4,4] (built in the minimum located at γ = 20 in the PES with (+, 0)) in 40 Ca. So the observed bands 1, 2 (partly) and 4 would be bands with rotation around the intermediate axis. The spin values, especially for the SD band, are higher enough and the negative values of γ are large enough, e.g. γ = 20 for the [4,4] and 60 40 for the [2,2] configurations confirming the rotation around the intermediate axis and the neglecting of the quantum fluctuations. Spectroscopic quadrupole moments are good signature and are sensitive for the nucleus obtaining a triaxial shape with rotation around the intermediate axis. [12] 3 Conclusions The observed normal deformed and SD band structures near yrast lines of the Z = N doubly magic nucleus 40 Ca have been investigated and the calculated bands reproduced well the observed bands at high spin. The calculated transition quadrupole moment Q t for the [4,4] configuration is in good agreement with that of the observed SD band 1 at high spin. The normal deformed terminating states were confirmed by the calculations. There are spherical states, normal-deformed (prolate, oblate and triaxial), and superdeformed bands, which shows shape coexistence even within the same configuration in 40 Ca. A comparison of the CNS calculations with shell model, HF, CRMF calculations concludes that the CNS approach could provide a better description in nuclear properties and band structure at high spin and works well in the A 40 mass region. Possible favored superdeformed band

514 DONG Bao-Guo, GUO Hong-Chao, and SHI Yi-Jin Vol. 43 terminations in 38 Ca and 38 Ar are predicted. The experimental results in 38 Ar are discussed simply. Possible normal deformed and SD bands with rotation around the intermediate axis in 40 Ca are discussed for several interesting configurations. Acknowledgments We would like to thank I. Ragnarsson for access to the CNS code and discussions. References [1] E. Ideguchi, et al., Phys. Rev. Lett. 87 (2001) 222501. [2] C.J. Chiara, et al., Phys. Rev. C67 (2003) 041303(R). [3] H.P. Leenhouts, Physica (Utrecht) 35 (1967) 290. [4] J.J. Simpson, et al., Phys. Rev. Lett. 35 (1975) 23. [5] W.J. Gerace and A. M. Green, Nucl. Phys. A93 (1967) 110; A123 (1969) 241. [6] D.C. Zheng, et al., Phys. Rev. C42 (1990) 1004. [7] C.E. Svensson, et al., Phys. Rev. Lett. 85 (2000) 2693. [8] C.E. Svensson, et al., Nucl. Phys. A682 (2001) 1c. [9] A.V. Afanasjev, et al., Phys. Rep. 322 (1999) 1. [10] T. Bengtsson and I. Ragnarsson, Nucl. Phys. A436 (1985) 14. [11] A.V. Afanasjev and I. Ragnarsson, Nucl. Phys. A591 (1995) 387. [12] A. Juodagalvis, I. Ragnarsson, and S. Aberg, Phys. Lett. B477 (2000) 66. [13] Bao-Guo Dong and Hong-Chao Guo, Eur. Phys. J. A17 (2003) 25. [14] D. Rudolph, et al., Phys. Rev. C65 (2002) 034305.