466 DNG Bin-Gang and SHEN Shui-Fa Vol. 38 ^P MK generates states of good angular momentum, thus restoring the necessary rotational symmetry violated i

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1 Commun. Theor. Phys. (Beijing, China) 38 (2002) pp. 465{469 c nternational Academic Publishers Vol. 38, No. 4, October 15, 2002 Level Structure of Rb in the Projected Shell Model DNG Bin-Gang 1 and SHEN Shui-Fa 2 1 Huzhou Teachers College, Huzhou , China 2 Shanghai nstitute of Nuclear Research, the Chinese Academy of Sciences, Shanghai , China (Received December 20, 2001 Revised March 12, 2002) Abstract The projected shell model is applied to the odd-proton nucleus Rb. The results of theoretical calculations about the excited positive-parity yrast states and the negative-parity ground-state band are compared with experimental data, and the best reproduction of the experiment has been given by this model. n addition, a band diagram calculated for the negative-parity g.s. band is also shown in order to extract physics out of the numerical results. PACS numbers: Cs, e, Hw Key words: projected shell model, yrast state, quadrupole deformation 1 ntroduction The neutron decient Rb nucleus having several protons and neutrons in the unlled shells exhibit quite a complex structure of excited states. Any of the simple models based on the known nucleon coupling schemes fails in the description of this nucleus. The possibility for protons and neutrons to occupy the same orbitals may further complicate the situation as the short-range protonneutron interaction is expected to become quite important. Nuclei around A = 80 have been studied extensively using -ray spectroscopy, since they exhibit abundant nuclear structure information. The high-spin scheme of Rb was studied by in-beam -ray spectroscopy in 1980 by Gast et al. [1] n that work, a stretched spin band built on the isomeric 9=2 + level at 42.3 kev, a 5=2 ; ground state quasirotational band, and a three-quasiparticle band were found. n the meantime, the asymmetric (symmetric) rotor model calculations for Rb positive (positive and negative) parity states are performed and compared with the experimental data. Recently the in-beam study of the nucleus Rb has been carried out by Doring et al. [2] in order to explore the onset of high-spin bands and collectivity in the N = 46 nuclei below the Z = 40 gap, and to investigate shape variations. Detailed information on states with low-spin values and low excitation energies can be obtained by using -ray (and conversion electron) spectroscopy following radioactive decay. The decay of Sr (T 1=2 = 32:41h) presents an excellent way to get information about the low energy levels of Rb. The decay of Sr was previously studied by Etherton et al., [3] Broda et al., [4] and Grutter. [5] Many inconsistencies appear within the decay scheme [6] as deduced from the experimental data due to poor energy resolution, low statistics or a lack of sucient coincidence data. Thus, it is needed to reinvestigate the decay of Sr. Because better experimental techniques are now available, this decay has been remeasured by our nuclear decay spectroscopy group of Shanghai nstitute of Nuclear Research in order to improve the Rb level scheme at low spins and to reveal a clearer connection with the high-spin scheme proposed by Gast et al., [1] and Doring et al. [2] Particular attention has been paid to many formerly unresolved - ray multiplets. Preliminary results of this work have been published in Ref. [7]. n the present work, the interpretation of the level scheme on the Rb nucleus in the framework of the projected shell model (PSM) is presented in order to investigate the interplay of collective and particle-hole excitations as well as deformation driving properties of the unpaired nucleons. n subsection 2.1, the projected shell model is briey described, and in subsection 2.2, the theoretical predictions are given and a comparison with the experimental results is presented. Finally, the paper is summarized in Sec Discussion 2.1 Summary of the Theory The PSM [8;11] employed in this paper is a microscopic theory, which solves the many-nucleon system in a full quantum mechanics method. The ansatz for the angularmomentum-projected wave function is given by jmi = X k f k ^P MKk j' k i (1) where k labels the basis states. ^P MK is the angular momentum projection operator which is explicitly given in Ref. [8]. Acting on an intrinsic state j' k i, the operator The project supported by the Major State Basic Research Development Program in China under Contract No. G , and the Natural Science Foundation of Shanghai under Grant No. 00ZA14078

2 466 DNG Bin-Gang and SHEN Shui-Fa Vol. 38 ^P MK generates states of good angular momentum, thus restoring the necessary rotational symmetry violated in the deformed mean eld. n this way the new shell model basis is constructed in which the Hamiltonian is diagonalized. This shell model basis taken in the present work is as follows ^P MK j' ki : (2) The basis states j' k i are spanned by the set fa y p j0i ay n 1 a y n 2 a y pj0ig (3) for odd proton nuclei. j0i denotes the quasiparticle vacuum state and a y n (a y p) is the neutron (proton) quasiparticle creation operator for this vacuum the index n 1(2) (p) runs over selected neutron (proton) quasiparticle states and k in Eq. (1) runs over the conguration of Eq. (2). The vacuum is obtained by diagonalizing a deformed Nilsson Hamiltonian [12] followed by a BCS calculation. n the calculations, we have used three major shells, i.e., N = 2 3, and 4 (N = 2 3, and 4) for neutrons (protons) as the conguration space. n this work we have used the Hamiltonian X [11] ^Q y ^Q y ; G M ^P ^P ; GQ ^P y ^P (4) ^H = ^H0 ; 1 2 X where ^H0 is the spherical single-particle shell model Hamiltonian, ^Q is the quadrupole moment operator, and ^P and ^P are monopole pairing operator and quadrupole pairing operator, respectively. Though the theory itself is not bound to any particular form of Hamiltonian, the advantage of using such a separable-force Hamiltonian is that the role of each interaction is well known and therefore the interpretation of the numerical result becomes easier. The interaction strengths are determined as follows: the strength of the quadrupole-quadrupole interaction is adjusted by the self-consistent relation so that the input quadrupole deformation 2 and the one resulting from the HFB (Hartree{Fock{Bogolyubov) procedure coincide with each other. [11] The monopole pairing strength constant is adjusted to give the known energy gap G M = h20:12 13:13 N ; Z i A ;1 (5) A where \;" is for neutrons and \+" for protons. Finally the quadrupole pairing strength G Q is simply assumed to be proportional to G M GQ G = GQ M n G = : (6) M p The proportionality constant is chosen to be 0.20 for all the bands calculated in the present work. The weights f k in Eq. (1) are determined by diagonalizing the Hamiltonian ^H in the basis given by Eq. (3) as outlined in Ref. [11]. Projection of good angular momentum onto each intrinsic state generates the rotational band associated with this intrinsic conguration j' k i. For example, ^P MK a y pj0i will produce a one-quasiproton band. The energies of each band are given by the normalized diagonal elements according to Ref. [8]. A diagram in which E k () for various bands are plotted against spin is referred to as a band diagram. [11] The lowest eigenvalue of the Hamiltonian for a given spin is named the yrast energy, and can be compared with the experiment. n this paper, we decide to compare the experimentally observed positive-parity yrast states and negative-parity groundstate band of Rb with the predictions of the PSM. The projected shell model has at least two advantages by this token: (i) The procedure of angular momentum coupling, which must be done troublesomely in the conventional shell model, is done automatically by the projector irrespective of the number of quasiparticles (qp) involved. (ii) t allows us to choose various multiquasiparticle bases according to physical importance. Unfortunately, our present computer code assumes axialsymmetry so that we cannot investigate those -deformed nuclei quantitatively. [13] The model has achieved considerable success when it was applied to the rare-earth region where the nucleus is well-deformed. n this paper, we try to apply this model to the A 80 region and to show the potential of this model via the study of low- and high-spin states of Rb. More recently, the shape coexistence in the even-even nucleus 72 Se has been studied by Palit et al. [14] (including Y. Sun). n that work, level energies E() and transition quadrupole moments Q t for the yrast positive-parity states in 72 Se are calculated using the PSM approach for a xed oblate deformation 2 = ;0:24 and a prolate deformation 2 = 0:29 corresponding to the minima obtained in the TRS calculations, and as a conclusion, for the yrast positive-parity band in 72 Se, the PSM calculations neatly conrmed the oblate deformation at lower spins with 6 and a transition to the prolate shape at higher spins. So for the present calculations, corresponding to dierent bands and a band before (after) the band crossing, the dierent values of quadrupole deformation parameter 2 are chosen, which is consistent with the work of Palit et al. [14] 2.2 Calculations and Comparison with Experimental Results n our calculations, the following formulae are used to calculate the pairing gap parameters p and n [15] p = 1 [B(N Z ; 2) ; 3B(N Z ; 1) 4 + 3B(N Z) ; B(N Z + 1)] (7) n = 1 4 [B(N ; 2 Z) ; 3B(N ; 1 Z) + 3B(N Z) ; B(N + 1 Z)] (8)

3 No. 4 Level Structure of Rb in the Projected Shell Model 467 where the values of the total nuclear binding energy B are taken from Ref. [16]. The results are p = 1:8 MeV and n = 1:08 MeV. The spin-orbit force parameters, k and, appearing in the Nilsson potential are taken from the compilation of Zhang et al., [17] which is a modied version of Bengtsson and Ragnarsson [18] and has been tted to the latest experimental data. t is supposed to apply over a suciently wide range of shells. These and are different for dierent major shells (N-dependent). According to total Routhian surface (TRS) calculations for the positive-parity states, [2] at very low frequencies, the nucleus Rb is calculated to be weakly deformed and very -soft, with a quadrupole deformation of about 2 = 0:13 (equivalent to 2 = 0:117) and a triaxiality parameter of = +17. At a rotational frequency of MeV (about spin 23=2 + ), a second minimum occurs which represents a well-deformed near-oblate shape with 2 = 0:25 (equivalent to 2 = 0:225) and = ;66. This second minimum persists up to high rotational frequencies. Therefore we take quadrupole deformations of 2 = 0:117 (5=2 + < 23=2 + ) and 2 = ;0:225 ( 23=2 + ), respectively, to calculate the positive-parity yrast states. The hexadecapole deformation parameter 4 = 0:007 is taken from the compilation of Moller et al. [16] n the calculations, the conguration space is constructed by selecting the quasiparticle states close to the Fermi energy in the N = 4 major shell for neutrons (protons) and forming multi-quasiparticle states from them. t should be pointed out that the connection mode between the calculated yrast energies based on the prolate (5=2 + < 23=2 + ) and oblate ( 23=2 + ) congurations is the same as Fig. 7 of Ref. [14]. The comparison of the experimentally observed positive-parity yrast levels of Rb with the predictions of the PSM is given in Fig. 1. The experimental 5=2 +, 7=2 +, 9=2 +, 11=2 +, and 13=2 + levels have been taken from our group's measurement, [7] while the rest have been taken from the in-beam study of the nucleus Rb by Doring et al. [2] n fact the 9=2 + {13=2 + (5=2 + {13=2 + ) levels had also been observed in the in-beam work by Doring et al. [2] and Gast et al. [1] The fair agreement of experimental and calculated level energies above the 21=2 + state indicates the alignment of a g 9=2 neutron pair in Rb, which drives the nucleus from a weakly deformed near-prolate shape to a medium deformed near-oblate shape. The reproduction of the relatively low-lying states (5=2 + < 23=2 + ) is not so satisfactory. t is because our present computer code assumes that the nucleus Rb is axially symmetric. But we still nd that the ordering of all calculated levels agrees with that of the experimental ones, especially the ordering of the 5/2, 7/2, 9/2 positive-parity yrast states, the so-called anomalous coupling states. From our calculations, it can be predicted that the 1=2 + and 3=2 + levels lie between the 11=2 + and 17=2 + levels, and the 19=2 + level lies above the 21=2 + level it is expected that some experiments will conrm it in the future. Fig. 1 Projected shell model calculations for positiveparity yrast states in Rb compared with the experimental data. n the negative-parity ground-state band calculations, the conguration space is constructed by selecting the quasiparticle states close to the Fermi energy in the N = 4 (N = 3) major shell for neutrons (protons) and forming multi-quasiparticle states from them. The negativeparity ground-state band (the 5=2 ; {13=2 ; levels are the yrast states), calculated also for a 2 = 0:117 deformation parameter, seems to give the best reproduction of the experiment. The theoretical energy E() and energy dierence E() ; E( ; 1) of the negative-parity groundstate band are compared with the experimental data as shown in Figs. 2 and 3, respectively. The experimental 5=2 ;, 7=2 ;, and 9=2 ; levels have been taken from our group's measurement, [7] and the rest have been taken from Ref. [1]. n fact the 5=2 ; {9=2 ; levels had also been observed in the in-beam work by Gast et al. [1] n the projected shell model calculation, by taking a quadrupole deformation of 2 = 0:117 and a hexadecapole deformation of 4 = 0:007, the ground-state spin and parity of Rb were calculated to be 5=2 ;, which is consistent with the experimental results. The nucleus Rb has been calculated to be almost spherical in its 5=2 ; ground-state with a small quadrupole deformation of 2 = 0:071 when

4 468 DNG Bin-Gang and SHEN Shui-Fa Vol. 38 a nite-range droplet macroscopic model and a folded- Yukawa single-particle microscopic model are used. [16] But when we use this quadrupole deformation to calculate the negative-parity ground-state states, the t is rather bad for these experimental data. So in the present work the deformation of this ground-state band is tentatively assigned as 2 = 0:117, and this is in accordance with the work of Maharana et al., [19] in which the ground state of Rb has been calculated using the relativistic mean-eld theory. Fig. 2 The comparison of the experimentally observed negative-parity ground-state band in Rb with the predictions of the PSM. Fig. 3 The theoretical energy dierence E() ; E( ; 1) of the negative-parity band based on the ground state 5=2 ; is compared with experiment. Fig. 4 A band diagram calculated for the negativeparity band in Rb based on the 5=2 ; ground-state (lled circles). Dierent multi-(1-quasiparticle and 3- quasiparticle) bands are shown in dierent lines. All the results (lled circles included) are extracted from PSM calculations. n fact in the present example of Rb, where the negative-parity band is based on the 5=2 ; ground-state, one nds that the neutron Fermi energy lies between the K = 5=2 and K = 7=2 quasiparticle states of the intruder 1g 9=2 subshell and the proton Fermi energy lies between the K = 3=2 and K = 5=2 quasiparticle states of the 1f 5=2 subshell. Therefore, orbitals around these levels are physically most important. To build a multi-quasiparticle basis, all neutron quasiparticle states of the intruder 1g 9=2 subshell are selected, from which 10 neutron 2-quasiparticle states are formed. Similarly, the K = 3=2 and K = 5=2 proton quasiparticle states of the 1f 5=2 subshell are selected. Furthermore, 40 3-quasiparticle states are constructed from these neutron 2-quasiparticle and proton 1- quasiparticle states. Consequently, the dimension of our conguration space nally becomes 42 (40+2). The shell model conguration space is constructed by projecting each of these multi-quasiparticle states onto a good angular momentum and the Hamiltonian is diagonalized in this space. The band diagram as mentioned in subsection 2.1 is shown in Fig. 4. Filled circles in this gure represent the negative-parity ground-state band numbers, which are obtained from the nal diagonalization procedures (band mixing) [20] and also have been shown in Figs. 2 and 3. t is this kind of diagrams that one can conveniently use to ex-

5 No. 4 Level Structure of Rb in the Projected Shell Model 469 tract physics out of the numerical results. [11] n the lowest spin region (3=2 11=2), the yrast states are represented by the 1-quasiparticle band, indicating that the 3-quasiparticle states have little eect on the yrast states in this spin range. The = 1=2 state has not yet certainly been observed in the experiment, so it will not be discussed in the present work. At spin ' 11=2, the rst band crossing is seen to take place. n the present case, there are accidentally four closely lying (3-quasiparticle) bands which cross the 1-quasiparticle band and after 15=2 there are some other 3-quasiparticle bands which come down in energy and contribute to the negative-parity band based on the 5=2 ; ground-state, thus complicating the structure. We thus want to know the congurations of these four closely lying (3-quasiparticle) bands. We see that the band #3 (#2) is a 3-quasiparticle band with K n1 = 5=2 and K n2 = ;7=2 coupled to K = ;1 and then coupled with K p = ;3=2 (K p = 5=2) to total K = ;5=2 (K = 3=2) while band #1 (#4) is a 3-quasiparticle band with K n1 = 5=2 and K n2 = ;7=2 coupled to K = ;1 and then coupled with K p = 3=2 (K p = ;5=2) to total K = 1=2 (K = ;7=2). 3 Summary n summary, the structure of excited positive-parity yrast states and the negative-parity ground-state band of Rb has been discussed in the framework of the projected shell model, and the best reproduction of the experiment has been given by this model. n the positive-parity yrast state calculations, the quadrupole deformations 2 = 0:117 (5=2 + 23=2 + ) and 2 = ;0:225 ( 23=2 + ), respectively, have been taken. We also take a 2 = 0:117 deformation parameter, which is in accordance with the work of Maharana et al., [19] to calculate the negativeparity ground-state band. n addition, a band diagram calculated for the negative-parity ground-state band has been shown. References [1] W. Gast, K. Dey, A. Gelberg, U. Kaup, F. Paar, R. Richter, K.O. Zell, and P. von Brentano, Phys. Rev. C22 (1980) 469. [2] J. Doring, G.D. Johns, R.A. Kaye, M.A. Riley, and S.L. Tabor, Phys. Rev. C60 (1999) [3] R.C. Etherton, L.M. Beyer, W.H. Kelly, and D.J. Horen, Phys. Rev. 168 (1968) [4] R. Broda, A.Z. Hrynkiewicz, J. Styczen, and W. Walus, Nucl. Phys. A216 (1973) 493. [5] A. Grutter, nt. J. Appl. Radiat. sot. 33 (1982) 456. [6] J. Muller, Nuclear Data Sheets 49 (1986) 579. [7] X.H. Yu, S.H. Shi, J.H. Gu, J.Y. Liu, W.X. Li, J.P. Zeng, J.Q. Tian, Y. Li, and J.Z. Zhou, High Energy Phys. Nucl. Phys. (in Chinese) 22 (1998) [8] K. Hara and Y. Sun, Nucl. Phys. A529 (1991) 445. [9] K. Hara and Y. Sun, Nucl. Phys. A531 (1991) 221. [10] K. Hara and Y. Sun, Nucl. Phys. A537 (1992) 77. [11] K. Hara and Y. Sun, nt. J. Mod. Phys. E4 (1995) 637. [12] C.G. Andersson, G. Hellstrom, G. Leander,. Ragnarsson, S. Aberg, J. Krumlinde, S.G. Nilsson, and Z. Szymanski, Nucl. Phys. A309 (1978) 141. [13] M.A. Rizzutto, E.W. Cybulska, L.G.R. Emediato, N.H. Medina, R.V. Ribas, K. Hara, and C.L. Lima, Nucl. Phys. A569 (1994) 547. [14] R. Palit, H.C. Jain, P.K. Joshi, J.A. Sheikh, and Y. Sun, Phys. Rev. C63 (2001) [15] Aage Bohr and Ben R. Mottelson, Nuclear Structure, Vol. 1, Benjamin, New York, Amsterdam (1969) p [16] P. Moller, J.R. Nix, W.D. Myers, and W.J. Swiatecki, At. Data Nucl. Data Tables 59 (1995) 185. [17] Jing-Ye Zhang, A.J. Larabee, and L.L. Riedinger, J. Phys. G13 (1987) L75. [18] T. Bengtsson and. Ragnarsson, Nucl. Phys. A436 (1985) 14. [19] J.P. Maharana and Y.K. Gambhir, Phys. Rev. C54 (1996) [20] Y. Sun and K. Hara, Comp. Phys. Commun. 104 (1997) 245.