Rotational Property of 249 Cm in Particle-Triaxial-Rotor Model

Transcription

1 Commun. Theor. Phys. (0) Vol., No., February, 0 Rotational Property of 4 Cm in Particle-Triaxial-Rotor Model ZHUANG Kai ( Ô), LI Ze-Bo (ÓÃ ), and LIU Yu-Xin ( ) Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 008, China (Received June 3, 0) Abstract We study the rotational energy spectrum and deformation feature of very heavy nucleus 4 Cm in the particle-triaxial-rotor model with variable moment of inertia. Such a nucleus is the unique one involving both multiband structure and high spin states and locating very near the superheavy region. By calculating the energy spectrum, we determine the configurations and quadrupole and triaxial deformation parameters β and γ of the nucleus. The calculated results indicate that the high spin band of 4 Cm is built upon the ν[60] + configuration with deformation parameters β = 0.6 and γ =. and the bands based on the ν[6] 3 +, ν[63] +, ν[0] configuration respectively are also the ones with quite large axial deformation but small triaxial deformation. PACS numbers:.0.+b,.60.ev, 3.0.Lv Key words: near superheavy nucleus, energy spectrum, deformation parameters, particle-triaxial-rotor model Introduction Studies of the structure, especially that of the high spin states, of odd-a nuclei near Z = 00 is highly significant and challenging since it can provide information for the stability and both the collective and the single particle properties of superheavy nuclei on one hand, but many difficulties exist in experiment [] on the other hand. Fortunately, after many years efforts, the band structure for some nuclei in that region, such as Cm,[] 6Cm,[ ] 8Cf,[] 8Cf,[3] Es,[4] 4 [] 4 [ 6] [ 8] [4,] 3 00Fm, 00Fm, 00Fm, 0Md, 0No, [0] 03 Lr,[4, ] have been established. Specifically, multi band structure with high spin states has been observed for 4 Cm. On theoretical side, the single particle states have been given together with the axial deformation parameters being proposed [3 ] for 4 Cm. It has also been shown that the superheavy nuclei may involve triaxial deformation [6] and the triaxial deformation may be significant in nuclear shape phase transitions. [] However, the collective band structure of the nucleus 4 Cm has not yet been analyzed in theory and whether there exists triaxial deformation has not been discussed either. It has been shown that particle triaxial-rotor model [8 0] is quite successful in studying the properties of nuclei related to triaxial deformation and still implemented widely in recent years (see for example, Refs. [ 8]). We then analyze the collective rotational property of heavy nucleus 4 Cm in the particle triaxial-rotor model with variable moment of inertia in this paper. The paper is organized as follows. In Sec., we describe briefly the particle triaxial-rotor model. In Sec. 3, we represent our calculated results and compare them with the recent experimental data. [ ] We determine, in turn, the intrinsic configuration and the deformation parameters β and γ of the bands of the nucleus. Finally, a summary is given in Sec. 4. Model In the particle triaxial-rotor model, the total Hamiltonian is usually written as [0] Ĥ = Ĥcore + Ĥs.p. + Ĥpair, () where Ĥcore describes the even-even core of the nucleus, Ĥ s.p. stands for the Hamiltonian of the valence nucleon in the potential of the deformed core, and Ĥpair denotes the residual interaction between the valence nucleons, which can be treated by the Bardeen Cooper Schrieffer(BCS) formalism. [0] In the case of triaxial deformation, one takes the Hamiltonian of the even-even core as [0,4] 3 R 3 i (I i j i ) Ĥ core = =, () J i J i i= i= where R, I, and j are the angular momentum of the core, the nucleus and the single particle, respectively. Each component of the moment of inertia, J i, is assumed to be decided by the equation of irrotational hydrodynamical type J κ = 4 ( 3 J 0 sin γ + π ) 3 κ, (3) Supported by the National Natural Science Foundation of China under Grant Nos and 00, the National Fund for Fostering Talents of Basic Science with Grant No. J03030, the Major State Basic Research Development Program under Grant No. G00CB8000 Corresponding author, yxliu@pku.edu.cn c 0 Chinese Physical Society and IOP Publishing Ltd

2 Communications in Theoretical Physics Vol. where γ is the triaxial deformation parameter, and an angular momentum dependent J 0 [0] J 0 (I) = J 00 [ + + bi(i + )] (4) can be taken to improve the calculation. [4] Ĥ s.p. can be given as [0,3] with Ĥ s.p. = Ĥ0 + Ĥε C l s D[l l N ], () Ĥ 0 = m + mω 0r, [ Ĥ ε = mω0r β cosγy 0 + ] sinγ(y + Y, ), where C, D can be transformed to the Nilsson parameter κ, µ, and Y i is the rank- spherical harmonic function. The Ĥpair is the Hamiltonian to describe the residual interaction, which mainly represents as pairing correlation here and can be treated by the BCS method. The single particle wave function can be diagonalized in the basis Nlj Ω as ν = CNlj ν Ω Nlj Ω, (6) lj Ω where ν is the sequence number of the Nilsson level which represents the corresponding Nilsson state. By diagonalizing the single particle Hamiltonian Ĥs.p. in this basis, we can obtain the single particle eigenvalue ε ν and the eigenstates of the single particle. With standard BCS treatment, the corresponding quasiparticle energy can be written as E ν = (ε λ) +, where λ and are, respectively, the Fermi energy and energy gap. The total Hamiltonian Ĥ in Eq. () can be rewritten in step operators as Ĥ = ( + ) [I I3 4 J J + j j3 (I +j + I j + )] + ( ) [I+ + I + j+ + j 8 J J (I + j + + I j )] + (I 3 j 3 ). () J 3 Then Ĥ can be diagonalized in the symmetrical strong coupling basis [] I + [ IMKν = D I 6π MK α ν 0 + ( ) I K D I M Kα ν 0 ], (8) where α ν is the creation operator of the quasiparticle in the Nilsson level ν, DMK I is the rotation matrix. 3 Calculation and Numerical Result In the present work, we take κ and µ as those quite close to the standard value, [3] κ = 0.04, µ = 0.4. For the energy gap parameter, considering the reduction of the pairing strength in very heavy nuclei as given quite recently, [] we take = 0.30 MeV. To improve the agreement between the calculated energy spectrum and the experimental one, a Coriolis attenuation factor ξ = 0. is introduced here. The free moment of inertia parameter J 00 in Eq. (4) is derived from the energy of the first excited state E + of the even-even core 48 Cm. In order to give an overall agreement with experimental data, we take 6 single particle orbitals near the Fermi surface to diagonalize the total Hamiltonian of the nucleus. Table Calculated main components of the single particle orbitals near the Fermi surface in terms of the deformed Nilsson levels in 4 Cm. Band ν Wave function in terms of Nlj Ω d i g i g 0. 6i 3 Band i g g d g g g i g i j h i d h p f 0.3 f Band j h 0.03 h h 0. f j j + 0. h 0.0 h j h j d i g i g i 3 Band i g g g i 0. 6i g d 0.3 6g g i d i g i d g 0.8 6i 3 Band i g g g i 0.6 6i d g g g i d3 In our calculation, a series of diagonalization of the total Hamiltonian with various values of β and γ are carried out. We assigned the set of (β, γ) as that minimize the calculation error χ = N j [ (E cal. j /E exp. j ) ], () where N is the number of levels in the band. It is remarkable that, in our calculation process, we did not assume that the bandhead energies of the excited bands equate

3 No. Communications in Theoretical Physics 3 to the experimental data so that only the relatively excitation in the bands can be analyzed, but fix only that of the ground state band and calculate those of the excited bands to check the results for the single particle configurations. It is also necessary to mention that the span of the deformation parameters β and γ we take here is the same as that used by Nilsson and Ragnarsson, [33] where β can be either positive or negative, γ varies from 0 to 30. Table Detailed values (in kev) of the calculated energy levels and the comparison with experimental data (taken from Refs. [ ]) and the main components of the calculated wavefunctions of 4 Cm in terms of the single particle levels. Band I π E cal. E exp. Wave function in terms of νk Band , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,. Band , , , , , , , , , , , , 0. Band , , , , , , , ,. 0. 3, , , , , , ,. Band , , , , , , , , , , , , , , ,.

4 4 Communications in Theoretical Physics Vol. According to Refs. [ ], the nucleus 4 Cm has three positive-parity rotational bands, whose bandhead angular momentum are (/) + (Band-, ground-state band), (3/) + (Band-3, with bandhead energy 08 kev) and (/) + (Band-4, with bandhead energy 4 kev), respectively. Besides, 4 Cm has also a negative-parity band with bandhead angular momentum (3/) and bandhead energy 40 kev (denoted as band-). By diagonalizing the single particle Hamiltonian we get the single particle orbitals near the Fermi surface in terms of the Nilsson levels. The obtained results for the four bands are listed in Table. By diagonalizing the total Hamiltonian, we obtain the energy spectrum and the corresponding wave functions of the nucleus. The obtained results are listed in Table. The obtained energy spectrum and the comparison with experimental data are also illustrated in Figs. to 4. The finally determined values of the deformation parameters (β, γ) of the rotational bands are listed in Table 3. 6d / / configuration but also the 6g / /, 6g / ( /) and other configurations. Since the percentage of the 6d / / configuration in the single particle level is slightly larger than those of others, such a band can be assigned as the [60] + band, just as that assigned in Refs. [, ]. By similar analysis, we know that the Band-, Band-3, Band-4 are the band generated mainly from the configuration [0], [6] 3+, [63] +, respectively, which are also consistent with those proposed previously. [] Table 3 Calculated deformation parameters β and γ of the four rotational bands of 4 Cm and the calculation error χ. Band β γ χ Band Band Band Band Fig. Calculated energy levels of the Band- of 4 Cm and the comparison with experimental data (taken from Ref. []). Fig. 3 The same as Fig. but for the Band-3. Fig. Calculated energy levels of the ground state band of nucleus 4 Cm and the comparison with experimental data (taken from Refs. [ ]). From Table and Figs. to 4, one can notice that the calculated results agree with experimental data quite well. From Table, one can find that the Band- of 4 Cm originates mainly from the 38th single particle orbital coupling with the even-even core. From Table, one can find that the 38th orbital mainly contains not only the Fig. 4 The same as Fig. but for the Band-4.

5 No. Communications in Theoretical Physics From Table 3, one can also find that the triaxial deformation parameter γ of the bands is quite small. However, the axial deformation parameter β of each of the bands is quite large. From the result, one can infer that such a very heavy nucleus is a well deformed nucleus with slightly triaxial deformation. 4 Summary In summary, we have calculated the properties of the rotational bands of very heavy nucleus 4 Cm in the particle-triaxial-rotor model with variable moment of inertia. The calculated energy spectrum fits the experimental data very well, especially the signature splitting of the ground state band and the bandhead energies of the excited bands are reproduced excellently. The calculation indicates that this near superheavy nucleus is the one with quite large axial deformation but very small triaxial deformation. References [] S.K. Tandel, et al., Phys. Rev. C 8 (00) 0430(R). [] T. Ishii, et al., Phys. Rev. C 8 (00) [3] I. Ahmad, J.P. Greene, E.F. Moore, E.G. Kondev, R.R. Chasman, C.E. Porter, and J.K. Felker, Phys. Rev. C (00) [4] A. Chatillon, et al., Eur. Phys. J. A 30 (006) 3. [] F.P. Hessberger, et al., Eur. Phys. J. A 3 (004) 4. [6] A. Lopez-Martens, et al., Phys. Rev. C 4 (006) [] F.P. Hessberger, et al., Eur. Phys. J. A (006) 6. [8] M. Asai, et al., Phys. Rev. C 83 (0) 043. [] A. Chatillon, et al., Phys. Rev. Lett. 8 (00) 303. [0] P. Reiter, et al., Phys. Rev. Lett. (00) 030. [] S. Ketelhut, et al., Phys. Rev. Lett. 0 (00) 0. [] H.B. Jeppesen, et al., Phys. Rev. C 80 (00) [3] I. Ahmad and R.R. Chasman, Phys. Rev. C 80 (00) [4] G.G. Adamain, N.V. Antonento, S.N. Kuklin, and W. Scheid, Phys. Rev. C 8 (00) [] Z.H. Zhang, J.Y. Zeng, E.G. Zhao, and S.G. Zhou, Phys. Rev. C 83 (0) 0304(R). [6] S. Cwios, P.H. Heenen, and W. Nazarewicz, Nature (London) 433 (00) 0. [] F. Iachello, Phys. Rev. Lett. (003) 30; M.A. Caprio and F. Iachello, Ann. Phys. 38 (00) 44. [8] J. Meyer-ter-Vehn, F.S. Stephens, and R.M. Diamond, Phys. Rev. Lett. 3 (4) 383; J. Meyer-ter-Vehn, Nucl. Phys. A 4 () ; A 4 () 4. [] S.E. Larsson, G.A. Leander, and I. Ragnarsson, Nucl. Phys. A 30 (8) 8. [0] P. Ring, and P. Scuck, The Nuclear Many-body Problem, Springer-Verlag, New York (80). [] I. Hamamoto, Phys. Rev. C 6 (00) [] K. Starosta, T. Koike, C.J. Chiara, D.B. Fossan, and D.R. LaFosse, Nucl. Phys. A 68 (00) 3c. [3] L. Fortunato, Phys. Rev. C 0 (004) 030(R); L. Fortunato, S. De Baerdemacker, and K. Heyde, Phys. Rev. C 4 (006) [4] G.M. Zeng and H.C. Song, High Energy Phys. Nuel. Phys. 6 (00) 0; H.C. Song, Y.X. Liu, and Y.H. Zhang, Chin. Phys. Lett. (00) 6; G.J. Chen, H.C. Song, and Y.X. Liu, Chin. Phys. Lett. (00) 0; G.J. Chen, Y.X. Liu, H.C. Song, and H. Cao, Phys. Rev. C 3 (006) ; X.M. Wu andy.x. Liu, Chin. Phys. Lett. (008) 300. [] Y.R. Shimizu, T. Shoji, and M. Matsuzaki, Phys. Rev. C (008) 043. [6] I. Sankowska, et al., Eur. Phys. J. A 3 (008) 6 (008). [] L. Chen, et al., Phys. Rev. C 83 (0) [8] D. Petrellis, A. Leviatan, and F. Iachello, Ann. Phys. 36 (0) 6. [] M.A.J. Mariscotti, G. Scharff-Goldhaber, and B. Buck, Phys. Rev. 8 (6) 864. [30] C.S. Wu and J.Y. Zeng, Commun. Theor. Phys. 8 (8). [3] S.G. Nilssom, Mat. Fys. Medd. Dan. Vid. Selsk. () 6. [3] T. Bengtsson and I. Ragnarsson, Nucl. Phys. 436 (8) 4. [33] S.G. Nilsson and I. Ragnarsson, Shapes and Shells in Nuclear Structure, Cambridge University Press, Cambridge, ().