Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Jafarizadeh, M. A.; Fouladi, N.; Sabri, H. Description of Even–Even 114–134Xe Isotopes in the Transitional Region of IBM Brazilian Journal of Physics, vol. 43, núm. 1-2, abril, 2013, pp. 34-40 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46425766010 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Braz J Phys (2013) 43:34–40 DOI 10.1007/s13538-013-0116-3 NUCLEAR PHYSICS Description of Even–Even 114–134Xe Isotopes in the Transitional Region of IBM M. A. Jafarizadeh & N. Fouladi & H. Sabri Received: 1 May 2012 /Published online: 18 January 2013 # Sociedade Brasileira de Física 2013 Abstract Properties of 114–134Xe isotopes are studied in the the anharmonic vibrator, axial rotor and γ-unstable ro- U(5)↔SO(6) transitional region of Interacting Boson Model tor, respectively. More generally, the Hamiltonian can be (IBM-1). The energy levels and B(E2) transition rates are expressed in terms of an invariant operator of that chain calculated via the affine SU(1,1) Lie Algebra. The agree- of symmetries, and a shape phase transition between the ment with the most recent experimental is acceptable. The dynamical symmetry limits results [6–8]. The analytic evaluated Hamiltonian control parameters suggest a spheri- description of the structural change at the critical point cal to γ-soft shape transition and propose the 130Xe nucleus of the phase transition being still an open problem, the as the best candidate for the E(5) symmetry. Hamiltonian must be diagonalized numerically. Pan et al. [9], proposed a new solution based on the affine SU Keywords Interacting Boson Model (IBM) . Infinite (1,1) algebraic technique, which determines the proper- dimensional algebra . Energy levels . B(E2) transition rates ties of nuclei in the U(5)↔SO(6) transitional region of IBM-1 [9, 10]. The xenon isotopes have been previously analyzed both theoretically and experimentally [11–29] with par- 1 Introduction ticular emphasis on describing the experimental data via collective models. The ground state properties of even– General algebraic group techniques, applied to the Inter- even Xe isotopes have been the subject to theoretical acting Boson Model (IBM), have rather successfully [11–20] and experimental studies [22] involving in-beam described the low-lying collective properties of a wide γ-ray spectroscopy. Recently, the nuclear structure of range of nuclei. In the relatively simple Hamiltonian of xenon isotopes have been investigated by Turkan in the the model, the collective states are described by a IBM-1 model [30], while the energy levels, electric system of interacting s-andd-bosons carrying angular quadrupole moments and B(E2) values of even-mass momenta 0 and 2, respectively, which define an overall nuclei such as Ba,Xe were studied within the framework U(6) symmetry [1–5]. The IBM Hamiltonian has exact of the IBM-2 [19, 31–34]. These descriptions suggest solutions in three dynamical symmetry limits [U(5), O these nuclei to be soft with regard to γ deformation with (6), and SU(3)], which are geometrically analogous to a nearly maximum effective trixiality of γ≅30° [11]. As M. A. Jafarizadeh pointed out by Zamfir et al. [11], Xe isotopes in the Department of Theoretical Physics and Astrophysics, mass region A~130 appear to evolve from U(5)-to O(6)- University of Tabriz, Tabriz 51664, Iran like structure in the IBM-1. It is very difficult to apply e-mail: [email protected] conventional mean-field theories to such structures, which M. A. Jafarizadeh are neither vibrational nor rotational. 114–134 Research Institute for Fundamental Sciences, Tabriz 51664, Iran Here we examine the even–even Xe isotopes in the U(5)↔SO(6) transition region and calculate the energy lev- : * N. Fouladi H. Sabri ( ) els and B(E2) transition probabilities in the frame work of Department of Nuclear Physics, University of Tabriz, Tabriz 51664, Iran IBM with the affine SU(1,1) algebraic technique. The esti- e-mail: [email protected] mated control parameters indicate a spherical to γ-soft shape Braz J Phys (2013) 43:34–40 35 transition. The same shape transition is revealed by the is generated by the operators Sv, v=0, and ±, which satisfies evolution of two-neutron separation energies S2n [31] de- the following commutation relations: rived from experimental results [19, 32–34]. Also, special Âà 130 Æ Æ þ À values are found for the Xe control parameter and R4/2, S0; S ¼S ; ½¼ÀS ; S 2S0 ð2:1Þ which suggest it as the best candidate for E(5) dynamical symmetry in this isotopic chain. This paper is organized as follows: Section 2 briefly The Casimir operator of SU(1,1) can be written as summarizes the theoretical aspects of transitional Hamilto- ÀÁ nian and the affine SU(1,1) algebraic technique. Section 3 b 0 0 þ À C2 ¼ S S À 1 À S S ; ð2:2Þ presents the numerical results obtained from applying the considered Hamiltonian to different isotopes. Finally, Sec- tion 4 summarizes our findings and the conclusions Representations of SU(1,1) are determined by a single κ extracted from the results in Section 3. number . The representation of the Hilbert space is hence spanned by orthonormal basis |κμ〉, where κ can be any positive number and μ=κ,κ+1,…. Therefore, 2 Theoretical Framework b 0 C2ðÞSUðÞ1; 1 jikμ ¼ kkðÞÀ 1 jikμ ; S jikμ ¼ μjikμ The phenomenological IBM in terms of U(5), O(6) and ð2:3Þ SU(3) dynamical symmetries has been employed to de- scribe the collective properties of several medium- and The bases of U(5)⊃SO(5) and SO(6)⊃SO(5) are simulta- d sd heavy-mass nuclei. These dynamical symmetries are geo- neously the bases of SU (1,1)⊃U(1) and SU (1,1)⊃U(1), metrically analogous to the harmonic vibrator, axial rotor respectively. For U(5)⊃SO(5) case, one has and γ-unstable rotor, respectively [1–5]. While these symmetries have already offered a fairly accurate descrip- jiNndnnΔLM tion of the low-lying nuclear states, attempts to analyti- d 1 5 d 1 5 cally describe the structure at the critical point of the ¼ N; k ¼ n þ ; μ ¼ nd þ ; nΔLM ; phase transition have only been partially successful. 2 2 2 2 Iachello [6, 7] established a new set of dynamical sym- ð2:4Þ metries, i.e. E(5) and X(5), for nuclei located at critical point of transitional regions. The E(5) symmetry, which Where N, n , v, L and M are quantum numbers of U describes a second-order phase transition, corresponds to d (6), U(5), SO(5), SO(3),andSO(2), respectively, while the transitional states in the region from the U(5) to the nΔ is an additional quantum number needed in the O(6) symmetries in the IBM-1. Different analyses of this reduction SO(5)↓SO(3) and κd and μd are quantum transitional region suggested certain nuclei, such as numbers of SUd(1,1) and U(1), respectively. On the 134Ba, 108Pd, as examples of that symmetry [9, 14]. other hand, in the IBM-1, the generators of the d-boson Elaborate numerical techniques are required to diag- pairing algebra created by onalize the Hamiltonian in these transitional regions and critical points. To avoid these problems, an algebraic 1 y y 1 solution based on the affine SU(1,1) Lie algebra has SþðdÞ¼ d : d ; SÀðdÞ¼ ed:ed ; been proposed by Pan et al. [9, 10]todescribethe 2 2 X ð2:5Þ properties of nuclei located in the U(5)↔SO(6) transi- 0ð Þ¼1 y þ y S d dn dn dndn tional region. The results of this approach are somewhat 4 n different from those obtained from the IBM, but as pointed out in Refs. [9, 10], there is a clear correspon- s dence with the description of the geometrical model for Similarly, the s-boson pairing forms another SU (1,1) this transitional region. algebra generated by the operators 1 y 1 SþðsÞ¼ s 2 ; SÀðsÞ¼ s2 ; 2.1 The Affine SU(1,1) Approach to the Transitional 2 2 Hamiltonian ð2:6Þ 1 y y S0ðsÞ¼ s s þ ss References [9, 10] describe the SU(1,1) algebra in detail. 4 Here, we briefly outline the basic ansatz and summarize the The infinite dimensional SUð1; 1Þ algebra is then gener- results. The Lie algebra corresponding to the group SU(1,1) ated by the operators [9, 10] 36 Braz J Phys (2013) 43:34–40 ÀÁ ÀÁ Æ þ Æ þ Æ " 2 þ 1 2 þ 5 ¼ 2n 1 ð Þþ 2n 1 ð Þ ; gcs ns 2 gcd n 2 Sn cs S s cd S d ¼ þ ð : Þ À 2 À 2 2 7 xi 1 cs xi 1 cdxi S0 ¼ c2nS0ðsÞþc2nS0ðdÞ n s d X 2 À for i ¼ 1; 2; ...; k ð2:13Þ where Cs and Cd are real parameters and n can be 0,± x À x i6¼j i j 1,±2,…. The generators in Eq. (2.7) satisfy the commutation rela- The eigenvalues E(k) of Hamiltonian (Eq. 2.9)canbe tions expressed in the form [9, 10] Âà Âà S0 ; SÆ ¼SÆ ; Sþ; SÀ ¼2S0 ð2:8Þ ðkÞ ¼ ðkÞ þ ðÞþþ ðÞþþ "Λ0 ; m n mþn m n mþnþ1 E h gn n 3 dLL 1 1 μ μ − … ð2:14Þ It follows that the {Sm , =0,+, ;±1,±2, } generate an Λ0 ¼ 1 2 þ 1 þ 2 þ 5 d 1 cs ns cd n affine Lie algebra SUð1; 1Þ without central extension.