Empirical evidence for the nucleus at the critical point of the U πν (5) - SU πν(3) transition in IBM2

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1 24th, July, 2010 /Ø( þfåæ0;k?ø Empirical evidence for the nucleus at the critical point of the U πν (5) - SU πν(3) transition in IBM2 Da-li Zhang ÜŒá Department of physics, Huzhou Teacher s College, Huzhou , Zhejiang, China Zhan-feng Hou ûó State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing , China Yu-xin Liu 4Œc State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing , China State Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou , China Page 1 of 16

2 Empirical evidence for the nucleus at the critical point of the U πν (5) - SU πν(3) transition in IBM2 Part I Part II PROPOSED PROCEDURE Part III CONCLUSION AND DISCUSSION Part IV Page 2 of 16

3 1 A.IBM(Interacting Boson Model) T hree Limits U(5) SU(3) O(6) Corresponding Spherioid Axially symmetric prolate γ unstable Add. SU(3) corresponding to the Axially symmetric oblate rotor phase. U(5) O(6) A seconder-order Critical point E(5)Symmetry U(5) SU(3) A first-order Critical point X(5)Symmetry SU(3) SU(3) A first-order Critical point Z(5)Symmetry B.IBM2 holds the symmetry U π (6) U ν (6) U πν (6). Involves: U πν (5), O πν (6), SU πν (3) (with χ π = χ ν = 7/2), SU πν (3) (with χ π = χ ν = + 7/2), SU πν(3) (with χ π = 7/2 and χ ν = 7/2, or χ π = 7/2 and χ ν = 7/2). Page 3 of 16

4 Corresponding to the collective motion mode of vibration,γ-unstable rotation, axially symmetric prolate rotation, axially symmetric oblate rotation, and, triaxial rotation, respectively. N = N ν + N π is the quantum number of the symmetric irreducible representation (IRREP) [N] of the group U πν (6). U πν (5) SU πν(3) second-order Phase transition (M.A.Caprio and F.Iachello PRL93,242502(2004), Ann. Phys.318(2005), Pavel Cejnar, Jan Jolie, Prog.Part.Nucl.Phys.62(2009). The empirical evidence of nucleus at a critical point has not yet been found! We try to show that nucleus 130 Xe may serves as such an evidence. Page 4 of 16

5 2 PROPOSED PROCEDURE Along the line of the effective order parameters established in IBM1 we take the observables as the energy ratios: R 42 = E(4 + 1 )/E(2+ 1 ), R 62 = E(6 + 1 )/E(2+ 1 ), R 82 = E(8 + 1 )/E(2+ 1 ), R 22 = E(2 + 2 )/E(2+ 1 ), R 60 = E(6 + 1 )/E(0+ 2 ), R = E(0 + 2 )/E(2+ 1 )and R = E(0 + 3 )/E(2+ 1 ) The B(E2) ratios: R B,42 = B(E2, )/B(E2, ), R B,22 = B(E2, )/B(E2, ), R B,02 = B(E2, )/B(E2, ). The values of these quantities of the states at the critical point of the phase transition for the nucleus with N π = N ν = 5 and the experimental data of nucleus 130 Xe are listed in Table 1 for comparison. U πν (5) SU πν(3) transition and the experimental data of 130 Xe are illustrated in Figure 1. Page 5 of 16

6 Table 1: Comparison of the key observables of the states at the critical point of the U πν (5) SU πν(3) transition (denoted as CPTh), the experimental data of 130 Xe (and labeled as 130 Xe) and the calculated ones in the IBM2 (referred to as IBM2). R 42 R 62 R 82 R 22 R 60 R 02 2 R 03 2 R B,42 R B,22 R B,02 CPTh Xe (20) 1.14(28) 0.45 IBM Page 6 of 16

7 a Figure1: Energy level schemes of the states at the critical point of the U πν (5) SU πν(3) transition (a), the experimental data of 130 Xe (b) and the calculated result in the IBM2 (c). All energies are normalized to the experimental E(2 + 1 ) value b c Page 7 of 16

8 The experimental data of both the characteristic energy ratios and the B(E2) ratios of 130 Xe agree with the theoretical values of those of the states at the critical point of the U πν (5) SU πν(3) transition well, except for those related to the state. 130 Xe can be a candidate of the empirical evidence of the the critical nucleus! The coherent state formalism are the ones of the system with infinite boson number. The practical nuclei are those with finite boson number. The properties of the states at the critical point in Fig. 1(a) are for a system with finite boson number( N π = 5 and N ν = 5). 130 Xe (with boson number N π = 2 and N ν = 3, respectively). To check the finite boson number effect in nucleus 130 Xe, we carry out a calculation in the framework of IBM2 with the commonly used Hamiltonian: Page 8 of 16

9 where Ĥ = ε d (ˆn dπ + ˆn dν ) + κ l ( ˆQ π ˆQ π + ˆQ ν ˆQ ν ) + κ nl ˆQπ ˆQ ν + ˆM πν, (1) ˆn dρ d ρ d ρ, (ρ = ν, π), ˆQ ρ (s d ρ ρ + d ρs ρ ) (2) + χ ρ (d d ρ ρ ) (2), (ρ = ν, π), ˆM πν λ 2 (s πd ν s νd π) (2) (s π dν s ν dπ ) (2) + λ k (d πd ν) (k) ( d π dν ) (k). k=1,3 The ε d is the energy of d-boson (of neutrons and protons ), κ l is the strength of the quadrupole-quadrupole interaction among like bosons, κ nl is the strength of the quadrupole-quadrupole interaction between neutron-boson s and protonboson s, λ k is the strength of the Majorana interaction. The Hamiltonian in Eq. (1) can be rewritten as Ĥ = [a 1,ρ C 1,Uρ (5) + a 2,ρ C 2,SUρ (3) + a 3,ρC 2,SUρ (3) + a 4,ρ C 2,Oρ (6) + a 5,ρ C 2,Oρ (5) ρ=π,ν +a 6,ρ C 2,Oρ (3)] + b 1 C 2,Uπν (6) + b 2 C 2,Uπν (5) + b 3 C 2,SUπν (3) + b 4 C 2,Oπν (6) +b 5 C 2,Oπν (5) + b 6 C 2,Oπν (3) + b 7 C 2,SUπν (3) + b 8C 2,SU πν (3), (2) where C k,g is the k-rank Casimir operator of group G, and the relation between the two forms of the Hamiltonian reads Page 9 of 16

10 a 1,π = ε d (λ 1 λ 3 ), a 2,π = κ l ( χ2 π + 14 χ π) 1 2 κ nl( χ πχ ν + 14 χ ν), a 3,π = κ l ( χ2 π 14 χ π) 1 2 κ nl( χ πχ ν + 14 χ ν 7 χ π), a 4,π = κ l (1 4 7 χ2 π) 1 2 κ nl[ (χ ν χ π ) 4 7 χ πχ ν ], a 5,π = κ l (1 4 7 χ2 π)+ 1 2 κ nl[ (χ ν χ π ) 4 7 χ πχ ν ], a 6,π = 1 2 κ nl[ 3 14 χ πχ ν (χ ν χ π )], (3) Page 10 of 16

11 b 1 = 1 2 λ 2, b 2 = 1 10 (λ 1 λ 3 ), 7 b 3 = κ nl ( 28 χ π 1 14 χ πχ ν ) 1 20 (λ 1 λ 3 ), b 4 = κ nl [ (χ ν χ π ) 2 7 χ πχ ν ] (λ 1 λ 3 ), b 5 = κ nl [ (χ ν χ π ) 2 7 χ πχ ν ] 3 10 (λ 1 λ 3 ), b 6 = κ nl [ (χ ν χ π ) χ πχ ν ] (λ 1 λ 3 ), 7 b 7 = κ nl ( 28 χ ν χ πχ ν ) 1 20 (λ 1 λ 3 ), b 8 = κ nl 7 28 (χ ν χ π ), and for ρ = ν, a i,ν (with i = 1, 2...6) takes the value of a i,π with the χ π and χ ν being interchanged. Page 11 of 16

12 Xenon isotopes are the nuclei in mass region A Their valence nucleons (holes) are in the major shell. The boson number N π = 2, N ν = 3 for 130 Xe in our calculation. The energy spectrum can be reproduced well with the parameters being taken as: ε d = MeV, κ l = MeV, κ nl = MeV, χ π = 0.700, χ ν = 0.600, λ 2 = MeV, λ 1 = λ 3 = MeV. The E2 transition rates with operator: ˆT (E2) = eπ ˆQπ + e ν ˆQν ; e π = eb, e ν = eb; χ π and χ ν as the same as mentioned above. On theoretical side, the parameters {b i } in Eq.(2) for 130 Xe (unit MeV) b 1 = , b 2 = , b 3 = , b 4 = , b 5 = ,b 6 = , b 7 = , b 8 = The dominant components of the dynamical symmetries of the Hamiltonian Page 12 of 16 describing the properties of 130 Xe are the U πν (5), SU πν(3) and O πν (6).

13 In previous studies has shown: a.the low-lying states of Xe isotopes are the ones in the between of γ-soft and rigid triaxial motions. b.the low-lying states of the nuclei in A 130 mass region involve a rich collective structure and shape coexistence, especially γ-soft rotor features exist in Xe isotopes, but with a dominancy of vibrational character. c.most of the nuclei in the Xe isotopes are close to the γ-unstable O(6) limit of IBM and no stable triaxial ground-state shape, but the three-body interactions possibly leading to triaxial deformation are necessary to describe the properties of the low-lying states well. All these results, the 130 Xe is shown to be in the intermediate of the transition from U πν (5) to SU πν(3) with considerable constituent of O πν (6) symmetry. Combining our present result with the former ones and the characteristic of the states at the critical point of the U πν (5) SU πν(3) transition. The nucleus 130 Xe is a promising empirical evidence for the nucleus at the critical point of the transition from vibration (with the U πν (5) symmetry) to triaxial rotation (with the SU πν(3) symmetry) with quite a large conbstituent of γ-unstable rotation (with the O πν (6) symmetry), similar to that of the vibration to axial rotation transition (with γ-vibration at the critical point with X(5) symmetry)and the axially oblate rotation to axially prolate rotation (with the γ-unstable rotation in O(6) symmetry as the critical point). Page 13 of 16

14 3 CONCLUSION AND DISCURSION In summary, by comparing the low-lying energy spectrum, certain energy ratios and E2 transition rate ratios, we show that the experimental data of nucleus 130 Xe agree with the predicted properties of the states at the critical point of the phase transition from U πν (5) to SU πν(3) well. By analyzing the calculated results in the framework of IBM2, we show that the nucleus 130 Xe is definitely a nucleus in the transitional region from the U πν (5) to the SU πν(3). This indicates that the 130 Xe provides a possible empirical evidence for the nucleus at the critical point of the phase transition from vibration to triaxial rotation and such a critical point contains considerable constituent of γ-unstable rotation. However, it should be mentioned that the agreement between the experimental data of the observables related to the state and the theoretical predictions is not as excellent as the other s and the experimental data of the values of B(M1)s is still in scarcity. More investigations on the experiment and the theory need focus on these aspects. This work was supported by the National Natural Science Foundation of China under Grant Nos , , , the Major State Basic Research Development Program under contract No. G One of the authors (DLZh) also thanks the support by the Natural Science Foundation of Zhejiang Province, China, under Grant No. KY Page 14 of 16

15 4 References [1] J. M. Arias, J. Dukelsky, J. E. García-Ramos, and J. Vidal, Phys. Rev. C 75, (2007). [2] Y. Zhang, Z. F. Hou, and Y. X. Liu, Phys. Rev. C 76, (R) (2007). [3] R. Fossion, C. E. Alonso, J. M. Arias, L. Fortunato, and A. Vitturi, Phys. Rev. C 76, (2007). [4] T. Nikšić, D. Vretenar, G.A. Lalazissis, and P. Ring, Phys. Rev. Lett. 99, (2007). [5] D. Bonatsos, E. A. McCutchan, R. F. Casten, and R. J. Casperson, Phys. Rev. Lett. 100, (2008). [6] D. Bonatsos, E. A. McCutchan, and R. F. Casten, Phys. Rev. Lett. 101, (2008). [7] L. M. Robledo, et al., Phys. Rev. C 78, (2008). [8] P. Cejnar and J. Jolie, Prog. Part. Nucl. Phys. 62, 210 (2009). [9] N. Turkan, J. Phys. G 34, 2235 (2007). [10] B. Sorgunlu, P. Van Isacker, Nucl. Phys. A 808, 27 (2008). Page 15 of 16

16 Thank you! Page 16 of 16