Isospin and Symmetry Structure in 36 Ar

Transcription

1 Commun. Theor. Phys. (Beijing, China) 48 (007) pp c International Academic Publishers Vol. 48, No. 6, December 15, 007 Isospin and Symmetry Structure in 36 Ar BAI Hong-Bo, 1, ZHANG Jin-Fu, 1 CAO Wan-Cang, 1 LÜ Li-Jun,1 Zhaorigt, 1 DONG Hong-Fei, 1 and LI Yan-Song 1 Department of Physics, Chifeng University, Chifeng 04001, China Department of Physics, Tsinghua University, Beijing , China (Received December 18, 006; Revised March 6, 007) Abstract The interacting boson model-3(ibm-3) has been used to study the isospin excitation states and electromagnetic transitions for 36 Ar nucleus. The mixed symmetry states and superdeformed band at low spin are also analyzed. The theoretical calculations are in agreement with experimental data, and the 36 Ar is superdeformed rotational nucleus close to the SU(3) limit. The present calculations indicate that the + 4 state is the lowest mixed symmetry state and the lowest isospin T = 1 excitation state and at about 6. MeV, and the bandhead of superdeformed band is 0 + state. PACS numbers: 1.60.FW, t Key words: IBM-3, energy level, isospin, mixed symmetry states, electromagnetic transitions In recent years, many experimental and theoretical works have been carried out for the investigation of the Z N lighter nuclei with some models. [1 5] The interacting boson model (IBM) is an algebraic model used to study the nuclear collective motions. In the original version (IBM-1), only one kind of boson is considered, and it has been successful in describing various properties of medium and heavy even-even nuclei. [6 10] In its second version (IBM-), the bosons are further classified into proton-boson and neutron-boson, and mixed symmetry in the proton and neutron degrees of freedom has been predicted. [11] For lighter nuclei, the valence protons and neutrons are filling the same major shell and the isospin should be taken into account, so the IBM has been extended to the interacting boson model with isospin (IBM-3). [1] In the IBM-3, three types of bosons including proton-proton (π), neutron-neutron (υ) and proton-neutron (δ) form the isospin T = 1 triplet. The microscopic foundation of IBM-3 is based on shell model. [13] In the lighter nuclei region the protons and neutrons are in the same major shell, the IBM-3 can describe the low-energy levels of some nuclei well and explain their isospin and F-spin symmetry structure. [3,14 16] The dynamical symmetry group for IBM-3 is U(18), which starts with U sd (6) U c (3) and must contain SU T () and O(3) as subgroups because the isospin and the angular momentum are good quantum numbers. The natural chains of IBM-3 group U(18) are [17] U(18) (U c (3) SU T ()) (U sd (6) U d (5) O d (5) O d (3)), U(18) (U c (3) SU T ()) (U sd (6) O sd (6) O d (5) O d (3)), U(18) (U c (3) SU T ()) (U sd (6) SU sd (3) O d (3)). The subgroups U d (5), O sd (6), and SU sd (3) describe vibrational, γ-unstable and rotational nuclei respectively. The 36 Ar is the even-even Z = N nucleus with A < 40, which is superdeformed nucleus with deformation parameter β = 0.46 by experiment. [18] In order to calculate the 8 + state, we assume 8 Si as the closed core, so there are four valence protons and four valence neutrons outside this closed core respectively. To prove that 8 Si can be taken as a closed core, an RMF calculation is carried out. The Fortran code in the calculation is a part of ReCAPS. [19,0] The RMF calculation tells that 8 Si is spherical. And the lowest valent orbit of neutron has a large gap of over 8.1 MeV, and is over 7.6 MeV for that of proton. The single particle energy levels are shown in Fig. 1. All these results support that our assumption is adequate. Recently, many researchers have studied the 36 Ar nucleus theoretically with different models such as shell model, cranked Nilsson Strutinsky, projected shell model and beyond-mean-field-model, [,4,5,1] and especially calculated and discussed the superdeformed band whose band head is 0 + state. This paper is principally aimed at calculating and discussing the isospin structure, mixed symmetry structure and superdeformed band at low spin in the IBM-3. Because lighter nuclei have isospin structure, the dynamical symmetry is not as typical as in medium and heavy nuclei. As the isospin is introduced, the dynamical symmetries become obvious. [15] By using IBM-3 model, it is easy to distinguish typical mixed symmetry states, which are caused by the The project supported by National Natural Science Foundation of China under Grant Nos and , and the Key Scientific Research Fund of Inner Mongolian Education Bureau under Grant Nos. NJ04116 and NJ hbbai@vip.sina.com

2 1068 BAI Hong-Bo, ZHANG Jin-Fu, CAO Wan-Cang, LÜ Li-Jun, Zhaorigt, DONG Hong-Fei, and LI Yan-Song Vol. 48 relative motions between proton bosons and neutron bosons. The mixed symmetry states have important significance in the study of nuclear structure. The isospin-invariant IBM-3 Hamiltonian can be written as [1] and H = ε sˆn s + ε dˆn d + H, (1) H = 1 C LT ((d d ) LT ( d d) LT ) + 1 B 0T ((s s ) 0T ( s s) 0T ) + A T ((s d ) T ( d s) T ) L T T T + 1 D T ((s d ) T ( d d) T ) + 1 G 0T ((s s ) 0T ( d d) 0T ), () T T (b 1 b )LT ( b 3 b4 ) LT = ( 1) (L+T) (L + 1)(T + 1)[(b 1 b )LT ( b 3 b4 ) LT ] 00, (3) blm,mz = ( 1) (l+m+1+mz) b l m mz, (4) T and L represent the two-boson system isospin and angular momentum. The parameters A, B, C, D, and G are the two-body matrix elements, A T = sd0 H sd0, T = 0,1,; B T = s 0T H s 0T, G T = s 0T H d 0T, D T = sdt H d T, D T = sdt H d T, and C LT = d L T H d L T, with T = 0,, L = 0,,4, C L1 = d L 1 H d L 1 with L = 1,3. The parameters A 1, C 11, and C 31 are Majorana parameters, which are similar to the ones in IBM- IBM-3 Hamiltonian can be expressed in Casimir operator form, i.e., H Casimir = λc Usd (6) + a T T(T + 1) + a 1 C 1Ud (5) + a 3 C SUSD(3) + a C Ud (5) + a 4 C Od (5) + a 5 C Od (3), (5) λ parameter can be used to determine the position of the mixed symmetry states. The parameters in the Hamiltonian can be determined by fitting the experimental spectra. The low-lying levels of 36 Ar can be described by the following Hamiltonians, H Casimir = 0.44C Usd (6)+1.46T(T +1)+0.05C 1Ud (5)+0.41C SUSD(3)+0.053C Ud (5)+0.111C Od (5) 0.08C Od (3). (6) The energy levels and wave function are given by the computation program written by Van Isacker. [] The parameters of the calculation are listed in Table 1. Table 1 The parameters of Hamiltonian of 36 Ar. ε dρ (ρ = π, υ, δ).637 ε sρ (ρ = π, υ, δ).180 A i (i = 0, 1, ) C i0 (i = 0,, 4) C i (i = 0,, 4) C i1 (i = 1, 3) B i (i = 0, ) D i (i = 0, ) G i (i = 0, ) The calculated and experimental energy levels [3] are exhibited in Fig. 1. When the spin value is below 8 +, the theoretical calculations are in agreement with experimental data. The IBM-3 calculation predicts that the bandhead of superdeformed band is 0 + state and its position is 4.39 MeV.[] The isospin value of superdeformed band is T = 0. The isospin value of state is T =, and the experimental and calculated energies of 0+ 3 state are MeV and 9.8 MeV respectively. Calculation predicts that the isospin values of + 4, 1+ 1, and 3+ states are T = 1. The main components of the wave function for these states are + 4 = d νs ν (d π) (d υ) d π s π d ν s ν (d π) (d υ) 4 d π s π s ν d π (d δ) d ν s π (d δ) = (d υ) (d π) (d υ) 4 (d π) d ν d π d δ = (d υ) (d π) (d υ) (d π) (d υ) 4 (d π) (d υ) 4 (d π) 4 +

3 No. 6 Isospin and Symmetry Structure in 36 Ar 1069 (d π) means that two d π bosons couple to L =. The main composition of the + 4 state is three d-bosons. The composition of the and 3+ states is four d-bosons, + 4 and 1+ 1 states contain a δ boson component. represents some smaller components. The C 11, C 31, and A 1 are Majorana parameters, the variation of which greatly affects the mixed symmetry states. Figure shows that the + 4, 1+ 1, and 3+ states are the mixed symmetry states because the variation of Majoeana parameters affects these states. The + 4 state is the lowest mixed symmetry state, whose experimental energy level is 6.63 MeV. Fig. 1 Single particle energy levels of 6 Si. Fig. Comparison between lowest excitation energy bands, T = T z, T z + 1, T z + of the IBM-3 calculation and experimental excitation energies of 36 Ar. Fig. 3 Variation in level energy of 36 Ar as functions of C 11, C 31, and A 1, respectively. All the other parameters are kept at their best-fit values. In the IBM-3 model, the quadrupole operator was expressed as [4] Q = Q 0 + Q 1, (7) The M1 transition is also a one-boson operator with an isoscalar part and an isovector part, Q 0 = α 0 3[s ˆd) 0 + (d ŝ) 0 ] + β 0 3[(d ˆd) 0 ], (8) Q 1 = α 1 [s ˆd) 1 + (d ŝ) 1 ] + β 1 [(d ˆd) 1 ]. (9) M = M 0 + M 1, (10) M 0 = g 0 3(d ˆd) 10 = 1 10 g 0 L, (11)

4 1070 BAI Hong-Bo, ZHANG Jin-Fu, CAO Wan-Cang, LÜ Li-Jun, Zhaorigt, DONG Hong-Fei, and LI Yan-Song Vol. 48 M 1 = g 1 (d ˆd) 11, (1) g 0 and g 1 are the isoscalar and isovector g-factors respectively, and L is angular momentum operator. For the 36 Ar, the parameters in the electromagnetic transitions are determined by fitting the experimental data, α 0 = 0.069, β 0 = 0.005, α 1 = 0.015, β 1 = 0.016, g 0 = 0, g 1 = 1, respectively. Table gives the electromagnetic transition rate calculated by IBM-3. [1] Table Experimental and calculated B(E) (e fm 4 ) and B(M1) (µ N) for 36 Ar. J + i J + f B(E) B(M1) Exp. Cal. Exp. Cal Table shows that the calculated B(E) values are quite close to the experimental ones. [1] As to the mixed symmetry states J + ms, their values of B(M1;J + ms J + s ) are larger, while the corresponding B(E; J + ms J + s ) values are small. [] Table shows that the + 4, 1+ 1, and 3 + states are mixed symmetry states as their values of B(E; + 4 J s + ), B(E;1 + 1 J s + ), and B(E;3 + J s + ) are smaller but the corresponding B(M1) values are larger. From the IBM-3 Hamiltonian expressed in Casimir operator form, we know that the 36 Ar is superdeformed rotational nucleus closer to the SU(3) limit because the interaction strength of C SUSD(3) is 0.41 and that of C 1Ud (5) is The calculated quadrupole moments of the + 1 state is eb, which is close to the experiment value +0.11(6) eb. By using the interacting boson model with isospin (IBM-3), we calculate the isospin excitation bands, superdeformed band at low spin, electromagnetic transitions, and mixed symmetry structure of 36 Ar. The results conclude that the IBM-3 description of the low-lying levels in the 36 Ar nucleus is satisfactory. Figure 3 shows that the + 4, 1+ 1, and 3+ states are the mixed symmetry states, and the + 4 state is the lowest mixed symmetry state. Moreover, these states are the isospin excitation states, whose isospin value is T = 1 determined by experiment, which supports our calculation. The calculated bandhead of β band should be 0 + state, which is not fully determined by experiment. Our calculation indicates that the energy level of isospin excited state of T = is around 10 MeV. For 36 Ar nucleus, the isospin excited state of T = is not found in experiment until now. By analyzing the IBM-3 parameters in the Casimir form, we know that the 36 Ar nucleus is close to the SU(3) limit and there exists some small vibration component. The mixed symmetry states of this nucleus appear in the position of higher energy level, so the rotational characteristic of collection motion is obvious. Summarizing our results we may conclude that the IBM-3 calculated results agree very well with available experimental data and the 36 Ar nucleus is a good rotational nucleus in low energy level, which is in accordance with the theoretical calculations by using other models. The present calculations also give the structures of the isospin and mixed symmetry states for 36 Ar nucleus. For this nucleus, the first isospin excited state is the same as the first mixed symmetry state, which is an interesting result. Acknowledgments The authors are greatly indebted to Prof. G.L. Long for his continuing interest in this work and his many suggestions.

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