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Isospin and mixed symmetry structure in 26 Mg

Isospin and mixed symmetry structure in 26 Mg. DONG Hong-Fei, BAI Hong-Bo LÜ Li-Jun, Department of Physics, Chifeng university. Introduction The IBM-3 Hamiltonian Energy levels Electromagnetic transition Conclusion. Introduction.

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Isospin and mixed symmetry structure in 26 Mg

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  1. Isospin and mixed symmetry structure in 26Mg DONG Hong-Fei, BAI Hong-Bo LÜ Li-Jun, Department of Physics, Chifeng university

  2. Introduction • The IBM-3 Hamiltonian • Energy levels • Electromagnetic transition • Conclusion

  3. Introduction Nuclei with Z≈N have been a subject of intense interest during the last few years [1-5] .The main reason is that the structure of these nuclei provides a sensitive test for the isospin symmetry of nuclear force. The interacting boson model (IBM) is an algebraic model used to study the nuclear collective motions.

  4. IBM • 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-2), 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)

  5. IBM-3 • whose microscopic foundation is shell model [12,13]. • The isospin T=1 triplet including three types of bosons :proton-proton(π) neutron-neutron(υ) proton-neutron(δ) • The IBM-3 can describe the low-energy levels of some nuclei well and explain their isospin and F-spin symmetry structure[3-5,14-16].

  6. The dynamical symmetry group for IBM-3 • The dynamical symmetry group for IBM-3 is U(18) , which starts with Usd(6)×Uc(3) and must contain SUT（2）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 the following[17] • U(18) (Uc(3) SUT(2))×(Usd(6) Ud(5) Od(5) Od(3)), • U(18) (Uc(3) SUT(2))×(Usd(6) Osd(6) Od(5) Od(3)), • U(18) (Uc(3) SUT(2))×(Usd(6) SUsd(3) Od(3)), • The subgroups Ud(5), Osd(6) and SUsd(3) describe vibrational,γ-unstable and rotational nuclei respectively.

  7. 26Mg lies in the lighter nuclei region and is one even-even nucleus. By making use of the interacting boson model (IBM-3), we study the isospin excitation states, electromagnetic transitions and mixed symmetry states at low spin for 26Mg nucleus. The main components of the wave function for some states are also analyzed respectively .

  8. The IBM-3 Hamiltonian • The IBM-3 Hamiltonian can be written as[13]

  9. , with =0，1，2； with =0，2， =0，2，4； • with =1，3。

  10. Casimir operator IBM-3 Hamiltonian can be expressed in Casimir operator form, i.e., Hamiltonians for the low-lying levels of 26Mg : From the IBM-3 Hamiltonian expressed in Casimir operator form, we know that the 26Mg is in transition from U(5) to SU(3) because the interaction strength of is 0.093 and that of is 0.175。　　　

  11. Energy levels Table 1. The parameters of the IBM-3 Hamiltonian of the 26Mg nucleus

  12. The calculated and experimental energy levels are exhibited in figure 1.When the spin value is below 8+, the theoretical calculations are in agreement with experimental data.

  13. Fig.1 Comparison between lowest excitation energy bands（T=1，T=2）of the IBM-3 calculation and experimental excitation energies of 26Mg

  14. The wave function of the , , , , and states

  15. We found that the main components of the wave function for the states above are sN, sN−1d, sN−2d2, sN−3d3 and so on configurations. The wave function of these states contain a significant amount of δ boson component, which shows that it is necessary to consider the isospin effect for the light nuclei. From the analysis of the component of wave function of and states, it is known that they are two-phonon states. The parameters C11 and C31 are Majorana parameter , which have a very large effect on the energy levels of mixed symmetry state. From Fig. 2, we see that the and states have a large change with the parameters C11 and C31 respectively, which shows that the and states are mixed symmetry states.

  16. Fig.2Variation in level energy of 26Mg as a function of C11 and C31 respectively

  17. Electromagnetic transition • In the IBM-3 model, the quadrupole operator was expressed as： • where • The M1 transition is also a one-boson operator with an isoscalar part and an isovector part • where M =

  18. For the26Mg, the parameters in the electromagnetic transitions are determined by fitting the experimental data, they are Table 2 gives the electromagnetic transition rate calculated by IBM-3[20]

  19. Experimental and calculated B(E2)( e2fm4) and B(M1)( ) for 26Mg

  20. Table 2 shows that the calculated B(E2) values are quite close to the experimental ones[21]. The calculated quadrupole moments of the state is Q( ) =0.59418eb. state is Q( ) =1.12365eb. state is Q( ) = 1.41749eb.

  21. Conclusion • The calculated results are in agreement with available experimental data. • 11+ and 32+ state is the mixed symmetry states. • the calculated quadrupole moments of the 21+ state is 0.59418eb. 22+ state is 1.12365eb. 41+ state is 1.41749eb. • 26Mg is in transition from U(5) to SU(3).

  22. Thanks • The authors are greatly indebted to Prof. G. L Long for his continuing interest in this work and his many suggestions.

  23. References • [1] R. Sahu and VKB Kota , Phys.Rev.C67(2003)054323. • [2] M. Bender, H. Flocard and P-H Heenen, Phys. Rev.C68 (2003)044321. • [3]H.Al-Khudair Falih, Li Yan-Song and Long Gui-Lu,J. Phys.G: Nucl.Part.Phys.30 • (2004)1287. • [4] E.Caurier，F.Nowacki and A.Poves，Phys.Rev.Lett.95(2005) 042502 • [5] Long G L and Sun Yang，Phys.Rev.C65(2001) R0712 (Rapid Communication) • [6] A.Arima and F. Iachello, Ann.Phys.(N.Y.)99(1976) 253. • [7] A.Arima and F. Iachello, Ann.Phys.(N.Y.)111(1978) 201. • [8] A.Arima and F. Iachello, Ann.Phys.(N.Y.)123 (1979)468. • [9] Liu Yu-xin, Song Jian-gang, Sun Hong-zhou and Zhao En-guang ,Phys. Rev.C 56(1997) 1370. • [10] Pan Feng, Dai Lian-Rong, Luo Yan-An, and J. P. Draayer,Phys. Rev.C 68 (2003)014308. • [11] F.Iachello and A. Arima, The Interacting Boson Model (Cambridge:Cambridge University Press) (1987). • [12] J. P. Elliott, A. P. White , Phys.Lett. B97(1980) 169. • [13] J. A. Evans, Long G L and J. P. Elliott , Nucl. Phys. A561(1993) 201-31. • [14] H Al-Khudair Falih, Li Y S and Long G L, High Energ Nucl Phys 28 (2004)370-376. • [15] HAK. Falih, Long G L ,Chin. Phys. 13 (8)( 2004)1230-1238. [16] Zhang Jin Fu, Bai Hong Bo , Chin. Phys. 13(11) (2004) 1843. • [17] Long G L , Chinese J. Nucl. Phys. 16(1994 )331. • [18] Li Y S,Long G L，Commun.Theor.Phys.41(2004) 579 • [19] P .Van Isacker ,et al., Ann. Phys.(N.Y.)171(1986) 253. • [20] R. B. Firestone , Table of Isotopes 8th edn ed V S Shirley (1998).

  24. Thank You

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