1. Isospin and mixed symmetry structure in 26 Mg 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 is U(18), which starts with Usd(6)×Uc(3) and must contain SU T （ 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) SU T (2))×(U sd (6) U d (5) O d (5) O d (3)), U(18) (Uc(3) SU T (2))×(U sd (6) O sd (6) O d (5) O d (3)), U(18) (Uc(3) SU T (2))×(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 dynamical symmetry group for IBM-3

7. 26 Mg 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 26 Mg 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 26 Mg : 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 ε dρ (ρ=π,υ,δ)4.763 ε sρ (ρ=π,υ,δ)1.171 Ai(i=0,1,2)-1.408-0.7580.758 Ci0(i=0,2,4)-0.1141.876-0.714 Ci2(i=0,2,4)2.0524.0421.452 Ci1(i=1,3)-0.832-2.232 Bi (i=0,2)-0.7261.440 Di(i=0,2)1.310 Gi(i=0,2)-1.525 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 s N, s N−1 d, s N−2 d 2, s N−3 d 3 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 C 11 and C 31 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 C 11 and C 31 respectively, which shows that the and states are mixed symmetry states.

16. Fig.2 Variation in level energy of 26 Mg as a function of C 11 and C 31 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 the 26 Mg, 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)( e 2 fm 4 ) and B(M1)( ) for 26 Mg B(E2)/B(M1)/ Exp.Cal. Exp.Cal..0061.006041.135223.17184.007923.000183.000209.0000020.2541.000101.0001098.000110

20. .058174.003396.064053.001891.0004918.000053.000011.003535.000000.013634.002759.023373.000004.002367.000074.040494.0018258.001820

21. .031531.028461.000000.001964.000009.179168.004965.0021.002283.000291.0064.000448.000217.000098.177053.009154

22. 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.

23. Conclusion The calculated results are in agreement with available experimental data. 1 1 + and 3 2 + state is the mixed symmetry states. the calculated quadrupole moments of the 2 1 + state is 0.59418eb. 2 2 + state is 1.12365eb. 4 1 + state is 1.41749eb. 26 Mg is in transition from U(5) to SU(3).

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

25. [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). References