Isospin and mixed symmetry structure in 26Mg

DONG Hong-Fei, BAI Hong-Bo LÜ Li-Jun,

Department of Physics, Chifeng university

IntroductionThe IBM-3 HamiltonianEnergy levelsElectromagnetic transitionConclusion

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.

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)

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].

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 ） a

s subgroups because the isospin and the angular momentum ar

e 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-u

nstable and rotational nuclei respectively.

The dynamical symmetry group for IBM-3

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 .

The IBM-3 Hamiltonian

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

2ˆ ˆs s d dH n n H 2 2 2 2

2 2

2 2

21

(( ) ( ) )2

L T L TL T

L T

H C d d dd 2 2

2

2

0 00

1(( ) ( ) )

2T T

TT

B s s ss

2 2 2 2

2 2

2 2

2 2 2 22 2

1(( ) ( ) ) (( ) ( ) )

2T T T T

T TT T

A s d ds D s d dd

2 2

2

2

0 00

1(( ) ( ) )

2T T

TT

G s s dd

, with =0 ， 1 ， 2 ；

with =0 ， 2 ， =0 ， 2 ， 4 ； with =1 ， 3 。

2 2 2 2 2 2 2 2 2 200

( )1 2 3 4 2 2 1 2 3 4( ) ( ) ( 1) (2 1)(2 1) ( ) ( )L T L T L T L T L Tb b b b L T b b b b

( 1 ), ( 1) zz z

l m mlm m l m mb b

2020 22sdHsdAT

22

222 00

2TsHTsBT 2

222

2 002

TdHTsGT

22

22 222

TdHTsdDT 222

2222

22TLdHTLdC TL

11 22

222

12LdHLdCL

2T

2T 2L

2L

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 。　　　

)3(5)5(24)5(22)3(23

)5(11)6(2 )1(

OdOdUdSUSD

UdTUsdCasimir

CaCaCaCa

CaTTaCH

62352

32525162

009.001.0126.0

175.0611.0093.01361.0359.0

ddd

sdddsd

OOO

SUUUUCasimir

CCC

CCCTTCH

Energy levels

εdρ(ρ=π,υ,δ) 4.763

εsρ(ρ=π,υ,δ) 1.171

Ai(i=0,1,2) -1.408 -0.758 0.758

Ci0(i=0,2,4) -0.114 1.876 -0.714

Ci2(i=0,2,4) 2.052 4.042 1.452

Ci1(i=1,3) -0.832 -2.232

Bi (i=0,2) -0.726 1.440

Di(i=0,2) 1.310 1.310

Gi(i=0,2) -1.525 -1.525

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

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.

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

0

2

4

6

8

10

1+

3+

4+

2+

0+

2+

3+

4+

5+

6+

4+

2+

0+

1+

3+

4+

2+

0+

5+

4+

3+

2+

6+

4+

2+

0+

T=2

T=1

Exp

26Mg

IBM-3

En

erg

y(M

ev)

The wave function of the , , , , and states

1012

1416

11

23

...2023.02980.03304.04215.05723.00 422222231

sssdssssddssss

...2324.02324.02649.04025.04929.02 222321

dsssdsssddssdssdss

...2221.02565.03848.05441.04 232221

dsddsssdssddss

...2384.02752.04129.05893.06 322221

dsdddssddsddss

...2460.02574.05022.05799.01 22321

ddsssddsddsddss

...2439.02587.05280.06097.03 2322

ddssddsssddsddss

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 C1

1 and C31 respectively, which shows that the and states are mixed symmetry states.

11

23

11

23

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

-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.00

1

2

3

4

5

6

7

8

9 61

+

32

+

51

+

42

+

11

+

31

+

22

+

41

+

21

+

ener

gy(M

ev)

C11

-3.0 -2.8 -2.6 -2.4 -2.2 -2.0 -1.8 -1.6 -1.40

1

2

3

4

5

6

7

8

9

22

+

32

+

11

+

61

+

41

+

21

+en

ergy

(Mev

)

C31

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 =

0 1Q Q Q

])~

[(3])~()~

[(3 200

20200

0 ddsddsQ

])~

[(2])~()~

[(2 210

21211

1 ddsddsQ

0 1M M10/)

~(3 0

100

0 LgddgM

1 111 2( )M g d d

For the 26Mg, 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]

119.00 037.01 672.00

581.01 000.00 g 301.01 g

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

B(E2)/ B(M1)/

Exp. Cal. Exp. Cal.

.0061 .006041

.135223 .17184 .007923

.000183 .000209

.000002 0.2541

.000101

.0001098 .000110

i fJ J

1 12 0

2 12 0 2 12 2

42 fme2N

2N

22 20

23 02 13 02

.058174 .003396

.064053 .001891

.0004918 .000053

.000011

.003535

.000000

.013634 .002759

.023373 .000004

.002367 .000074

.040494 .0018258 .001820

13 22 23 22 12 20

22 20 11 01 21 01 11 21 21 21 31 21 11 23

.031531 .028461 .000000

.001964 .000009

.179168 .004965

.0021 .002283

.000291

.0064 .000448

.000217

.000098

.177053 .009154

21 23 31 23 11 43 11 24 21 24 12 24 22 24 31 24 11 44

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.

1212

2222

14 14

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).

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

Thanks

[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) 137

0. [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 Pr

ess) (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 B

o , 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