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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. - PowerPoint PPT Presentation
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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