npsc-2003gabriela popa microscopic interpretation of the excited k = 0 +, 2 + bands of deformed...
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NPSC-2003 Gabriela Popa
Microscopic interpretation of the excited K = 0+, 2+ bands
of deformed nuclei
Gabriela PopaRochester Institute of Technology
Collaborators:J. P. Draayer
Louisiana State University
J. G. HirschUNAM, Mexico
A. GeorgievaBulgarian Academy of Science
NPSC-2003 Gabriela Popa
Outline
IntroductionIntroductionWhat we know about the nucleusWhat we know about the nucleusCharacteristic energy spectraCharacteristic energy spectraTheoretical Model Theoretical Model Configuration spaceConfiguration spaceSystem HamiltonianSystem HamiltonianResults Results Conclusion and future workConclusion and future work
Nuclear Physics Summer School June 15-27, 2003 Gabriela Popa
Chart of the nuclei
N (number of neutrons)
Z (protons)
NPSC-2003 Gabriela Popa
Nuclear vibrations
-vibration
3
1
2
3
-vibration1
2
0, 2, 4
00246
8
6
8
20246
23456
spherical deformed
E = n E = I(I+1)/(2)
Schematic level schemes of spherical and deformed nuclei
NPSC-2003 Gabriela Popa
Experimental energy levels
0
0.5
1
1.5
2
0
0.5
1
1.5
2
10 12 14 16 18 20 22
152Nd156Sm160Gd164Dy 168Er172Yb176Hf
2
21
02
0g.s.
Number of Pairs of Particles
Nuclear Physics Summer School June 15-27, 2003 Gabriela Popa
ExperimentalExperimental energy spectra of 162Dy
-0.5
0
0.5
1
1.5
2
2.5
3
162Dy
g.s.
K = 2
K = 0K = 0
2+4+
6+
8+
0+
2+4+6+
0+
4+
7+
2+
4+
6+
8+
3+
5+
7+
6+
5+1+
1+
1+
2+4+6+8+
0+
K = 4Ene
rgy
[MeV
]
NPSC-2003 Gabriela Popa
Single particle energy levels
1s
1p
2s
1g
N = 3
N = 2
N = 1
N = 0
N = 4
1d
l·l l·s
d5/2
d3/2s1/2
1f2p
3s2d
1hN = 5
s1/2
p1/2p3/2
f7/2f5/2
p5/2
f9/2p1/2
d3/2d5/2
h11/2s1/2
g7/2
NPSC-2003 Gabriela Popa
Valence space: U() U()
total normal unique
=32 n =20 u
=12 =44 n
=30 u=14
Particle distributions: protons: neutrons: total 152Nd 10 6 4 10 6 4 (12,0) (18,0) (30, 0)156Sm 12 6 6 12 6 6 (12,0) (18,0) (30, 0) 160Gd 14 8 6 14 8 6 (10,4) (18,4) (28, 4) 164Dy 16 10 6 16 10 6 (10,4) (20,4) (30, 4) 168Er 18 10 8 18 10 8 (10,4) (20,4) (30, 4) 172Yb 20 12 8 20 12 8 (36,0) (12,0) (24, 0) 176Hf 22 14 8 22 14 8 (8,30) (0,12) (8, 18)
Particle distribution
NPSC-2003 Gabriela Popa
Wave Function
| = Ci | i
|i = |{; } (,)KL S; JM
( = , ) = n [f ] ( , ), S
( , ) ( , ) = (, )
NPSC-2003 Gabriela Popa
Coupling proton and neutron irreps to total (coupled) SU(3):
+++- ++++-+-+-
m,l +-2m+l++m-2l)k
with the multiplicity denoted by k = k(m,l)
Direct Product Coupling
NPSC-2003 Gabriela Popa
(10,4) (18,4) (28,8) (29,6) (30,4) (31,2) (32,0) (26,9) (27,7)(10,4) (20,0) (30,4)(10,4) (16,5) (26,9) (27,7)(10,4) (17,3) (27,7)(12,0) (18,4) (30,4)(12,0) (20,0) (32,0)(8,5) (18,4) (26,9) (27,7)(9,3) (18,4) (27,7)(7,7) (18,4) (25,11) (26,9) (27,7)(7,7) (20,0) (27,7)(4,10) (18,4) (22,14)
(,)(,) (,)
21st SU(3) irreps corresponding to the highest
C2 values were used in 160Gd
NPSC-2003 Gabriela Popa
Tricks
Invariants InvariantsRot(3) SU(3)Tr(Q2) C2
Tr(Q3) C3 ~+ + + + =tan+
NPSC-2003 Gabriela Popa
SU(3) conserving Hamiltonian:
H = Q•Q + a L2 + b KJ2 + asym C2 + a3 C3
this Hamiltonian becomes ...
+ one-body and two-body terms
+ Hs.p.Hs.p.
G HPG HP
Rewriting
H = Hsp
+Hsp+G HP
+G HP+Q•Q
+ a L2 + b KJ2 + asym C2 + a3 C3
.
System Hamiltonian
NPSC-2003 Gabriela Popa
Parameters of the Pseudo-SU(3) Hamiltonian
Fit to experiment(fine tuning): a, b, asym and a3
= 0.0637 = 0.60
= 0.0637 = 0.60
From systematics: = 35/A 5/3 MeVG= 21/A MeVG= 19/A MeVh = 41/A 1/3 MeV
NPSC-2003 Gabriela Popa
0
0.5
1
1.5
2
2.5
K0+
K+
K0
2
+
160Gd
0+
6+
0+
2+
2+
2+
4+
4+
8+
6+
3+
8+
7+
6+
4+
5+
8+
Exp. Th. Exp. Th. Exp. Th.
Energy spectrum for 160Gd
NPSC-2003 Gabriela Popa
Coupling proton and neutron irreps to total (coupled) SU(3):
+++- ++++-+-+-
m,l +-2m+l++m-2l)k
Twist
Scissors
Scissors + Twist
… Orientation of the - system is quantized with the multiplicity denoted by k = k(m,l)
Protons
Neutrons
Direct Product Coupling
NPSC-2003 Gabriela Popa
Nucleus B(M1) mode
160 ,162Dy (10,4) (18,4) (28,8) ( 29, 6) 0.56 t( 26, 9) 1.77 s(27; 7)1 1.82 s+t(27; 7)2 0.083 t+s
164 Dy (10,4) (20,4) (30,8) (31,6) 0.56 t(28,9) 1.83 s(29,7) 1.88 s+t(29,7) 0.09 t+s
() () () ()1+
M1 Transition Strengths [ 2N]
in the Pure Symmetry Limit of the
Pseudo SU(3) Model
NPSC-2003 Gabriela Popa
0
0.2
0.4
0.6
0.8
1
1.2
1.4
2.2 3.0 3.7 4.5
Th.Exp.
164Dy
B(M
1,
0+ -
> 1+)
[n2 ]
Energy [MeV]
f)
-0.5
0
0.5
1
1.5
2
2.5
164 Dy
e)K = 2
K = 0 K = 0
2+4+
6+
0+2+4+
6+
0+
2+4+6+
0+
2+4+
0+
2+4+
0+
2+4+6+
0+
2+
4+6+
3+5+
2+4+6+
3+5+
g.s.
c)
Ene
rgy
[MeV
]Exp. Th. Exp. Th. Exp. Th. Exp. Th.
Energy levels of 164Dy
… and M1 Strengths
NPSC-2003 Gabriela Popa
0
0.2
0.4
0.6
0.8
1
2.0 2.5 3.0 3.5 4.0 4.5
Th.168Er
B(M
1,
0+ -
> 1+)
[n2 ]
Energy [MeV]
b)
0
0.5
1
1.5
2
2.5
3
168 Er
K = 0
K = 0K = 2
0 +2 +
8 +
6 +
4 +
0 +2 +
8 +
6 +
4 +
0 +2 +
8 +
6 +
4 +
0 +2 +4 +
2 +
8 +
6 +
4 +
7 +
3 +5 +
2 +
8 +
6 +
4 +
7 +
3 +5 +
0 +2 +6 +
6 +
4 +0 +2 +
6+
4 +
g.s.
Ene
rgy
[MeV
]
Exp. Th. Exp. Th.Exp. Th. Exp. Th.
a)
Energy Levels of 168Er
… and M1 Strengths
NPSC-2003 Gabriela Popa
Nucleus
ExperimentCalculated
Pure S U (3) Theory
160 Dy 2.48 4.24 2.32162 Dy 3.29 4.24 2.29164 Dy 5.63 4.36 3.05
B(M1)[N2]
Total B(M1) strength (N2)
NPSC-2003 Gabriela Popa
First excited K=0+ and K=2+ states
0
0.5
1
1.5
2
0
0.5
1
1.5
2
10 12 14 16 18 20 22
152Nd 156Sm 160Gd 164Dy 168Er 172Yb 176Hf
2
21
02
0g.s.
Number of pairs of particles
NPSC-2003 Gabriela Popa
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
0gs
2 02
0gs
2 02
156Sm
(30,0)(12,0)x(18,0)(26,6)(12,0)x(14,2)(26,2)(8,2)x(18,0)(24,6)(12,0)x(12,6)
SU
(3)
con
tent
[%
]
NPSC-2003 Gabriela Popa
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
0gs
2 02
0gs
2 02
160Gd
a = (28,8)(10,4)x(18,4)b = (30,4)(10,4)x(18,4)c = (30,4)(10,4)x(20,4)d = (30,4)(12,0)x(18,4)e = (32,0)(10,4)x(18,4)f = (32,0)(12,0)x(20,0)
NPSC-2003 Gabriela Popa
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0gs
2 02
0gs
2 02
164Dy
a = (30,8)(10,4)x(20,4)b = (32,4)(10,4)x(20,4)c = (32,4)(12,0)x(20,4) d = (32,4)(10,4)x(20,4)e = (34,0)(10,4)x(20,4)f = (34,0)(12,0)x(22,0)g = (24,14)(10,4)x(14,10)
NPSC-2003 Gabriela Popa
0
20
40
60
80
100
0
20
40
60
80
100
0gs
2 02
0gs
2 02
168Er
a = (30,8)(10,4)x(20,4)b = (32,4)(10,4)x(20,4)c = (32,4)(12,0)x(20,4)d = (32,4)(10,4)x(22,0)e = (34,0)(10,4)x(20,4)f = (24,14)(10,4)x(14,10)
NPSC-2003 Gabriela Popa
0
20
40
60
80
100
0
20
40
60
80
100
0gs
2 02
0gs
2 02
a = (36,0)(12,0)x(24,0)b = (28,10)(12,0)x(16,10)c = (20,20)(4,10)x(16,10)
SU(3) content in 172Yb
NPSC-2003 Gabriela Popa
Conclusions
•B(E2) transitions within the g.s. band wellreproduced
ground state, , first and second excited K = 0+ bands well described by a few representations
•calculated results in good agreement with the low-energy spectra
•1+ energies fall in the correct energy range
•fragmentation in the B(M1) transition probabilities correctly predicted
NPSC-2003 Gabriela Popa
Conclusions
A proper description of collective properties of the first excited K = 0+ and K=2+ states must take into account the mixing of different SU(3)-irreps, which is driven by the Hamiltonian.
A microscopic interpretation of the relative position of the collective band, as well as that of the levels within the band, follows from an evaluation of the primary SU(3) content of the collective states. The latter are closely linked to nuclear deformation.
If the leading configuration is triaxial (nonzero ), the ground and bands belong to the same SU(3) irrep; if the leading SU(3) configuration is axial (=0), the K =0 and bands come from the same SU(3) irrep.
NPSC-2003 Gabriela Popa
Future work
• In some nuclei total strength of the M1 distributionis larger than the experimental value
•Investigate the energy spectra in super-heavy nuclei•Investigate the M1 transitions in light nuclei
• There are new experiments that determine the inter-band B(E2) transitions• Improve the model to calculate these transitions
• Consider the abnormal parity levels• Consider the states with J = 1 in the configuration space
NPSC-2003 Gabriela Popa