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Applications of Nilsson Mean-field Applications of Nilsson Mean-field Plus Extended Pairing Model for Plus Extended Pairing Model for
Well-Deformed NucleiWell-Deformed Nuclei关鑫,潘峰
辽宁师范大学物理学辽宁师范大学物理学
Liaoning Normal UniversityLiaoning Normal University
Introduction
Extended Pairing Model
Summary
OutlineOutline
Numerical results and discussions
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IntroductionIntroduction
Pairing is an important residual interaction that is used in nuclear physics.
The Bardeen-Cooper- Schrieffer (BCS) and Hartree-Fock-Bogolyubov (HFB) methods suffer from serious difficulties.
Extended Extended Pairing Pairing ModelModel
1 2 1 2 2
1 2 2
1 , 1
22
ˆ
1( )
( !)
p p
j j i jj i j
i i i i i ii i i
H n G a a
G a a a a a a
The standard pairing Hamiltonian
Feng Pan et al., PRL. 92.112503(2004)
The total number of Nilsson levels
The pairing strength
The single-particle
energies
The extended pairing Hamiltonian
Extended Pairing ModelExtended Pairing Model
Feng Pan et al., PRC. 80, 044306 (2009)
The extended pairing model
The standard pairing model
The first step approximation
Only the lowest energy eigenstate is taken into consideration
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Extended Pairing ModelExtended Pairing Model
1 2 1 2
1 2
( )1 1
1
; ; ,..., ,...,m i i i i i i mi i i p
j j C a a a j j
1 2
( )
( )
1
1
(1 )i i i
i
C
k-pair eigenstates
The expansion coefficient can be expressed as
The additional quantum number
The undetermined variable
1 2
1 21
2; ;0 ( ; ;0 0 )j j i i i
j i i i p
n a a a
1 2 1 2 2
1 2 2
1 2 1 2
1 2 1 2
22
( )
1 1
1( ) ; ;0
( !)
( ) 0 ( 1) ; ;0
p
i i i i i i i ii i i i
i i i i i ii i i p i i i p
a a a a a a a a
C a a a
mean-field
pairing
Extended Pairing ModelExtended Pairing Model
The eigenvalue equation of the model
( )( )
2( 1)E G
1 2
( ) ( )1
1
20
(1 )i i i p i
G
The undetermined variable is given by( )
Extended Pairing ModelExtended Pairing Model
k-pair excitation energies
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Numerical results and discussions
The average binding energy per valence neutron
Er isotope = - 7.475 Mev
Yb isotope = - 7.476 Mev
Hf isotope = - 7. 71 Mev
0jj The neutron single particle energy is expressed as
Neutron single-particle energies calculated
from the Nilsson model
The average binding energy of per valence
neutron in a major shell
The contribution from the core
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Numerical results and discussions
P(A)=E(A-1)+E(A+1)-2E(A)
1
2exp 2[ ]thE E N
The definition of odd-even mass difference
Mean-square deviation of binding energies
The total number of the nuclei fitted
All nuclei fitted in a chain
Numerical results and discussionsB
indi
ng
ene
rgy
Odd-even m
ass difference
Numerical results and discussions
BCS approximation
Extended Extended Pairing ModelPairing Model
Nearest-orbit pairing model
Er EXP TH Er EXP TH
154 ———— 0.893 155 ———— 0.368
156 0.930 0.850 157 0.110 1.030
158 0.806 1.087 159 ———— 0.914
160 0.894 0.891 161 0.725 0.292
162 1.087 0.854 163 0.164 0.164
164 1.217 1.514 165 0.296 0.096
166 1.460 1.813 167 0.643 0.134
168 1.217 1.915 169 0.562 0.561
Numerical results and discussions
The first pairing excitation energy of The first pairing excitation energy of Er
Er EXP TH Er EXP TH
154 ———— 1.587 155 ———— 0.543
156 ———— 2.040 157 0.242 1.352
158 1.387 1.364 159 ———— 0.897
160 ———— 0.967 161 ———— 0.397
162 1.421 1.901 163 0.440 0.604
164 1.417 2.459 165 0.384 0.397
166 2.189 3.083 167 0.873 0.809
168 1.422 5.255 169 1.094 4.381
Numerical results and discussions
The second pairing excitation energy of The second pairing excitation energy of Er
Hf EXP TH Hf EXP TH
166 0.695 0.726 167 ———— 0.528
168 0.942 1.111 169 0.059 0.179
170 0.88 0.721 171 0.555 0.134
172 0.871 0.799 173 ———— 0.041
Numerical results and discussions
Hf EXP TH Hf EXP TH
166 0.909 1.163 167 ———— 0.883
168 ———— 1.995 169 ———— 0.411
170 ———— 1.633 171 0.789 0.789
172 1.295 0.867 173 ———— 0.632
The first pairing excitation energy of The first pairing excitation energy of Hf
The second pairing excitation energy of The second pairing excitation energy of Hf
Numerical results and discussions
Yb EXP TH Yb EXP TH
160 1.086 0.659 161 0.211 0.147
162 0.606 1.815 163 0.058 0.188
164 0.976 1.777 165 0.174 0.469
166 1.043 1.590 167 0.213 0.121
168 1.154 2.063 169 0.647 0.131
170 1.069 1.851 171 0.953 0.698
The first pairing excitation energy of The first pairing excitation energy of Yb
The second pairing excitation energy of The second pairing excitation energy of YbYb EXP TH Yb EXP TH
160 ———— 1.497 161 ———— 0.371
162 1.006 2.638 163 ———— 0.511
164 ———— 2.150 165 0.400 1.427
166 ———— 2.715 167 0.239 0.369
168 1.197 3.190 169 0.832 1.064
170 1.229 1.064 171 0.988 1.148
Summary and perspectiveSummary and perspectiveSummary
Mean-field plus extended pairing interaction for the description of well-deformed nuclei in Rare-earth Region is adopted. Freezing the excitation of the proton pairs, binding energies, pairing excitation energies, even-odd mass differences of 152-169Er , 154-171Yb, 153-173Hf are calculated and compared with the corresponding experimental data.
There is a little deviation of the pairing excitation energies.
Perspective In the future work we will calculate in both proton-proton and
neutron-neutron pairing excitation cases, and also calculate and moment of inertia of low-lying excitation states and the electric quadrupole moment.
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关鑫,潘峰辽宁师范大学物理学辽宁师范大学物理学
Liaoning Normal UniversityLiaoning Normal University
Numerical results and discussions
The ground The ground state energy state energy
The second The second pairing excitation pairing excitation
energyenergy
The first pairing The first pairing excitation excitation
energyenergy
K=0
K=0
K=0
K=2