Applications of Nilsson Mean-field Plus Extended Pairing Model for Well-Deformed Nuclei

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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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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


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( )


k-pair excitation energies

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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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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


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


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


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


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


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


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

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Applications of Nilsson Mean-field Plus Extended Pairing Model for Well-Deformed Nuclei