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同济大学TONGJI UNIVERSITY
Nucleon-pair coupling in rotational nuclei
GuanJian Fu
Tongji University
Jan. 13, 2020 原子核结构理论研讨会, 绵阳
Introduction: rotational nuclei in the NSM, IBM, and NPA
The Elliott’s SU(3) limit
For realistic nuclei
Summary
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Nuclear collective behaviorNuclear collective behavior
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Nuclear collective behaviorNuclear collective behavior
Models based on deformation
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Rotational motion in the nuclear shell modelRotational motion in the nuclear shell model
Elliott’s SU(3) model:
Quadrupole interaction
One or many H.O. major shells
a new perspective: the SU(3) symmetry
Q operator → quadrupole deformation
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Yrast band in Cr-48
Superdeformed band in Ca-40
0
0
Rotational motion in the nuclear shell modelRotational motion in the nuclear shell model
E. Caurier et al., Rev. Mod. Phys. 77, 427 (2005).
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Rotational motion in terms of the shell modelRotational motion in terms of the shell model
T. Dytrych et al., Phys. Rev. Lett. 111, 252501 (2013).
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Elliott’s SU(3) model:
Quadrupole interaction
One or many H.O. major shells
a new perspective: the SU(3) symmetry
Q operator → quadrupole deformation
For heavy-mass regions:
strong spin-orbit couplings, SU(3) broken
one HO major shell is not enough,
intruder orbits, pseudo- and quasi-SU(3)
configuration space is too gigantic
Need truncation!
Rotational motion in the nuclear shell modelRotational motion in the nuclear shell model
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Truncations of the model spaceTruncations of the model space
The NPA is a truncation of the shell-model configuration space;
1 1 2 20
n nj m j m j mC C C
Cooper pairs with good angular momentum r:
Nucleon-pair approximation (NPA):
1 2† † †
p 0Nrr rA A A
Shell model, full configuration interaction
† † † † † †
b 0s s s d d d
Cooper pair => boson; Interacting boson model (IBM):
further mapping to IBM.
A. Arima and F. Iachello, Ann. Phys. (NY) 99, 253 (1976); 111, 201 (1978); 123, 468 (1979); J. Q. Chen, Nucl. Phys. A 626, 686 (1997); Y. M. Zhao and A. Arima, Phys. Rep. 545, 1 (2014).
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Microscopic foundation of the IBMMicroscopic foundation of the IBM
e.g. OAI mapping
Sph.
Deformed
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Microscopic foundation of the IBMMicroscopic foundation of the IBM
For deformed nuclei, require manually adda term to reproduce moment of inertia…
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For nearly spherical nuclei:
compare NSM wave functions with NPA wave functions,
overlap > 90% for g.s. with, e.g., the SD-pair truncation
Validity of the NPAValidity of the NPA
Y. Lei et al., Phys. Rev. C 82, 034303 (2010); 84, 044301 (2011);Y. Y. Cheng et al., Phys. Rev. C 94, 024321 (2016).
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For nearly spherical nuclei:
compare NSM wave functions with NPA wave functions,
overlap > 90% for g.s. with, e.g., the SD-pair truncation
For rotational nuclei:
can reproduce rotational band (parameters)
Validity of the NPAValidity of the NPA
Structure coefficients of the SD pairs?
Higher-spin pairs?Exact SDthe pf shell
Elliott’s SU(3)
under the same interaction: NPA predicts much smaller
moment of inertia and B(E2) than NSM does
Difficulty: Selecting good pairs is a long-standing problem;
no one has successfully reproduced rotational bands by
the NPA in a large space with effective interactions.
Y. M. Zhao et al., Phys. Rev. C 62, 014316 (2000).
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Pair-structure coefficientPair-structure coefficient
In early applications, one determines pair-structure coefficients using the
generalized seniority (GS) states of the SM.
the S pair is chosen so that the expectation value of Hamiltonian in the S-
pair condensate is minimized:
D, G,… pairs obtained by diagonalizing the Hamiltonian matrix in the space
spanned by the generalized-seniority-two (i.e., one-broken-pair) states
it does not work for deformed nuclei;
it does not tell us which pairs are important in advance
simple; works well for nearly-spherical nuclei
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Pair-structure coefficientPair-structure coefficient
In early applications, one determines pair-structure coefficients using the
generalized seniority (GS) states of the SM.
Conjugate gradient method (CG):
1. treat pair coefficients as free parameters;
2. minimize g.s. energy by iterative NPA calculations.
heavy calculation;
it does not tell us which pairs are important in advance
numerically optimal solution;
works well for deformed nuclei;
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0 2 4 6 8 10 12 14 16 18 20 22 24
0
50
100
150
0 2 4 6 8 10 12 14 16 18 20 22 24
0
400
800
1200
I
Exact
SDGS
Ex
(M
eV)
(b)
I
B(E
2)
(e2
fm4)
(a)
0 2 4 6 8 10 12 14 16 18 20 22 24
0
50
100
150
0 2 4 6 8 10 12 14 16 18 20 22 24
0
400
800
1200
I
Exact
SDGS
SDGGS
Ex
(M
eV)
(b)
I
B(E
2)
(e2
fm4)
(a)
Elliott’s SU(3) limitElliott’s SU(3) limit
0 2 4 6 8 10 12 14 16 18 20 22 24
0
50
100
150
0 2 4 6 8 10 12 14 16 18 20 22 24
0
400
800
1200
I
Exact
SDGS
SDGGS
SDCG
Ex
(M
eV)
(b)
I
B(E
2)
(e2
fm4)
(a)
0 2 4 6 8 10 12 14 16 18 20 22 24
0
50
100
150
0 2 4 6 8 10 12 14 16 18 20 22 24
0
400
800
1200
I
Exact
SDGS
SDGGS
SDCG
SDS'D'CG
Ex
(M
eV)
(b)
I
B(E
2)
(e2
fm4)
(a)
0 2 4 6 8 10 12 14 16 18 20 22 24
0
50
100
150
0 2 4 6 8 10 12 14 16 18 20 22 24
0
400
800
1200
I
Exact
SDGS
SDGGS
SDCG
SDS'D'CG
SDGCG
Ex
(M
eV)
(b)
I
B(E
2)
(e2
fm4)
(a)
6p6n in the pf shell under
generalized-seniority based method(GS)
conjugate gradient method (CG)
Precisely reproduce the pf-shell SU(3) in the SDG-pair truncation!
G. J. Fu et al., to be published.
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0 2 4 6 8 10 12 14 16 18 20 22 24
0
100
200
300
0 2 4 6 8 10 12 14 16 18 20 22 24
0
1000
2000
3000
4000
I
Exact
SDGS
SDGIGS
SDCG
SDGCG
SDGICG
Ex
(M
eV)
I
(b)
B(E
2)
(e2
fm4)
(a)
Elliott’s SU(3) limitElliott’s SU(3) limit
generalized-seniority based method(GS)
conjugate gradient method (CG)
Precisely reproduce the sdg-shell SU(3) in the SDGI-pair truncation!
6p6n in the sdg shell under
G. J. Fu et al., to be published.
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unitary transformation:
Pairs obtained from the HF basisPairs obtained from the HF basis
1. Solve the HF equation in the shell-model space with effective interactions.
(SHERPA)
NSM s.p. basisHF s.p. basis
2. The HF state of 2N valence protons is written by a pair-condensate state.
the phase of pair is arbitrary
3. HF pair => pairs with good spin in the NSM basis
selecting pairs with large
yJM, it tells us what pairs
are important in advance!
G. J. Fu and C. W. Johnson, arXiv:1909.08785.
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Pairs obtained from the HF basisPairs obtained from the HF basis
e.g., 6p6n in the pf shell under
Solving the HF equation + angular momentum projection = exact solution
HF pair => SDG pairs
SDGI pairs for 6p6n, 10p10n(?), 12p12n(?) in the sdg shell…
SU(3) boson mapping
G. J. Fu et al., to be published.
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0.0 0.5 1.0 1.5
0
1
2
3
4
0.0 0.5 1.0 1.5
0
2
4
6
8
x
Exact
SDGS
SDGGS
SDCG
SDGCG
Ex
(M
eV)
x
(a')
Shape evolution driven by interactionsShape evolution driven by interactions
Iπ = 2+ Iπ = 4+
CG works for rotational
εjα: s.p. energy from kb3g;
VP: pairing interaction;
VQ: SU(3) quadrupole interaction.GS works for spherical
6p6n in the pf shell with
G. J. Fu et al., to be published.
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Shape evolution driven by interactionsShape evolution driven by interactions
Iπ = 2+ Iπ = 4+
6p6n in the pf shell with
εjα: s.p. energy from kb3g;
VP: pairing interaction;
VQ: SU(3) quadrupole interaction
0.0 0.5 1.0 1.5
1
2
3
0.0 0.5 1.0 1.51
3
5
7
Exact
SDGS
SDGGS
SDCG
SDGCG
RI+
2
x
(b')
xG. J. Fu et al., to be published.
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Shape evolution driven by interactionsShape evolution driven by interactions
Iπ = 2+
Iπ = 4+
6p6n in the pf shell with
εjα: s.p. energy from kb3g;
VP: pairing interaction;
VQ: SU(3) quadrupole interaction
0.0 0.5 1.0 1.50
400
800
0.0 0.5 1.0 1.5
0
400
800
1200 Exact
SDGS
SDGGS
SDCG
SDGCG
B(E
2)
(e2
fm4)
x
(c')(c)
x
Iπ = 4+
CG works for rotational
GS works for spherical
G. J. Fu et al., to be published.
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N = 26 isotonesN = 26 isotones
NPA:
pf shell + KB3G interaction
G. J. Fu et al., to be published.
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pf shell + KB3G interaction
From HF to NPAFrom HF to NPAHF: selecting pairs from HF states
G. J. Fu et al., to be published.
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From HF to NPAFrom HF to NPA
1p1/21p3/20f5/20g9/2 shell + JUN45 interaction
oblate prolate
G. J. Fu and C. W. Johnson, arXiv:1909.08785.
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NSM
From HF to NPAFrom HF to NPA
1p1/21p3/20f5/20g9/2 shell + JUN45 interaction
I : two bands are calculated
based on two HF solutions,
HF1 and HF2, respecitively;
II: we mixed the states obtained
from HF1 and HF2.
G. J. Fu and C. W. Johnson, arXiv:1909.08785.
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2s1/21d3/21d5/20g7/20h11/2 (50-82) shell + Bonn potential
From HF to NPAFrom HF to NPA
NSM NSM
G. J. Fu and C. W. Johnson, arXiv:1909.08785.
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We study deformed nuclei using the NPA in the framework of the shellmodel, by selecting good pairs using the CG method and from the HF.
We find the SDGCG pair truncation precisely reproduces the Elliott’s SU(3)of the 6p6n system in the pf shell, and the SDGICG pair truncationreproduces the Elliott’s SU(3) in the sdg shell.
The CG method works very well for the rotational motion, and thetraditional GS method works for nearly spherical nuclei. CG + GSreproduces nuclear shape evolution.
We can select good pairs from the HF. Our preliminary results show thatthe SDGHF pair truncation can well reproduce low-lying rotational bands.
SummarySummary
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• Generally, for given J one obtains 2J+1 different pairs for different M.
• A HF state can have arbitrary orientation with the same physical meaning, but yJM will be changed.
Triaxially and octupole deformed nucleiTriaxially and octupole deformed nuclei
The truth is the 2J+1 pairs are not linearly independent.
orthogonalization: diagonalize the norm matrix of pair
Numerically, axially deformed: 1 pair
triaxially deformed: 2 pairs
octupole deformed: parity-nonconserving pairs
HF pair => pairs with good spin in the NSM basis
yJM are rotation invariant
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Improved pairing in the HF basisImproved pairing in the HF basis
The HF state of 2N valence protons is written by a pair-condensate state.
the phase of pair is arbitrary
Consider pairing in the HF basis: