Post on 09-Aug-2020
Shell model calculations of nuclei around 208Pb
E.Teruya１ N.Yoshinaga１ K.Higashiyama2
1Department of Physics, Saitama University 2 Department of Physics, Chiba Institute of Technology
15/09/2016 INPC2016 in Adelaide
1. introduction
2. framework
3. results
4. summary
Outline of talk
・Shell model framework
・Energy levels
・E2, magnetic moments, and quadrupole moments
・ Backgrounds
・ About nuclei around 208Pb
・Truncation scheme
1. Introduction
introduction framework results summary
2
2
8
8
20
20
28
28
50
50
82
82
126
Half-life Range
Unknown
<0.1 s
0.1 - 5 s
5 - 100 s
100 s - 1 h
1 h - 1 y
1 y - 1 Gy
Stable
http://ie.lbl.gov/systematics/chart2000g.pdf
Introduction
◆ Nuclei around 208Pb
Neutron 126 Proton 82 Super-heavy
208Pb
126
82
◆ Systematic shell-model calculations
A～80, 100, 130, and so on
2
2
8
8
20
20
28
28
50
50
82
82
126
Half-life Range
Unknown
<0.1 s
0.1 - 5 s
5 - 100 s
100 s - 1 h
1 h - 1 y
1 y - 1 Gy
Stable
http://ie.lbl.gov/systematics/chart2000g.pdf
Introduction
◆ Nuclei around 208Pb Neutron 126 Proton 82 Super-heavy
208Pb
126
82
◆ Systematic shell-model calculations
A～80, 130, 200
208Ra 209Ra 210Ra 211Ra 212Ra 213Ra 214Ra 215Ra 216Ra 217Ra 218Ra 219Ra
207Fr 208Fr 209Fr 210Fr 211Fr 212Fr 213Fr 214Fr 215Fr 216Fr 217Fr 218Fr
206Rn 207Rn 208Rn 209Rn 210Rn 211Rn 212Rn 213Rn 214Rn 215Rn 216Rn 217Rn
205At 206At 207At 208At 209At 210At 211At 212At 213At 214At 215At 216At
204Po 205Po 206Po 207Po 208Po 209Po 210Po 211Po 212Po 213Po 214Po 215Po
203Bi 204Bi 205Bi 206Bi 207Bi 208Bi 209Bi 210Bi 211Bi 212Bi 213Bi 214Bi
202Pb 203Pb 204Pb 205Pb 206Pb 207Pb 208Pb 209Pb 210Pb 211Pb 212Pb 213Pb
201Tl 202Tl 203Tl 204Tl 205Tl 206Tl 207Tl 208Tl 209Tl 201Tl 201Tl 201Tl
Enlargement
2
2
8
8
20
20
28
28
50
50
82
82
126
Half-life Range
Unknown
<0.1 s
0.1 - 5 s
5 - 100 s
100 s - 1 h
1 h - 1 y
1 y - 1 Gy
Stable
http://ie.lbl.gov/systematics/chart2000g.pdf
Introduction
◆ Nuclei around 208Pb Neutron 126 Proton 82 Super-heavy
208Pb
126
82
◆ Shell model calculations
A～80, 130, 200
208Ra 209Ra 210Ra 211Ra 212Ra 213Ra 214Ra 215Ra 216Ra 217Ra 218Ra 219Ra
207Fr 208Fr 209Fr 210Fr 211Fr 212Fr 213Fr 214Fr 215Fr 216Fr 217Fr 218Fr
206Rn 207Rn 208Rn 209Rn 210Rn 211Rn 212Rn 213Rn 214Rn 215Rn 216Rn 217Rn
205At 206At 207At 208At 209At 210At 211At 212At 213At 214At 215At 216At
204Po 205Po 206Po 207Po 208Po 209Po 210Po 211Po 212Po 213Po 214Po 215Po
203Bi 204Bi 205Bi 206Bi 207Bi 208Bi 209Bi 210Bi 211Bi 212Bi 213Bi 214Bi
202Pb 203Pb 204Pb 205Pb 206Pb 207Pb 208Pb 209Pb 210Pb 211Pb 212Pb 213Pb
201Tl 202Tl 203Tl 204Tl 205Tl 206Tl 207Tl 208Tl 209Tl 201Tl 201Tl 201Tl
EDM
225Ra
EDM
Octupole
deformation
Magnetic
rotation
Low-lying νg9/2πh9/2 band
Ground state 1- (not 0-)
Enlargement
V. Margerin,
et al., PRC93
064309 (2016)
N.Cieplicka,
et al., PRC 86
054322 (2012)
L. Chen,
et al., PRL110
122502 (2013)
Introduction
・Shell model
D. Zwarts and P. W. M. Glaudemans, Z. Phys. A 320, 487 (1984).
E. Caurier, M. Rejmund, and H. Grawe, Phys. Rev. C67, 054310 (2003).
L. Coraggio, et. al., Phys. Rev. C 60, 064306 (1999).
◆ Systematic studies in theory
N=126
Z=82 L. Coraggio, et. al., Phys. Rev. C 58, 3346 (1998).
Single-closed nuclei
J. B. McGrory and T. T. S. Kuo, Nucl. Phys. A 247, 283 (1975).
Z=82 and N=126
Introduction
・Shell model
・Nucleon pair approximation Z. Y. Xu , et. al., Phys. Rev. C 79, 054315 (2009).
◆ Systematic studies in theory
Po isotopes T. R. McGoram, et. al., Nucl. Phys. A 637, 469 (1998).
Pb, Bi, Po, At, Rn, Fr
206Pb, 210Po, 208Bi C. W. Ma and W. W. True, Phys. Rev. C 8, 2313 (1972).
Po and Rn isotopes A. Zemel and J. Dobes, Phys. Rev. C 27, 2311 (1983).
Open-shell nuclei
・Interacting-boson approximation plus quasi-particle
2
2
8
8
20
20
28
28
50
50
82
82
126
Half-life Range
Unknown
<0.1 s
0.1 - 5 s
5 - 100 s
100 s - 1 h
1 h - 1 y
1 y - 1 Gy
Stable
http://ie.lbl.gov/systematics/chart2000g.pdf
82
Introduction
◆ What about A ～200 region
Neutron holes under 126 Proton particles over 82
Super-heavy nuclei
Around 208Pb
A～200
126 208Ra 209Ra 210Ra 211Ra 212Ra 213Ra 214Ra 215Ra 216Ra 217Ra 218Ra 219Ra
207Fr 208Fr 209Fr 210Fr 211Fr 212Fr 213Fr 214Fr 215Fr 216Fr 217Fr 218Fr
206Rn 207Rn 208Rn 209Rn 210Rn 211Rn 212Rn 213Rn 214Rn 215Rn 216Rn 217Rn
205At 206At 207At 208At 209At 210At 211At 212At 213At 214At 215At 216At
204Po 205Po 206Po 207Po 208Po 209Po 210Po 211Po 212Po 213Po 214Po 215Po
203Bi 204Bi 205Bi 206Bi 207Bi 208Bi 209Bi 210Bi 211Bi 212Bi 213Bi 214Bi
202Pb 203Pb 204Pb 205Pb 206Pb 207Pb 208Pb 209Pb 210Pb 211Pb 212Pb 213Pb
201Tl 202Tl 203Tl 204Tl 205Tl 206Tl 207Tl 208Tl 209Tl 201Tl 201Tl 201Tl
N≦126
Enlargement
N≧126
N=126
2
2
8
8
20
20
28
28
50
50
82
82
126
Half-life Range
Unknown
<0.1 s
0.1 - 5 s
5 - 100 s
100 s - 1 h
1 h - 1 y
1 y - 1 Gy
Stable
http://ie.lbl.gov/systematics/chart2000g.pdf
82
Introduction
◆ What about A ～200 region
Neutron holes under 126 Proton particles over 82
Super-heavy nuclei
Around 208Pb
A～200
126 208Ra 209Ra 210Ra 211Ra 212Ra 213Ra 214Ra 215Ra 216Ra 217Ra 218Ra 219Ra
207Fr 208Fr 209Fr 210Fr 211Fr 212Fr 213Fr 214Fr 215Fr 216Fr 217Fr 218Fr
206Rn 207Rn 208Rn 209Rn 210Rn 211Rn 212Rn 213Rn 214Rn 215Rn 216Rn 217Rn
205At 206At 207At 208At 209At 210At 211At 212At 213At 214At 215At 216At
204Po 205Po 206Po 207Po 208Po 209Po 210Po 211Po 212Po 213Po 214Po 215Po
203Bi 204Bi 205Bi 206Bi 207Bi 208Bi 209Bi 210Bi 211Bi 212Bi 213Bi 214Bi
202Pb 203Pb 204Pb 205Pb 206Pb 207Pb 208Pb 209Pb 210Pb 211Pb 212Pb 213Pb
201Tl 202Tl 203Tl 204Tl 205Tl 206Tl 207Tl 208Tl 209Tl 201Tl 201Tl 201Tl
N≦126
Enlargement
E. Teruya, et al., PRC93, 064327 (2016)
N≧126
Now in progress
N=126
2. Framework
introduction framework results summary
Shell model framework
13/ 20 0.065 1.698SPE i N
13/ 20 0.050 1.659SPE i N
9/ 2 7/ 2 13/ 2 3/ 2 5/ 2 1/ 20 , 1 , 0 , 2 , 1 , 2h f i p f p
N
number of valence particles
◆ neutrons and protons in the shell 82 – 126 and 126 –
◆ particle number dependence on orbitals
82 – 126 (6 orbitals):
126 – (7 orbitals) : 9/2 11/2 15/2 5/2 1/2 7/2 3/21 , 0 , 0 , 2 , 3 , 1 , 2g i j d s g d
/2
7/21 0.896 2N
SPE f
15/ 20 0.150 1.423SPE j N N
N
：proton particles
： neutron particles
： neutron holes
(in units of MeV)
5/ 21 f
3/ 22 p
1/ 22 p
7/ 21 f
9/ 20h
0.000
0.570
0.898
2.340
3.415
3.634
2.826
3.119
0.000
proton
126
13/ 20i
82
13/ 20SPE i
13/ 20SPE i
neutron
2.578
0.779
neutron
9/ 21g
11/ 20i
15/ 20 j5/ 22d
1/ 23s
7/ 21g3/ 22d
126
0.000
1.567
2.491
2.032
5/ 21 f
3/22p
1/22p
7/21 f
9/20h
13/20i
126
82
15/ 20SPE j
0.869
5/ 21 f
3/ 22 p
1/ 22 p
7/ 21 f
9/ 20h
0.000
0.570
0.898
2.340
3.415
3.634
2.826
3.119
0.000
126
13/ 20i
82
13/ 20SPE i
13/ 20SPE i
2.578
0.779
9/ 21g
11/ 20i
15/ 20 j5/ 22d
1/ 23s
7/ 21g3/ 22d
0.000
1.567
2.491
2.032
5/ 21 f
3/22p
1/22p
7/21 f
9/20h
13/20i
126
82
15/ 20SPE j
neutron proton
126
neutron
N<126
0.869
5/ 21 f
3/ 22 p
1/ 22 p
7/ 21 f
9/ 20h
0.000
0.570
0.898
2.340
3.415
3.634
2.826
3.119
0.000
126
13/ 20i
82
13/ 20SPE i
13/ 20SPE i
2.578
0.779
9/ 21g
11/ 20i
15/ 20 j5/ 22d
1/ 23s
7/ 21g3/ 22d
0.000
1.567
2.491
2.032
5/ 21 f
3/22p
1/22p
7/21 f
9/20h
13/20i
126
82
7/ 21SPE f
15/ 20SPE j
neutron proton
126
neutron
N>126
ˆ ˆ ˆ ˆH H H H
(2) (2)H Q Q
(N 126) 0.060,
Shell model framework
strengths 𝑮𝟎 𝑮𝟐 𝑮𝟒 𝑮𝟔 𝑮𝟖 𝑮𝟏𝟎
Protons 0.145 0.013 0.400 0.400 -0.60 0.00
Neutrons N<126 0.145 0.013 0.500 0.500 1.10 2.00
Neutrons N>126 0.102 0.008 0.400 0.300 0.00 0.450
(N 126) 0.080
9/2 7/2
†
,
† 0 0 † 2 2 †(L) (L)
0 2
, 4,6,8,10
8 † 8 8
9/2 7/2 9/2 7/2
j jm jm
jm
L
L
h f
H c c
G P P G P P G P P
G P h f P h f
9/2 7/2
8† 8 † †
9/2 7/2M h f MP h f c c
9/2 7/2
80.50,h fG
Shell model framework
Neutron system
Diagonalization
Results
Proton system
Neutron-Proton system
500 500
Energy cut-off
Diagonalization (Lanczos)
Diagonalization
Energy cut-off
1. Diagonalization of the neutron (proton) system [Complete diagonalization] energy sort
2. Energy cut-off (500～levels from them)
3. Diagonalization of the neutron-proton system [Lanczos method]
3. Results
introduction framework results summary
Pb isotope (even)
0 2 4 6 8 10 120
1
2
3
4
5
206Pb
(MeV
)E
expt. (positive)expt. (negative)calc. (positive)calc. (negative)
I0 2 4 6 8 10 12 14 16
0
1
2
3
4
5
204Pb
I
(Me
V)
E
0 2 4 6 8 10 12 14 16
0
1
2
3
4
5
210Pb
I
(MeV
)E
( )
( )
( )
( )
( )( )
( )( )
( )( )
( )
( )
( )
( )
( )
( )
0 2 4 6 8 10 12 14 16
0
1
2
3
4
5
212Pb
I
(MeV
)E
( )( )( )
( )
( )
N≦
126
N>
126
204
82 122Pb 206
82 124Pb
expt. (positive)expt. (negative)calc. (positive)calc. (negative)
210
82 128Pb 212
82 130Pb
Pb isotope (even)
0 2 4 6 8 10 120
1
2
3
4
5
206Pb
(MeV
)E
expt. (positive)expt. (negative)calc. (positive)calc. (negative)
I0 2 4 6 8 10 12 14 16
0
1
2
3
4
5
204Pb
I
(Me
V)
E
0 2 4 6 8 10 12 14 16
0
1
2
3
4
5
210Pb
I
(MeV
)E
( )
( )
( )
( )
( )( )
( )( )
( )( )
( )
( )
( )
( )
( )
( )
0 2 4 6 8 10 12 14 16
0
1
2
3
4
5
212Pb
I
(MeV
)E
( )( )( )
( )
( )
N≦
126
N>
126
204
82 122Pb 206
82 124Pb
expt. (positive)expt. (negative)calc. (positive)calc. (negative)
210
82 128Pb 212
82 130Pb
3- core
excitation
3- 3-
3-
Pb isotope (odd)
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 310
1
2
3
4
5
205Pb
2I
(Me
V)
E
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 310
1
2
3
4
5
203Pb
2I
(Me
V)
E
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
0
1
2
3
4
5
211Pb
2I
(MeV
)E
( )( )
( )
( )( )
( )
( )
( )
( )
( )
( )
( ) ( )
N≦
126
N>
126 Low-lying states
are excellently
reproduced
203
82 121Pb205
82 123Pb
211
82 129Pb
expt. (positive)expt. (negative)calc. (positive)calc. (negative)
At isotope (odd)
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 310
1
2
3
4
5
211At
2I
(MeV
)E
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 310
1
2
3
4
5
207At
2I
(MeV
)E
expt. (positive)expt. (negative)calc. (positive)calc. (negative)
N≦
126
N>
126
207
85 122At 211
85 126At
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
0
1
2
3
4
5
215At
2I
(MeV
)E
( )( )
( )( )( )( )
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
0
1
2
3
4
5
213At
2I
(MeV
)E
( )
( )
( )( )
215
85 130At213
85 128At
At isotope (doubly-odd)
0 2 4 6 8 10 12 14 160
1
2
3
4
5
210At
I
(MeV
)E
0 2 4 6 8 10 12 14 160
1
2
3
4
5
206At
I
(MeV
)E
expt. (positive)expt. (negative)calc. (positive)calc. (negative)
N≦
126
N>
126
0 2 4 6 8 10 12 14 16
0
1
2
3
4
5
212At
I
(MeV
)E
( )( ) ( )
( ) ( )
( ) ( )( )
( )
( )( )
( )( )
( ) ( )( )( )
( ) ( ) ( )( )( )
206
85 121At210
85 125At
212
85 127At
0 2 4 6 8 10 12 14 16
0
1
2
3
4
5
214At
I
(MeV
)E
( )( ) ( ) ( ) ( ) ( )
( )
214
85 129At
E2 transition rates
210Po 𝐞𝐱𝐩𝐭. [W.u.] 𝐜𝐚𝐥𝐜. [W.u.]
2+→0+ 0.56(12) 4.123
4+→2+ 4.46(18) 4.453
6+→4+ 3.05(9) 3.028
8+→6+ 1.12(4) 1.055
10+→8+ 0.0002
212Rn 𝐞𝐱𝐩𝐭. [W.u.] 𝐜𝐚𝐥𝐜. [W.u.]
2+→0+ 6.237
4+→2+ 1.04(4) 0.688
6+→4+ 0.40(5) 0.373
8+→6+ 0.115(6) 0.144
10+→8+ 3.950
12+→10+ 4.4(2) 2.966
Effective charge
211At 𝐞𝐱𝐩𝐭. [W.u.] 𝐜𝐚𝐥𝐜. [W.u.]
13/2－→9/2－ >0.039 4.264
3/2－→5/2－ 12.8(14) 9.800
3/2－→7/2－ 0.40(4) 0.619
15/2－→11/2－ 1.7(3) 2.271
15/2－→13/2－ 0.37(8) 0.598
21/2－→17/2－ 2.66(10) 2.119
0.85e e
1.50e e
Magnetic and Quaqrupole moments
212Rn 𝐞𝐱𝐩𝐭. [μN] 𝐜𝐚𝐥𝐜. [μN]
2+ +1.714
4+ 4.0(2) +3.369
6+ 5.45(5) +5.058
8+ +7.15(2) +6.726
14+ 15.0(4) +13.207
17－ 17.0(8) +16.205
208Bi 𝐞𝐱𝐩𝐭. [μN] 𝐜𝐚𝐥𝐜. [μN]
5+ +4.57(13) +3.819
10－ 2.671(14) +2.379 210At 𝐞𝐱𝐩𝐭. [μN] 𝐜𝐚𝐥𝐜. [μN]
11+ +9.79(3) +8.832
19+ 13.26(13) +12.746
11－ 15.675(17) +13.864
205Pb 𝐞𝐱𝐩𝐭. [eb] 𝐜𝐚𝐥𝐜 .[eb]
5/2－ +0.226(37) +0.206
13/2+ 0.33(5) 0.304
Magnetic moments
Quadrupole moments
209Po 𝐞𝐱𝐩𝐭. [eb] 𝐜𝐚𝐥𝐜. [eb]
13/2－ 0.126(5) -0.130
17/2－ 0.659(7) -0.703
0.85 , 1.50e e e e
0.00, 1.00, 1.91, 2.79s sg g g g
Summary
◆ We have carried out the shell model calculation for nuclei around 208Pb.
◆ Energy levels and electromagnetic properties were calculated. ◆ We have obtained good agreements with the experimental data.
Target nuclei : Even-even, odd-mass, and doubly-odd nuclei for Pb, Bi, Po, At, Rn, and Fr isotopes
“Large-scale shell-model calculations of nuclei around mass 210”
E. Teruya, K. Higashiyama, N. Yoshinaga, Phys Rev C 93, 064327 (2016). N≦
126
◆ We will analyze nuclear structure in future.
Back up
Truncation
Shell model framework
208Rn : 4 neutron-holes and 4 proton-particles system
◆ Neutron system
Jπ=0+ : 113 states Jπ=4+ (M=4) : 560 states
◆ Proton system
Jπ=0+ : 113 states Jπ=4+ (M=4) : 560 states
Number of all positive parity states : 4985
Number of all positive parity states : 4985
◆ Neutron-Proton system (after cut-off)
Jπ=0+ : 21422 states Jπ=6+ : 174171 states
without cut-off Jπ=0+ : 1953785 states c.f.
Shell model framework
208Rn : 4 neutron-holes and 4 proton-particles system
0 200 400 600 800 1000
101
102
103
104
105
106
0 200 400 600 800 1000
0
1
2
3
4
5
6
7
[10+5
]
cut−off
0+
6+
208Rn
cut−off
0+
6+
0 200 400 600 800 10000
1
2
3
4
5
208Rn
cut−off
(MeV
)E
0+
2+
4+ 6
+8
+
10+
12+
/
Number of states in each cut-off for the neutron-proton system
Convergence of Energy
◆ Validity of the energy cut-off
the necessity of the interaction 9/2 7/2h fH
About the necessity of the interaction 9/2 7/2h fH
9/2 7/2 9/2 7/2
8 † 8 8
9/ 2 7/ 2 9/ 2 7/ 2h f h fH G P h f P h f
9/2 7/2
8† 8 † †
9/2 7/2M h f MP h f c c
0 2 4 6 8 10 12 14 160
1
2
3
4
5
210Po
I
(MeV
)E
0 2 4 6 8 10 12 14 160
1
2
3
4
5
210Po
I
(MeV
)E
without with
28
28
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 310
1
2
3
4
5
211At
2I
(MeV
)E
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 310
1
2
3
4
5
211At
2I
(MeV
)E
0 2 4 6 8 10 12 14 160
1
2
3
4
5
212Rn
I
(MeV
)E
0 2 4 6 8 10 12 14 160
1
2
3
4
5
210Rn
I
(MeV
)E
without
without
with
with
114
Neutron-proton interaction
A necessity of the hexadecapole
interaction
The neutron-proton interaction
208Bi : 1 neutron-hole and 1 proton-particle system
13/ 2 9/ 2i h configuration: 2 ,3 , ,11
states
0 2 4 6 8 10 12 14 160
1
2
3
4
5
208Bi
I
(Me
V)
E
□ QQ only ◇ QQ + hexadecapole
2 3 4 5 6 7 8 9 10 11 12
0
0.4
0.8
1.2
1.6
2
expt. SM
I
E(M
eV
)
208Bi12583 SM (multipole)
The neutron-proton interaction
□ QQ only
◇ QQ + hexadecapole
132Sb : 1 neutron-hole and 1 proton-particle system
11/ 2 7/ 2h g configuration
2 ,3 , ,9 states
2 3 4 5 6 7 8 9 10
0
0.4
0.8
1.2
1.6
2
expt. SM
I
E(M
eV
)
132Sb8151 SM (multipole)