SH nuclei – structure, limits of stability & high-K ground-states/isomers
1. Equilibrium shapes 2. Fission barriers 3. Q alpha of Z=98-126 (with odd and odd-
odd) nuclei.4. K-isomers or high–K ground states of odd &
odd-odd nuclei - a chance for longer half-lives
5. Predictions for SHE with Z>126
P.Jachimowicz (UZ), W.Brodziński, M.Kowal, J.Skalski (NCBJ)
ARIS 2014, Tokyo, Japan
Mostly results of the Woods-Saxon micro-macro model; some Skyrme HFBCS results.
136 144 152 160 168 176 184 192
98100102104106108110112114116118120122124126 min
20
-0.50-0.45-0.40-0.35-0.30-0.25-0.20-0.15-0.10-0.0500.050.100.150.200.250.30
N
Z
SDO
OBLATE
PROLATE
SPHERICAL
Ground state shapes, even-even
20
:
0.5
3.
2
SDO
Axis ratio
Micro-macro results
In contrast to many Skyrme forces, Woods-Saxon micro-macro model gives lower barriers and mostly oblate ground states for Z>=124,126 (no magic gap for 126 protons).
P. Jachimowicz, M. Kowal, and J. Skalski, PRC 83, 054302 (2011).
SLy4, M. Bender, P-H. Heenen,to be published(inverted colors)
Gogny force, M. Warda
L. Próchniak
Possible alpha-decay hindrance: the 14- SD oblate ground state in parent. The G.S. to G.S. transition inhibited; SDO to SDO has the Q value smaller by 2 MeV.
Fission barriers calculated using micro-macro model (e-e nuclei)
Performance for even-even actinides:
1-st barriers, 18 nuclei
rms : 0.5 MeV
2-nd barriers, 22 nuclei
rms : 0.69 MeV
Even-even SH nuclei: barries decrease for Z>114 The highest barrier for Z=114, N=178
P. Jachimowicz, M. Kowal, and J. Skalski, PRC 85, 084305 (2012).M. Kowal, P. Jachimowicz and A. Sobiczewski, PRC 82, 014303 (2010) .
Heaviest even-even fissioning nuclei:112, 170 0.8 ms (old calc. 71 ms) 112, 172 97 ms (old calc. 4 s)114, 172 130 ms (old calc. 1.5 s) (for Z=114, the local minimum in barrier at N=168)
Old calculation: Smolańczuk, Skalski, Sobiczewski (1995)
-3.0
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3.04.0
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7.08.0
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282112; min:
20
30
162 164 166 168 170 172 174 176 178 180 1820
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8
9
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Z=112
B
f [M
eV]
N
HN FRLDM RMF SkM*
164 166 168 170 172 174 176 178 180 1820
1
2
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4
5
6
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8
9
10
Z=114
Bf [
MeV
]
N
HN FRLDM RMF SkM*
168 170 172 174 176 178 180 182 184 1860
1
2
3
4
5
6
7
8
9
10
Z=120
Bf [
MeV
]
N
HN FRLDM RMF SkM*
HN – Woods-SaxonFRLDM – P. Moller et al.SkM* - A.Staszczak et al.RMF – H.Abusara et al.
FRDLM & RMF also perform well in actinides!
Comparison of various models: some must be wrong.
SHE masses (including odd & odd-odd)
• A fit to exp. masses Z>82, N>126, • number of nuclei: 252• For odd and odd-odd systems there are 3
additional parameters – macroscopic energy shifts (they have no effect on Q alpha).
>>Predictions for SHE:88 Qalpha values, Z=101-118, 7 differ from exp. by more than 0.5 MeV; the largest deviation: 730 keV (blocking).
Slight underestimate for Z=108;Overestimate: Z=109-113
P. Jachimowicz, M. Kowal, and J. Skalski, PRC 89, 024304 (2014)
Statistical parameters of the fit to masses in the model with blocking in separate groups of even-even, odd-even, even-odd and odd-odd heavy nuclei:
The same but for the method without blocking.
Q alpha
204 nuclei in the fit region
blocking q.p.method
mean 326 keV 225 keV error
rms 426 keV 305 keV
88 nuclei Z=101-118
mean 217 keV 196 keVerror
rms 274 keV 260 keV
High-K states: a chance for longer half-lives.
< Candidates for high-K g.s. in odd or odd-odd SHN in the W-S model
In even-even systems one should block high-Kclose-lying orbitals,like:9/2+ and 5/2- protonsbelow Z=108 or 11/2- and 9/2+ neutronsbelow N=162
Z N Omega(n) Omega(p) K113 173 5/2+ 7/2- 6-112 173 15/2- 15/2-111 170 11/2+ 11/2+ 169 5/2+ 9/2- 7- 163 13/2- 3/2- 8+110 163 13/2- 13/2-109 All 11/2+ > 11/2 169 9/2+ „ 10+ 161 „ „ „ 159 „ „ „ 163 13/2- „ 12-108 163 „ 13/2- 157 11/2- 11/2-107 163 13/2- 5/2- 9+ 157 11/2- „ 8+106 163 13/2- 13/2- 157 11/2- 11/2-105 157 11/2- 9/2+ 10- 151 9/2- 9/2+ 9-104 157 11/2- 11/2-105 157 11/2- 7/2- 9+ 151 9/2- 7/2- 8+ 149 7/2+ 7/2- 7-101 157 11/2- 1/2- 6+
protons
neutrons
156 158 160 162 164 166 168 170
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log 10
[T 1/
2 (s)
]
N
GS->GS GS->EX
Z=109
156 158 160 162 164 166 168 170 172 174-0.2
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Q
[MeV
]
N
Z=113 Z=111 Z=109 Unique blocked orbitals may hinder
alpha transitions. The effect of a reduced Q alpha for g.s. -> excited state (left panel) on the life-times (below)according to the formula by Royer.
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40272Mt
G.S. configuration:P:11/2+ [6 1 5]N:13/2- [7 1 6]
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7.0
6.0
5.0
4.0
3.0
2.0
8.0
1.0
8.0
9.0
0
10.0
9.0
7.0
11.012.0
6.0
13.0
14.0
5.0
10.0
15.0
10.011.0
4.0
16.0
12.0
17.0
12.0
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272Mt
Fixing the g.s. configuration rises the barrier by 6 MeV. Even if configuration is not completely conserved, a substantial increase in fission half-life is expected.
Microscopic-macroscopic method• Shape parametrization:
}
1{})({),()(
4444)(
4242)(
2222
80806060404020200
YYY
YYYYRcR
• β20 & β22 on the mesh, minimalization in {β40 β60 β80 β42 β44 }.
Hartree-Fock-BCS with SLy6 force – an „upper limit” for barrier
• 180 neutron & 110 proton levels• Pairing: delta interaction of time-reversed pairs with a
smooth energy cutoff, Vn= 316 MeV fm3 , Vp= 322 MeV fm3
Stability for Z>126W. Brodziński, J. Skalski, Phys. Rev C 88, 044307 (2013)
Macroscopic energy
vs axial elongation
in the beta-gamma plane
200 300
Spherical shell correction with the SLy6 force;
W-S gives a very similar pattern for Z>126
In both W-S and SLy6 models-doubly magic spherical system.
In the W-S model:
Q alpha = 14.3 MeV.From the formula by Royeret al. T alpha = 100 s.
B eff > 700 hbar^2/MeV, along a stright path (axially symmetric) one obtains T fission > 10^7 s.
Next doubly magic nucleus??
β-stable, HFBCS: Qα≈10 MeV, T alpha = 0.1 s, T fission (rough estimate) = 10^{-6} s; more for odd & odd-odd systems
W-S minimum: SD-oblate
Fission barrier: 2 MeV
HFBCS minimum: spherical/SD-Oblate, fission barrier: 4.2 MeV
Micro-macro Hartree-Fock-BCS
N=228 region:
• W-S micro-macro model predicts reasonable barriers for actinides and SH nuclei;
• Q alpha also seem reasonable;
• Large differences in barriers between our model and the FRDLM or Skyrme-type; nobody knows what happens for Z>=120;
• High-K ground states of some odd and odd-odd nuclei, with blocked intruder orbitals, may be the longest-lived SHE;
• Z>126 systems – rather pessimistic predictions: nonaxiality ruins stability; no stability in the W-S model, while SLy6, known to give too high barriers (by up to 2.5 MeV), leads to estimated (roughly) fission half-lives:10^-6 s & alpha half-lives of 0.1 s. This does not promise much stability.
Conclusions