SH nuclei – structure, limits of stability &
high-K ground-states/isomers
P.Jachimowicz (UZ),
W.Brodziński, M.Kowal, J.Skalski (NCBJ)
ARIS 2014, Tokyo, Japan
1. Equilibrium shapes
2. Fission barriers
3. Q alpha of Z=98-126 (with odd and oddodd) nuclei.
4. K-isomers or high–K ground states of odd &
odd-odd nuclei - a chance for longer halflives
5. Predictions for SHE with Z>126
Mostly results of the Woods-Saxon micro-macro model; some
Skyrme HFBCS results.
Ground state shapes, even-even
SDO :
 20  0.5
Z
Axis.ratio 
126
124
122
120
118
116
114
112
110
108
106
104
102
100
98
Micro-macro results
3
2
min
20
OBLATE
SDO
SPHERICAL
PROLATE
136
144
152
160
N
168
176
184
192
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0
0.05
0.10
0.15
0.20
0.25
0.30
In contrast to many Skyrme
forces, Woods-Saxon micromacro 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).
L. Próchniak
Gogny force, M. Warda
SLy4,
M. Bender, P-H. Heenen,
to be published
(inverted colors)
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 micromacro model (e-e nuclei)
Even-even SH nuclei: barries decrease for Z>114
The highest barrier for Z=114, N=178
Performance for
even-even actinides:
1-st barriers, 18 nuclei
rms : 0.5 MeV
2-nd barriers, 22 nuclei
rms : 0.69 MeV
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)
0.4
282
112; min:
-4.0
-3.0
30
0.3
8.0
-7.0
-1.0
6.0
7.0
-2.0
-5.0
4.0
5.0
0.2
2.0
9.0
-2.0
3.0
1.0
0.1
0
-2.0
-3.0
0.0
-4.0
-6.0
-1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
20
Comparison of various models: some must be wrong.
10
10
HN
RMF
9
8
8
7
7
Bf [MeV]
Bf [MeV]
9
FRLDM
SkM*
6
5
4
6
5
4
3
3
2
2
1
Z=112
1
0
162 164 166 168 170 172 174 176 178 180 182
0
164
HN
RMF
Z=114
166
168
170
172
174
176
178
FRLDM
SkM*
180
182
N
N
10
HN – Woods-Saxon
FRLDM – P. Moller et al.
SkM* - A.Staszczak et al.
RMF – H.Abusara et al.
9
8
Bf [MeV]
7
6
5
4
FRDLM & RMF also perform well
in actinides!
3
2
1
0
168
HN
RMF
Z=120
170
172
174
176
178
N
180
FRLDM
SkM*
182
184
186
SHE masses (including odd & odd-odd)
P. Jachimowicz, M. Kowal, and J. Skalski, PRC 89, 024304 (2014)
• 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
Statistical parameters of the fit to masses in the
model with blocking in separate groups of eveneven, odd-even, even-odd and odd-odd heavy
nuclei:
Q alpha
204 nuclei in the fit region
blocking
q.p.method
mean 326 keV
error
225 keV
rms
305 keV
426 keV
88 nuclei Z=101-118
The same but for the method without blocking.
mean 217 keV
error
196 keV
rms
260 keV
274 keV
Z
113
112
111
110
109
108
107
106
105
104
103
101
N
173
173
170
169
163
163
All
169
161
159
163
163
157
163
157
163
157
157
151
157
157
151
149
157
Omega(n)
5/2+
15/25/2+
13/213/29/2+
„
„
13/2„
11/213/211/213/211/211/29/211/211/29/27/2+
11/2-
Omega(p)
7/211/2+
9/23/2-
11/2+
„
„
„
„
5/2„
9/2+
9/2+
7/27/27/21/2-
K
615/211/2+
78+
13/2> 11/2
10+
„
„
1213/211/29+
8+
13/211/210911/29+
8+
76+
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-K
close-lying orbitals,
like:
9/2+ and 5/2- protons
below Z=108 or
11/2- and 9/2+ neutrons
below N=162
protons
neutrons
2.2
2.0
Z=113
Z=111
Z=109
1.8
1.6
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.
1.2
1.0
0.8
0.6
0.4
0.2
0.0
158
160
162
164
166
168
170
172
174
N

-0.2
156
log10[T1/2 (s)]
Q [MeV]
1.4
5
4
3
2
1
0
-1
-2
-3
-4
-5
156
GS->GS
GS->EX
Z=109
158
160
162
164
N
166
168
170
40
0.1
0.0
-0.1
-0.2
9.0
-1.0
7.0
2.0
-2.0
Mt 8.0
-2.0
6.0
0
4.0 5.0
-4.0
3.0 1.0
-6.0
-1.0 0 1.0
-5.0
-1.0
-5.0 -4.0
2.0
-3.0
-7.0
8.0
-2.0
7.0 5.0
-6.0
-3.0
10.0
4.0
9.0
13.0 12.0
3.0
11.0
6.0
14.0
16.0
8.07.0
15.0
272
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
20
G.S. configuration:
P:11/2+ [6 1 5]
N:13/2- [7 1 6]
0.2
12.0 10.0
272
Mt
0
6.0 4.0
7.0
7.0
0.0
-0.2
8.0
10.0
9.0
0.1
-0.1
12.0
11.0
5.0

0.2
-3.0
-1.0
-7.0-5.0 4.0
2.0
-4.0 -6.0
1.0
-2.0
6.0
3.0
9.0
8.0
9.0
5.0
12.0 10.0
13.011.0
15.0
14.0
16.0
17.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

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.
Stability for Z>126
W. Brodziński, J. Skalski, Phys. Rev C 88, 044307 (2013)
Microscopic-macroscopic method
• Shape parametrization:
R(, )  c({ }) R0{1   20Y20   40Y40   60Y60  80Y80 
  22Y22(  )   42Y42(  )   44Y44(  ) }
• 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
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
Next doubly magic nucleus??
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 Royer
et al. T alpha = 100 s.
B eff > 700 hbar^2/MeV,
along a stright path (axially
symmetric) one obtains
T fission > 10^7 s.
Micro-macro
Hartree-Fock-BCS
N=228 region:
W-S minimum: SD-oblate
HFBCS minimum: spherical/SDOblate, fission barrier: 4.2 MeV
Fission barrier: 2 MeV
β-stable, HFBCS: Qα≈10 MeV, T alpha = 0.1 s, T fission (rough
estimate) = 10^{-6} s; more for odd & odd-odd systems
Conclusions
• 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.