Effects of high order deformation on superheavy high-$K$ isomers

Abstract

Using, for the first time, configuration-constrained potential-energy-surface
calculations with the inclusion of $\beta_6$ deformation, we find remarkable
effects of the high order deformation on the high-$K$ isomers in $^{254}$No,
the focus of recent spectroscopy experiments on superheavy nuclei. For shapes
with multipolarity six, the isomers are more tightly bound and,
microscopically, have enhanced deformed shell gaps at $N=152$ and $Z=100$. The
inclusion of $\beta_6$ deformation significantly improves the description of
the very heavy high-$K$ isomers.

Full text

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RAPID COMMUNICATIONS
PHYSICAL REVIEW C 83, 011303(R) (2011)
Effects of high-order deformation on high-K isomers in superheavy nuclei
H. L. Liu,1F. R. Xu,2P. M. Walker,3and C. A. Bertulani1
1Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, Texas 75429-3011, USA
2School of Physics, Peking University, Beijing 100871, China
3Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom
(Received 4 November 2010; published 24 January 2011)
Using, for the first time, configuration-constrained potential-energy-surface calculations with the inclusion of
β6deformation, we find remarkable effects of the high-order deformation on the high-K isomers in254No, the
focus of recent spectroscopy experiments on superheavy nuclei. For shapes with multipolarity six, the isomers
are more tightly bound and, microscopically, have enhanced deformed shell gaps at N = 152 and Z = 100. The
inclusion of β6deformation significantly improves the description of the very heavy high-K isomers.
DOI: 10.1103/PhysRevC.83.011303PACS number(s): 21.10.−k, 21.60.−n, 23.20.Lv, 27.90.+b
By overcoming the strong Coulomb repulsion between the
large number of protons, shell effects can lead to the so-called
“island of stability” centered on a doubly magic nucleus
beyond208Pb that has yet to be identified. On the way to the
predicted island, new chemical elements up to Z = 118 [1,2]
have been synthesized, while the transfermium nuclei have
been studied in detail through spectroscopy experiments [3].
Of special note in spectroscopy studies are multi-quasiparticle
(multi-qp) high-K (K is the total angular momentum projec-
tion onto the symmetry axis) isomers whose decay to low-K
states is inhibited because of K forbiddenness [4]. They
provide a probe into the underlying single-particle structure
aroundtheFermisurface.Forexample,thesystematicobserva-
tion of Kπ= 8−isomers in A ≈ 250 nuclei demonstrates the
existence of N = 152 and Z = 100 deformed shell gaps [5].
Such information is vital for determining the nuclear potential
that can then be used to predict properties of superheavy
nuclei. Furthermore, superheavy high-K isomers can have
enhanced stability against α decay and spontaneous fission
fromunpairednucleons[6],perhapsservingassteppingstones
toward the “island of stability.”
Among the A ≈ 250 nuclei in which high-K isomers have
been discovered,254No was the focus of recent experiments
because of its relatively high production rate. Two-qp and
four-qp high-K isomers were first established by Herzberg
et al. [7], Tandel et al. [8], and Kondev et al. [9]. Later these
isomers were extensively studied by Heßberger et al. [10] and
Clark et al. [11], with emphasis on the spectrum above the
two-qp isomer. All the experiments agree on the existence
of a four-qp isomer with a half-life in the region of 200 µs,
but the suggested configurations are controversial. Heßberger
etal.[10]andClarketal.[11]deriveddifferentlevelsbridging
the four-qp and two-qp isomers. More work is required, both
experimental and theoretical, to confirm the254No high-spin
level structure.
Theoretical descriptions of superheavy nuclei have made
continuous progress [12] along with experiments. One im-
portant finding is that high-order deformation, especially β6,
is significant in modeling very heavy nuclei [13,14]. The
inclusion of β6 deformation can give extra binding energy
in excess of 1 MeV, resulting in improved reproduction
of experimental masses [13]. The254No moment of inertia
calculated with the addition of β6deformation is 17% larger
than the calculation with only β2and β4deformations [15].
Remarkable β6deformations were predicted in the A ≈ 250
massregion,withthelargestmagnitude(β6≈ −0.05)in254No
[15]. In this work, we investigate the high-order deformation
effects on254No high-K isomers.
Configuration-constrained potential-energy-surface (PES)
calculations [16] have been applied to the three-dimensional
deformation space (β2, β4, β6) to determine the deformations
and excitation energies of multi-qp states. Other frequently
used deformation degrees of freedom such as γ and β3are
excluded as they are calculated to be negligible in254No.
The observation of large hindrance in K-forbidden γ-ray
transitions (which indicates approximately good K quantum
numbers) in254No has confirmed that the nucleus is well
deformed and axially symmetric [11]. Reflection asymmetry
can significantly reduce the outer barrier beyond the second
potential well of a prolate superheavy nucleus, but does not
affect the first well [17]. In addition,254No has no indication
of β8 deformation [15]. Deformations with multipolarity
higher than eight have been demonstrated to be negligible
in calculations [14]. Therefore, it is justified for us to limit the
calculations to the (β2, β4, β6) deformation space.
We employ the axially deformed Woods-Saxon potential
with the set of universal parameters [18] to provide single-
particle levels. To reduce the unphysical fluctuation of the
weakened pairing field (from the blocking effect of unpaired
nucleons)anapproximateparticle-numberprojectionwasused
by means of the Lipkin-Nogami method [19], with pairing
strengths determined by the average gap method [20]. In the
configuration-constrained PES calculation, it is required to
adiabatically block the unpaired nucleon orbits that specify
a given configuration. This was achieved by calculating and
identifying the average Nilsson quantum numbers for every
orbit involved in a configuration [16]. The good quantum
numbers of parity and ? (the individual angular momentum
projection onto the symmetry axis) facilitate the configuration
constraint in (β2, β4, β6) deformation space. The total energy
of a state consists of a macroscopic part that is obtained with
the standard liquid-drop model [21] and a microscopic part
011303-1
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H. L. LIU, F. R. XU, P. M. WALKER, AND C. A. BERTULANI PHYSICAL REVIEW C 83, 011303(R) (2011)
0.240.28
β2
–0.04
0.00
β6
0.240.28
β2
(a) (b)
FIG. 1. Calculated PESs for
Kπ=16+{ν9/2−[734]⊗ν7/2+[613]⊗π7/2−[514]⊗π9/2+[624]}
state (b). At each point (β2,β6), the energy is minimized with respect
to β4. The energy interval between neighboring contours is 200 keV.
254No ground state (a) and
that is calculated by the Strutinsky shell-correction approach,
includingblockingeffects.Theconfiguration-constrainedPES
calculation can properly treat the shape polarization from
unpaired nucleons.
In Fig. 1, we display the calculated PESs for254No ground
state (g.s.) and four-qp high-K state relevant to experiments
(see below). The PESs show that the states have remarkable
β6deformations. The g.s. β6deformation −0.029 is smaller
in magnitude than −0.05 that was calculated by Muntian
et al. [15]. This is because we employ the standard liquid-drop
model with a sharp surface for the macroscopic energy,
whereasMuntianetal.[15]usedtheYukawa-plus-exponential
model with a diffuse surface that is relatively soft against
deformation.Becausethelattertreatmentseemsmorerealistic,
our calculations may slightly underestimate the magnitude
of the β6 deformation and hence its effects. Figure 1 also
shows that the shape of254No is robust against multi-qp
excitations, which verifies that the increase in moment of
inertia of the high-K bands with respect to the g.s. band
0.2 0.30.4
β2
0.50.6
-8
-6
-4
-2
0
2
4
6
8
E (MeV)
Kπ=8−{ν9/2[734]⊗ν7/2[613]}
Kπ=8−{π7/2[514]⊗π9/2[624]}
g.s.
{ν9/2[734]⊗ν7/2[613]⊗
π7/2[514]⊗π9/2[624]}
Kπ=16+
FIG. 2. (Color online)254No potential energy curves calculated
with (solid lines) and without (dashed lines) β6deformation. The
energy for each β2point is minimized with respect to deformations
β4and β6.
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
E (MeV)
g.s.
{π1/2[521]⊗π7/2[514]}
{ν9/2[734]⊗ν7/2[613]}
{ν9/2[734]⊗ν11/2[725]}
{π7/2[514]⊗π9/2[624]}
{ν9/2[734]⊗ν7/2[624]}
{ν9/2[734]⊗ν7/2[613]⊗
π7/2[514]⊗π9/2[624]}
{ν9/2[734]⊗ν3/2[622]⊗
π7/2[514]⊗π9/2[624]}
{ν9/2[734]⊗ν7/2[624]⊗
π7/2[514]⊗π9/2[624]}
Kπ
16+
14+
16+
8-
10+
8-
8-
3+
Kπ
16+
10+
8-
3+
0+
0+
Exp. Cal.
with β6
Cal.
without β6
FIG. 3. Calculations of254No multi-qp states with and without β6
deformation, compared with experimental data [7,11].
is from the reduction of pairing rather than a change of
deformation [11]. The influence of the high-order deformation
on the stability is significant. The g.s. obtains an extra binding
energyof0.8MeVfromβ6deformation.Themulti-qphigh-K
states also have deeper potential wells than those calculated
withoutβ6deformation,asshowninFig.2.Thedepthincrease
for the Kπ= 8−{π7/2−[514] ⊗ π9/2+[624]} state reaches
0.856 MeV. Importantly, our calculations indicate that the β6
deformation has no influence on the barrier peaks (see Fig. 2),
so that the extra binding energy results in a net increase in
fissionbarrierheight.ItisseeninFig.2thatthemulti-qpstates
have wider and higher fission barriers than the g.s., implying
enhancedstabilityagainstfissionfromunpairednucleons.This
is consistent with the observed very small spontaneous fission
branch of ≈10−4for the two isomers in254No [10].
The multi-qp states calculated with and without β6defor-
mation are compared with experimental data in Fig. 3. (Note
that because the excitation energy data for the Kπ= 3+,8−
states from different experiments [7–11] are similar, we adopt
the earliest accurate data [7]; the detailed data from the
most recent experiment [11] are used for the other states.)
The Kπ= 3+state is firmly assigned the proton two-qp
configuration π1/2−[521] ⊗ π7/2−[514] through g-factor
measurement [7,8,11]. The K = 3 coupling is energetically
favored over the K = 4 coupling because of the residual spin-
spin interaction between the quasiparticles [22,23]. According
to the Gallagher-Moszkowski (GM) rule [22,23], the spin-
antiparallelcouplingisenergeticallyfavoredfortwoquasineu-
trons or two quasiprotons, whereas the spin-parallel coupling
is lower in energy for the combination of a quasineutron and
a quasiproton. The splitting energies for the A ≈ 180 nuclei
are found to be in the range of ≈100–400 keV [24]. The
energy is too small to substantially change the calculation
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EFFECTS OF HIGH-ORDER DEFORMATION ON HIGH-K ...
PHYSICAL REVIEW C 83, 011303(R) (2011)
-9.0
-8.5
-8.0
-7.5
-7.0
-6.5
-6.0
-5.5
E (MeV)
-5.5
-5.0
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
5/2[642]
3/2[521]
7/2[633]
7/2[514]
1/2[521]
9/2[624]
Z=100
1/2[631]
5/2[622]
7/2[624]
9/2[734]
7/2[613]
1/2[620]
11/2[725]
3/2[622]
N=152
β6=−0.029 β6=0.000β6=−0.029
β2=0.247, β4=0.011
β6=0.000
β2=0.247, β4=0.011
(a) (b)
FIG. 4.254No neutron (a) and proton (b) single-particle levels
calculated using the Woods-Saxon potential with the universal
parameter set.
of a multi-qp state. Our model in its present version does
not include the residual spin-spin interaction. The calculations
usuallywellreproducetheenergeticallyfavoredcoupling(see,
e.g., Refs. [6,16]).
Our calculation of the π1/2−[521] ⊗ π7/2−[514] config-
uration with β6 deformation gives an excitation energy of
0.965 MeV, in very good agreement with the experimental
data0.988MeV[7].Thelowexcitationenergyimpliesthatthe
π1/2−[521] and π7/2−[514] orbits must be close in energy.
In Fig. 4, we present the single-particle levels calculated with
and without β6deformation. One can see in Fig. 4 that the two
orbits become nearly degenerate because of β6deformation so
that we obtain an improved reproduction of the state with the
inclusion of the high-order deformation. It is worth noting that
β6deformation leads to an enlarged Z = 100 deformed shell
gap,consistentwiththatpredictedinRef.[13].Experiment[5]
has confirmed the existence of the gap together with the
strongerN = 152gap.TheKπ= 3+stateisofspecialinterest
because the π1/2−[521] orbit originates from the spherical
orbit 2f5/2whose position relative to the spin-orbit partner
2f7/2determines whether Z = 114 is a magic number for the
“islandofstability.”Thegoodagreementbetweenexperiments
and our calculations with β6 deformation demonstrates the
importance ofthehigh-order deformationinveryheavynuclei
and the validity of the Woods-Saxon potential in this mass
region.
Unlike the Kπ= 3+state with its configuration unam-
biguously assigned, the observed 266 ms Kπ= 8−isomer
has its configuration controversially assigned in the literature.
The proton two-qp configuration π7/2−[514] ⊗ π9/2+[624]
is suggested for the isomer in Refs. [7,8,10], whereas the
most recent experiment [11] favors a neutron two-qp con-
figuration. There are two possible Kπ= 8−neutron two-qp
configurations, ν9/2−[734] ⊗ ν7/2+[613] and ν9/2−[734] ⊗
ν7/2+[624]. Our calculation of the latter indicates that the
state is too high in energy to be the isomer. The high energy
is because both orbits lie below the large N = 152 shell
gap. Therefore, it requires two neutrons to cross the gap
to form the state. The configuration favors the formation
of an isomer in N = 150 nuclei where the Fermi surface
is between the two orbits. Indeed, low-energy isomers with
this configuration were systematically observed in N = 150
isotones [3]. For the other Kπ= 8−neutron two-qp configu-
ration, ν9/2−[734] ⊗ ν7/2+[613], the energy calculated with
β6deformation is very similar to that of the proton two-qp
configuration π7/2−[514] ⊗ π9/2+[624] (see Fig. 3). Both
the calculated Kπ= 8−states are in better agreement with
experiments than those calculated without β6 deformation.
This is attributed to the β6deformation that enhances the N =
152 and Z = 100 deformed shell gaps, leading to increased
separation of the ν9/2−[734] and ν7/2+[613] orbits and
decreased separation of the π7/2−[514] and π9/2+[624]
orbits. It should be noted that the K = 8 coupling for the
neutron two-qp configuration is not the energetically favored
one of the GM doublet. When considering the residual
spin-spin interaction, the proton two-qp state, instead of the
neutron two-qp state, could be the lowest Kπ= 8−state.
Nevertheless, they remain close to each other because the
GM splitting energy is small. Experimental information such
as the g factor is needed to distinguish between the two
configurations for the Kπ= 8−isomer.
The two low-energy Kπ= 8−configurations can couple
to form a four-qp Kπ= 16+state, analogous to the well-
known Kπ= 16+isomer in178Hf [4]. Indeed, a four-qp
184-µs isomer was observed. However, its configuration
is less clear than those of the two-qp states. Two possi-
ble configurations, Kπ= 16+{ν9/2−[734] ⊗ ν7/2+[624] ⊗
π7/2−[514] ⊗ π9/2+[624]} and Kπ= 14+{ν9/2−[734] ⊗
ν3/2+[622] ⊗ π7/2−[514] ⊗ π9/2+[624]}, were suggested
in Ref. [7] and Refs. [8,9], respectively. The most recent
experiment [11] preferred a spin-parity assignment of Kπ=
16+. Our calculations shown in Fig. 3 indicate that the
configuration suggested in Ref. [7] is much higher than the
four-qp Kπ= 16+configuration involving the ν7/2+[613]
orbit. This is because of the high energy of the ν9/2−[734] ⊗
ν7/2+[624]coupling,asdiscussedpreviously.TheKπ= 14+
configurationwithalow-?orbitν3/2+[622]involvediscalcu-
lated to be also higher than the ν9/2−[734] ⊗ ν7/2+[613] ⊗
π7/2−[514] ⊗ π9/2+[624] configuration. Consequently, the
calculated lowest-lying Kπ= 16+state is likely the 184-µs
isomerbecauseofitslowenergyandhighK value,compatible
with the experimental evidence of a Kπ= 16+spin-parity as-
signment [11]. The excitation energy calculated with β6defor-
mation is 2.722 MeV, which is close to the measured value of
2.928MeV[11].InFig.3,itcanbeseenthattheinclusionofβ6
deformation increases the calculated energy, making it closer
to the experimental value. Furthermore, the neutron compo-
nent of unfavored residual interaction is expected to further
increase the energy.
In addition to all the multi-qp states observed before, a
two-qpKπ= 10+statewasobservedinthemostrecentexper-
iment[11],withtheconfigurationν9/2−[734] ⊗ ν11/2−[725]
suggested. Figure 3 shows that the calculated excitation
energy is 1.479 MeV, much lower than the experimental data
2.013 MeV [11]. However, the Kπ= 10+state has unfavored
spin-spin coupling that would increase the excitation energy.
The energy increment could reach ≈400 keV as our calculated
excitation energy can be taken as the value for the favored
coupling.
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H. L. LIU, F. R. XU, P. M. WALKER, AND C. A. BERTULANIPHYSICAL REVIEW C 83, 011303(R) (2011)
TABLE I. Theoretical deformations and excitation energies of
multi-qp states in254No.
Kπ
Configurationa
β2
β4
β6
Ex(keV)
0+
3+
8−
8−
6−
10+
7−
8+
7+
9−
8−
9−
16+
14+
18−
17+
16+
25−
24−
25+
g.s.
ab
AB
bc
AE
AD
bd
cd
BC
CD
AC
BD
ABbc
AEbc
ADbc
ABCD
ACbc
ABCDbc
ABCDbd
ABCDcd
0.247
0.247
0.241
0.245
0.247
0.244
0.246
0.244
0.242
0.246
0.243
0.238
0.240
0.245
0.242
0.239
0.241
0.238
0.239
0.236
0.011
0.011
0.012
0.009
0.010
0.010
0.010
0.009
0.014
0.012
0.014
0.010
0.010
0.008
0.008
0.013
0.012
0.011
0.013
0.011
−0.029
−0.030
−0.024
−0.028
−0.029
−0.027
−0.028
−0.027
−0.026
−0.028
−0.025
−0.022
−0.024
−0.028
−0.026
−0.023
−0.025
−0.023
−0.023
−0.021
0
965
1357
1378
1427
1479
1481
1658
1774
1881
2032
2237
2722
2803
2845
3158
3407
4522
4631
4774
aNeutron orbits 9/2−[734], 7/2+[613], 7/2+[624], 11/2−[725], and
3/2+[622] are represented by A, B, C, D, and E, respectively.
Proton orbits 1/2−[521], 7/2−[514], 9/2+[624], and 7/2+[633] are
represented by a, b, c, and d, respectively.
AsshowninFig.4,thereexistseveralhigh-?orbitsaround
the254No Fermi surface that can couple to many other high-K
states. Table I summarizes the calculations with the inclusion
of β6deformation. The calculated excitation energy of the six-
qp Kπ= 25−state is 4.522 MeV, comparable to 3.942 MeV,
the excitation energy of the observed 24+g.s. band member
[7].TheKπ= 25−statecouldbeclosetotheyrastline(where
the state has the lowest energy among the states with the same
angular momentum), possibly forming an isomeric state.
In summary, the effects of the high-order deformation, β6,
on the high-K isomers in254No are investigated by applying
configuration-constrained PES calculations in (β2,β4,β6)
deformation space. The isomers gain extra binding energy
from the β6deformation, implying enhanced stability against
fission. The high-order deformation rearranges the single-
particle levels, leading to strengthened deformed shell gaps at
N = 152 and Z = 100, which influences the properties of the
multi-qp states. These effects are found to be significant. All
the observed multi-qp states in254No are better reproduced
by the calculations with β6deformation. This indicates the
importanceofthehigh-orderdeformationincalculatingmulti-
qp states in very heavy nuclei.
We are grateful to T.L. Khoo and F.G. Kondev for
suggesting the present work. This work was supported in
part by the US Department of Energy under Grants No. DE-
FG02-08ER41533 and No. DE-FC02-07ER41457 (UNEDF,
SciDAC-2), and the Research Corporation; the Chinese Major
State Basic Research Development Program under Grant No.
2007CB815000; the National Natural Science Foundation of
China under Grants No. 10735010 and No. 10975006; and
the Science and Technology Facilities Council and Atomic
Weapons Establishment plc (UK).
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