Prog. Theor. Exp. Phys. 2015, 073D03 (14 pages)
DOI: 10.1093/ptep/ptv099
Nuclear stiffness evolutions against axial and
non-axial quadrupole deformations in even- A
osmium isotopes
Hua-Lei Wang1,∗ , Sha Zhang1 , Min-Liang Liu2 , and Fu-Rong Xu3,4
1
School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
3
State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University,
Beijing 100871, China
4
Center of Theoretical Nuclear Physics, National Laboratory for Heavy Ion Accelerator of Lanzhou,
Lanzhou 730000, China
∗
E-mail: [email protected]
2
Received April 19, 2015; Revised June 7, 2015; Accepted June 11, 2015; Published July 28 , 2015
...............................................................................
Systematic potential energy surface calculations for the ground states of even–even 162–200 Os
isotopes have been performed in (β2 , γ , β4 ) deformation space. The shape instabilities are evaluated by analyzing the potential energy curves with respect to β2 and γ deformation degrees of
freedom. Moreover, based on a simple harmonic approximation, the nuclear stiffnesses against
β2 and γ deformations are quantitatively investigated. The present results are compared with previous calculations and available experiments. In addition, inspired by a recent study [N. Wang
et al., Phys. Lett. B 734, 215 (2014)], we, taking the near-drip-line nucleus 162 Os as an example,
investigate the effects of potential parameter modifications (e.g., the strength of the spin–orbit
potential, λ, and the nuclear surface diffuseness, a) on the deformation energy curve, showing
the possible onset of a relatively low fission barrier (if a similar phenomenon occurs in a heavy
or superheavy nucleus, the survival probability will be strongly affected).
...............................................................................
Subject Index
1.
D11, D12, D13
Introduction
In the body-fixed system, the wave function of a nuclear many-body system is usually not an eigenstate of fundamental symmetry operators, though the fundamental quantum numbers (e.g., angular
momentum, parity, baryon number, etc.) associated with the basic space-time symmetries are conserved in the laboratory system. Since the well-known concept of spontaneous intrinsic symmetry
breaking in atomic nuclei was introduced in the 1950s [1], considerable efforts have been made to
reveal its mechanism and to obtain conclusive evidence for the existence of different nuclear intrinsic
shapes [2].
So far, it has been recognized that the mechanism responsible for the appearance of various nuclear
shapes depends crucially on the delicate balance between the symmetry-violating vibronic Jahn–
Teller interaction and the symmetry-restoring pairing force [3]. In general, an intrinsic nuclear
shape can be described by the parametrization of the nuclear surface or the nucleon density distribution [4]. For instance, multipole expansion with spherical harmonics Yλμ (θ, φ) is often used to
describe the nuclear surface in mean-field calculations [5]. It is more suitable for describing low-spin
© The Author(s) 2015. Published by Oxford University Press on behalf of the Physical Society of Japan.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/),
which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
PTEP 2015, 073D03
H.-L. Wang et al.
and ground-state shapes, as applied in the ε (Nilsson perturbed-spheroid parametrization) [6] and
β parametrizations [7]. Moreover, the shape degrees of freedom with lower-order multipolarity
λ are suggested to be important, and an abundance of experimental phenomena connected with
such degrees of freedom are actually observed in nuclei [2]. As is known, the axially symmetric
spheroid can describe the majority of nuclear shapes, which is confirmed by the observation of rotational band structures and measurements of their properties (e.g., quadrupole moments) [8]. The
triaxial γ deformation manifests itself by a wobbling motion, chiral doublets, and it may also play
an important role in signature splitting (or inversion) [9–13]. The effects of the higher-order multipole deformations (e.g., β3 , β32 , β4 , even up to β6 ) have also been investigated experimentally
and theoretically [2,3,14–17]. Nevertheless, there is almost no conclusive experimental evidence on
high-multipolarity deformations with λ > 6 to date.
Nowadays, the progress of radioactive beam facilities has provided us with marvelous findings in
nuclear structure, such as neutron halos and neutron skins [18–20], pygmy resonances [21], changes
of magic numbers [22], etc. Also, much new information on the shapes and structures of nuclei far
from stability is being revealed. One expects to know where and why exotic shapes appear and how
they change along the isotopic and isotonic chains. In the osmium isotopes, 42 members from 161 Os85
to 202 Os126 (between the N = 82 and 126 closed shell) have so far been discovered experimentally [23,24]; these include 7 stable, 24 neutron-deficient, and 11 neutron-rich isotopes. According
to the HFB-14 model [25], in this isotopic chain the last odd–even and even–even neutron-rich isotopes are theoretically predicted to be 257 Os and 260 Os, respectively. On the proton-rich side, it is
pointed out that three more particle stable osmium isotopes are predicted 158–160 Os , and in addition seven more isotopes could possibly still have half-lives longer than 10−9 s [26]. As mentioned
above, with the development of the radioactive beam facility, heavy-ion accelerator, and highly effective detector systems, there is increasing interest in the structure evolution properties of these nuclei
far from stability.
However, all the theoretical results often need to be confronted with existing experimental data.
It has been noticed that some basic experimental quantities, such as ground-state binding energies,
half-lives, different excited states, and so on, have to date been measured and studied in known
161–202 Os nuclei, which do not depend on nuclear models. In particular, the systematic observations [23,27–29] of the relatively low-lying quasi-β and -γ bands or states over a large range
present an excellent opportunity for comprehensive experimental and theoretical investigations on
the evolution of ground-state axial β2 and non-axial γ quadrupole deformation degrees of freedom in the osmium isotopic chain. It is also found that various theoretical approaches have been
applied in the description of the ground-state nuclear properties [7,30,31]. For instance, Möller
et al. [7,31] and Aboussir et al. [32] have, respectively, calculated the global systematics of groundstate deformations by using a macroscopic–microscopic model [the folded Yukawa (FY) singleparticle potential and the finite-range droplet model (FRDM)] and the extended Thomas–Fermi
plus Strutinsky integral (ETFSI) method based on microscopic Skyrme-type forces. Nevertheless,
as is known, since the nuclear wave functions are a superposition of states corresponding to different shapes for soft nuclei, the potential energy surface (PES) may be very flat over a large
deformation domain. The equilibrium deformations identified from the minimum of such PES is
strongly affected both by nucleonic pairing and the deformed mean field, while the shape stiffness
is generally model independent. In our previous work [33–35], similar stiffness evolution both in
ground states and rotational states has been qualitatively studied in some quadrupole and octupole
soft nuclei.
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With the above facts in mind, we have performed a systematic investigation on stiffness evolution
in even–even 162–200 Os and quantitatively obtained the stiffness parameters according to a simple
harmonic approximation. Such investigation is so far scarce and somewhat necessary due to the large
uncertainty in equilibrium deformations, in particular, for soft nuclei. Part of the aim of this work is
to test the parameter reliability of the single-particle potential and the predictive power of the present
model when extrapolating towards the drip-line nuclei.
2.
Theoretical description
The present PES method applied here is based on the macroscopic–microscopic (MM) model, which
can usually give relatively high accuracy on nuclear properties as compared to the microscopic
models employing effective nucleon–nucleon interactions [36]. It has been widely used to give
the right deformation and energy of a many-body nucleus system [37] in the drip-line and superheavy mass regions as well as the medium-heavy mass region [17,38,39]. In the MM model, the
total potential energy E total Z , N , β̂ of a nucleus with deformation β̂ is the sum of a macroscopic
bulk-energy term E mac Z , N , β̂ , being a smooth function of Z , N , and deformation, and a micro
scopic term δ E mic Z , N , β̂ representing the quantum correction based on some phenomenological
single-particle potential [37,40].
Several phenomenological liquid-drop (LD) models with slightly different properties are able to
be used for calculating the smoothly varying part, in which the dominating terms are mainly associated with the volume energy, the surface energy, and the Coulomb energy. In present work, the
macroscopic energy is obtained from the standard LD model with the parameters used by Myers and
Swiatecki [41]. Since our attention is just on the PES, the nuclear potential energy relative to that of
a spherical LD is adopted in the calculations, which is [5,41]
E LD Z , N , β̂ = Bs (β̂) − 1 + 2χ Bc (β̂) − 1 E s(0) ,
(1)
where the relative surface and Coulomb energies Bs and Bc are only functions of nuclear shape. The
(0)
spherical surface energy E s and the fissility parameter χ are dependent on Z and N . Though such a
sharp-surface LD model does not consider the surface diffuseness and the finite range of the nuclear
interaction, it provides a rather good description of nuclear ground-state properties and collective
excitations.
The microscopic correction part δ E mic Z , N , β̂ consists of the shell correction δ E shell Z , N , β̂
and the pairing correction δ E pair Z , N , β̂ . These two terms can be evaluated from a set of
calculated single-particle levels by using the well-known Strutinsky method [42–45] and the Lipkin–
Nogami (LN) approach [43,45]. The two constants, namely the polynomial order p = 6 and the
smoothing range γ = 1.20ω0 ω0 = 41/A1/3 MeV , are used in the Strutinsky shell-correction
calculations. For the pairing correlations, the particle-number projection is approximated by the
LN technique [43,45] and thus the spurious pairing phase transition encountered in the usual
Bardeen–Cooper–Schrieffer (BCS) calculation can be avoided. The monopole pairing strength G
is determined by the average gap method [46,47]. The LN equations are solved in a sufficiently large
space of single-particle states (e.g. about 80 single-particle levels are included in the present pairing
windows for both protons and neutrons). The LN pairing energy can be given by [7,37]
E LN =
k
2vk 2 ek −
2
N
−G
vk 4 + G − 4λ2
u k 2 vk 2 ,
G
2
k
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(2)
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where vk 2 , ek , , and λ2 represent the occupation probabilities, single-particle energies, pairing gap,
and number-fluctuation constant, respectively.
The single particle energies, as needed above, and the single particle wave functions are calculated
by solving the Schrödinger equation of the stationary states for a deformed Woods–Saxon (WS)
potential including a central field, a spin-orbit interaction, and the Coulomb potential for the protons.
The deformed WS potential with the set of universal parameters [5,48] is generated numerically at
each (β2 , γ , β4 ) deformation lattice. The representative matrix of the one-body Hamiltonian is built
by means of the axially deformed harmonic oscillator basis in the cylindrical coordinate system, and
then diagonalized.
The pairing-deformation self-consistent PES is finally obtained by interpolating between the lattice points in the multi-dimensional deformation space. The ground-state equilibrium deformations
of a nucleus are related to the absolute minimum of the PES. Note that in present calculations
the symmetry breaking achieved through the minimization of the total energy over the shape variables is different from the spontaneous symmetry-breaking mechanism, but they show equivalent
results [36].
3.
Calculations and discussions
The nuclear shape plays an important role in determining the structure properties, which are experimentally estimated from the observed level energies (e.g., the use of the empirical Grodzins’ rule [49]
or from interpretation of B(E2) values from 2+ state lifetime measurements [50]) and theoretically
from equilibrium deformation predictions (e.g., the Strutinsky-type calculations with the modified
oscillator (Nilsson) potential, the Woods–Saxon potential, etc., or the microscopic self-consistent
mean-field calculations, such as the Skyrme Hartree–Fock and Gonny Hartree–Fock calculations).
In the present work, the PES calculations in the (β2 , γ , β4 ) deformation spaces have been performed
for even–even 162–200 Os which have at least been identified experimentally. In the actual calculations
the Cartesian quadrupole coordinates X = β2 cos(γ + 30◦ ) and Y = β2 sin(γ + 30◦ ) were used,
where the parameter β2 specifies the magnitude of the quadrupole deformation, while γ specifies
the asymmetry of the shape.
As we know, all the theoretical results need to be confronted with experiments or other accepted
theories. Therefore, the calculated ground-state β2 , β4 deformations obtained from the PES minima
are given in Table 1 for even–even 162–200 Os isotones ranging from N = 86 to 124, together with
the results based on the folded Yukawa (FY) single-particle potential and the finite-range droplet
model (FRDM), and the partial experimental values obtained from reduced transition probabilities
B(E2) [50] for comparison. As can be seen, these three theoretical predictions with slight differences
have similar trends, which are basically in agreement with the experimentally measured results. For
instance, the β4 deformations between theories show, at least, a consistent sign for most of the nuclei.
The β2 deformations increase as the neutron number N moves away from the closed shell and, as
expected, it reaches a maximum near N = 104 180 Os situated at the neutron midshell, half way
between the N = 82 and N = 126 major shell closures. Comparatively speaking, our calculations
systematically underestimate the β2 values. Indeed, such shape inconsistency between theory and
experiment has previously been noticed by Dudek et al. [51] and a modified relationship is suggested
between the calculated potential parameters and the nucleonic distributions (labelled by ρ), e.g., for
ρ
protons in the WS case, β2 1.10β2 − 0.03(β2 )3 . It should be noted that our calculated |γ | value,
∼3◦ , for 194 Os indicates that this nucleus possesses a prolate shape (β2 > 0), without any doubt.
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Table 1. The ground-state equilibrium deformation parameters β2 and β4 obtained from the calculated PES
minima for even–even 162–200 Os, compared with the FY+FRDM [7] and the ETFSI [32] calculations and partial
experimental values (EXP) obtained from reduced transition probabilities B(E2) [50].
FY + FRDM
PESa
Nuclei
162
76 Os86
164
76 Os88
166
76 Os90
168
76 Os92
170
76 Os94
172
76 Os96
174
76 Os98
176
76 Os100
178
76 Os102
180
76 Os104
182
76 Os106
184
76 Os108
186
76 Os110
188
76 Os112
190
76 Os114
192
76 Os116
194
76 Os118
196
76 Os120
198
76 Os122
200
76 Os124
β2
EXPc
ETFSI
β4
β2
β4
β2
β4
β2
0.098
0.003
0.045
−0.008
0.13
0.01
—
0.118
0.002
0.107
−0.004
0.15
0.00
—
0.139
0.002
0.134
−0.002
0.17
−0.01
—
0.154
0.000
0.162
−0.006
0.19
0.00
—
0.173
−0.001
0.171
−0.014
0.20
−0.01
—
0.191
−0.002
0.190
−0.011
0.21
−0.02
0.225
0.215
0.000
0.226
−0.006
0.23
−0.03
0.266
0.223
−0.006
0.246
−0.011
0.29
0.01
0.221
−0.019
0.247
−0.027
0.29
−0.01
—
0.217 b
−0.031
0.238
−0.045
0.25
−0.04
0.226
0.212
−0.041
0.239
−0.062
0.25
−0.04
0.234
0.205
−0.049
0.229
−0.071
0.23
−0.05
0.213
0.194
−0.055
0.220
−0.082
0.20
−0.08
0.200
0.179
−0.059
0.192
−0.086
0.20
−0.07
0.186
0.164
−0.058
0.164
−0.080
0.19
−0.06
0.178
0.146
−0.055
0.155
−0.081
0.17
−0.08
0.167
0.127
−0.050
0.145
−0.082
−0.16
−0.01
—
0.112
−0.038
−0.156
−0.028
−0.12
−0.01
—
0.097
−0.027
−0.096
−0.028
−0.08
−0.01
—
0.039
−0.009
−0.061
−0.037
0.00
0.01
—
—
The calculated |γ | values of 162–194 Os are less than 6◦ . Nevertheless, 196,198,200 Os nuclei, respectively, have
the |γ | values of 35◦ , 44◦ , and 48◦ , indicating triaxially oblate shapes.
b
The bold italic denotes that this value among these three theoretical ones is relatively close to experimental
data.
c
The uncertainties are less than 0.008 except for 174,180,182 Os between 0.011 and 0.025; see Ref. [50] for details.
a
This result is consistent with the FY + FRDM calculation but in conflict with the ETFSI calculation
with negative β2 , as shown in Table 1. In addition, our calculations show that 196,198,200 Os nuclei
with the large |γ | values of 35◦ , 44◦ , and 48◦ , respectively, are in triaxially oblate shapes, which are
basically in agreement with other calculations.
Relative to the equilibrium deformations strongly affected by the mean-field and pairing potential
parameters, especially in the soft nuclei, the deformation energy curves along different deformation
degrees of freedom are generally model independent and may relatively describe the nuclear-shape
properties better. To more reasonably display the shape evolutions in the β2 and γ directions, we
show such deformation energy curves in Fig. 1. As pointed out above, the actual calculations are
performed in the (X, Y )—equivalent to (β2 , γ )—lattice with hexadecapole deformation β4 variation.
Thus the β2 value is always positive and the triaxiality parameter covers the range −120◦ γ 60◦ .
In principle, one of the three sectors [−120◦ , −60◦ ], [−60◦ , 0◦ ], and [0◦ , 60◦ ] is enough to describe
the ground-state nuclear shape, since such three sectors represent the same triaxial shapes though
they represent rotation about the long, medium, and short axes, respectively (cranking calculation is
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β
γ
Fig. 1. Deformation energy curves against β2 (left) and γ (right) for even–even 162–200 Os nuclei, calculated in
(β2 , γ , β4 ) deformation space. At each β2 (γ ) point, the energy has been minimized with respect to the γ (β2 )
and β4 deformations.
beyond the scope of this work). For convenience, the γ range is shown from −60◦ to 60◦ , as show
in Fig. 1. From this figure, one can qualitatively evaluate the minimum stiffnesses and minimum
depths, as well as the equilibrium deformations from these deformation energy curves. It can be
seen that most of these Os isotopes are soft both in β2 and γ directions, which is in agreement with
the experimental observations of the quasi-β and -γ vibration states, as discussed below.
It is difficult to directly measure the triaxial parameter γ , and even the quadrupole deformation β2
cannot be determined from the experimental B(E2) value when the axial symmetry breaks. Some
phenomenological or empirical laws used to evaluate the nuclear properties are, therefore, important.
For instance, the ratio of the excitation energy of first and second excited states, E 4+ /E 2+ , provides
1
1
a test of the axial assumption; there is a very general empirical relationship between E 2+ and β2
1
E 2+ ≈ 1225/A7/3 β22 MeV that essentially all even–even nuclei follow [49,52]; deviations of the
nuclear shape from axial symmetry can sensitively affect the second-lowest 2+
2 states (generally, the
+
quasi-γ bandheads) of even–even nuclei; the second lowest 02 states are related to the β softness,
and so on.
+
+ +
As seen in Fig. 2, the excited energies of the 2+
1 , 41 and 02 , 22 states identified to date are shown
for Os isotopes whose spectroscopic information now extends down to 162 Os and up to 198 Os. From a
simple perspective on nuclear structure, it can be considered that nuclei in the middle of the spherical
shell closures have maximum collectivity where interactions between valence nucleons are maximized, and thus the excitation energies of the first 2+ states may have local minima. To reveal
different collective properties, two phenomenological ratios, R4/2 ≡ E 4+ /E 2+ and E S /E 2+
1
1
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1
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H.-L. Wang et al.
(a)
(b)
(c)
+
+ +
Fig. 2. Energies of the first excited 2+
1 , 41 and the second 02 , 22 states (a), the ratio R4/2 (b), and the quantity
E S /E 2+ (c) in even–even Os isotopes as a function of the neutron (mass) number. See text for more details.
The available experimental data are taken from Refs. [23,27–29].
+ E S = E 2+
2 − E 41 , are also shown in Fig. 2(b) and (c), respectively. As is well known, the
energy ratio R4/2 is 3.33 for an axially symmetric rotor, 2.5 for a γ -soft limit, and 2.0 for a harmonic
vibrator. Furthermore, it is pointed out that 3.0 is the shape/phase transition point to quadrupole
deformed nuclei [53,54] and 1.82 is the separatrix between single-particle and collective characteristics, namely, the Mallmann critical point [55]. One can see that all the nuclei with R4/2 > 1.82 show
collective characteristics. Six nuclei, 178–188 Os, are above the shape/phase transition point and others
mainly lie between 2.0 and 3.0, agreeing with the facts of soft nuclei. The quantity E S /E 2+
1 can be,
empirically, a global signature of the structural evolution involving axial asymmetry [56]. Due to the
+
+
completely degenerate 2+
2 and 41 states in the extreme γ -unstable limit, the value of E S /E 21 is
close to zero [57]. For a rigid-triaxial rotor (RTR) with 25◦ γ 30◦ [58], the 2+
2 state goes under
+
◦
the 41 level and reaches the bottom at the extreme of triaxiality with γ = 30 E S /E 2+
1 = −0.67 .
+
The negative values of E S /E 21 between these two extremes 0 and −0.67 indicate likely γ -soft
potentials with shallow minima at the average γ value close to 30◦ . However, the positive value of
+
E S /E 2+
1 indicates that the nucleus possesses an axially symmetric shape, because the 22 state lies
+
+
at a high excitation energy relative to the 2+
1 and 41 states. Figure 2(c) shows the available E S /E 21
ratios are all larger than zero except for that of 192 Os (∼−0.44). The smallest value (∼−0.44) in
192 Os is slightly smaller than the empirical value of E /E 2+ ≈ 0.5, which is characteristic of
S
1
the critical-point nuclei in terms
of
maximum
γ
softness
between
prolate and oblate shapes. Accord
ing to the formula R2/2 = 1 + 1 − 89 sin2 (3γ ) / 1 − 1 − 89 sin2 (3γ ) , R2/2 ≡E 2+ /E 2+ , as
2
1
described by the RTR model [59], the γ parameter is approximately estimated to be about 25◦ , which
cannot be reproduced by present calculations. In addition, one can also see that the largest E S /E 2+
1
value,
∼0.05,
almost
locating
the
γ
-unstable
limit.
value appears at 184 Os and 190 Os with E S /E 2+
1
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β
(a)
γ
(b)
Fig. 3. Calculated stiffness coefficients Cβ (a) and Cγ (b) towards β2 and γ deformation degrees of freedom
for even–even 162–200 Os nuclei.
Figures 1 and 2 can qualitatively describe the evolutions of nuclear stiffnesses along the β2 and
γ deformation directions by using the deformation energy curves and available data. To present a
quantitative description, based on a simple harmonic approximation, we calculate the corresponding
stiffness parameter Cβ and Cγ for these 20 even–even Os isotopes ranging from 162 Os to 200 Os, as
shown in Fig. 3. Note that the stiffness constants Cβ are determined numerically from the deformation
energy curves with respect to β2 , which is defined from the equation [60]
2
(3)
E = E min + 12 Cβ β2 − β2min ,
where E min denotes the the minimum energy and the constant Cβ can be extracted from the energies
at β2 = β2min and β2 = β2min ± β2 . A step size β2 ∼
= 0.05 is used here. Such a step size yields
results in the harmonic approximation that are close to those calculated with the Wentzler–Kramers–
Brillouin (WKB) approximation, in which anharmonicities in the potential energy are taken into
account [40]. Similarly, the stiffness constants Cγ are obtained from
E = 12 Cγ γ 2 .
(4)
The corresponding step size γ ∼
= 10◦ is adopted in the Cγ calculations (note that a slightly different
step size gives a similar result). Although the calculated stiffness coefficients exhibit irregular oscillating behaviors, several facts are consistent with the data. For instance, the variation trend of E 0+
2
[see Fig. 2(a)] agrees well with that of Cβ , the largest (smallest) Cβ and E 0+ values appearing near
2
188 Os 176 Os at the same time. The available E /E 2+ ratios also have a consistent trend with C .
s
γ
1
However, it is time to note that the stiffness constant Cγ < 0 corresponds to a permanent γ deformation, as discussed in Ref. [3]. Our calculations show 196,198,200 Os nuclei may have permanent triaxial
γ deformations, which await experimental confirmation.
In order to test the validity of the WS potential (mean-field) parameters to some extent, as shown
in Fig. 4, we calculate Fermi energy levels for proton and neutron with universal parameter set. One
sees that the trend of the calculated Fermi energy levels is expected to be consistent with that of
the two-proton and -neutron separation energies and that of the lifetime data. The stable isotopes
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(a)
(b)
(c)
Fig. 4. (a) Calculated neutron and proton Fermi energy levels for even–even 162–200 Os nuclei with the calculated equilibrium shapes as input quantities. (b) Available two-proton and neutron separation energies for
known Os isotopes. (c) Available half-lives of Os isotopes [23], indicating the nuclear instabilities. Note that
the half-lives of 184,186 Os are, respectively, greater than 5.6 × 1013 and 2.0 × 1015 years.
actually appear at the position where protons and neutrons simultaneously have low Fermi levels and
two-nucleon separation energies. In three Os isotopes with A < 168, the proton Fermi energy levels
have positive energies, indicating such protons are possibly quasibound or unbound. For such weakly
bound nuclei, the particle continuum usually becomes important and the standard way of extracting
the shell correction may break down [61]. However, the positive-energy spectrum was approximated
by quasibound states in this investigation. That is, even the pairing window includes positive energy
states, particles do not scatter into the continuum by the pairing force. The single-particle picture
does not give the true nuclear ground or excited states and it only serves as the set of basis functions
for the shell and pairing calculations. Such a procedure should not be considered as satisfactory and
the Wigner–Kirkwood expansion and Green function method (beyond the scope of our work) are
suggested to deal with the continuum states [61,62].
Besides the mean-field, the LN pairing approach has been tested crudely. Figure 5 shows the proton
and neutron pairing gaps calculated in terms of such an approach for even–even 162–200 Os nuclei,
compared with the FY + FRDM calculations [47] and the experimental values extracted from experimental masses [63] by use of fourth-order finite-difference expressions [64]. In the present method,
a pairing gap and number-fluctuation constant λ2 are obtained as solutions of the pairing equations. The LN values representing the sum + λ2 can be crudely compared with the experimental
odd–even mass differences. It can be seen that the LN trend calculated here is consistent with that
of the FY + FRDM results, in particular for protons. However, these two theories underestimate the
experimental LN of both protons and neutrons in the lighter nuclei. Relative to the FY+FRDM calculation, it seems that our calculated LN trend of neutrons is closer to that of the experimental data,
which decreases with N approaching the 126 magic number. As shown in the insets of Fig. 5, it is
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(a)
(b)
Fig. 5. Calculated LN pairing gaps of proton and neutron for even–even 162–200 Os nuclei, compared with the
FY+FRDM calculations [47] and experiments (see text for more details).
found that the overall trends of proton and λ2 slightly fluctuate but in opposite directions, which
results in a relatively smooth constant of LN . The individual neutron contribution decreases considerably with increasing N , and such a decrease is slightly compensated for by a rather slow increase
in neutron λ2 . More appropriately, as discussed in Ref. [47], the pairing gap should be determined
directly from odd–even mass differences based on theoretical masses, where the nonsmooth contributions given by spherical (even deformed) gaps and shape transitions may cancel out to some extent.
It turns out that, even so, the corresponding theoretical mass differences are still inconsistent with the
experimental ones [64] (in principle, they should be identical). The deformed mean-field and oddnucleon effects are pointed out to be responsible for this discrepancy [65,66]. Actually, when such
effects are taken into account, pairing gaps can be changed due to changing pairing strength [64].
In the present work focusing on the stiffness evolutions, we have not performed the pairing strength
adjustment since such an operation affects the ground-state deformation very slightly [37].
It was recently revealed that the isospin dependence of the spin–orbit potential and the nuclear
surface diffuseness is important for accurate descriptions of the ground state properties of nuclei, in
particular near the drip lines [67,68]. Also, the angular dependence of surface diffuseness is discussed
by Adamian et al. [69]. It is thus of interest to examine whether and how and to what extent the present
results will be affected by the model parameters like the strength of the spin–orbit potential, λ, and the
nuclear surface diffuseness, a. Taking the near-drip-line nucleus 162 Os as an example, we investigate
the effects of a and λ parameter modifications on the total potential energy, E LD + δ E shell + δ E pair ,
the shell plus pairing corrections, δ E shell + δ E pair , and shell correction, δ E shell . Note that according
to isospin-dependent function relationships of a and λ parameters discussed in Ref. [67], the possible
variation ranges (δλ, δa) in 162 Os can be crudely estimated to be (1, 0.2) and (−1, 0.0) for proton
and neutron, respectively. Based on such estimation, the initial universal parameters (λ, a), namely
(36, 0.70) for proton and (35, 0.70) for neutron, are changed to be (37, 0.72) for proton and (34, 0.70)
for neutron. All other potential parameters are identical with those of the universal values [5,48]. As
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(a)
(b)
Fig. 6. Deformation energy curves of E LD + E shell + δ E pair , δ E shell + δ E pair , and δ E shell against β2 (top) and
γ (bottom) for even–even 162 Os nuclei, calculated in (β2 , γ , β4 ) deformation space. At each β2 (γ ) point, the
energy has been minimized with respect to the γ (β2 ) and β4 deformations. The solid and dashed lines denote
the calculated results by using the initial and modified universal parameters, respectively. See text for details.
shown in Fig. 6, one can see that the equilibrium deformation is mainly determined by the shell
correction. The pairing correction can affect the deformation energy strongly but the equilibrium
deformation weakly. It can also be found that modification of the universal potential parameters
hardly affects the equilibrium deformations of this nucleus, though it produces non-zero γ minima
and can very slightly decrease the nuclear stiffness related to the curvature at the minimum of the
energy curve. Certainly, the depth of the minimum may increase to some extent, which results in
the improvement of the binding energy (mass), agreeing with the discussion in Ref. [67]. It is worth
mentioning that, as seen in this figure, such slight parameter modifications may have some influence
on the barrier (the energy difference between the minimum and the maximum in the deformation
energy curve). Although the barrier is not important in light nuclei, it will strongly affect the fission
probability for the heavier actinides and superheavy nuclei [70–72]. However, detailed parameter
fitting is beyond the scope of this work. Such study is certainly desired since the Hamiltonian depends
on the parameters. Also, in recent years special attention has been paid to the predictive power and
theoretical uncertainties of the Hamiltonian (mathematical modelling) by Dudek et al. [73,74] in
nuclear physics.
4.
Summary
In summary, pairing-deformation self-consistent PES calculations in the (β2 , γ , β4 ) deformation
space have been carried out for the ground states of 20 even–even Os isotones ranging from N = 86
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to 124, paying attention to the stiffness evolutions in the β2 and γ directions. The calculated equilibrium deformations are compared with previous theoretical results and available experimental data.
The somewhat large uncertainty of the calculated equilibrium deformations in soft nuclei may, to a
large extent, be attributed to the difference of model parameters. The relatively model-independent
deformation energy curves in the β2 and γ directions are analyzed in detail, and are helpful in
understanding the structural evolutions of these nuclei. We also systematically present the stiffness
parameters Cβ and Cγ , which are basically in agreement with data. In addition, the WS potential
parameters and the LN pairing method are crudely tested and discussed, indicating special attention
should be paid when extrapolating to drip-line nuclei. It is also found that the isospin dependence
of the model parameters, like the spin–orbit potential and the nuclear surface diffuseness, should be
considered, especially when those nuclei far from stability and/or the barriers are focused on. Based
on the present predictions, the further identification of low-lying β- and γ -vibration states would
be of interest both in neutron-deficient and neutron-rich directions, especially the γ correlations in
196–200 Os nuclei. New experimental devices and techniques, such as radioactive ion beam accelerators and γ -ray tracking arrays, are certainly desired due to the low production cross-sections when
approaching the drip-line nuclear regions.
Acknowledgements
This work is supported by the Natural Science Foundation of China (Grant Nos. 10805040 and 11175217), the
Foundation and Advanced Technology Research Program of Henan Province (Grant No. 132300410125), and
the S & T Research Key Program of Henan Province Education Department (Grant No. 13A140667). One of
the authors (H.L.W.) would like to thank Professor Jerzy Dudek for helpful suggestions on the manuscript.
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