Scientific and Educational Reports of the Facultyof Science and Technology, Kochi University
Vol. 1 (2018), No. 2

How nuclear incompressibility affects shell structure of superheavy nuclei
Kei Iida,1, 2, ∗ Mari Adachi,2 and Kazuhiro Oyamatsu3
1

Department of Mathematics and Physics, Kochi University, Kochi 780-8520, Japan2
Department of Natural Science, Kochi University, Kochi 780-8520, Japan3
Department of Human Informatics, Aichi Shukutoku University, 9 Katahira, Nagakute, Aichi 480-1197, Japan
We systematically examine how the structure of superheavy nuclei depends on the equation of stateof nuclear matter. In doing so, we first describe the neutron and proton distributions of such nuclei
within a simplified Thomas-Fermi framework in such a way as to reproduce empirical masses andcharge radii of stable nuclei. We then calculate the central and Coulomb parts of the single-particle
potential, which are complemented by the spin-orbit part that makes the resultant single-particle
spectra for 208 Pb consistent with the empirical behavior. We find that the incompressibility ofsymmetric nuclear matter significantly affects the calculated Z = 28, 50, 82 gaps for 278 113 and
294
118.

Recent experiments to synthesize superheavy nuclei
not only produced several new elements, but also led
to implications for the island of stability on the chartof nuclides [1]. Even in relatively lighter nuclides suchas 208 Pb, however, it is poorly known how the singleparticle levels look like well below the Fermi energy. Thisis related to uncertainties in the spin-orbit part of thesingle-particle potential, which scales as the gradient of
the nuclear density distribution. This distribution is inturn connected with the equation of state (EOS) of nuclear matter in a way clearer for heavier nuclei [2]. It isthus important to ask the question of how the EOS of nuclear matter affects the internal shell structure of heavy
and superheavy nuclei through the spin-orbit potential.
So far, the EOS of nuclear matter has yet to be welldetermined even at subnuclear densities [3]. Empirical
data such as nuclear masses and radii are not sufficient
to severely constrain the EOS, while many-body calculations depend on what kind of nuclear forces and approximate procedures are adopted even in the simple caseof pure neutron matter. We thus start with model EOSs
that account for possible uncertainties. In previous works[2, 4, 5], two of the present authors address the questionsof how such EOSs affect the macroscopic prediction ofradii and masses of unstable neutron-rich nuclei. In thiscase, the poorly known density dependence of the symmetry energy controls the radii and masses through thebulk and surface properties, respectively. In the present
work, we consider the EOS effect on the single-particle
levels of superheavy nuclei 278 113 and 294 118, which weresynthesized in laboratories [6, 7] and recently given anelement name as nihonium (Nh) and oganesson (Og), respectively.
For the purpose of clarifying such effects, we first calculate, within a simplified Thomas-Fermi model of a nucleus, the neutron and proton density distributions in amanner that is dependent on the EOS while reproducing
empirical masses and charge radii of stable nuclei. From
the optimal neutron and proton density distributions, weevaluate the central, Coulomb, and spin-orbit parts of theneutron and proton single-particle potentials, in which
we determine the effective range of the nuclear force andthe strength of the spin-orbit potential in such a way as
to reproduce empirical single-particle energies of 208 Pb.
We then solve the Schrödinger equations to obtain theneutron and proton single-particle levels of superheavy
nuclei for various EOSs. In this work, for simplicity, we
assume that the nuclear density distributions are spherically symmetric.
We begin by describing the total energy of a nucleus ofcharge number Z and neutron number N in terms of thelocal neutron and proton density distributions nn (r) andnp (r) by using the following energy density functional:
E = Eb + Eg + EC + N mn c2 + Zmp c2 ,

(1)

where
Eb =

Z

d3 rn(r)w [nn (r), np (r)](2)

is the bulk energy with n(r) = nn (r) + np (r) and theenergy w per nucleon of uniform nuclear matter,Z
Eg = F0 d3 r|∇n(r)|2(3)
is the gradient energy with adjustable constant F0 ,ZZe2
np (r)np (r′ )3EC =d r d3 r ′(4)2|r − r′ |is the Coulomb energy, and mn (mp ) is the neutron (proton) mass.
The bulk energy (2) dominates Eq. (1), while being
controlled by the energy w of uniform nuclear matterwithin the framework of the local density approximation.
For w, just like the previous works [2, 4, 5] based on Eq.
(1), we adopt the phenomenological EOS models in theform of [8]w(nn , np ) = [t(nn , np ) + v(nn , np )] /n(5)with the kinetic energy density,∗ Electronic address: iida@kochi-u.ac.jp

Received: January 14, 2018
Reviewed by anonymous referee, and accepted: January 27, 2018
Published: March 31, 2018t=

3~2 (3π 2 )2/3 5/3(nn + n5/3
p )
10mn

(6)

2
and the potential energy density,22

v = (1 − α )vs (n) + α vn (n),

(7)

where
vs = a1 n2 +a2 n3
1 + a3 n(8)

vn = b1 n2 +b2 n3
1 + b3 n(9)andare the potential energy densities for symmetric nuclear
matter and pure neutron matter, and α = (nn − np )/nis the neutron excess.
It is useful to note that near the saturation point ofsymmetric nuclear matter, the energy per nucleon given
by Eq. (5) reduces to a familiar expansion, namely [9],


K0
L
2
(n − n0 ) + S0 +w = w0 +(n − n0 ) α2 . (10)
18n20
3n0
Here w0 , n0 , and K0 are the saturation energy, the saturation density, and the incompressibility of symmetric nuclear matter. L and S0 are associated with thedensity dependent symmetry energy coefficient S(n) via
S0 = S(n0 ) and L = 3n0 (dS/dn)n=n0 . As the neutron
excess increases from zero, the saturation point moves
from (n0 , w0 ) in the density versus energy plane. Up tosecond order in α, the saturation energy ws and density
ns are given by
ws = w0 + S0 α 2

(11)

and
ns = n0 −3n0 L 2
α .
K0

(12)

The six parameters a1 , · · · , b3 are related to the EOS
parameters n0 , w0 , K0 , S0 , and L. As in Refs. [2, 4, 5],
we assume various sets of K0 and L in the range of
0 < L < 160 MeV and 180 MeV ≤ K0 ≤ 360 MeV,
while the others are determined in such a way that themass and charge radius of stable nuclei that can be calculated from optimization of Eq. (1) are consistent withthe empirical values. We remark that the parameter b3 ,which controls the EOS of matter for large neutron excess
and high density while having little effect on the saturation properties of nearly symmetric nuclear matter, is set
to the typical value 1.58632 fm3 , which was obtained by
one of the authors [8] in such a way as to reproduce theneutron matter energy of Friedman and Pandharipande[10].
For fixed (K0 , L), we optimize Eq. (1) by assuming thenucleon distributions ni (r) (i = n, p) of a nucleus of given
Z and N , where r is the distance from the center of thenucleus, to be"  ti # 3


r

in
 ni 1 −, r < Ri ,Rini (r) =
(13)




0,
r ≥ Ri .

Here, Ri roughly represents the nucleon distribution radius, ti the relative surface diffuseness, and nin
i the central number density. We then determine n0 , w0 , S0 , andF0 by fitting the resultant masses and charge radii tothe experimental values, which leads to about 200 setsof the EOS models as shown in Ref. [2]. Note that thepresent analysis can be regarded as a simplified versionof the Thomas-Fermi theory, which assumes the form ofni (r) but still can well reproduce the empirical charge
distribution of stable nuclei.
By building the EOS models described above into Eq.
(1), we have calculated the nucleon density distributionsfor 208 Pb, 278 113, and 294 118, from which we shall calculate the single-particle energies. As clarified in Ref. [2],the results show the tendency that the central density
decreases with L, while the surface diffuseness decreases
with K0 . The latter feature plays a vital role in affecting
the internal shell structure via the spin-orbit potential.
We now move on to the single-particle energies by constructing the single-particle potential from the optimalnucleon density distributions obtained above. We first
calculate the single-particle potential U0(i) (i = n, p) for
uniform nuclear matter as
U0(i) =∂v(nn , np )
.
∂ni

(14)

By substituting the optimal distributions into Eq. (14),
we obtain the central part of the single-particle potential
within the local density approximation in which the rangeof the nuclear force is set to zero. Here, as in Ref. [11],we incorporate the finite range of the nuclear force into
the potential U0(i) (r) via
3 Z



|r − r′ |2
d3 r′ U0(i) (r′ ) exp −
,
κ2
(15)
where κ is the adjustable parameter that corresponds to
the range of the nuclear force. Note that in the ThomasFermi framework as described above, the gradient energy(3) represents the correction term due to the finite rangeof the nuclear force.
The Coulomb potential, which takes effect solely for
protons, reads as
Ui (r) =



1
√
πκ

VC (r) =δEC [np (r)],δnp (r)(16)

where the density distributions are the optimal ones.
Finally, we set the spin-orbit part of the single-particle
potential as


1
dn(r)
d
VLS (r) =λ1± λ2 [nn (r) − np (r)] l · s,
r
dr
dr
(17)
where l is the orbital angular momentum, s is the spin
angular momentum, λ1 and λ2 are the adjustable parameters that control the strength of the isoscalar and isovector parts of the spin-orbit potential, and ± are taken for3
neutrons (+) and protons (−). This part is crucial to theinternal shell structure, while being sensitive to the EOS
of nuclear matter, the incompressibility in particular.
We proceed to calculate the single-particle energies εby solving the Schrödinger equations for neutrons andprotons,
~2−∆ + Un (r) + VLS (r) ψ(r) = εψ(r), (18)
2mn


~2
−
∆ + Up (r) + VC (r) + VLS (r) ψ(r) = εψ(r).(19)2mp
To determine the parameters κ, λ1 , and λ2 in such a way
as to reproduce the experimental single-particle energies
of 208 Pb near the Fermi energies, we solve Eqs. (18) and(19) for each EOS model. As a result of this fitting, theparameters κ, λ1 , and λ2 are constrained as κ = 1.15–
1.35 fm, λ1 = 170–210 MeV fm5 , and λ2 = 10–170 MeV
fm5 .
In Fig. 1, we illustrate the EOS dependence of thesingle-particle energies of 208 Pb. Here, we adopt the five
EOS models with (K0 , L) = (180, 5.66), (180, 52.2), (230,42.6), (360, 12.8), and (360, 146) in MeV; the EOS of(230, 42.6) is a typical case, while the others are extreme
ones. Hereafter, we refer to these five EOS sets as G,
A, E, I, and C, according to Fig. 1 in Ref. [4]. The
corresponding values of κ, λ1 , and λ2 are tabulated inTable I. We can observe that the neutron and protonlevels near the respective Fermi level are well constrained
from the fitting for all the EOS models, while the Z =28, 50 gaps decrease with K0 to an extent in which they
are no longer main gaps. We remark in passing that theneutron internal shell structure is comparatively rather
insensitive to the EOS models (see Table II) and that the
proton and neutron low-lying levels go up with increasein K0 . The latter reflects the fact that the kinetic energy
decreases with increasing K0 , a feature related to thecorrelation between K0 and n0 that comes from fitting
to empirical charge radii of stable nuclei [2].
Let us now show the results for 278 113 and 294 118.
The Z = 28, 50, 82 gaps decrease with K0 and, more importantly, are no longer main gaps even for the smallest
K0 , as illustrated in Figs. 2 and 3 as well as in Table
II. The N = 50, 82, 126 gaps also decrease with K0 , but
remain main gaps except for the largest K0 . Such reduction in the gaps can be seen, e.g., in the previous work

TABLE I: The spin-orbit and range parameters for various
EOS sets.
EOS
G A
E
I
C
K0 (MeV) 180 180 230 360 360
L (MeV)
5.66 52.2 42.6 12.8 146
κ (fm)
1.15 1.17 1.22 1.31 1.34
λ1 (MeV fm5 ) 177 181 184 188 202
λ2 (MeV fm5 ) 10.7 20.5 28.2 34.4 138

114

82

50

28

EOS G EOS A EOS E

EOS I

EOS C

EXP.

EOS I

EOS C

EXP.

184

126

82

50
EOS G EOS A EOS E

FIG. 1: 208 Pb proton (upper) and neutron (lower) singleparticle energy levels calculated for various EOS sets. The
Fermi levels are drawn in thick red lines. The last column ineach panel denotes the experimental data [12].

by Dudek [13] who used various sets of the Woods-Saxon
potentials that reproduce the experimental data for the
single-particle level spectra of doubly magic nuclei, although the K0 dependence was not discussed therein.
Finally, we emphasize that the present analysis is not
relevant to the prediction of the location of the stability

4

114

114

82

82

50

50

28

28

EOS G EOS A EOS E

EOS I

EOS C

184

EOS I

EOS C

EOS I

EOS C

184

126

126

82

82

50
EOS G EOS A EOS E

EOS G EOS A EOS E

50
EOS I

EOS C

FIG. 2: 278 113 proton (upper) and neutron (lower) singleparticle energy levels calculated for various EOS sets. The
Fermi levels are drawn in thick red lines.

island. This is because the fitting to the experimental
single-particle energies of 208 Pb and the adopted formof the spin-orbit potential (17) severely constrain the location to around (Z, N ) = (114, 184), in contrast with,e.g., the previous work based on various mean-field calculations [14]. We also remark that the present analysis
neglects possible deformations, which are predicted to be

EOS G EOS A EOS E

FIG. 3: Same as Fig. 2 in the case of 294 118.

relatively weak for 294 118 but significant for 278 113 (e.g.,
Ref. [15]).
In summary, we have constructed the single-particle
potential of superheavy nuclei in a manner that is dependent on the EOS of nuclear matter. We have foundthe sensitivity of the internal proton shell structure to
the incompressibility K0 . Deduction of K0 from data on
the breathing modes of stable nuclei ranges 190–315 MeV

5TABLE II: The energy gaps (MeV) for various EOS sets.
EOS
G A
E
I
C
Z = 28 (208 Pb) 2.54 2.48 2.41 2.27 2.13
Z = 50 (208 Pb) 3.82 3.74 3.67 3.53 3.34
N = 50 (208 Pb) 3.56 3.50 3.38 3.17 3.19
N = 82 (208 Pb) 4.97 4.90 4.79 4.61 4.61
Z = 28 (278 113) 1.88 1.84 1.79 1.69 1.61
Z = 50 (278 113) 2.68 2.67 2.46 2.06 2.12
Z = 82 (278 113) 1.94 1.96 1.69 1.21 1.31
N = 50 (278 113) 2.58 2.55 2.45 2.26 2.25
N = 82 (278 113) 3.41 3.25 3.01 2.65 2.43
N = 126 (278 113) 2.88 2.73 2.48 2.11 1.94
Z = 28 (294 118) 1.78 1.74 1.69 1.60 1.51
Z = 50 (294 118) 2.45 2.43 2.33 1.84 1.89
Z = 82 (294 118) 1.65 1.65 1.39 0.92 1.00
N = 50 (294 118) 2.43 2.39 2.29 2.12 2.11
N = 82 (294 118) 3.18 3.01 2.78 2.44 2.21
N = 126 (294 118) 2.61 2.45 2.21 1.84 1.67

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due to large systematic errors [16, 17]. Once uncertainties in K0 are reduced, a better prediction is expected
to be made of the structure of superheavy nuclei. In thepresent analysis, we have confined ourselves to spherical
nuclei. Deformations could play a role in modifying thestructure drastically in a far region from the island of stability. The location of the island of stability itself would
thus be desired to be specified.

Acknowledgments

We are grateful to H. Koura for useful discussions.
This work was supported in part by Grants-in-Aid for
Scientific Research on Innovative Areas through No.
24105008 provided by MEXT.

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