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The highest limiting Z in the extended periodic table
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2015 J. Phys. G: Nucl. Part. Phys. 42 125105
(http://iopscience.iop.org/0954-3899/42/12/125105)
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The highest limiting Zin the extended
periodic table
Y K Gambhir
1,2
, A Bhagwat
3
and M Gupta
1
1
Manipal Centre for Natural Sciences, Manipal University, Manipal 576104,
Karnataka, India
2
Department of physics, IIT-Bombay, Powai, Mumbai-400076, India
3
UM-DAE Centre for Excellence in Basic Sciences, Mumbai 400 098, India
E-mail: yogy@phy.iitb.ac.in
Received 6 July 2015, revised 5 October 2015
Accepted for publication 21 October 2015
Published 17 November 2015
Abstract
The problem of finding the highest limiting Zin the extended periodic table is
discussed. The upper limit suggested by the atomic many body theory at
Z=172 may be reached much earlier due to nuclear instabilities. Therefore,
an extensive set of calculations based on the relativistic mean field formulation
are carried out for the ground state properties of nuclei with Z=100 to 180
and N/Zratio ranging from 1.19 to 2.70. This choice of Zand Nextends far
beyond the corresponding values of all the known heavy to superheavy ele-
ments. To facilitate the analysis of the huge quantity of calculated results,
various filters depending upon the pairing energies, one and two nucleon
separation energies, binding energy per particle (BE/A)and α-decay plus
fission half lives, are introduced. The limiting value of Zis found to be 146.
For the specificfilter with =
B
EA 5.5 MeV a few nuclei with Z=180 also
appear. No evidence for the limiting Zvalue 172 is found. We stress the need
to bridge the atomic and nuclear findings and to arrive at an acceptable lim-
iting value of highest Z(or rather combination of Zand N)of the extended
periodic table.
Keywords: relativistic mean field model, limiting atomic number in periodic
table, superheavy nuclei
(Some figures may appear in colour only in the online journal)
1. Introduction
The number of physically possible elements is unknown. To find the highest limiting value of
Zin the extended periodic table is a challenging and difficult task. This problem is primarily
Journal of Physics G: Nuclear and Particle Physics
J. Phys. G: Nucl. Part. Phys. 42 (2015)125105 (8pp)doi:10.1088/0954-3899/42/12/125105
0954-3899/15/125105+08$33.00 © 2015 IOP Publishing Ltd Printed in the UK 1
of academic interest because it is highly unlikely that any nucleus will be stable or quasi
stable for such high values of Z. Therefore the answer to this problem is expected to emerge
only through theoretical investigations. The relativistic quantum many body theory developed
and used in atomic physics has reached a high level of sophistication (e.g. see [1,2])and is
able to describe and predict atomic properties to a very high degree of precision. This
approach involves two steps: one first generates a single particle basis in the Dirac–Fock
mean field framework and then the leftover interactions (correlations)are incorporated either
perturbatively or through the multi-configuration mixing procedure. Using a relativistic
Hartree–Fock program Fricke et al [3]carried out calculations of chemical elements up to
Z=172. Recently, based on the latest version of the Multi Configuration Dirac–Fock
(MCDF)program by Desclaux and Indelicato [4], Pyykkö [5]proposed an extended periodic
table up to
Z17
2
along with their respective electronic shell configurations. This has been
recently validated by Indelicato et al [6]. The criterion used in these studies for determining
the highest value of Zis that the binding energy of the deepest (most bound), i.e. 1 s, electron
should dive into the Dirac continuum at ∼−2
Mc
e2(M
e
being the mass of the electron).
Around this situation the nuclear structure effects will be important and may even play a
decisive role. The only place where the nuclear structure effects enter in the atomic calcu-
lations is through the use of extended nuclear charge distribution (in place of the conventional
point charge (Ze)) in which the nuclear charge radius depends upon the mass number A
(
=+
A
ZN,
Nbeing the neutron number). This prescription for incorporating nuclear effects
may not be adequate specially for such heavy (high Z)elements. It is expected that the upper
limit projected by the atomic theory at Z=172 may be an over-estimation due to the nuclear
instabilities. It is therefore very important and desirable to bridge the atomic and nuclear
findings and to arrive at an acceptable limiting value of highest Z(or combination of Zand N)
of the extended periodic table.
The relativistic mean field theories, also termed as the effective mean field or density
functional theories, have been astonishingly successful in accurately describing the ground
state properties of nuclei spread over the entire periodic table. Motivated by the findings
discussed above, an extensive set of calculations based on the relativistic mean field (RMF)
formulation are carried out for the ground state properties of nuclei with Z=100 to 180 and
N/Zratio ranging from 1.19 to 2.70. This choice of Zand Nextends much beyond the
corresponding values of all the known heavy to superheavy elements. A brief sketch of the
RMF formulation is presented in the following section.
2. Formulation
The RMF theory starts with a Lagrangian describing Dirac spinor nucleons interacting only
through the meson (
s
w
r
,,)and electromagnetic (photon)fields. The classical variation
principle leads to the equations of motion. The mean field approximation is then introduced as
a result the fields are not quantised and are treated as c-numbers. For the static case along with
the charge conservation and the time reversal invariance one is left with a Dirac-type equation
for the nucleons with potential terms involving the fields and Klein–Gordon type equations
for meson and electromagnetic fields with sources depending upon baryon currents and
densities.
The important pairing effects are incorporated either through constant pairing gap (Δ)
approximation (BCS type)or self-consistently. The latter leads to relativistic Hartree–
Bogolyubov (RHB)equations involving the self-consistent field describing the particle–hole
correlations and the pairing field (
D
ˆ
)which describes particle–particle (pp)correlations. The
J. Phys. G: Nucl. Part. Phys. 42 (2015)125105 Y K Gambhir et al
2
pairing field
D
ˆ
involves the matrix elements of the two-body nuclear interaction (V
pp
)in the
pp-channel. In the constant gap approximation
D
ˆ
(ºD)becomes diagonal and decouples into
a set of diagonal matrices resulting in the BCS type expressions for the occupation prob-
abilities. As a result the RHB equations reduce to the BCS type equations with a constant
pairing gap Δ. This nonlinear coupled set of equations, known as RMF/RHB equations, are
to be solved self-consistently.
3. Details of the calculation
The parameters (meson masses and various coupling constants)appearing in the RMF
Lagrangian are usually fixed through a
c2
fit to some of the observed ground state properties
(like binding energies, radii etc)of a few selected even–even spherical nuclei. This set of
parameters is then kept fixed and is used in all the calculations for any nucleus (given N, Z).
We employ here the most widely used set, NL3 [7]. We solve RHB equations to incorporate
pairing correlations self-consistently. Here, finite range Gogny D1S interaction [8,9]which
has the right content of pairing for V
pp
is used. The calculation yields Dirac nucleon spinors,
the meson and electromagnetic fields and the total ground state binding and pairing energies
(Tr
kD2,
ˆ
here, κis anomalous density, see [10]), single particle states and their energies
along with their respective occupancies. The other ground state properties, such as nucleon
density distributions, nuclear sizes, radii, etc, can then be calculated. Extending the calcu-
lations to neighbouring nuclei, one can then calculate other physical quantities like one/two
nucleon separation energies, Q-values for alpha decay, etc. For details, see [10–14].
The spin–orbit and the pairing parts of the mean field are known to dictate the evolution
of the shell structure, leading to shell closures (magic numbers). The essential characteristic of
the magicity (shell closure)is the existence of a large gap between the first unoccupied state
and the last occupied state reflecting an extra stability. The variation of neutron/proton
pairing energies exhibits peaks at these nucleon numbers corresponding to the respective shell
closures. This is also manifested as sudden jumps in the variation of single and two nucleon
separation energies, i.e. appearing as kinks at the corresponding nucleon numbers [15–18].
These details are important for the present investigation, as one of the principal goals is to
quantify the limit of nuclear stability as a function of Z.
The establishment of the stability of nuclei also demands that one knows the nuclear
decay half lives. The αdecay and fission half lives, which are also important in the nuclear
stability studies, are calculated using empirical formulae available in the literature [19–23]
and discussed.
4. Results and discussion
An extensive set of calculations based on the RHB framework are carried out for the ground
state properties of 15,375 nuclei with Z100 180 and N/Zratio ranging from 1.19 to
2.70. Several relativistic as well as non-relativistic mean field calculations for nuclei up to
Z=120 have already been reported (e.g. see [14–18])and discussed. The calculations
reproduce the experimental findings including the αdecay Q-values and half lives rather well.
The next conventional magic number at Z=126, N=184 have also been studied and
discussed. One of the important findings of these studies indicated that for the nuclei in the
super heavy region, the extra stability or magicity depends upon a specific combination of
neutron number (N)and the proton number (Z)of the nucleus, rather than the conventional
J. Phys. G: Nucl. Part. Phys. 42 (2015)125105 Y K Gambhir et al
3
single value of Nor Zalone. This is manifestation of the fact that the superheavies are
stabilised by shell effects alone [18].
In view of the above remarks, we shall no longer discuss the results for nuclei with
<Z100 126.
Therefore, we shall now present and discuss the results for 10,904 nuclei
with Z126 180 and N/Zratio ranging from 1.19 to 2.70.
The variation of the calculated binding energy per particle (BE/A)with neutron number,
for nuclei with Z126 150 (Z150 180 )is displayed in figure 1. The figures reveal
that the BE/A, in general, smoothly decreases with the increase in the neutron number Nand
(or)proton number Z. Interestingly, all the nuclei with Z126 150 (Z150 180 )
have almost the same value of BE/Aaround 6.3 (5.7)MeV at neutron number ~
N
264
(
~
N
340
). These nuclei, if they exist, may be very useful in the energy generation through
fission process.
The variation of the pairing energy E
pair
for neutrons with neutron number N, for
Z126 150 and Z150 180 is shown in figure 2(left and right panels, respectively).
The pairing energy as a function of neutron number N, for Z126 180 shows various
peaks corresponding to the maximum pairing energy indicating extra stability at specific
neutron number Nfor a set of Zvalues. For example a sharp peak at N=216 for
=-Z144 150 and at N=308 for
=-Z168 180
is visible at almost zero pairing energy.
Also a modest peak appears at N=238 for
=-Z168 180
and at N=320 for
=-Z168 180.
at pairing energy around −5 MeV. The deviation of the pairing energy from
zero is attributed to the extent of robustness of shell closures or the magicity. These features
are clearly visible in the variation of the calculated one and two neutron separation energies
S
1n
and S
2n
(see figures 3and 4below)and the neutron Fermi energy
l,
nas sudden jumps or
kinks, at these specific values of Zand the corresponding neutron numbers N.
Similarly, the variation of the calculated proton pairing energy indicates a prominent
peak representing extra stability or shell closures at Z=132 for neutron numbers
=-
N
214 244
and 294−344 around −8 MeV pairing energy and a sharp peak at Z=154
for neutron numbers
=-
N
240 294
at around −4 MeV pairing energy. The analysis of the
calculated single and two proton separation energies S
1p
and S
2p
do not show prominent kinks
at Z=132 or 154. However, a very minor kink does appear around neutron number N=260
for
=-Z126 132.
The variation of the proton Fermi energy (
l
p
)with neutron number also
do not show any significant specific jumps.
For proper perspective, clarity and to simplify the analysis of a large amount of calcu-
lated results, we introduce various filters. The most crucial filter is on the single-, two-nucleon
separation energies and the corresponding Fermi energies. For the bound state solutions,
single- and two-nucleon separation energies (
S
SSS,,,
pnp n112 2
)have to be positive. The
introduction of this filter, designated as F
0
, reduces the number of nuclei to be considered
from 10,904 to 7,489. This filter F
0
is basic and is included in all the rest of the filters and
therefore will not be explicitly mentioned. The next filter, labeledF
p
, restricts both the
neutron and proton pairing energies between 0 to −20 MeV. The inclusion of this filter F
p
(automatically includes F
0
)further reduces the number of nuclei to be studied to 1523,
terminating at Z=161 along with a few isotopes of Z=180. The number of possible
elements existing can be further restricted by studying behaviour of the experimental BE/A
values for all the known nuclei. A detailed study of the experimental BE/Avalues [24]
reveals that most of the known nuclei with
Z8
and
N
8
have BE/Avalues above
7.4 MeV. The number of known nuclei goes on decreasing with decreasing BE/A, and only
four nuclei (
20
Mg,
25,26
O and
28
F)are known to exist with BE/Avalues less than 6.8 MeV,
with the smallest known BE/Aat about 6.4 MeV. We therefore set a lower cutoff on BE/A
values to 6.5 MeV. The corresponding filter is denoted by F
65
. In order to investigate the
J. Phys. G: Nucl. Part. Phys. 42 (2015)125105 Y K Gambhir et al
4
nuclei with larger Zvalues (upto 180), we lower this restriction to 5.5 MeV, and the corre-
sponding filter is denoted by F
55
. The filter F
65
(F
55
)restricts the number of nuclei to be
considered to 878 (3073), where the highest Zallowed is 146 (180).
To study deformation effects, axially deformed RMF calculations have been carried out
for nuclei which show an extra stability. It is found that the value of the deformation para-
meter (β)for these systems turns out to be very small (
0.
1
). As a result the deformed RMF
results are very similar to the corresponding RHB results. Therefore, the conclusions drawn
here may not be affected, at most the predicted Zvalues may change by amaximum oftwo
units.
It is known that the non-relativistic mean field calculations with Skyrme type interactions
yield results very similar to the corresponding relativistic mean field results (see, for example,
the discussion in [25]). Therefore, the conclusions arrived here are expected to be unaltered.
The α-decay and fission also play an important role in the nuclear stability study. As
mentioned before, we have calculated halflives for α-decay and fission, using empirical
formulae available in the literature [19–23]. Next, the filters which restricts the halflives to
Figure 1. The calculated BE/Afor a)Z126 150 (left panel)and b)150
Z180
(right panel).
Figure 2. The calculated neutron pairing energy for a)Z126 150 (left panel)and
b)Z150 180 (right panel).
J. Phys. G: Nucl. Part. Phys. 42 (2015)125105 Y K Gambhir et al
5
Figure 3. The calculated single neutron separation energies for a)Z126 150 (left
panel)and b)Z150 180 (right panel).
Figure 4. The calculated two neutron separation energies for (a)Z126 150 (left
panel)and (b)Z150 180 (right panel).
Table 1. Summary of the calculated results obtained with various filters. All the filters
include the common filter F
0
automatically. The entries in parenthesis correspond to the
results obtained by including an additional pairing filter F
p
.
The highest Restricted number
Filters value of Zof nuclei
F
65
146 (146)878 (843)
F
55
180 (180)3073 (1523)
+
a
FF
65
146 (146)878 (843)
+
a
FF
55
180 (180)3073 (1523)
++
a
FFF
f65 146(146)433 (433)
++
a
FFF
f55 180 (180)1467 (550)
J. Phys. G: Nucl. Part. Phys. 42 (2015)125105 Y K Gambhir et al
6
10
−10
seconds or more for both α-decay and fission are introduced which are indicated by a
F
and F
f,
respectively. We have explicitly checked that the α-decay half lives obtained by using
the Viola Seaborg and Dasgupta formulae are very close to those obtained by using the
double-folded potential within WKB approximation (for details, see [14]). Similar checks are
difficult to achieve for fission half lives. Thus, comparatively, the filter on α-decay half lives
is much more stringent as compared to that on the fission half lives. The inclusion of the filter
a
Fin addition to the filter F
65
(F
55
)restricts the number of nuclei to 878 (3073). Incorporating
the additional filter F
f
further reduces the number of nuclei to be studied to 433 (1467).
For easy reference all the results with various filters are summarised in the following
table. The corresponding results obtained with an additional pairing filter F
p
are shown in
parenthesis.
5. Summary and conclusions
The problem of finding the highest limiting Zin the extended periodic table is addressed.
Based on the sophisticated atomic relativistic quantum many body theory Z=172 has been
proposed as the highest limiting value of Z. This highest limit at Z=172 may be reached
much earlier due to the nuclear instabilities. In atomic calculations, the nuclear structure
effects are taken into account only through the use of extended nuclear charge distribution (in
place of the conventional point charge (Ze)). This prescription may not be adequate. The
relativistic mean field theories indicate that in the super heavy elements region the extra
stability or magicity depends upon a specific combination of neutron number (N)and the
proton number (Z)of the nucleus, rather than the conventional single value of Nor Zalone.
To find which combination of Nand Zforms a bound nucleus, an extensive set of calculations
based on the RMF formulation are carried out, for the ground state properties of nuclei with
Z=100 to 180 and N/Zratio ranging from 1.19 to 2.70. To facilitate the analysis of the
results, various filters depending upon the pairing energies, one and two nucleon separation
energies, binding energy per particle (BE/A)and α-decay plus fission half lives are intro-
duced. The limiting value of Zis found to be 146. For the specificfilter with
=B
EA 5.5 MeV
a few nuclei with Z=180 do appear. No evidence for the limiting Zvalue 172 is found. We
advocate that it is important and highly desirable to bridge the atomic and nuclear findings
and to arrive at an acceptable limiting value of highest Z(or combination of Zand N)of the
extended periodic table.
Acknowledgments
The authors wish to thank Peter Ring, Gottfried Münzenberg and Alok Shukla for their
interest in this work. Part of this work was carried out under the program Dynamics of
Weakly Bound Quantum Systems (DWBQS)under FP7-PEOPLE-2010-IRSES (Marie Curie
Actions People International Research Staff Exchange Scheme)of the European Union. AB
acknowledges financial support from DST, Govt. of India (grant number DST/INT/SWD/
VR/P-04/2014).
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