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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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