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The problem of finding the highest limiting Z in 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/Z ratio ranging from 1.19 to 2.70. This choice of Z and N extends far beyond the corresponding values of all the known heavy to superheavy elements. 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 Z is found to be 146. For the specific filter with = BE A 5.5 MeV a few nuclei with Z=180 also appear. No evidence for the limiting Z value 172 is found. We stress the need to bridge the atomic and nuclear findings and to arrive at an acceptable limiting value of highest Z (or rather combination of Z and N) of the extended periodic table.
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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 nding 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 eld 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 lters depending upon the pairing energies, one and two nucleon
separation energies, binding energy per particle (BE/A)and α-decay plus
ssion half lives, are introduced. The limiting value of Zis found to be 146.
For the speciclter 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 ndings 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 eld model, limiting atomic number in periodic
table, superheavy nuclei
(Some gures may appear in colour only in the online journal)
1. Introduction
The number of physically possible elements is unknown. To nd the highest limiting value of
Zin the extended periodic table is a challenging and difcult 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 rst generates a single particle basis in the DiracFock
mean eld framework and then the leftover interactions (correlations)are incorporated either
perturbatively or through the multi-conguration mixing procedure. Using a relativistic
HartreeFock program Fricke et al [3]carried out calculations of chemical elements up to
Z=172. Recently, based on the latest version of the Multi Conguration DiracFock
(MCDF)program by Desclaux and Indelicato [4], Pyykkö [5]proposed an extended periodic
table up to
Z17
2
along with their respective electronic shell congurations. 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
ndings and to arrive at an acceptable limiting value of highest Z(or combination of Zand N)
of the extended periodic table.
The relativistic mean eld theories, also termed as the effective mean eld 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 ndings
discussed above, an extensive set of calculations based on the relativistic mean eld (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)elds. The classical variation
principle leads to the equations of motion. The mean eld approximation is then introduced as
a result the elds 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 elds and KleinGordon type equations
for meson and electromagnetic elds 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 eld describing the particlehole
correlations and the pairing eld (
D
ˆ
)which describes particleparticle (pp)correlations. The
J. Phys. G: Nucl. Part. Phys. 42 (2015)125105 Y K Gambhir et al
2
pairing eld
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 xed through a
c2
t to some of the observed ground state properties
(like binding energies, radii etc)of a few selected eveneven spherical nuclei. This set of
parameters is then kept xed 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, nite 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 elds 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 [1014].
The spinorbit and the pairing parts of the mean eld 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 rst unoccupied state
and the last occupied state reecting 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 [1518].
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 ssion half lives, which are also important in the nuclear
stability studies, are calculated using empirical formulae available in the literature [1923]
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 eld calculations for nuclei up to
Z=120 have already been reported (e.g. see [1418])and discussed. The calculations
reproduce the experimental ndings 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 ndings of these studies indicated that for the nuclei in the
super heavy region, the extra stability or magicity depends upon a specic 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 gure 1. The gures 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
ssion process.
The variation of the pairing energy E
pair
for neutrons with neutron number N, for
Z126 150 and Z150 180 is shown in gure 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 specic
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 gures 3and 4below)and the neutron Fermi energy
l,
nas sudden jumps or
kinks, at these specic 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
=-
214 244
and 294344 around 8 MeV pairing energy and a sharp peak at Z=154
for neutron numbers
=-
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 signicant specic jumps.
For proper perspective, clarity and to simplify the analysis of a large amount of calcu-
lated results, we introduce various lters. The most crucial lter 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 lter, designated as F
0
, reduces the number of nuclei to be considered
from 10,904 to 7,489. This lter F
0
is basic and is included in all the rest of the lters and
therefore will not be explicitly mentioned. The next lter, labeledF
p
, restricts both the
neutron and proton pairing energies between 0 to 20 MeV. The inclusion of this lter 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 lter 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 lter is denoted by F
55
. The lter 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 eld calculations with Skyrme type interactions
yield results very similar to the corresponding relativistic mean eld results (see, for example,
the discussion in [25]). Therefore, the conclusions arrived here are expected to be unaltered.
The α-decay and ssion also play an important role in the nuclear stability study. As
mentioned before, we have calculated halflives for α-decay and ssion, using empirical
formulae available in the literature [1923]. Next, the lters 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 lters. All the lters
include the common lter F
0
automatically. The entries in parenthesis correspond to the
results obtained by including an additional pairing lter 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 ssion 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
difcult to achieve for ssion half lives. Thus, comparatively, the lter on α-decay half lives
is much more stringent as compared to that on the ssion half lives. The inclusion of the lter
a
Fin addition to the lter F
65
(F
55
)restricts the number of nuclei to 878 (3073). Incorporating
the additional lter F
f
further reduces the number of nuclei to be studied to 433 (1467).
For easy reference all the results with various lters are summarised in the following
table. The corresponding results obtained with an additional pairing lter F
p
are shown in
parenthesis.
5. Summary and conclusions
The problem of nding 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 eld theories indicate that in the super heavy elements region the extra
stability or magicity depends upon a specic combination of neutron number (N)and the
proton number (Z)of the nucleus, rather than the conventional single value of Nor Zalone.
To nd 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 lters depending upon the pairing energies, one and two nucleon separation
energies, binding energy per particle (BE/A)and α-decay plus ssion half lives are intro-
duced. The limiting value of Zis found to be 146. For the speciclter 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 ndings
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 nancial support from DST, Govt. of India (grant number DST/INT/SWD/
VR/P-04/2014).
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8
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What are the limits of the existence of nuclei? What are the highest proton numbers $Z$ at which the nuclear landscape and periodic table of chemical elements cease to exist? These deceivably simple questions are difficult to answer especially in the region of hyperheavy ($Z\geq 126$) nuclei. We present the covariant density functional study of different aspects of the existence and stability of hyperheavy nuclei. For the first time, we demonstrate the existence of three regions of spherical hyperheavy nuclei centered around ($Z\sim 138, N\sim 230$), ($Z\sim 156, N\sim 310$) and ($Z\sim 174, N\sim 410$) which are expected to be reasonably stable against spontaneous fission. The triaxiality of the nuclei plays an extremely important role in the reduction of the stability of hyperheavy nuclei against fission. As a result, the boundaries of nuclear landscape in hyperheavy nuclei are defined by spontaneous fission and not by the particle emission as in lower $Z$ nuclei. Moreover, the current study suggests that only localized islands of stability can exist in hyperheavy nuclei.
Chapter
Nach der Entdeckung der letzten Elemente der 7. Periode kam die Wissenschaft beim Oganesson an, das die Kernladungszahl 118 besitzt. Aber wohin geht die Reise danach? Sind bereits Versuche unternommen worden, noch schwerere Atomkerne zu erzeugen? Was sagen theoretische Berechnungsmodelle über die Möglichkeiten, die überhaupt noch bestehen? Dieses Kapitel gibt Ihnen einige Antworten auf diese spannenden Fragen.
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This paper is the second part of the new evaluation of atomic masses, Ame2012. From the results of a least-squares calculation, described in Part I, for all accepted experimental data, we derive here tables and graphs to replace those of Ame2003. The first table lists atomic masses. It is followed by a table of the influences of data on primary nuclides, a table of separation energies and reaction energies, and finally, a series of graphs of separation and decay energies. The last section in this paper lists all references to the input data used in Part I of this Ame2012 and also to the data included in the Nubase2012 evaluation (first paper in this issue).
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This paper is the first of two articles (Part I and Part II) that presents the results of the new atomic mass evaluation, Ame2012. It includes complete information on the experimental input data (including not used and rejected ones), as well as details on the evaluation procedures used to derive the tables with recommended values given in the second part. This article describes the evaluation philosophy and procedures that were implemented in the selection of specific nuclear reaction, decay and mass-spectrometer results. These input values were entered in the least-squares adjustment procedure for determining the best values for the atomic masses and their uncertainties. Calculation procedures and particularities of the Ame are then described. All accepted and rejected data, including outweighed ones, are presented in a tabular format and compared with the adjusted values (obtained using the adjustment procedure). Differences with the previous Ame2003 evaluation are also discussed and specific information is presented for several cases that may be of interest to various Ame users. The second Ame2012 article, the last one in this issue, gives a table with recommended values of atomic masses, as well as tables and graphs of derived quantities, along with the list of references used in both this Ame2012 evaluation and the Nubase2012 one (the first paper in this issue).
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A new parametrization for an effective nonlinear Lagrangian density of relativistic mean field (RMF) theory is proposed, which is able to provide a very good description not only for the properties of stable nuclei but also for those far from the valley of beta stability. In addition the recently measured superdeformed minimum in the 194Hg nucleus is reproduced with high accuracy.
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The authors review the present status of self-consistent mean-field (SCMF) models for describing nuclear structure and low-energy dynamics. These models are presented as effective energy-density functionals. The three most widely used variants of SCMF's based on a Skyrme energy functional, a Gogny force, and a relativistic mean-field Lagrangian are considered side by side. The crucial role of the treatment of pairing correlations is pointed out in each case. The authors discuss other related nuclear structure models and present several extensions beyond the mean-field model which are currently used. Phenomenological adjustment of the model parameters is discussed in detail. The performance quality of the SCMF model is demonstrated for a broad range of typical applications.
Book
Relativistic Methods for Chemists, written by a highly qualified team of authors, is targeted at both experimentalists and theoreticians interested in the area of relativistic effects in atomic and molecular systems and processes and in their consequences for the interpretation of the heavy element’s chemistry. The theoretical part of the book focuses on the relativistic methods for molecular calculations discussing problems such as relativistic two-component theory, density functional theory, pseudopotentials and correlations. These chapters are mostly addressed to experimentalists with only general background in theory and to computational chemists without training in relativistic methods. The experimentally oriented chapters describe the use of relativistic methods in different applications focusing on the design of new materials based on heavy element compounds, the role of the spin-orbit coupling in photochemistry and photobiology, and its relations to relativistic description of matter and radiation. This part of the book includes subjects of interest to theoreticians and experimentalists working in different areas of chemistry. Relativistic Methods for Chemists is written at an intermediate level in order to appeal to a broader audience than just experts working in the field of relativistic theory. The book is aimed at individuals not highly versed in these methods who want to acquire the rudiments of relativistic computing and the associated problems of importance for the heavy element chemistry. Relativistic Methods for Chemists is written for graduate students, academics, and researchers in theoretical chemistry as well as experimentalists in materials chemistry, inorganic chemistry, and physical chemistry.
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Systematic investigations of the pairing and two-neutron separation energies which play a crucial role in the evolution of shell structure in nuclei, are carried out within the framework of relativistic mean-field model. The shell closures are found to be robust, as expected, up to the lead region. New shell closures appear in low mass region. In the superheavy region, on the other hand, it is found that the shell closures are not as robust, and they depend on the particular combinations of neutron and proton numbers. Effect of deformation on the shell structure is found to be marginal.
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Relativity in atomic and molecular physics.- Relativity in atomic and molecular physics.- Foundations.- Relativistic wave equations for free particles.- The Dirac Equation.- Quantum electrodynamics.- Computational atomic and molecular structure.- Analysis and approximation of Dirac Hamiltonians.- Complex atoms.- Computation of atomic structures.- Computation of atomic properties.- Continuum processes in many-electron atoms.- Molecular structure methods.- Relativistic calculation of molecular properties.- Frequently used formulae and data.- Frequently used formulae and data.- Supplementary mathematics.- Supplementary mathematics.
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