How many elements can there be?

(1)

M. Gupta,¹ Y. K. Gambhir,¹ Eva Lindroth,² and G. M¨unzenberg^{1, 3}

1Manipal Centre for Natural Sciences, Manipal University, Manipal 576104, India

2Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, Sweden

3GSI Helmholtzzentrum f¨ur Schwerionenforschung mbH, Planckstr. 1, 64291 Darmstadt, Germany

(Dated: May 27, 2015)

The question of a possible limit for the existence of the chemical elements is discussed in the context of recent findings in the fields of nuclear and atomic physics and chemistry. At the time of this writing, there is no evidence to support the oc- currence of the heaviest elements in Nature. The upper end of the Periodic Table has been reached only through the artificial synthesis of a new element using heavy ion accelerators. The synthesis of Element 118 is at the present day limit of experimental sensitivity. Severe constraints on the availability of suitable target-projectile combinations pending the upgrade of existing facilities makes even the production of Elements 119 and 120 uncertain. Calculations using the most precise relativistic atomic many body quantum theory predict that elements upto Z=172 can exist. It is likely however that the natural limit will be determined by nuclear rather than atomic effects. It is advocated that the answer may lie at the interface of atomic and nuclear theory. Concurrently, keeping in mind the possible upgrade of experimental facilities in the foreseeable future, new reaction mechanisms coupled with improved measurement techniques for the simultaneous detection of new, and possibly rarer, decay modes is suggested together with a more accurate determination of mass and charge. This would enable the experimental verification of limiting mass (A) as a combination of Z and N (nuclear decay modes) and limiting charge (Z) (signature atomic radiations). A conceptual design of a next generation in-flight separator is proposed in this context.

(2)

I. INTRODUCTION

The question of how many elements can “exist” has enticed physicists and chemists since the 1930’s. Subsequent decades have seen many related advances in both atomic and nuclear physics but the question still remains.

One may well ask - what constitutes “existence”? An interesting working definition has been suggested by chemists in a somewhat narrow context. Taking into account the time it takes for an atom to acquire its outer electrons, a working definition of “existence” is synonymous with a decay half-life of not less than 10⁻¹⁴seconds [1]. This definition is based on the fact that chemical properties exhibit themselves through valence electrons and in that regard may serve chemists and experimentalists well. In a more absolute sense, it is not obvious whether an element can be considered to exist in the absence of its outermost electrons and it remains to be seen whether this criterion can be usefully embraced in other contexts. For the moment, the question is set aside.

Theory predicts an atomic limit at Z=172 (e.g. see [2]) although it may be reached much earlier due to nuclear instabilities. While a well defined nuclear limit is much harder to predict, the answer may lie closer to Zv125 assuming a classical liquid drop and a balance of Coulomb forces versus surface tension at the nuclear surface. Whatever the case, the experimental challenge lies in exploring uncharted territory. The successful search for new elements hinges on both a reliable method of synthesisas well as the accurate identification of these nuclides which comprise the very top end of the nuclear chart. As a result, the quest for new elements unites the fields traditionally labeled “reactions” and “structure” in a common cause. A few general comments may serve to provide a perspective.

Following the discovery of induced fission [3, 4], Meitner and Frisch [5] suggested Z=100 as the upper limit. This was based upon the assumption that the bulk properties of the nucleus could be likened to a charged nuclear droplet. A heavy system with a large number of protons could break into (two or more) fragments as a result of the repulsive Coulomb force overcoming the attractive nuclear force, thus revealing a delicate balance between cohesive and disruptive forces. The experimental search for new, and inevitably heavier, elements was in fact the quest for stability at the very limits of mass and charge. Strutinsky [6] proposed that microscopic effects, which incorporate individual degrees of freedom, when added to the nuclear droplet could serve as a bulwark against disintegration. This begged the question of

(3)

whether extremely heavy nuclei beyond the limit imposed by Meitner and Frisch could exist only as a result of microscopic effects i.e.shell stabilization. If so, the nucleus could remain bound with many more than 100 protons, its disintegration being influenced by factors other than simply the macroscopic limit and allowing the possibility of additional decay modes.

α decay would then be expected to predominate over spontaneous fission (SF).

The implications of this were far reaching. SF was well known to occur in the actinides with half-lives decreasing sharply with increasing Z. However, fission barriers remained approximately the same atv6 MeV for the elements beyond uranium. This meant that for the superheavy elements (those above Z=103), the decay half-life would be governed by shell effects alone. Moreover, localised regions of greater stability, the so-called “shell closures”, suddenly became possible. Even more amazingly, such shell closures could also be possible in highly deformed nuclides.

The first postulate of an “island of stability” was made by Myers and Swiatecki for Z=126 and N=184 [7]. There were other “firsts” associated with this seminal work: the term

“superheavy nuclei” was used for the first time when proposing the microscopic-macroscopic approach with nuclear deformations to calculate fission barriers.

Since then many theoretical formalisms have been developed with varying results as predictions of closed shells are highly model dependent. Proton “magic” numbers vary from Z=108-110, 114, 126 (see for instance [8, 9]) and beyond Z=120 [10] though all theories agree that the next closed neutron shell would be at N=184.

Clearly, such a great increase in nuclear mass could not be without atomic (and chemical) repercussions. With the addition of more protons the electrostatic field provided by the nucleus increases and consequently the innermost electronic orbitals shrink and squeeze in closer to the nucleus. Relativistic effects such as the contraction of electronic shells within the atom, could become very important. For instance, the radius of the inner 1s electron in Copernicium (element 112) is shrunk by 43% and results in the stabilisation of the s and p_1/2 orbitals [11]. Unexpectedly, the effect of direct relativistic stabilisation is large even for the outer valence orbitals [12] influencing its chemical behaviour. If indeed element 112 has a closed shell configuration with a well stabilised 7s orbital, it would behave chemically like the volatile, noble gas radon. This prediction is at variance with its expected place in the Periodic Table as a group 12 element and the heavier homologue of mercury. Chemists have been attempting to settle this question although results have remained inconclusive.

(4)

A large number of chemical studies have concentrated on attempts to place new elements in the Periodic Table through the understanding of properties including oxidation states and complex formation. Simultaneously, atomic theorists have made increasingly accurate calculations of characteristic radiations such as X-rays which would assist in the reliable identification of new and unknown elements produced in the future.

Experimentally, the test of these theories lies in finding innovative ways to artificially synthesise heavier elements in the laboratory even as their conclusive identification becomes increasingly challenging. Extremely rare and short lived as they are, the nuclear and chemical properties have to be deduced from the decay time and energy often measured from the synthesis of a single atom at a time. Indeed, as we shall see, the mass assignment of an unknown element relies heavily on the correctinterpretationof scarce experimental data and ambiguities have often lead to conflicting and sometimes erroneous results.

It may be said that the art and science of heavy element synthesis stands at a cross- roads and the way forward is not clear. Being as we are at the very limit of experimental sensitivity, new techniques need to be identified to bridge the gap and reach the postulated magic region(s). Even then, clinching evidence to support mass assignments may remain as elusive as a true estimate of the number of elements that can exist.

Theoretically speaking, a fresh assessment from a new perspective will be equally timely.

Just as the search for new elements has blurred the traditionally accepted boundaries between “structure” and “reaction” physics, the interface between atomic and nuclear theory could be further explored. Even though such a pursuit may throw up some surprises and raise new questions and challenges, the clue to the fundamental question of how many elements can exist, could well lie in this region of overlap.

The object of the present study is to explore the theoretical interface between atomic and nuclear physics in an attempt to revisit this question. It is also recognised that the ability to reliably identify the unknown nucleus is inextricably linked to the experimental process.

State-of-the-art calculations quantifying signature atomic radiations such as X-rays to the heaviest mass regions will help to guide the design of next generation experimental facilities as well as to assist in the identification of these elusive and exotic elements which define the very top end of the nuclear chart as we know it.

(5)

II. GENERAL CONSIDERATIONS

To find the highest limiting Z in the extended Periodic Table is a challenging task. For the investigation of this fundamental and important problem we first turn to our theoretical understanding of microcosmos and in the following we discuss some aspects of the physics that determines nuclear stability as well as the forming of the electron shell structure, and how this will affect the number of possible elements.

A. Nuclear physics aspects

Relativistic mean field theories have been astonishingly successful in describing the nuclear ground state properties of the nuclei spread over the entire Periodic Table including the super heavy elements (SHE). It is found that in the SHE region the evolution of shell structure shows that extra stability or ‘magicity’ depends upon a specific combination of neutron (N) and proton number (Z) of the nucleus, rather than the conventional single value of N or Z alone. Proton (neutron) drip line defines the limit at which nuclei become un- bound due to the emission of protons (neutrons) from their ground state. These drip lines are known experimentally only for light nuclei. Low Z nuclei lying beyond this limit only exists as short lived resonances and cannot be detected directly. It is therefore important to address the question: what combinations of N and Z form a bound nucleus? In the absence of experimental information, only, the theoretical investigations may indicate the position of these drip lines in the nuclear landscape. The conventional criteria used for this purpose is the limit where single nucleon separation energy (Sp for proton orSn for neutron) or more accurately two nucleon separation energies (S_2p or S_2n) just approach zero. Different theoretical investigations predict the location of these drip lines within a few units or a small nuclear band. In the SHE region or beyond it is therefore very important to find which combination of N and Z forms a bound nucleus? Here, the nuclear structure will play a dominating and maybe even a decisive role.

B. Atomic physics aspects

Assuming the nucleus can be formed, we can start to ponder over how the bound electron states change when we consider a superheavy nucleus. The electronic states in light atoms

(6)

are well described by the Schr¨odinger equation, but when the nuclear charge is increased the kinetic energy of the electrons will also increase to a point where a relativistic framework is needed to describe them. Such a framework is provided by the Dirac equation [13], which can be solved exactly for one electron bound to a point charge. Now, these solutions indicate that the energy includes a factor: √

1−α²Z² (this precise expression refers to the ground state), where α is the so-called fine-structure constant (α ≈ 1/137). It is clear that the factor √

1−α²Z² will become complex for Z >1/α. Does this then mean that it would be impossible for the electron ground state to form around a nucleus of such a high charge?

No, the real nucleus will have a finite charge distribution which smoothes the electrical field.

When this is accounted for much higher charge states can be contemplated. When replacing the point nuclear charge with a more realistic charge distribution the limit will instead be found when the lowest bound state dives into the so called negative energy continuum. The Dirac equation provides two types of solutions. In addition to the states with an energy close tomc² (for free electrons as well as electrons bound to light nuclei), there are solutions with an energy around −mc². We interpret these states as positron states. More precisely:

vacuum is thought of as being the state of matter where all negative energy states are filled, and a positron is the physical manifestation of the presence of one unoccupied state. The positron will again have an energy close to mc², but a positive electric charge. An electron can make a transition to such an empty negative energy state, thereby giving away the excess energy as an energetic photon. This is the process of electron-positron annihilation.

In the absence of external fields the energy gap between the positive and negative energy states is 2mc², but in the presence of a strong electric field, e.g. from the nucleus, this gap diminish, and at some point (for large Z) it will disappear. At this point there is no more any stable electronic states and spontaneous electron-positron pair creation is possible. For which nuclear charge will this limit be hit? In recent compilation by Pyykk¨o [2] an extension of the periodic table up to Z = 172 was proposed based on the so called Dirac-Fock model of the electron-electron calculation and a nuclear potential from an extended nucleus. This may be far beyond the region where we expect that a nucleus can be formed and it is thus most probably the physics of the nucleus will decide where the Periodic Table terminates and not that of the electron shells.

(7)

FIG. 1: Upper end of the Chart of the Nuclides (From: Ref. 14)

III. EXPERIMENTAL STATUS

We have seen that the heaviest elements exist only due to shell effects. These effects were the basis for postulating new magic islands and regions of stability setting off the race for the discovery of new elements. The key to discovery is a conclusive mass identification which is a complex process requiring that the results be independently confirmed. To date, superheavy elements upto Z=118 have been produced artificially through the complete fusion of heavy ions. Of these, the synthesis of elements 113, 115, 117 and 118 has yet to be confirmed and named. As we can see from Figure 1, the uppermost end of the nuclear chart still looks quite sparse.

A very brief overview of experimental aspects is provided here and the reader is referred to [14] for a recent comprehensive review of the subject. The artificially synthesised product of interest is created through a two stage process where the excess energy imparted to the fused compound nucleus (CN) is evaporated through the loss of neutrons. The resulting evaporation residue (EVR) is expected to be the (unknown) super heavy nucleus which decays by the emission of a sequence of α particles. For less n-rich nuclides, the decay pro-

(8)

ceeds through a previously studied mass region until it terminates by long-lived daughters, spontaneous fission (SF), or β decay the last of which cannot be measured by the detec- tors which are in use today. Advantage is taken of the fact that the decay of the entire sequence is spatially well localised within a small area of the detection system so that decay energies and life times can be measured “in situ”. The time signals and parent-daughter α-α correlations provide the criteria for identification and mass assignments. In general, the observed nuclides are extremely short lived and sophisticated techniques are used to distinguish between true correlations and random sequences.

Equally conclusive mass identification is however not always possible when the decay chain transits and terminates in an unknown mass region as is the case with very neutron rich parents where the sequence may end with SF. First discovered by Petrzhak and Flerov [15], SF appears as an additional and competing decay mode over alpha emission in the trans- Fermium region. Historically speaking, and central to the development of techniques and experimental facilities optimised for heavy element synthesis, was the debate over whether α decay or SF was more reliable as proof of production. It was already clear that α-α correlations could provide a robust means for mass identification when decay chains end in known regions. By analogy, SF would also be expected to provide a reliable indication of the mass of the parent if the (typically two) fission fragments could be properly identified.

This would require the simultaneous measurement of A and Z of both fragments as well as the number of emitted neutrons and other decay-specific observables. These quantities have not been adequately measurable so far, prompting the development of the somewhat analogous method of α−SF or α− α−SF correlations which has yielded satisfactory results. When used in combination with other techniques to cross check observations, it has become possible to synthesise and identify previously unknown and highly n-rich nuclides [16]. These, it was hoped, would lead the way to the next magic number predicted at N=184. A comprehensive critical evaluation of experimental data for the heaviest elements is provided in [17, 18] and updated through [19]. These decay properties are also retreivable through the “LiveChart” of the nuclides [20] and select cross-section information is provided through [21].

With the number of neutrons greatly exceeding the number of protons in SHE, the neutron-to-proton (N/Z) ratio is a handy index. For the heaviest nuclides, N:Z ∼ 1.5 [17, 18]. To reach the predicted doubly magic nucleus at Z=114 and N=184 (N/Z∼1.6) for

(9)

instance, reactions must be capable of producing even more neutron rich species. In the face of steeply decreasing cross-sections with the addition of every proton, the synthesis of a new super heavy element is synonymous with the production of a single atom. The highest production yields (a few atoms) for SHE are obtained with the most stable neutron-rich projectiles available in nature such as ⁴⁸Ca. It is at once obvious that one seriously limiting factor is the availability of appropriate projectile-target combinations which are rapidly becoming scarce. As it happens, we have reached the limits of experimental sensitivity and new techniques will have to be formulated to reach the predicted “magic” shell at N=184.

It is interesting to note that of the heaviest elements synthesised to date, odd nuclides comprise the majority. Extremely short lived as they are, their decay half-lives are typically longer than their even-even neighbours and the predominant decay mode is through the emission of an alpha particle. An excellent overview is provided by Seaborg [22]. In practice, whileαdecay energies can be very accurately measured, decay times have large uncertainties due to poor statistics. As may be expected of odd nuclides, α decay proceeds through excited levels (long range transitions) which are very closely spaced above the ground state.

Experimentally however, the ground state of odd-nuclides is itself difficult to identify. Since theoretical predictions are usually made only for ground states, any attempt at comparison with experimental data is therefore accompanied by some degree of ambiguity. Furthermore, neither the decay energies nor emission times can be considered to be signatures of the decaying nucleus and a conclusive identification of the newly discovered species remains a challenge. In the absence of direct mass measurements, the availability of elemental signatures will become increasingly important for the successful identification of the heaviest species in next generation experiments.

Since alpha decay has become synonymous with SHE synthesis and has been extensively discussed in the literature, we will not deal with it any further here. Instead, some considerations regarding both nuclear and atomic gamma radiation and its measurement, relativistic effects and a comparison of competing decay modes will be discussed. We end the section with a few comments on transactinide chemistry and the naming of new elements followed by the some thoughts on the future of heavy element synthesis in the context of next generation experimental facilities.

(10)

A. Gamma radiations from Nuclear and Atomic processes

Upon being formed, the compound nuclear system is usually in an excited state. Any excited nuclear level can return to a lower energy or to its ground state through gamma radiation or via the emission of electrons following internal conversion (IC) and Auger electrons.

The transition from an excited level of a nucleus to a lower level of the same nucleus without the emission of a photon is due to IC. The energy involved in the nuclear transition is transferred directly to a bound electron within the same atom. The nuclear energy difference is E_{dif f} is “converted” to the energy of the atomic electron which is immediately ejected such that: E_i = E_{dif f} −B_electron, where E_electron is the atomic binding energy (BE) of the electron. The ejected electron will have a kinetic energy less than the gamma energy by the binding energy of the K-, L- etc. orbits. The process competes with γ emission and a nucleus may decay by both modes. It is important to recognise that the two are independent processes. This distinction is important for the experimentalist who must ideally take into account both components for the correct estimation of the total decay time.

The capture of an atomic electron by a nucleus is also an important mode of radioactive decay. It competes directly with positron decay and predominates when β⁺ decay is not energetically possible as the case may be for very n-rich SHE. If a vacancy is created following EC in an atomic K- or L- shell, X-rays are emitted in the corresponding K- or L- series or by the emission of Auger electrons.

Specifically, the “hole” may be filled by an electron from the higher L-shell followed by the emission of a characteristic K_α X-ray (in heavy elements) or after K-capture by the emission of an Auger electron (in the lighter elements) [23]. In fact, a K- X-ray is expected to be emitted with near certainty for large Z elements such as the super heavies.

IC and K capture are quite similar except that in the case of the former, the X-ray energy is from a nuclear transition whereas in the case of the latter with the emission of an Auger electron, the energy is from an atomic transition. To assist in identifying an unknown nucleus, it should be noted that in all cases except that of IC and gamma ray emission, the photon radiation is characteristic of the daughter nucleus. For IC and gamma ray emission, the X-rays are characteristic of the parent in which the original nuclear transition took place.

Related to these processes are the existence of isomers, which are longer lived excited

(11)

states which decay by IC or delayed gamma emission. X-rays are useful in identifying nuclear isomers in transitions which compete with β emission [23]. Accurate theoretical predictions of these quantities are important in guiding experimental work.

B. Relativistic Effects

Segre and Helmholtz [24] adopted the condition that the velocity of the K electron will be small when compared to the conversion electron and proposed the relationα_k=2^5/2Z³α⁴ν^−7/2 where ν is the energy of the gamma ray in units of mc². From here it is evident that the rapid increase as a function of Z³ indicates that IC will become increasingly important for the heaviest nuclides. As to the magnitude of the conversion coefficients, their non- relativistic treatment was inadequate as the velocity of the electron in either the bound or ionised state is large and may even increase further with the addition of more nucleons in the heaviest elements. The earliest calculations using the Dirac formalism by Taylor and Mott [25] showed that the incorporation of relativistic effects provides better agreement with experimental data.

In fact the inclusion of relativistic effects reveals some other surprising facts. With every increase in Z, the electronic orbitals will move even closer to the core thereby causing the electrons to move faster resulting directly in a relativistic mass increase given by m_rel = m₀ p

1−v²/c². The corresponding decrease in the Bohr radius for the s and p_1/2 electrons is then a_B = ~²/m_relc² = a⁰_Bp

1−v²/c². This contraction of the orbitals was shown to originate from the inner K- and L- shells [26, 27] and could be understood. However, the somewhat large effect on the outer s and p1/2 valence electrons was quite unexpected.

For instance, the contraction as a percentage difference of the relativistic (rel) and non- relativistic (nr) radial distribution in the 7s orbital in Db (Element 105) is found to be

∆_Rhri= (hri^nr− hri^rel/hri^nr)=25% [28]. Even more interestingly, these orbital contractions reach their maximum in the 6th period with Gold (17.3%) and in the 7th period with Copernicium (31%) and are associated with changes in chemical properties and remain the subject matter of vigorous investigations. An analysis of higher order effects is provided in [28].

From this discussion, it should be sufficient to note that the reduction (or stabilisation) of orbitals due to relativistic effects has a direct impact on the phenomenon of internal con-

(12)

version where with increasing Z, the increase in the conversion coefficient is influenced by boththe decrease in the radius of the Bohr orbit as well as the transition energies due to the more closely spaced levels. Fortunately, since the calculation of internal conversion coefficients is essentially an atomic rather than a nuclear problem, it is possible to calculate these quantities more accurately than determining the lifetimes (widths) of gamma transitions for which detailed information of electromagnetic matrix elements and structure information is required. In the absence of supporting structure information, an attempt to “fit” observed gamma energies into a decay scheme, could lead to erroneous results.

We conclude by pointing out that as with nuclear levels, the energy differences in observed atomic spectra also reflect the differences between the binding energies of atomic electrons in the different shells. The ratio of conversion between K, L and M shells associated with the emission may be directly compared with predictions of the multipolarity of the levels.

Meitner [5] especially emphasised that the fact that the K-L difference has to be the same as the K_α X-ray photon energy and hence, is a direct measure of the atomic number of the atom in which the nuclear transition is taking place.

Practically speaking, given the paucity of data for the SHE, the selection of atomic shell binding energies could have a significant impact on estimating the conversion coefficients, especially for cases when the kinetic energy of the emitted electrons is very low. A precise knowledge of the binding energies is very essential to the experimental identification of conversion electrons and X-ray radiations as well as to determine the atomic number of the emitting nucleus [29].

Recent reports [30, 31] of the very accurate measurement of the first ionization potential for Element 103 (Lr) are very promising and will be very useful to our understanding of the region.

C. Comparison of competing decay modes

As we have seen, it is possible that the discovery of new shell closures around N=184 could result in longer lived nuclides in very neutron rich mass regions. It is likely that energy levels will be extremely closely spaced and it is not clear whether evidence for these states will be discernable within the existing experimental sensitivities. Their presence may be deduced indirectly by realising that in cases where an excited state is longer lived, IC, SF

(13)

or β decay could provide competing modes for de-excitation which could serve to shorten the expected half-period. α decay may also co-exist as a strong possibility closer perhaps to regions of local stability where it is conjectured that SF may proceed at a slower rate.

In general however, SF is expected to increasingly become the most frequently observed decay mode depending also on the method of synthesis. There are compelling reasons to believe that new modes of fission would also exhibit themselves. We will not discuss these here. However, we shall make a brief attempt to understand which decay mode might have the highest likelyhood of being associated with detectable gamma emission in future experiments.

The most general order-of-magnitude comparison may be put forward extending the arguments proposed by Fermi [32]. Taking into account transition rates of beta decay and life-times of alpha decay, we see that the exponentially varying alpha decay rate with energy would overshadow the comparatively slower transition rates for beta decay (∼E⁵τ).

Consequently, gamma decays from a nucleus created following beta emission would be more likely than if the daughter was produced through alpha decay. The fact that gammas are most elusive following alpha decay is borne out through contemporary experiments which are optimised to detect rare α events. Indeed as an experimental rule of thumb for superheavy synthesis, oneγis expected to be emitted for approximately every 1000αdecays.

We have seen that for superheavy elements, the decay half-lives are governed by shell effects alone and spontaneous fission appears as an additional decay mode.

In fact, with the high N/Z ratios typically associated with superheavy elements, fission could predominate over particle emission. SF could be likened toαdecay in the limited sense that both processes involve surmounting their respective barriers and are extremely sensitive to energy considerations although the numerous factors that govern fission are quite distinct from the other processes. For our purposes here, it can be clearly seen from these arguments that SF emerges as a direct competitor to alpha particle emission in the superheavy region.

Indeed it appears as an additional decay mode for the heaviest elements.

(14)

IV. CHEMICAL PROPERTIES AND PLACEMENT IN THE PERIODIC TABLE OF ELEMENTS

In the above discussions, we have neglected the effect of electron screening with the contraction of atomic orbitals. Broadly speaking, this effect is useful to the chemist who in order to place new elements in the Periodic Table, requires information on chemical properties which manifest themselves through the interactions of the outermost valence electrons. We have seen that as the number of electrons increases, atomic orbitals get closer together with the innermost ones squeezing in closer to the centre. Screening due to this dense electronic “cloud” would affect the behaviour of the outermost electrons which may experience decreased binding and in turn influence their chemical properties. The interplay between relativistic effects and screening are not well understood and chemical investigations employ varied methods to measure different chemical properties of these heavy systems.

Chemists have found it useful to synthesise the heaviest elements using well studied reactions employed by the physicist. Details will not be provided here and the reader is referred to excellent reviews of the subject such as [33]. For specific instances of interest, a comparative study of data from both chemical and physical studies can be found in [17].

The placement of new elements in the Periodic Table has been possible through chemistry experiments. However, only a few nuclides lend themselves to such studies since a large number of them have half lives shorter than the few seconds necessary for their chemical detection. Where possible, chemical procedures are employed to enable the separation of

“atomic number” which is facilitated when a given isolated element exhibits special properties [34]. Over the past decade isotopes of Bh and Hs have been chemically investigated (see for instance [33] and references therein) and a combination of both physical and chemical techniques have been successfully used to approve the discovery of elements above Rf. All the elements upto 112 have been named (see [14] for a history) and the most recent addi- tions are elements 114 and 116, named Flerovium (Fl) and Livermorium (Lv) respectively [35]. The heaviest element produced to date is ²⁹⁴118. We stand at the limit of present day experimental capabilities and are about 7 neutrons away from the closed shell predicted at N=184.

(15)

V. THE FUTURE OF HEAVY ELEMENT SYNTHESIS

The artificial synthesis of elements heavier than Z=118 will require facility upgrades as well as the development of new methods of synthesis beyond heavy ion fusion as it is known today. There is every indication that the key to a successful future would involve probing the intensity frontier. Multinucleon transfer reactions are emerging as a viable path to the discovery of lower Z, neutron rich superheavies. Methods to reach the ”magic island” at N=184 however, remain elusive.

Assuming new methods of production are found, future experimental facilities would have to be designed to measure an increasing richness of decay modes associated with a surplus of neutrons expected in highly unstable ultra heavy mass regions. Noting that X-ray emission from IC is near certain with higher Z, future detection systems must be designed to adequately measure electromagnetic radiations following both beta decay and IC. The detection of signature radiations will be crucial to accurate mass identifications. It can be stressed that since IC does not involve a change in nuclear charge, X-ray spectra are characteristic of the element in which the nuclear transition took place. X-rays from daughters will also be very helpful in identifying the decaying nucleus in hitherto unstudied regions through which mass chains will inevitably transit and end. The inherent merits of both can be seen at once in the hypothetical context of a fissioning nuclide where the fragments are observed within a decay sequence. If direct mass measurements are not possible, X-rays from the two (or more) fission products could provide a reliable elemental signature for the fragments from which an approximate mass of the fissioning parent can be estimated. Indeed legacy studies have shown that the determination of both elemental and isotopic signatures will be important to next generation experiments.

To reach the heaviest regions as a test of existing theories, physics at the intensity frontier will require a substantial upgrade for existing facilities such as SHIP at GSI. The conceptual design of the next generation separator is underway as part of a collaborative project between Manipal University, GSI and the University of Giessen (Figure 2) [36].

In the final analysis, it is highly probable that the upper limit of the Periodic Table will be dictated by nuclear, rather than atomic, considerations. While the theorised upper limit may never be reached experimentally, there is reason to believe that new phenomena at the interface between atomic and nuclear physics will bring more clarity to the debate on this

(16)

FIG. 2: Conceptual plan of a next generation separator for super heavy elements. From: G.

M¨unzenberg, EXON 2014.

important question which has eluded an unequivocal answer over the better part of the last century.

VI. NUCLEAR THEORY

The Relativistic Mean Field (RMF) or the effective Lagrangian theory [37, 38] is now well established and is known to accurately describe the ground state nuclear properties of nuclides across the entire periodic table. It starts with a Lagrangian describing the Dirac spinor nucleons interacting only via the exchange of mesons and the photon. The mesons considered generally are: (1) the isoscalar - scalarσ (2) isoscalar - vectorω and (3) isovector - vectorρ mesons. Theσ (ω) meson produces long range attraction (short range repulsion), whereas the ρmeson is necessary for the isospin dependence of the nuclear properties. The photon, as usual, produces the Coulomb interaction. The Lagrangian therefore consists of free baryon and meson terms and the interaction terms involving coupling constants and the meson masses. This choice of the Lagrangian is not unique. Some authors have used different Lagrangian by including additional terms or dropping some of the existing (non linear) terms and even introduced density dependent coupling constants.

The Euler - Lagrange variational principle yields the equations of motion. At this stage the mean field approximation is introduced by replacing the fields by their expectation values

(17)

(c - numbers) as a result of which one ends up with a set of non-linear coupled equations:

1. The Dirac equation with potential terms involving meson and electromagnetic (e.m.) fields describing the nucleon dynamics.

2. A set of Klein-Gordon type equations with sources involving nucleonic currents and densities, for the mesons and the photon.

This scheme is schematically shown in Figure 3. Here, m_σ (g_σ), m_ω (g_ω) and m_ρ (g_ρ) represent the masses (coupling constants) of σ, ω and ρ fields respectively; g2 (g3) is the coupling constant for the cubic (quartic) self interaction term(s) for the σ field; ande is the electronic charge.

The pairing correlations, essential for the description of open shell nuclei, are incorporated either by simple BCS prescription, or self consistently through the Bogoliubov transforma- tion. The latter leads to the Relativistic Hartree Bogoliubov (RHB) equations. The RHB equations have two distinct parts: the self consistent field (h_D) that describes the long range particle-hole correlations and the pairing field ( ˆ∆) that accounts for the correlations in the particle-particle (pp) channel. The pairing field ˆ∆ involves the matrix elements of the two body nuclear potential in thepp-channel. In the case of the constant gap, ˆ∆ (≡ ∆) becomes diagonal as a result, the RHB equations reduce to the RMF equations with a constant gap (BCS - approximation). The BCS type expressions for the occupation probabilities (v²) [37, 38] is given by:

v_k² = 1 2



1 − _k − λ q

(k − λ)² + ∆²





where _k is the energy of the single particle state k and the Lagrange multiplier λ (fermi energy) is determined through the BCS number equation. This set of non-linear coupled equations, known as RMF equations with constant gap or RHB equations is to be solved self -consistently.

The sources (nuclear currents and densities) appearing in the above Klein - Gordon equations involve super spinors or in the constant gap approximation simply the corresponding BCS occupation probabilities. In practice, the sum is taken over the positive energy states (no-sea approximation).

(18)

FIG. 3: Flow chart depicting RMF formulation.

Nucleons Interact With

Fields Meson

σ, ω, ρ

Masses: M, m

_σ

,m

_ω

, m

_ρ

Lagrangian

C − Numbers Approx:

Equations of Motion

Operators Field Classical

Principle Variation

Relativistic Mean Field: RMF γ

and

Dirac Equation: Nucleons

K. G. Equation: Mesons & Photon

{ _Couplings: _g

_, _,

σ

g

_ω ρ ,

g

₃

g

₂

For: + +

Densities and Currents Sources: Baryon

(RMF Equations) To be solved Self − Consistently Closed Set of Equations

Static Case

Charge

Conserv Reversal Time

Dirac Eqn:

K. G. Eqn:

Scalar + Vector Pot.

M

)

= M +

* σ g

_σ

)

ω , ρ

o ^oo^,

A

^o

(

(19)

A reliable and satisfactory inclusion of nuclear interaction in pp channel (pairing interaction) V^pp is not yet achieved in RMF [38, 39]. Therefore, in practice, it is customary to adopt a phenomenological approach while solving the RHB equations. Therefore, one often uses for V^pp, the finite range Gogny-D1S [40, 41] interaction, which is known to have the right pairing content. In the case of the constant gap approximation, the required gap parameters are either introduced phenomenologically or are determined so as to reproduce the corresponding RHB pairing energies obtained by using Gogny D1S pairing interaction.

The parameters (nucleon and meson masses and the respective coupling constants) appearing in the Lagrangian, are fixed by reproducing the observed properties (e.g. binding energies, radii, isotopic shifts etc.) of a few selected spherical nuclei through a chi-squares fit.

These are then frozen and are used in the calculation for any nucleus spherical or deformed spread of the entire periodic table. The explicit values of parameters so determined are not unique and depend upon the specific experimental data included in the fit and the form of the Lagrangian used. Therefore, a number of set of Lagrangian parameters exist. Basically, all the sets reproduce the nuclear ground state properties (total binding energies, nucleon distributions and nuclear radii (sizes), deformations, single and two nucleon separation energies etc.) rather well. For example, the binding energies are reproduced with in 0.25%

and radii up to second decimal place of Fermi.

The RMF has also been employed extensively to nuclei in the super heavy region. The interested reader is referred to [8] and references therein. Calculated ground state properties including α-decay Q-values and half lives agree reasonably well with experimental data.

The shell evolution, extra stability (”magicity”) in this region were also investigated. We have found that extra stability or magicity depends upon a specificcombination of neutron number (N) and the proton number (Z) of the nucleus, rather than the conventional single value of N or Z alone.

The reader may note that the RMF equations can be solved either by using the basis expansion method or directly in the coordinate space itself. The former method employing the oscillator basis for the expansion of nucleon spinors and the meson fields has been extensively used. One of the main criticisms of this procedure is that it does not incorporate the correct asymptotic behavior which may play a vital role specially for the very loosely bound nuclei or those at the drip line. The problem requires further investigation.

(20)

VII. CONCLUSIONS

The question of how many elements can exist in nature has been discussed from the point of view of contemporary thought in atomic and nuclear theory. Nuclear physics effects are likely to determine the natural upper limit to the Periodic Table, perhaps long before Z=172, as predicted by atomic theory, is reached.

The limit of experimental sensitivity has been reached before this question can be adequately addressed. As we stand today at the technological limit of heavy element synthesis, it is argued that the solution to this fundamental and challenging problem must emerge through theoretical investigations guiding both the design of next generation experiments as well as new methods of synthesis for heavier species.

There is reason to believe that a better theoretical understanding at the interface between atomic and nuclear physics could greatly assist in resolving this most fundamental and important question. For instance, in present day atomic theory, nuclear structure effects are taken into account only through the use of an extended charge distribution which may not be adequate.

Preliminary experimental results based on data from multi-nucleon transfer reactions are encouraging and provide some hope that the method could be extended to heavier mass regions. It is clear that the design of future experimental facilities will need to be optimised to accommodate the exacting requirements dictated by new techniques of synthesis for the heaviest elements.

Next generation in-beam detection systems will also allow for the sensitive measurement of an increasing richness of decay- as well as fission- modes associated with the large surplus of neutrons expected in highly unstable ultra heavy mass regions. Noting that X-ray emission from IC is near certain with higher Z, new facilities must be designed to adequately measure electromagnetic radiations following both beta decay and IC. It would not be surprising if the efficacy of future detection systems would be determined primarily by the efficiency with which such decays are measured.

(21)

Most importantly, the capability for simultaneously measuring nuclear and atomic radiations and decays as well as accurate masses from next generation experimental facilities is seen as an inevitable and inescapable requirement for providing conclusive observational evidence for the discovery of new elements. This would enable the experimental verification of limiting mass (A) as a combination of Z and N (nuclear decay modes) and limiting charge (Z) (signature atomic radiations) which would go a long way in answering the very fundamental question of how many elements there can be.

Acknowledgments

This work was carried out under the program Dynamics of Weakly Bound Quantum Systems (DWBQS) under FP7-PEOPLE-2010-IRSES (Marie Curie Actions People Inter- national Research Staff Exchange Scheme) of the European Union. Work done under the Manipal University-GSI-University of Giessen collaboration is also acknowledged.

[1] R. C. Barber et al., Prog. Part. Nucl. Phys. 29, 453 (1992).

[2] P. Pyykk¨o, Phys. Chem. Chem. Phys.13, 161 (2011).

[3] O. Hahn and F. Strassmann, Die Naturwissenschaften27, 11 (1939).

[4] O. Hahn and F. Strassmann, Die Naturwissenschaften27, 89 (1939).

[5] L. Meitner and O. R. Frisch, Nature143, 239 (1939).

[6] V. M. Strutinsky, Nucl. Phys. A 95, 420 (1967).

[7] W. D. Myers and W. J. Swiatecki, Nucl. Phys. 81, 1 (1966).

[8] Y. K. Gambhir, A. Bhagwat, and M. Gupta, Ann. Phys. 320, 429 (2005).

[9] A. Sobiczewski and K. Pomorski, Nucl. Phys. 58, 292 (2007).

[10] G. G. Adamian, N. V. Antonenko, and V. V. Sargasyan, Phys. Rev. C79, 054608 (2009).

[11] V. Pershina, inRelativistic Electronic Structure Theory, edited by P. Schwerdtfeger (Elsevier, Amsterdam, 2004), vol. 14 of Theoretical and Computational Chemistry, p. 1.

[12] P. Pyykk¨o, Chem. Rev. 88, 563 (1988).

(22)

[13] P. A. M. Dirac, Proc. R. Soc. A 117, 610 (1928).

[14] G. M¨unzenberg and M. Gupta, inHandbook of Nuclear Chemistry, edited by A. Vrtes, S. Nagy, Z. Klencsr, R. Lovas, and F. Rsch (Springer US, 2011), pp. 877–923, ISBN 978-1-4419-0719-6, URLhttp://dx.doi.org/10.1007/978-1-4419-0720-2_19.

[15] K. A. Petrzhak and G. N. Flerov, Die Naturwissenschaften28, 500 (1940).

[16] Y. Oganessian, Journal of Physics G Nuclear Physics 34, 165 (2007).

[17] M. Gupta and T. W. Burrows, Nuclear Data Sheets 106, 251 (2005).

[18] M. Gupta and T. W. Burrows, Erratum: Nuclear Data Sheets 107, 789 (2006).

[19] ENSDF,www.nndc.bnl.gov/ensdf.

[20] LiveChart of Nuclides,http://www-nds.iaea.org.

[21] EXFOR,http://www-nds.iaea.org.

[22] G. T. Seaborg and W. D. Loveland, John Wiley and Sons (1990).

[23] R. Evans, The Atomic Nucleus (1955).

[24] E. Segr`e and A. C. Helmholz, Reviews of Modern Physics21, 271 (1949).

[25] H. M. Taylor and N. F. Mott, PRSA, London 138, 215 (1933).

[26] W. H. E. Schwarz, E. M. van Wezenbeek, E. J. Baerends, and J. G. Snijders, Journal of Physics B Atomic Molecular Physics 22, 1515 (1989).

[27] E. J. Baerends, W. H. E. Schwarz, P. Schwerdtfeger, and J. G. Snijders, Journal of Physics B Atomic Molecular Physics 23, 3225 (1990).

[28] D. Hoffman, D. Lee, and V. Pershina, inThe Chemistry of the Actinide and Transactinide Ele- ments, edited by L. Morss, N. Edelstein, and J. Fuger (Springer Netherlands, 2006), pp. 1652–

1752, ISBN 978-1-4020-3555-5, URL http://dx.doi.org/10.1007/1-4020-3598-5_14.

[29] T. Kib´edi, M. B. Trzhaskovskaya, M. Gupta, and A. E. Stuchbery, Atomic Data and Nuclear Data Tables 98, 313 (2012), 1103.0594.

[30] T. K. Sato, M. Asai, et al., Nature Letter 520, 209 (2015), URL http://www.nature.com/

nature/journal/v520/n7546/full/nature14342.html.

[31] T. K. Sato, N. Sato, M. Asai, K. Tsukada, A. Toyoshima, K. Ooe, S. Miyashita, M. Schdel, Y. Kaneya, Y. Nagame, et al., Review of Scientific Instruments 84, 023304 (2013), URL http://scitation.aip.org/content/aip/journal/rsi/84/2/10.1063/1.4789772.

[32] E. Fermi, Nuclear Physics (1950).

[33] M. Sch¨adel, Angewandte Chemie International Edition45, 368 (2006), ISSN 1521-3773, URL

(23)

http://dx.doi.org/10.1002/anie.200461072.

[34] H. G¨aggeler, The European Physical Journal A - Hadrons and Nuclei 25, 583 (2005), ISSN 1434-6001, URL http://dx.doi.org/10.1140/epjad/i2005-06-202-2.

[35] R. D. Loss and J. Corish, Pure and Applied Chemistry84, 1669 (2012), ISSN 1365-3075, URL http://dx.doi.org/10.1351/pac-rec-11-12-03.

[36] S. Heinz, W. Barth, B. Franczak, H. Geissel, M. Gupta, S. Hofmann, S. Mickat, G. Mnzenberg, W. Plass, C. Scheidenberger, et al., Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 317, Part B, 354 (2013), ISSN 0168-583X, {XVIth} International Conference on ElectroMagnetic Isotope Separators and Techniques Related to their Applications, December 27, 2012 at Matsue, Japan, URL http:

//www.sciencedirect.com/science/article/pii/S0168583X1300699X.

[37] Y. K. Gambhir, P. Ring, and A. Thimet, Ann. Phys. 198, 132 (1990).

[38] P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996).

[39] H. Kucharek and P. Ring, Z. Phys. A339, 23 (1991).

[40] J. F. Berger, M. Girod, and D. Gogny, 428, 32 (1984).

[41] T. Gonzalez-Llarena et al., Phys. Lett. B379, 13 (1996).