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The Entropy Reduction Laboratory

Double-Shell PSE: Extended Discussion
Prolog The history and new developments of the periodic sytem are layed out. Theoretical and experimental aspects related to the new Double-Shell Periodic System are presented.

Table of Contents

  • Chemistry of Transuranium Elements
  • From Empedokles to Mendeleev
  • Electronic Structure of the Hydrogen Atom and the Periodic System
  • The Double-Shell structured Periodic system
  • The PSE in Chemistry
  • Resolving Periodic Anomalies
  • Symmetry in 3D Space - The Tetrahedral Sphere
  • References
  • See also - links to further pages of this site
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  • Chemistry

    The Chemistry of the Transuranium elements

    Recently, chemical reactions with the transuranium element 106, Seaborgium, have been reported by Schaedel (Institut of Heavy Ion Accelerator Research , GSI Darmstadt) and Kratz (Institut for Nuclear Chemistry, University Mainz) (1). They succeeded for the first time to study chemically the very small number of Seaborgium atoms available. They demonstrated, that this element may be identified by fast chemical reactions in combination with liquid- and gas-chromatic methods analogous to the stable elements Molybdenum and Wolfram of the group 6 elements. Thus, Seaborgium behaves entirely different from the pseudohomologuous element Uranium which chemically is homologuous to Neodym. The authors point out that with this identification the fundamental question on the validity of the the current periodic system of the elements also for very heavy elements has come closer to an answer.Further, since these special methods applied to single atoms have been proven to be successful, new ways in extending traditional chemical studies to short living elements are opened.

    At this turning point of transuranium element chemistry, it is appropriate to take up anew the question on the structure of periodic system of the elements, especially with respect to the position of the very heavy elements (2) in the system. Instead of asking " What predicts the periodic system as the position of current and future artificial elements? ", the question may be posed also from the opposite direction: " What can be learnt about the structure of periodic system from the chemical properties of current and future artificial elements? And What are the chemical properties of the first elements of new subshells, i.e. of the elements 113 and 119 ? " and especially " What are the new properties of element 121, which is expected to start with a shell of type g ( l = 4 ) ? "

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    From Empedokles to Mendeleev

    The notion of elements as basic building stones for matter we owe Empedokles (495 - 435 BC). In his view, there are four such essential entities which, in a multitude of mixtures, form the totality of the visible world.

    Aristoteles (384-322 BC) reiterated this idea of four fundamental elements. An ordered frame of reference was introduced by Platon (427-347 BC). He envisaged a more concrete picture and related the shape of the building stones of matter and of the universe to the shape of the five regular polyhedra (the Platonic bodies):

    • earth relates to the shape of the hexahedron
    • water relates to the shape of the ikosahedron
    • air relates to the shape of the octahedron
    • fire relates to the shape of the tetrahedron
    Three of these assignments reflect the most striking differences between the physics of solids, liquids, and gases. The exclusive property of fire is attributed to the exclusive polyhedron, the self related tetrahedron. The fifth polyhedron, the dodekahedron, Platon relates to the function of the mysterious quinta essentia, the ether of heaven.

    In the 10th century the scripts of the mysterious Arabic scientists Dschabir became known in Europe and introduced alchemy and the art of goldmaking. These and the ideas of Aristoteles dominated the activities in physics and chemistry all through the middle-ages. The goal of alchemy concentrated on the search for the elixir of life to create immortality and on the search for the philosopher's stone to tranform base matter into higher matter. Even eminent personalities like bishop Albert Bollstaedt (1193 - 1280), known as Albertus Magnus and widely admired scientist of his time, claimed to have found the transition into gold and silver to be possible. As a byproduct he discovered Arsenium.

    Even though the methods were unsystematic, many useful techniques of modern chemistry like glassworking and destillation were developed. In 1789 Lavoisier was able to set up a list of 23 identified elements. In 1829 the idea of periodicity in the chemical properties of the elements was perceived by Doebereiner who proposed for a subset of 15 of the then known elements a system of 5 triads. An elaborous version elucidating periodicity, based on increasing atomic weight, was constructed in the form of spirals by Chancourtois (1862). A corner stone periodic system and widely accepted until today, became the periodic system of the elements by Dimitri Ivanowitch Mendeleev (1869, 1871) and Lothar Meyer (1870), systematizing the 63 then known elements with well located gaps for elements still to be discovered. Since this system was still based on the atomic weight, several mysterious inconsistencies would have remained, if not resolved intuitively by so called inversions. For a detailed account of the vivid activities to construct alternative periodic systems in the subsequent decades, see the survey of Mazurs (3).

    To get an impression of the state of knowledge 90 years back from today, the article on the elements in Meyers Konversations-Lexikon, Bibliographisches Institut, Leipzig und Wien, 1907, may be consulted. Here we find the periodic system to be still an infant state, but also the suggestion, that the atomic weight may not be the clue for setting up the final system. It was stated that systematic chemical behaviour of the elements was important, but also that the elementary property of these elements had to be seriously questioned: " Man muss annehmen, dass den verschiedenen chemischen Elementen ein und dieselbe Ursubstanz zugrunde liegt, dass sie Kondensate oder Aggregate, auf bestimmte Gesetzmässigkeiten zurückzuführende Gruppierungen derselben darstellen. Auf jeden Fall steht fest, dass die Qualitäten der Elemente auf Quantitäten zurückzuführen sind." The electron had at that time already been discovered by Thompson (1907); apparently, it was still too new a particle to be taken into consideration as an atomic ingredient.

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    of the
    Hydrogen Atom
    and the

    Electronic Structure of the Hydrogen Atom and the Periodic System

    Shortly after, the work of Rutherford (1911) on the atomic nucleus and Moseley (1912) on the X-ray spectra suggested, that not the atomic weight, but the atomic charge, being - in the neutral atom - equal and of opposite sign in the nucleus and in the atomic shell, is the decisive parameter for ordering the elements.

    In 1922 Niels Bohr (4) gave the first quantummechanical interpretation of the periodicity of the elements. According to the Bohr-Sommerfeld "Aufbau" principle, elliptical planetary orbits are filled one by one with electrons each time the nuclear charge is increased by one positive charge.

    With the great success of Schrödinger's formulation of quantummechanics in the derivation of the energy levels of the hydrogen atom, taking into account also Pauli's exclusion principle (5), this model advanced to become the theoretical basis for the "Aufbau" principle. Accordingly, the subshell energies increase by the so called (n, l) rule and by doubling the number of states due to the electron's two possible spin directions, this model explains the electronic structure underlying the periodic system. Does it really? As well recognized, beginning with Z=19, a vivid struggle between normal filling and exeptional filling of subshells starts(14).

    Despite the many exeptions from the rule, the convincing derivation of the hydrogen spectrum as a solution of the Schrödinger equation did leave no room for casting doubt on the validity of the consequential (n, l) buildup rule as the basis for the formation of the periodic system. This appears rather surprising, considering that Bohr (3) and Madelung (6) had found quite early that the "Aufbau" principle works according to the (n+l,n) - the Madelung - rule rather than to the (n, l) - the hydrogen spectrum - rule. It may be recalled, that by the (n, l) rule the energy increases with increasing n, and for fixed n, with increasing l, while by the Madelung rule it increases with increasing n+l, and for fixed n+l, with increasing n. In other words, hydrogen states have the same energy for constant n = nr+l+1, but in the PSE this is true for states with constant N = nr+2l+1. The format of one of the periodic systems designed by Janet (7) reflects the Madelung rule already in 1927.

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    The Double-Shell structured Periodic system

    For a recent review on theories about the structure of the periodic system we refer to the paper " The Periodic System of the Chemical Elements: Old and New Developments" by M.Kibler (8). Clearly, the electrons in outer shells of an atom do experience not only a central force due to the positively charged nucleus, but also an average - self-consistent - repulsive force due to all the other electrons. Sofar electron-correlation theory has not been shown to be related to the Madelung rule. However, various group theoretical interpretations of the rule have been proposed. On the other extreme, the Thomas-Fermi atom - a statistical model neglecting any fine structure - predicts the order of filling of electron shells in accordance with the Madelung rule.

    The periodic system proposed by Neubert(9) consists of only four shells, each shell being a double-shell with twice as many electrons per shell as in the conventional periodic system. In addition, the PSE as presented up side down compared to the conventional one, reflects in a natural way the filling of energy levels of an attractive potential well. Plots of physical properties of the elements, among them especially the ionization energies of various degrees, exhibit a high correlation to the structure of the PSE.

    In the PSE, the filling of a new shell starts with states of highest angular momentum of the shell and then continues with decreasing l. After filling of the lower half of a double-shell, this process is repeated for the upper half. Due to the magnetic quantum number m, there are 2l+1 states for each l. In addition to the spin quantum number s = ± 1/2, which for fixed l results in a doubling of states according to spin up and spin down electrons, a newly introduced topical quantum number c = ± 1/2 results in a further doubling of states according to electrons in the lower and upper halves of the double-shells. It is pointed out that the Schrödinger equation for the Coulomb potential contains a free parameter and consequently there are two systems, one system with the filling of energy levels according to the hydrogen spectrum (PSH), and another one according to the double-shell structure (PSE). In quantum number space, there exists a continuous transformation from the PSE-spectrum into the PSH-spectrum, if the effective strength of the Coulomb attraction is continuously increased, while the number of electrons is kept constant (9). If Z is the nuclear charge and Z' the number of electrons which can be bound by Z,

    Z' = kappa × Z,

    then kappa is related by

    kappa = cos(delta)

    to the rotation by the angle delta in quantum number space. For the PSE holds kappa=1, for the PSH kappa=1/2. Thus, the binding power per unit nuclear charge is much weaker in case of the PSH than in case of the PSE. In order to arrive from the atoms of the PSE at a periodic system corresponding to the hydrogen spectrum, by this interpretation at each constant number of electrons in the shell, the nuclear charge has to be increased by a factor two.

    Group theoretical investigations of the (n, l) rule and the Madelung rule reveal the inherent differences in the level structure of the two resulting systems. According to Barut (10), the relevant chains of subgroups are:

    Ionized atoms  : SO(4,2)   SO(4,1)  SO(4)
    Neutral atoms  : SO(4,2)   SO(3,2)  SO(3) x O(2)

    Kibler (8) derives the periodic table on the basis of a chain of subgroups starting with

    SU(2) x SO(4,2).

    Both authors conclude that a double-shell structure of the PSE results as a natural consequence. Furthermore, Barut confirms an earlier remark (9), that the PSE resembles the levels of a three dimensional harmonic oscillator with alternating parity. He demonstrates, how, by redefinition of the quantum numbers, the oscillator shell structure can be made to look like that of the hydrogen atom.

    On the other extreme, as discussed in detail by Essén (11), the predictions of the - statistical - Thomas-Fermi atom with respect to the order of completing the filling of subshells do not agree with the predictions of the (n, l) rule but with those of the Madelung rule and thus lead to a double-shell structure.

    Another semiclassical consequence of the Madelung rule has been pointed out by Wheeler (12). According to this rule, the electron energy at the top of a filled distribution stays constant, so long as

    dN = dnr + 2dl = 0

    which implies that the corresponding atomic orbit has the shape of a "necklace " which closes after a deflection of of 4pi and not "as popularized throughout the world " the shape of an "ellipse " which closes after a deflection of 2pi as in case of a pure Colomb potential. Demkov and Ostrowski (13) have introduced a " focussing " potential which conforms to this necklace condition and to Madelung rule.

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    The PSE

    The PSE in Chemistry

    The chemical properties of the elements have been critically reviewed from time to time (15) , especially also with respect to the role of the Lanthanides. The PSE complies with these considerations. Recently, Weir et al. (16) have produced a metallic form of liquid hydrogen and thus the placement of hydrogen in the group of the Alkalahides was put onto a firm basis also from the chemical point of view. The fact that He, with two outer s - electrons, behaves chemically like a noble gases with outer p - electrons, is to be looked at as a fortuitous coincidence. As for the chemical properties of the transuranium elements, it has been noted in (1) that Dubnium, element 105, of group 5, reveals chemical similarities to Nb, element 41, but not to Ta, element 73, of the same group. This is not a contradiction, considering the fact that certain properties, if plotted (9) "vertically" for the elements of a given group, tend to show a discontinuity at a double-shell boundary. More examples are given below in the section Resolving Periodic Anomalies.

    Many properties depend clearly on the number of core electrons and not on the proton number of the nucleus, as seen most clearly from plotting the ionization energies "horizontally" for the elements of the subshells. Incidentically, if only certain properties are taken into account, the dream of the alchemists is not so hard to fullfil after all: e.g. Hg, element 80, once ionized, resembles with respect further ionization the property of gold and so do other isolelectronic ions!

    Finally, a remarcable combination of extreme properties is exhibited by the element beryllium. Even though very light, it is a very stable metal. Its elasticity is larger than that of steel, its strength four times that of aluminum. The melting point at nearly 1600 K is very high. In the gas phase beryllium is monoatomic. As a possible reason for these extreme properties we observe, that the element Be has four electrons, which migtht provide an explanation on the basis of the hypothesis presented in the chapter on the symmetry in 3D space.

    The formation of diatomic molecules is modelled in a natural way by the PSE. The group 18 elements are the center for ionic compounds: Starting e.g. from Ne, element 10, we get as possible diatomic compounds NaF, MgO, AlN up to the very hard SiC of strong ionic binding, but already with a lattice of the zinc blende structure. It looks like a trick of nature: with respect to Ne as the origin, there aree 2 s-electrons and 6 p-electrons involved to form SiC. Adding 8 protons to each of the C atoms, the Si - crystal evolves, which is of the same lattice structure, but of covalent bonding, each bond being a sp3 hybrid bond as predicted by modern solid state theory. Now, starting from any of the group 14 elements as the center, the typical III-V and IV-VI semiconductors with predominanly covalent bonding result in agreement with the evidence exhibited by the conventional periodic system.

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    Resolving Periodic Anomalies

    It has been pointed out by Neubert(9) that the double-shell structure of the PSE suggests that the properties of elements along a given group will not change "continuously" in passing from the lower halve to the upper halve of a double-shell and passing between double-shells. The resulting discontinuities in the plots of properties were demonstrated to exist indeed for ionization energy and electronegativity as typical examples. Barut(10) took up this idea and proposed to plot separate property curves for the elements of the lower and upper halves of double-shells. These two curves run about parallel, but don't coincide, as would to be expected for the conventional atomic shell structure.

    A number of anomalies in chemical reactions along the chemical groups in the conventional periodic system have been reviewed by Huheey, Keiter and Keiter(17). Some typical examples of such reactions are:

    1. The reduction of group 14 halides with zinc and hydrocloric acid shows
      • germanium resembles carbon
      • tin resembles silicon
    2. Alternating behaviour of group 14 elements in organolithium reactions:
      • Ph3CH (triphenylmethane)  + LiR
                --> LiCPh3 + RH
      • Ph3SiH (triphenylsilane)     + LiR
                --> SiPh3R + LiH
      • Ph3GeH (triphenylgermane) + LiR
                --> LiGePh3 + RH
      • Ph3SnH (triphenylstannane) + LiR
                --> SnPh3R +LiH
    3. For the pentaclorides of the group 15 elements holds:
      • PCl5 is stable
      • AsCl5 is unstable
      • SbPCl5 is stable
    4. A further illustrating example is given on page 259 of the book of Huheey et al. by a figure relating crystal structure to atomic charge. The sum of charge coefficients is plotted over the difference in electronegtivity of the participating atoms. Among others there are shown property curves for the alkali halides. For X= I, Br, Cl, and F, it is seen that
      • the curves LiX and NaX lie close together; than there is a gap.
      • the curves KX and RbX lie close together; than there is a gap.
      • the curve CsX lies isolated; the curve FrX has not yet been measured;
      The data point to the effect, that the gap lies not between the halides of different double-shells, but rather between the halides of the lower and upper halve of a given double-shell.
    By the double-shell structure of the PSE these four examples of "irregularities" appear to be rather regular.

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    in 3D Space

    Symmetry in 3D Space - The Tetrahedral Sphere

    The AUFBAU principle according to the hydrogen spectrum is widely accepted, exeptions are explained by relativistic spin orbit coupling (14), also modification of the Coulomb potential due to interaction of the shell electrons has been considered. On the other hand, the Madelung rule provides an AUFBAU principle not burdened with exeptions. In addition, this rule has been used successfully as the basis in various theoretical treatments explaining the structure of the periodic system. Each of these different theories stands isolated by itself and there remains the question for a common connecting link.

    Here we take as the guiding principle the notion that nature prefers to choose simple and symmetric solutions and arrive at the double-shell structured PSE as introduced in (9). Despite the fact, that the PSE has double-shells instead of single shells and thus the main quantum number n looses its significance, blocks of elements with a given angular momentum l are identical in both the PSE and PSH. In the PSH, electron energies are expected to be proportional to the square of n, but in fact this is only partially supported by experimental values and is corrected in practice by introduction of the so called quantum defect. Thus, from this point of view, it is not very relevant, which " main quantum number " is assigned to a shell or double-shell.

    If we accept the double-shell structure, the question remains " What produces the fourfold degeneracy in the PSE compared to the twofold degeneracy in the PSH? " For large Z, there are, as discussed above, many possible solutions (8-13). For small Z, these loose their relevance.

    Requesting complete symmetry in threedimensinal space, by elementary geometry we are confronted to the following situation: Two points define a line, which definitely gives weight to a special direction in space. Similarly, three points define a plane. Since the plane may have different orientations in space, it does not define an isotropic space. It needs at least four points to define threedimensional space.

    Now we formulate the following hypothesis: Disregardig the nucleus, it takes four electrons to define three dimensional space. By symmetry, the four electrons define a TETRAHEDRON. Each two of the four electrons at the corners are viewed as having opposite spin and being placed on one of two orthogonal edges. If the electron charges are imagined to become evenly distributed, in this highly symmetric static model they each cover a spherical triangle on the surface of the TETRAHEDRAL SPHERE. The resulting element 4, Beryllium, is expected to show exeptional stability. - Surprisingly, at the end of our journey through the development of the periodic system, we are back at the magic number 4 of Greek philosophers!

    It remains to be investigated, if and how this principle might extend to higher Z. In the PSE, the nuclear charge does provide the overall confinement of the electrons, but does not determine their mutual position in threedimensional space. The requirement of spatial symmetry as a possible driving force for structuring the PSE awaits further confirmation.

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    1. M. Schaedel, J. V. Kratz
      Chemie des Seaborgium - Chromatografie mit einzelnen Atomen
      Phys. Bl. 53 (1997), p. 865
    2. S. A. Goudsmit and P. I. Richards
      Proc. Nat. Acad. Sci. US, 51 (1964), p. 664
    3. Edward G. Mazur
      Graphic Representations of the Periodic System during One Hundred Years
      The University of Alabama Press, Alabama 1974
    4. Niels Bohr
      Z. Physik, 9 (1922), p. 1
    5. Wolfgang Pauli
      Z. Phys. 31 (1925 ), p.765
    6. E. Madelung
      Die mathematischen Hilfsmittel des Physikers, 3rd ed.
      Springer, Berlin, 1936
      See footnote p.670 in 2) for the early reference in 1926
    7. Charles Janet
      La structure du noyau de l'atome, considérée dans
      la classification périodique des éléments chimique
      Beauvais: novembre, 1927, p. 15
    8. M. Kibler
      The Periodic System of the Elements: Old and new Developments
      Journal of Molecular Structure (Theochem), 187 (1989), p. 83
    9. D. Neubert
      Double Shell Structure of the Periodic System of the Elements
      Z. Naturforschung, 25a (1970), p. 210
    10. A. O. Barut, Group Structure of the Periodic System
      in Wybourne, Ed., The Structure of Matter
      (Rutherford Centennial Symposium, Christchurch 1971)
      University of Canterburyy Press, Christchurch 1972, p. 126
    11. Hanno Essén
      Periodic Table of the Elements and the Thomas-Fermi Atom
      Int. Journal of Quantum Chemistry, 21 (1982), p. 717
    12. J. A. Wheeler
      Analysis Illuminating the Connection between
      Potential and Bound States and Scattering
      in E. H. Lieb, B. Simon, and A. S. Wightman Eds.
      Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann
      Princeton University Press 1976, p. 351
    13. Y. N. Demkov and V. N. Ostrovski
      n+l filling rule in the periodic system and focussing potentials
      Soviet Physics JETP 35, p. 66
    14. Richard P. Feynman, Robert B. Leighton, Matthew Sands
      The Feynman Lectures on Physics, Vol. III, Quantum Mechanics
      Addison Wesley 1965, p. 19-6
    15. Christian K. Jørgensen
      The Periodic Table and Induction as a Basis of Chemistry
      Journal de Chemie Physique, 76 (1979), p.630
    16. S. T. Weir, A. C. Mitchell, and W. J. Nellis
      Metallization of Fluid Molecular Hydrogen at 140 GPa
      Phys. Rev. Letters, 76 (1996), p. 1860
    17. James E. Huheey, Ellen A. Keiter, and Richard L. Keiter
      Inorganic Chemistry, Principles of Structure and Reactivity
      HarperCollins College Publishers 1993, p.857
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