Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 13 (2017), 038, 15 pages      arXiv:1706.02535      https://doi.org/10.3842/SIGMA.2017.038

Mendeleev Table: a Proof of Madelung Rule and Atomic Tietz Potential

Eugene D. Belokolos
Department of Theoretical Physics, Institute of Magnetism, National Academy of Sciences of Ukraine, 36-b Vernadsky Blvd., Kyiv, 252142, Ukraine

Received February 27, 2017, in final form May 22, 2017; Published online June 07, 2017

Abstract
We prove that a neutral atom in mean-field approximation has ${\rm O}(4)$ symmetry and this fact explains the empirical $[n+l,n]$-rule or Madelung rule which describes effectively periods, structure and other properties of the Mendeleev table of chemical elements.

Key words: Madelung rule; Mendeleev periodic system of elements; Tietz potential.

pdf (363 kb)   tex (19 kb)

References

  1. Abel N.H., Solution de quelques problémes à l'aide d'intégrales définies, in Œuvres completes de Niels Henrik Abel, Editors L. Sylow, S. Lie, Christiania, Norway, 1881, 11-27.
  2. Allen L.C., Knight E.T., The Löwdin challenge: origin of the $n+l,n$ (Madelung) rule for filling the orbital configurations of the periodic table, Int. J. Quantum Chem. 90 (2002), 80-88.
  3. Arnol'd V.I., Mathematical methods of classical mechanics, 3rd ed., Nauka, Moscow, 1989.
  4. Bacry H., Ruegg H., Souriau J.M., Dynamical groups and spherical potentials in classical mechanics, Comm. Math. Phys. 3 (1966), 323-333.
  5. Bargmann V., Zur Theorie des Wasserstoffatoms. Bemerkungen zur gleichnamigen Arbeit von V. Fock, Z. Phys. 99 (1936), 576-582.
  6. Belokolos E.D., The integrability and the structure of atom, J. Math. Phys. Anal. Geometry 9 (2002), 339-351.
  7. Demkov Y.N., Ostrovskii V.N., Internal symmetry of the Maxwell ''fish eye'' problem and the Fock group for the hydrogen atom, Sov. Phys. JETP 33 (1971), 1083-1087.
  8. Fermi E., Über die Anwendung der statistischen Methode auf die Probleme des Atombaues, in Quantentheorie und Chemie, Editor H. Falkenhagen, Leipziger Vorträge, S. Hirzel-Verlag, Leipzig, 1928, 95-111.
  9. Fock V., Zur Theorie des Wasserstoffatoms, Z. Phys. 98 (1935), 145-154.
  10. Fradkin D.M., Existence of the dynamic symmeries ${\rm O}(4)$ and ${\rm SU}_{3}$ for all classical central potential problems, Progr. Theoret. Phys. 37 (1967), 798-812.
  11. Hakala R., The periodic law in mathematical form, J. Phys. Chem. 56 (1952), 178-181.
  12. Ivanenko D., Larin S., Theory of periodic system of elements, Dokl. Akad. Nauk USSR 88 (1953), 45-48.
  13. Kibler M.R., On a group-theoretical approach to the periodic table of chemical elements, quant-ph/0408104.
  14. Kibler M.R., From the Mendeleev periodic table to particle physics and back to the periodic table, quant-ph/0611287.
  15. Klechkovski V.M., The distribution of atomic electrons and the filling rule for $(n + l)$-groups, Atomizdat, Moscow, 1968.
  16. Landau L.D., Lifshits E.M., Theoretical physics, Vol. 1, Mechanics, Nauka, Moscow, 1973.
  17. Landau L.D., Lifshits E.M., Theoretical physics, Vol. 3, Quantum mechanics: nonrelativistic theory, Nauka, Moscow, 1974.
  18. Latter R., Atomic energy levels for the Thomas-Fermi and Thomas-Fermi-Dirac potentials, Phys. Rev. 99 (1955), 510-519.
  19. Lenz W., Zur Theorie der optischen Abbildung, in Probleme der mathematischen Physik, Leipzig, 1928, 198-207.
  20. Lieb E.H., Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys. 53 (1981), 603-641.
  21. Lieb E.H., Simon B., The Thomas-Fermi theory of atoms, molecules and solids, Adv. Math. 23 (1977), 22-116.
  22. Madelung E., Die mathematischen Hilfsmittel des Physikers, Die Grundlehren der Mathematischen Wissenschaften, Vol. 4, Springer-Verlag, Berlin, 1936.
  23. Maxwell J.C., Solutions of problems, in The Scientific Papers of James Clerk Maxwell, Editor W.D. Niven, Dover Publications, New York, 1952, 76-79.
  24. Meek T.I., Allen L.C., Configuration irregularities: deviations from the Madelung rule and inversion of orbital energy levels, Chem. Phys. Lett. 362 (2002), 362-364.
  25. Mukunda N., Dynamical symmeries and classical mechanics, Phys. Rev. 155 (1967), 1383-1386.
  26. Ostrovsky V.N., Dynamic symmetry of atomic potential, J. Phys. B: At. Mol. Phys. 14 (1981), 4425-4439.
  27. Ostrovsky V.N., What and how physics contributes to understanding the periodic law?, Found. Chem. 3 (2001), 145-182.
  28. Pauli W., Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik, Z. Phys. 36 (1926), 336-363.
  29. Reed M., Simon B., Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York - London, 1978.
  30. Tietz T., Über eine Approximation der Fermischen Verteilungsfunktion, Ann. Physik 450 (1955), 186-188.
  31. Tietz T., Elektronengruppen im periodischen System der Elemente in der statistischen Theorie des Atoms, Ann. Physik 460 (1960), 237-240.
  32. Tietz T., Über Eigenwerte und Eigenfunktionen der Schrödinger-Gleichung für das Thomas-Fermische Potential, Acta Phys. Acad. Sci. Hungar. 11 (1960), 391-400.
  33. Torrielli A., Classical integrability, J. Phys. A: Math. Theor. 49 (2016), 323001, 31 pages, arXiv:1606.02946.
  34. Wong D.P., Theoretical justification of Madelung's rule, J. Chem. Educ. 56 (1979), 714-717.

Previous article  Next article   Contents of Volume 13 (2017)