By prof. LEFTERIS KALIAMBOS (Λευτέρης Καλιαμπός) T.E. Institute of Larissa, Greece

( February 2014 )

Despite the enormous success of the Bohr model and the quantum mechanics of Schrodinger based on the well-established laws of electromagnetism in explaining the principal features of the hydrogen spectrum and of other one-electron atomic systems, so far, under the abandonment of natural laws neither was able to provide a satisfactory explanation of the two-electron atoms. In atomic physics a two-electron atom is a quantum mechanical system consisting of one nucleus with a charge Ze and just two electrons. This is the first case of many-electron systems. The first few two-electron atoms are:

Z =1 : H^{-} hydrogen anion. Z = 2 : He helium atom. Z = 3 : Li^{+} lithium atom anion. Z = 4 : Be^{2+} beryllium ion. Z = 5 : B^{3+} boron.

Prior to the development of quantum mechanics, an atom with many electrons was portrayed like the solar system, with the electrons representing the planets circulating about the nuclear “sun”. In the solar system, the gravitational interaction between planets is quite small compared with that between any planet and the very massive sun; interplanetary interactions can, therefore, be treated as small perturbations.

However, In the helium atom with two electrons, the interaction energy between the two spinning electrons and between an electron and the nucleus are almost of the same magnitude, and a perturbation approach is inapplicable.

In 1925 the two young Dutch physicists Uhlenbeck and Goudsmit discovered the electron spin according to which the peripheral velocity of a spinning electron is greater than the speed of light. Since this discovery invalidates Einstein’s relativity it met much opposition by physicists including Pauli. Under the influence of Einstein’s invalid relativity physicists believed that in nature cannot exist velocities faster than the speed of light. So great physicists like Pauli, Heisenberg, and Dirac abandoned the natural laws of electromagnetism in favor of wrong theories including qualitative approaches under an idea of symmetry properties between the two electrons of opposite spin which lead to many complications. Thus in the “Helium atom-Wikipedia” one reads: “Unlike for hydrogen a closed form solution to the Schrodinger equation for the helium atom has not been found. However various approximations such as the Hartree-Fock method ,can be used to estimate the ground state energy and wave function of atoms”.

It is of interest to note that in 1993 in Olympia I presented at the international conference “Frontiers of fundamental physics” my paper “Impact of Maxwell’s equation of displacement current on electromagnetic laws and comparison of the Maxwellian waves with our model of dipolic particles ", according to which LAWS AND EXPERIMENTS INVALIDATE FIELDS AND RELATIVITY . At the same time in Larissa I tried to find not only the nuclear force and structure but also the coupling of two electrons under the application of the abandoned electromagnetic laws. For example in the photoelectric effect the absorption of light contributed not only to the increase of the electron energy but also to the increase of the electron mass because the particles of light have mass m = hν/c^{2}

( See my DISCOVERY OF PHOTON MASS ).

However the electron spin which gives a peripheral velocity greater than the speed of light cannot be affected by the photon absorption. Thus after 9 years I presented at the 12th symposium of the Hellenic nuclear physics society my paper “Nuclear structure is governed by the fundamental laws of electromagnetism (NCSR “Demokritos, 2002), and I showed not only my DISCOVERY OF NUCLEAR FORCE AND STRUCTURE but also that the peripheral velocity (u >> c) of two spinning electrons with opposite spin gives an attractive magnetic force F_{m} stronger than the electric repulsion F_{e} when the two electrons of mass m and charge (-e) are at a very short separation r < 578.8 /10^{15 } m. Because of the antiparallel spin along the radial direction the interaction of the electron charges gives an electromagnetic force F_{em} = F_{e} –F_{m} . Therefore in my research the integration for calculating the mutual F_{em} led to the following relation:

F_{em} = F_{e} – F_{m} = Ke^{2}//r^{2} – (Ke^{2}/r^{4})(9h^{2}/16π^{2}m^{2}c^{2})

Of course for F_{e} = F_{m} one gets the equilibrium separation r_{o} = 3h/4πmc = 578.8/10^{15} m.

That is, for r < 578.8/10^{15} m the two electrons of opposite spin exert an attractive electromagnetic force, because the attractive F_{m} is stronger than the repulsive F_{e} . Here F_{m} is a spin-dependent force of short range. As a consequence this situation provides the physical basis for understanding the pairing of two electrons described qualitatively by the Pauli principle, which cannot be applied in the simplest case of the deuteron in nuclear physics, because the binding energy between the two spinning nucleons occurs when the spin is not opposite (S=0) but parallel (S=1). According to the experiments in the case of two electrons with antiparallel spin the presence of a very strong external magnetic field gives parallel spin (S=1) with electric and magnetic repulsions given by

F_{em} = F_{e} + F_{m}

So according to the well-established laws of electromagnetism after a detailed analysis of paired electrons in two-electron atoms I concluded that at r < 578.8/10^{15} m a motional EMF produces vibrations of paired electrons. Unfortunately today physicists in the absence of a detailed knowledge believe that the two electrons of two-electron atoms under the Coulomb repulsion between the electrons move not together as one particle but as separated particles possessing the two opposite points of the diameter of the orbit around the nucleus. In fact, the two electrons of opposite spin behave like one particle circulating about the nucleus under the rules of quantum mechanics forming two-electron orbitals in helium, beryllium etc. In my paper “Spin-spin interaction of electrons and also of nucleons create atomic molecular and nuclear structures” published in Ind. J. Th. Phys. (2008) I showed that the positive vibration energy (E_{v}) described in eV depends on the Ze charge of nucleus as

E_{v} = 16.95Z - 4.1

Of course in the absence of such a vibration energy E_{v} it is well-known that the ground state energy E described in eV for two orbiting electrons could be given by the Bohr model as

E = -27.2 Z^{2.}

So the combination of the energies of the Bohr model and the vibration energies due to the opposite spin of two electrons led to my discovery of the ground state energy of two-electron atoms given by

E = -27.2 Z^{2} +16.95 Z - 4.1

For example the laboratory measurement of the ionization energy of H^{-} yields an energy of the ground state E = - 14.35 eV

In this case since Z = 1 we get E -27.2 + 16.95 - 4.1 = -14.35 eV

In the same way writing for the helium Z = 2 we get

E = - 108.8 + 32.9 - 4.1 = -79.0 eV

which is equal to the laboratory measurement. In the same way we can calculate the ground state energies for the Z = 3 : Li^{+} ion , Z = 4 : Be^{2+} beryllium ion, and Z = 5 : B^{3+} boron.

The discovery of this simple formula based on the well-established laws of electromagnetism was the first fundamental equation for understanding the energies of many-electron atoms, while various theories based on qualitative symmetry properties lead to complications.