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Journal of Applied Mathematics and Physics

A Spherical Atom Model of Helium Based on Well-Defined Electron Trajectories

Thomas Allmendinger

Published Jan 1, 2022• 1 citation
Publisher

Abstract

The present approach is an advancement of the author’s former attempts to develop an atom model of Helium with well-defined electron trajectories. Thus it calls in question the traditional quantum mechanics which assume—in contrast and as a consequence of Heisenberg’s uncertainty principle—electronic probabilities of presence. Its basic idea consists of the assumption that the motions of the two electrons are influenced by their spins exhibiting the value h/2π, but in two different ways: on the one hand, one spin induces a rotation; and on the other hand, the other spin induces a harmonic oscillation. A second important relation is given by the fact that the retroactive force of the oscillation process is due to the centrifugal force when the process runs along the surface of a sphere, whereas in usual oscillation processes—such as the one of a spring pendulum—it is due to a permanent shift between potential and kinetic energy. Therefore, in the present case, the potential energy remains constant since the distance between the nucleus and the—diametrically positioned—electrons remains constant. Considering these two conditions and the usual physical relations such as Coulomb attraction, centrifugal force, and the conservation laws of the angular momentum and of the energy, it was possible to compute the respective key values. Thereby, the deflection of the oscillation angle ψ = 45˚ is remarkable. Finally, the process is described using a Cartesian coordinate system with z as the rotation axis, a variable oscillation distance d and variable rotation velocities rrot. Thereby, the projections onto the x-axis and on the y-axis are not identically equal, leading to an elliptic projection shape. Thus this system is anisotropic, in contrast to the isotropic array of the conventional quantum mechanics according to Schrödinger, where the 1s-orbital is spherically symmetrical. This anisotropy explains the existence of interatomic Van der Waals forces, which enable the condensation of Helium, even though the condensation temperature is very low. But in particular, it exhibits well-defined electron waves, thus finally delivering the explanation of the hypothesis of Louis de Broglie, which has been established 100 years ago.

DOI
10.4236/jamp.2022.106136

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The present approach is an advancement of the author’s former attempts to develop an atom model of Helium with well-defined electron trajectories. Thus it calls in question the traditional quantum mechanics which assume—in contrast and as a consequence of Heisenberg’s uncertainty principle—electronic probabilities of presence. Its basic idea consists of the assumption that the motions of the two electrons are influenced by their spins exhibiting the value h/2π, but in two different ways: on the one hand, one spin induces a rotation; and on the other hand, the other spin induces a harmonic oscillation. A second important relation is given by the fact that the retroactive force of the oscillation process is due to the centrifugal force when the process runs along the surface of a sphere, whereas in usual oscillation processes—such as the one of a spring pendulum—it is due to a permanent shift between potential and kinetic energy. Therefore, in the present case, the potential energy remains constant since the distance between the nucleus and the—diametrically positioned—electrons remains constant. Considering these two conditions and the usual physical relations such as Coulomb attraction, centrifugal force, and the conservation laws of the angular momentum and of the energy, it was possible to compute the respective key values. Thereby, the deflection of the oscillation angle ψ = 45˚ is remarkable. Finally, the process is described using a Cartesian coordinate system with z as the rotation axis, a variable oscillation distance d and variable rotation velocities rrot. Thereby, the projections onto the x-axis and on the y-axis are not identically equal, leading to an elliptic projection shape. Thus this system is anisotropic, in contrast to the isotropic array of the conventional quantum mechanics according to Schrödinger, where the 1s-orbital is spherically symmetrical. This anisotropy explains the existence of interatomic Van der Waals forces, which enable the condensation of Helium, even though the condensation temperature is very low. But in particular, it exhibits well-defined electron waves, thus finally delivering the explanation of the hypothesis of Louis de Broglie, which has been established 100 years ago.
The original intention of the author’s preoccupation with the quantum-mechanical behaviour of simple atoms and molecules such as Hydrogen and Helium was, on the one hand, the elegant simplicity of Niels Bohr’s atom model for Hydrogen, describing the metastable states of the excited electrons by planar concentric electron orbits, and, on the other hand, the hardly intelligible wave mechanical approach of Heisenberg, Schrödinger and others, describing mainly atoms with multiple electrons by three-dimensional orbitals which were characterized by probabilities of presence. Thereby the question arose whether it would be possible to find alternative atom models with well-defined electron trajectories. Therein, Louis de Broglie’s thesis of the wavy nature of electron motion implicating standing waves would have to be implemented. Nevertheless, as reviewed in the introduction, the orthodox three-dimensional concept influenced the own thinking in such a way that three-dimensional constellations for the electronic excited states were conceived. The break-through was achieved for the electronic ground state in the form of the spin-orbit coupling where the spin acts as a perpetuum mobile, inducing the orbital angular momentum h/2π. Furthermore, the insight was gained that a circularly rotating electron intrinsically corresponds to a harmonic oscillator, thus fulfilling the condition of a standing wave. Based on this concept, a double planar model was established for the H2-molecule which could be empirically verified by X-ray data from literature. However, for the two electrons containing Helium a 2D-array seemed impossible since the Pauli-principle seemed to be violated. After a long stepwise succession of 3D-attempts which turned out to be impossible—not least since eccentric forces are not possible in such a system—the here presented 2D-version for Helium was found, composed by two imaginary orthogonal electron orbits. It will enable in a subsequent publication the quantum mechanical interpretation of the thermal-radiative behaviour of Helium which was reported in the author’s publication nine years ago.
Proceeding from the double-cone model of Helium, based on Bohr’s theorem and recently published in [13], a spherical modification could be made by introducing a second electron rotation which exhibits a rotation axis perpendicular to the first one. Thereby, each rotation is induced by the spin of one electron. Thus the trajectory of each electron represents the superposition of two separate orbits, while each electron is always positioned opposite to the other one. Both electron velocities are equal and constant, due to their mutual coupling. The 3D electron orbits could be 2D-graphed by separately projecting them on the x/z-plane of a Cartesian coordinate system, and by plotting the evaluated x-, y- and z-values versus the rotation angle. Due to the decreased electron velocity, the resulting radius is twice the size of the one in the double-cone model. Even if distinct evidence is not feasible, e.g. by means of X-ray crystallographic data, this modified model appears to be the more plausible one, due to its higher cloud coverage, and since it comes closer to Kimball’s charge cloud model.
The atom model of Niels Bohr is commonly considered as antiquated even if it describes the atomic spectrum of hydrogen quite accurately. The later published relation of Louis De Broglie could arithmetically be implemented into Bohr’s formulation, leading to the concept of standing waves as the existence cause of excited electron states. However, at that time it was not possible to find well defined electron trajectories being classically describable. As a consequence, the »quantum mechanics« were developed by several authors, particularly by Schrodinger and by Heisenberg, but delivering a hardly under¬standable formalism where the classical physical laws appear being abrogated while abstract terms replace concrete and imaginable ones, abandoning the particle perception of mass points. In particular, Heisenberg’s »uncertainty principle« and the assumption of state probabilities seem to be in a striking variance to the idea of standing waves. In contrast, a formulation is given here which exactly describes the electron trajectories in the exited states solely by applying classical physical laws. Firstly, the original Bohr-model - in combination with De Broglie’s relation - is rolled up. From this formula system a vibration frequency - corresponding to the De-Broglie frequency – is deduced which is n-times larger than the rotation frequency of the Bohr-model. Furthermore, a direct coherence between that vibration-frequency of the electron and the frequency of the involved light is evident being explainable as a resonance effect. Then, a three-dimensional model is proposed where the electron oscillates and pulses perpendicularly to a virtual rotation plane i.e. rotating around a vertical axis, accompanied by a perpetual energy exchange between potential and kinetic energy. This leads at the excited, metastable energy states to well-defined, three-dimensional and wavy trajectories winding up on a surface similar to the one of a hyperboloid, whereas at the ground state the trajectory is planar and stable.
The purpose of this contribution was to evaluate a recently published atom model for Helium, characterized by a double rotation of the electrons which exhibit perpendicular rotation axes. Thereby, each rotation is induced by the spin of one electron [1]. Hereto, a tangible mechanical model was used which facilitated to derive the mathematical formulae as the basics for two-dimensional projections, and—not least—for a digital animation yielding freeze images from different perspectives. The resulting shape of the electron shell turned out to be not spherical. In particular, the total velocity of the electrons is variable since the relative running direction may change—in contrast to the initial assumption—, even leading to an intermittent standstill, and implying a variable kinetic energy. Thus it can be concluded that this model describes a rotating rotor but not the Helium atom, and that it must be abandoned.
This chapter illustrates Niels Bohr's ideas in some detail. In the formulation of his model, Bohr assumed that the electron in the hydrogen atom moves under the influence of the Coulomb attraction between it and the positively charged nucleus, as assumed in classical mechanics. The Bohr model was very successful in explaining the discrete atomic spectra of one-electron atoms. The Bohr model is, of course, a theoretical model of an atom. Using Bohr's quantization postulate, find the linear velocity of the electron's sphere. To find the radius of the orbit, Bohr postulated that the angular momentum of the electron is quantized, i.e., it can only take values that are integer multiples of ħ. The chapter begins with considering the classical equation of motion for the electron in a circular orbit, which is based on Newton's laws of motion and Coulomb's law of electric force.
It is shown that the approximation of a strong Coulomb interaction between electrons results in a new model of the atom with a spatial quantization of electrons accompanied by their quantization in energy. This model implies that electrons rotate in circular orbits centered outside the atomic nucleus and only orbit axes pass through it. The Coulomb interaction between electrons leads to a spherically symmetric distribution of their orbits on the surfaces of equipotential spheres of a spherically symmetric electrostatic field of the nucleus. The distribution is similar to “inscribing” electron orbits into faces of regular nucleus-centered polyhedra so each polyhedron corresponds to a certain electron state (s, p, d, f), and a certain set of polyhedra corresponds to a certain period of the Mendeleev Table. It is shown that a spherically symmetric distribution of electron orbits gives rise to the formation of electron pairs in which electron orbits with a common axis are located symmetrically with respect to the nucleus and the orbital magnetic moments of the electrons are oppositely directed. The physical meaning of the electron spin concept becomes clear. The spin turns out to be related to the orbital magnetic moment of an electron and reflects the fact that two electrons of a pair rotate in opposite directions relative to their common axis. So the spin is one of characteristics of the electron state in the atom associated with electron rotation in the orbit centered outside the nucleus. The atomic model gives an insight into the periodicity of changes in the atomic properties with increasing nuclear charge and the reasons for an electron double energy quantization associated with different states and periods. The model shows that the atomic structure and properties can be explained by using concepts of classical mechanics and classical electrodynamics which regard the electron as a particle.