According to the conventional theory of quantum mechanics, the electron trajectories in atoms and in molecules are assumed as indeterminable and solely describable by probabilities of presence, implying Heisenberg’s uncertainty principle. In the case of Helium—the element with the most simple singular atoms—, the symmetry of the double-occupied 1s orbital is assumed as strictly spherical, exhibiting no asymmetry or anisotropy. The charge cloud model proposed by Kimball [2] is clearer but not exactly computable.

In contrast to this atom- and molecule-model where probabilities of presence are assumed, the author’s alternative approach is based on the assumption of well-defined electron trajectories. It proceeds from Bohr’s planar hydrogen model, published in 1913 [3] , wherein a fixed orbital angular momentum is postulated for the ground state, determined by Planck’s constant h/2π. But in addition, the electron spin—discovered in 1925/26 by Uhlenbeck and Goudsmith [4] [5] —is adduced to explain the existence of this ground state, inducing an invariable orbital angular momentum of the electron (=radius × velocity × mass), due to a resonance effect, and leading to the—empirically detectable—spin-orbit coupling.

As the author could point out in [6] , it is possible to precisely compute the bond distance in the H₂-molecule known from X-ray-spectroscopic measurements, using normal physical laws and the condition of the constant orbital momentum for each electron. In the next step, a 3D atom model for the noble gases Helium and Neon was conceived [7] in light of the fact that a Helium model is more speculative than the H₂-model since no precise empiric data exist for its verification. Therein, the two electron orbits of the first shell were arranged as double cones, cf. Figure 1. Surpassing this, a further approach was made which promised to be more spherical [1] , cf. Figure 2. However, that electron shell turned out to be not easily describable. As a consequence, a more profound evaluation of this approach seemed to be indicated, using a tangible

mechanical model which facilitated to derive the mathematical formulae as the basics for two-dimensional projections, and—not least—for a digital animation yielding freeze frames from different perspectives.

As outlined in [1] , the here proposed three-dimensional atomic model of Helium is characterized by a double rotation of the electrons which exhibit perpendicular rotation axes. 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 (Figure 2). Both electron velocities were assumed as counter-current and identically equal, due to the mutual coupling, implicating a permanent total velocity according to (u_tot)² = 2(u_rot)².

In order to visualise the resulting electron trajectories, it was advantageous to construct a tangible mechanic model which can be manually stirred. Thereto, the kit of Stokys was used. The here applied model is shown from two different perspectives and at two different rotation angles in Figure 3 and Figure 4. The definition of the axis-directions is given in Figure 5.

In order to 2D-describe the electron trajectories in a Cartesian coordinate system (defined in Figure 5) with the rotation angle φ as a variable, their x-, y- and z-projections on the x/y-plane may be used as parameters. Applying descriptive geometry and using the tangible mechanic model as an aid, for one electron the analysis yields

x = R ⋅ sin φ ( 1 + cos φ ) , y = − R ⋅ cos φ , z = R ( cos φ − ( sin φ ) 2 )

For the other electron, the formulas are identical equal except the sign. The respective diagrams for the two electrons are shown in the Figure 6 and the Figure 7.

Note: These formulas are analogous to those given in [1] , but they differ with respect to the signs and the mappings due to another definition of the coordinate system.

The availability of the x/y/z-parameters as a function of the rotation angle enables to easily determine the trajectory projections on the three possible planes, namely the x/y-, the x/z- and the y/z-plane. They are displayed in Figures 8-10. The respective shapes are exceptional and do not resemble to any known geometric figures. Obviously, they exhibit sharpened points which indicate standstills. Thus the motion is not continuous and constant.

In order to visualize the three dimensional shape of the electron trajectories, 3D-animations were made using a VPython computer program. The resulting images strongly depend on the—virtual—camera positions, implying different perspectives. From the numerous possibilities, the four examples shown in Figures 11-14 as freeze images were chosen.

The here presented figures yield a deepened and clearer comprehension of the recently published atom model of Helium [1] . For tracing the electron motions, a tangible mechanic model was used, enabling the derivation of formulas for the Cartesian coordinate-intercepts by applying descriptive geometry. The formulas provide 2D-depictions of the electron trajectories and, in particular, 3D-ones as freeze images of animations. The resulting shape of the electron shell is three-dimensional but 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.

I thank Dr Andreas Rüetschi-Isler for his critical objections enabling a distinct assessment and David Kummer for performing the 3D-animation and their freeze images.

The author declares no conflicts of interest regarding the publication of this paper.

Allmendinger, T. (2019) The Revision of the Alleged Spherical Atom Model of Helium. Journal of Applied Mathematics and Physics, 7, 1212-1219. https://doi.org/10.4236/jamp.2019.75081