_{1}

^{*}

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 hydrogen atom model of Niels Bohr, published in 1913 [

Ten years later, the hypothesis of Louis de Broglie allowed taking a step forward, assuming the wavy nature of electron trajectories, and implying standing electron waves in their excited states. However, such well-defined trajectories could not be vividly evaluated at that time. As a consequence, Heisenberg postulated the so-called “uncertainty principle”, implying for each electron probabilities of presence, instead of well-describable trajectories. This assumption was adopted by the leading physicists, especially by Born, Schrödinger and Dirac. 90 years later, it still represents the “official” quantum mechanical doctrine, even if it contradicts the fundamental scientific principle of causality, ignoring the existence of an angular momentum in the ground state, disregarding the fact that standing waves represent the epitome of accuracy, and hazarding the lacking vividness and unintelligibility of that model. After all, the charge cloud model of Kimball, proposed in 1940 [

Based on the multiplicities of spectral lines found in the presence of magnetic fields, Uhlenbeck and Goudsmith postulated in 1925 the electron spin [

Induced by these contradictions, the author searched and found a vivid solution for the excited electron states implying the De Broglie phenomenon and starting from Bohr’s original approach of the H-atom-model [

Initially, the existence of the (stable) planar ground state could not be explained. But after having found its real cause―namely the electron spin―the way was clear for modelling molecules and atoms in the ground state. In particular, it was possible to develop a vivid model for the H_{2}-molecule, exhibiting planar electron orbits [

X-ray measurements, verification was possible by empirical evidence, delivering an accurate result (_{2}-molecule―, in contrast to the conventional theory.

Extending this approach, atom models for the noble gases Helium and Neon were developed, keeping in mind Kimball’s perception [

The essential idea of the original approach consisted in the assumption of an eccentric structure of the electron orbits leading to a three-dimensional double-cone (

the radius r yielded the value

r = 8 h 2 ⋅ ε 0 7 π ⋅ m e ⋅ e 2 = 0.60477 × 10 − 10 m

(Ad [

However, this electron array does not resemble a sphere which would be desirable according to Kimball’s perception, and which would explain the atomic cores of the higher elements as well as the close-packing of ions in crystal lattices. Moreover, it appears odd that the spins of the two electrons refer in the same manor to the nucleus, inducing the same angular momentum. As a consequence, this model has been modified in the following way.

The double-cone model of Helium can be morphed into a spherical one 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 mutual coupling (

In order to 2D-describe the electron trajectories in a Cartesian coordinate system with the rotation angle φ as a variable, their x-, y- and z-projections on the x/z-plane may be used as parameters. Applying trigonometric relations, for one electron the analysis yields

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

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

The numerical computation of R and r can be made analogously to the one for the double-cone case, described in [

u t o t 2 = 2 u r o t 2 .

Coulomb-attraction between the nucleus and an electron: 2 K r 2

whereby K = e 2 4 π ε 0

Coulomb-repulsion between the two electrons: K 4 r 2

Centrifugal force: 2 m e ⋅ u r o t 2 r

Force equilibrium: 2 K r 2 = K 4 r 2 + 2 m e ⋅ u r o t 2 r

Quantum condition for the angular momentum of an electron: m e ⋅ u r o t ⋅ R = h 2 π

Since R = r 2 , the calculation yields r = 16 h 2 ⋅ ε 0 7 π ⋅ m e ⋅ e 2 = 1.21 × 10 − 10 m ,and R = 0.855 × 10 − 10 m .

Due to the increased electron velocity, the result is twice as high as the one in the double-cone model.

Proceeding from the double-cone model of Helium, based on Bohr’s theorem and recently published in [

Due to the increased 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. Contrary to the conventional quantum mechanical approach assuming a diffuse 1s-orbital with temporally variable electron radii, this spherical Helium atom model exhibits precise electron trajectories. Thus it delivers a further proof that Bohr’s concept, combined with the spin-theorem, and implicating the spin-orbit coupling, can be employed on any case, even if for more complicated atoms or molecules the calculation seems quite intricate. Moreover, it represents a vivid explanation of the Pauli Principle.

The deformation of this atomic electron shell, which is to be expected in the case of collisions due to thermal motion in the gas state, has not been studied so far. At least, the author’s recent investigations about the thermal behaviour of gases under the influence of infrared-radiation gave evidence enough that even at noble gases absorption as well as emission of thermal radiation takes place [

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

Allmendinger, T. (2019) The Spherical Atom Model of Helium Based on the Theorem of Niels Bohr. Journal of Applied Mathematics and Physics, 7, 172-180. https://doi.org/10.4236/jamp.2019.71015