_{1}

^{*}

In a previous, primary treatise of the author the mathematical description of electron trajectories in the excited states of the H-atom could be demonstrated, starting from Bohr’s original model but modifying it three dimensionally. In a subsequent treatise, Bohr’s theorem of an unalterable angular momentum
h
/2π, determining the ground state of the H-atom, was revealed as an inducement by the
—unalterable—electron spin. Starting from this presumption, a model of the H
_{2}
-molecule could be created which exhibits well-defined electron trajectories, and which enabled computing the bond length precisely.
In the present treatise, Bohr’s theorem is adapted to the atom models of he
lium and of neon. But while this was feasible exactly in the case of helium, the neon atom turned out to be too complex for a mathematical modelling. Nevertheless, a rough ball-and-stick model can be presented, assuming electron rings instead of electron clouds, which in the outer shell are orientated as a tetrahedron. It entails the principal statement that the neon atom does not represent a static construction with constant electron distances and velocities, but a pulsating dynamic one with permanently changing internal distances. Thus, the helium atom marks the limit for precisely describing an atom, whereby at and under this limit such a precise description is feasible, being also demonstrated in the author’s previous work. This contradicts the conventional quantum mechanical theory which claims that such a—locally and temporally—precise description of
any
atom or molecule structure is generally not possible, also not for the H
_{2}
-molecule, and not even for the H-atom.

In 1913, the Dane Niels Bohr published an article entitled “On the Constitution of Atoms and Molecules” [

However, several questions remained: Firstly, the existence of a minimal ground state (permanent state) could not be explained, i.e. it was not plausible why the electron does not tumble on the nucleus. Secondly, the intrinsic cause for the existence of such exited―meta-stable―energy states could not be found. Thirdly, Bohr did not deliver a model of a molecule like the H_{2}-molecule, in contrast to the notification in the title of his article. And fourthly, the structure of atoms with higher atomic number was not given―not even the one of helium―, and likewise the “Aufbau-Principle” of the periodic system of the elements.

A step forward was made in 1924 by Louis De Broglie in his thesis, assuming a wavy electron motion, and leading to the term “wave mechanics”. It not only explained the deflection of electron beams on thin metal foils (cf. for instance [

An escape from this problem was delivered by Heisenberg postulating the «uncertainty principle» which implied the statement that the location and the momentum of a particle could not simultaneously be determined. This principle promised to explain the wavy motion of the electron, as well as the fact that it cannot tumble on the nucleus. Based on this theorem, Schrödinger and others developed a complicated theory, based on statistical probability rules, and yielding cloudy orbitals instead of well-defined electron trajectories. In spite of considerable doubts, even expressed by Einstein, this theory was well established and forwarded, in particular implementing the spin phenomenon by Pauli. That was discovered in 1925 by Uhlenbeck and Goudsmith [

Nevertheless, this conventional theory exhibits crucial contradictions in its terms which already are included within its foundations, and which could not really be cleared away since then. A cardinal intrinsic contradiction is given by Heisenberg’s uncertainty principle which is incompatible with De Broglie’s standing wave concept implying well-defined electron-trajectories in the excited states. Thereby, the uncertainty of measurability is erroneously equalized to an uncertainty of real states. Moreover, Bohr’s hypothesis is not fulfilled for the ground state of the H-atom since a single electron cannot exhibit a constant (vectored) angular momentum when it simultaneously describes a spherical cloudy trajectory, as it would be the case for the s-orbital.

Induced by these contradictions, the author searched and found an H-atom-model implying the De Broglie phenomenon and starting from Bohr’s original approach [

However, the existence of the (stable) planar ground state could not be explained so far. The only plausible explanation appeared when the spin of the electron was taken into account, inducing a respective angular momentum leading to the―well-known―spin/orbit coupling. Indeed, the spin hypothesis was known neither to Bohr nor to De Broglie, while the common wave mechanics originally disregarded the spin phenomenon, too. It was solely implemented afterwards, according to the Pauli principle. The electron spin cannot be explained classically, but must be accepted as a natural constant. Since it cannot be annihilated, it delivers the explanation for the ground state at the H-atom (as well as at other atoms or molecules).

Starting from this assumption, it was possible to develop a vivid model for the H_{2}-molecule, exhibiting planar electron orbits [_{2}-molecule―, questioning the conventional theory, questioning the conventional theory.

Extending this model-approach, analogous atom models for the noble gases helium and neon are presented here. However, empirical evidence can hardly be delivered, solely plausibility. In view of the complexity of the problem, for neon an exact model computation cannot be delivered, solely an intuitive design.

Starting from Bohr’s H-atom model in the ground state, and regarding the constant angular momentum induced by the constant electron spin, for the He-atom the assumption of a simple concentric structure according to

However, the resulting disc-shaped atom structure is not plausible, particularly when the He-atom is taken as the basic model for the atomic core of the higher elements where a tetrahedral array of the external electrons must be assumed, being evident from the CH_{4}-molecule (methane). Rather a three-dimensional structure should be envisaged, potentially enabling a tetrahedral array of the outer electrons. Opposed to this, a spherical, ideally three dimensional structure, as it is claimed within the s-orbital of the conventional quantum mechanics, must be excluded, regarding the above alleged arguments.

Alternatively, the eccentric structure shown in

R = r 2 (1)

and

z = r 2 (2)

Moreover, for Coulomb’s law the abbreviation

K = e 2 4 π ε 0 (3)

is used.

(e = elementary charge, ε 0 = permittivity)

The Coulomb attraction force between each electron and the nucleus is given by the relation 2 K r 2 , while the Coulomb repulsion force between the electrons is K 4 r 2 .

The determination of the concentric centrifugal force of each electron is of particular interest, since―on one hand―it acts in the same direction as the Coulomb forces do, while―on the other hand―solely the eccentric centrifugal force is evident, given by the expression

m e ⋅ u r o t 2 R

(m_{e} = electron mass, u_{rot} = rotation velocity of the electron)

However, a vector splitting is feasible, delivering the concentric portion of the

centrifugal force, namely m e ⋅ u r o t 2 ⋅ z R 2 . When R and z are substituted by r, according to (1) and to (2), the resulting value for the concentric centrifugal force is m e ⋅ u r o t 2 r and thus identically equal to the centrifugal force in the concentric

model structure according to

The balance of forces has to be focused on one electron, related to the nucleus and to the other electron. It is reached when the Coulomb attraction force is equal to the Coulomb repulsion force plus the centrifugal force of the electron, yielding Equation (4):

2 K r 2 = K 4 r 2 + m e ⋅ u r o t 2 r → 7 K 8 r = m e ⋅ u r o t 2 (4)

Now, the quantum condition has to be regarded, being identically equal for any electron:

m e ⋅ u r o t ⋅ R = h 2 π → m e ⋅ u r o t 2 = h 2 4 π 2 ⋅ m e ⋅ R 2 = h 2 2 π 2 ⋅ m e ⋅ r 2 (5)

The combination of (4) and (5) yields the exact value for r:

r = 8 h 2 7 π 2 ⋅ K ⋅ m e = 8 h 2 ⋅ ε 0 7 π ⋅ m e ⋅ e 2 = 0.60477 × 10 − 10 m (6)

As a consequence, the distance R between the nucleus and the electron rotation centre is r / 2 = 0.42767 × 10^{−10} m (10^{−10} m = 1 Å). These values cannot be verified empirically, since the effective atomic radius is not identical with the distance between the nucleus and the electrons. Moreover, this eccentric model is not spherical and thus anisotropic, letting assume that both radii―namely r and R―have to be taken into account. At least it is striking that the average value of these two radii (0.516 Å) is similar to the value of the atomic radius of helium found in the literature (0.49 Å). However, the original sources for the empiric data concerning helium could not be found since they are part of the common physical-chemical data base.

In order to assess the interactions with other He-atoms, i.e. the interatomic forces, not only the local positions of these particles are relevant but also their electric fields. The electric field-strength distribution around the helium atom―according to this model―is quite complicated, and, because of the rotating electrons, mostly oscillating. A detailed, three-dimensional computation is beyond the scope of this treatise. However, the following special constellation is exemplary and thus worth to be discussed in detail, namely the one along the straight line where the field strength is temporally constant, given by the line across the nucleus and the rotation centres of the electrons.

The respective computation is easily feasible according to the schedule given in

a 2 = ( d − R ) 2 + R 2 = d 2 − 2 d R + 2 R 2 , b 2 = ( d + R ) 2 + R 2 = d 2 + 2 d R + 2 R 2

The respective electric field intensities F are:

F nucleus = 2 K e ⋅ d 2 , F electron1 = − K e ⋅ a 2 , F electron2 = − K e ⋅ b 2

yielding the total value

F t o t = K e ( 2 d 2 − 1 a 2 − 1 b 2 ) (7)

Thereby, a positive value means that a negative charged particle is attracted, while a positive one is pushed away. According to Formula (7) and assuming invariability of r and R, the relative field strengths can be plotted versus d as a function of a multiple of R (

However, at least two restrictions have to be made: Firstly, this field model is solely valid for dot-like charges but not for whole atoms containing several charged particles, and being three-dimensionally extended. And secondly, it is only valid along the straight line nucleus―rotation centres of the electrons. At any other points, the field strength is temporally not constant, so that temporal fluctuation of the interatomic forces could be expected, which may explain the existence of zero-point-oscillations. Overall, the conditions are too complex for computing intermolecular forces and distances, even if it were possible to use averaged values due to the inertia of the nuclei. As a consequence, distinct coherences between the atom structure and the macro-physical properties such as boiling point (4.215 K), melting point (0.95 K), and the (hexagonal) crystallographic structure (

Neon exhibits the atomic number 10 and thus ten electrons. They are placed within the first and the second atomic shells, comprising 2 and 8 electrons. The former ones represent―together with the nucleus―the core of the atom, while the latter ones are spherically arrayed in a tetrahedron of four electron couples. As an additional condition, for each electron Bohr’s theorem of a constant angular momentum h/2π must be fulfilled.

In view of the large amount of correlative electrons it does not seem feasible finding an exact mathematical solution for this problem, necessitating at least probability functions due to the perpetually changing situations. Already the fact

that the Cartesian coordinate system is orthogonal let suppose considerable difficulties for describing the tetrahedral system of the outer electron shelf, while the inner electron shelf―corresponding to the structure of the helium atom―can be described by an orthogonal system. Thus the two coordinate systems are not compatible. But even in case of the inner shelf, considerable difficulties may arise since the respective rotation trajectories of the two electrons are probably not flat but wavy, requiring the introduction of a polar coordinate system. In any case it must be assumed that the neon atom does not represent a static construction with constant electron velocities, but a dynamic one with permanently changing internal distances.

Therefore, within

electron rings instead of spherical electron clouds, thus the latter one may still be used for practical purposes in chemistry.

Applying Bohr’s theorem of a constant angular momentum h/2π for electrons as a consequence of the spin phenomenon, and assuming an eccentric electron rotation instead of a concentric one, the radii for a three dimensional―but not spherical―atom model of helium could easily be computed. The result appears plausible with respect to the known atomic radius even if exact empiric data are not available. Above all, precise statements about interatomic forces are not possible. In particular, a stringent reliance between the atomic or molecular structure and the crystallographic structure is not evident.

In contrast to the atom model of helium, the one of neon is not exactly describable and computable since too many interdependencies should be taken into account. It allows solely a rough idea of a vivid model which is characterized by rotating electron rings instead of spherical electron clouds. With respect to a mathematical modelling, one sticking point is founded by the tetrahedral structure of the outer electron shell which cannot be easily described using the orthogonal Cartesian coordinate system. Moreover, the fact that the neon atom does not represent a static construction with constant electron velocities, but a dynamic one with permanently changing internal distances, appears to be considerably complicating. Thus, the helium atom marks the limit for precisely describing an atom, whereby at and under this limit such a precise description is feasible, being also demonstrated in the author’s previous work. This contradicts the conventional quantum mechanical theory which claims that such a―locally and temporally―precise description of any atom or molecule structure is generally not possible, also not for the H_{2}-molecule, and not even for the H-atom.

Allmendinger, T. (2018) The Atom Model of Helium and of Neon Based on the Theorem of Niels Bohr. Journal of Applied Mathematics and Physics, 6, 1290-1300. https://doi.org/10.4236/jamp.2018.66108