Chapter 9. Motion of Orbital Electrons, Atoms and Bodies of Atoms in the Epola

9.1 Epola explanation of the Bohr – de Broglie orbit stability

In Section 3.8, we discussed Bohr's theory of the hydrogen atom, and in Section 3.9 we introduced a united "Bohr – de Broglie condition" for the stability of electron orbits in an atom, stating that circumferences of allowed atomic orbits must contain integral numbers of de Broglie electron wavelengths, or

Here, $r$ is the radius of the circular orbit, $\lambda_B$ is the de Broglie wavelength of the orbital electron and $n$ is the main quantum number.

We are now able to explain the postulated Bohr – de Broglie condition as resulting from the epola structure of space. The orbit, containing an integral number of de Broglie waves of the orbital electron, contains an integral number of wavelengths of the accompanying epola wave of the orbital electron, or an even number of half-wave epola deformation clusters of such wave. In the frame of reference of the orbital electron, the accompanying wave is, therefore, a standing wave with circular boundary conditions, with no energy flow into it or out of it (Section 8.4). Relative to the epola, this accompanying wave is a closed circular chain of half-wave clusters, one of which contains the orbital electron. The chain rotates in its plane with constant linear velocity, equal to the velocity of the orbital electron; the atomic nucleus is positioned in the plane, at the center of the chain. Mechanically, this is a rotation of a balanced, wave-shaped solid ring or toroid. The centripetal accelerations of the particles of such rotating ring are provided by the internal bonds between the particles and are a matter of internal concern. In the rotation of the ring as a whole, there is no centripetal acceleration. Hence, the orbital electron as a particle of this rotating ring is not centripetally accelerated with respect to the outside epola and it does not radiate its rotational energy into the epola.

9.2 Stability of elliptic orbits

Based on the epola model, it is also possible to explain the stability of any orbit and the meaning of all quantum numbers which define them. On plane elliptic orbits the speed of the electron is changing, and so is the wavelength of its accompanying wave. At points of orbit closest to the atomic nucleus, the velocity of the orbital electron is largest and the size of the half-wave cluster around the electron is smallest. At points of the orbit farthest from the nucleus. the velocity of the orbital electron is smallest and the size of the half-wave cluster around the electron is largest.

For orbit stability, energy must not be radiated into the epola, i.e., the energy of the system "orbital electron – its accompanying wave" must be constant. This requires that for each section of the orbit on which energy is pumped into one part of the system, there is an appropriate section on which this energy is given back. The pumping of energy occurs between the half-wave deformation clusters of the orbit. When the cluster containing the electron approaches the points where its velocity is largest, energy must flow into it from the half-wave deformation cluster(s) on other parts of the orbit. When the cluster containing the electron approaches the points of the orbit where the velocity is smallest, energy is pumped from this cluster to the one(s) on the other parts of the orbit.

The described way of maintaining constant the energy of the system "orbital electron – its accompanying wave" is possible only on certain special elliptic orbits. Such orbits must not only contain an integral number $n$ of wavelengths of the accompanying wave but must allow an appropriate positioning of the half-wave deformation clusters of the wave with respect to the axes of the ellipse. Therefore, the allowed elliptical orbits may have only certain quantized values of the ratio of their short axes to the long ones.

The condition, that for the stability of the orbit, the energy of the system "orbital electron – its accompanying wave" must remain constant, results also in a limitation or quantization of the positions, which the plane of the orbit may assume in the atom. These allowed positions, as well as the allowed ratio of the axes of the elliptical orbit, can be derived on the basis of the epola model and pure physical causality. The epola derivation can be directed to lead, as in quantum mechanics, to three more quantum numbers, determining the energy of the orbit: the orbital quantum number $l_q$, the magnetic quantum number $m_q$ and the electron spin quantum number $s_q$. The four quantum numbers $n$, $l_q$, $m_q$ and $s_q$, fully determine the energy and momentum state of an orbital electron in the atom. Spectral lines observed in the atomic spectrum are always due to a transition of the orbital electron from the orbit identified by these four numbers, to an orbit in which at least one quantum number has a different allowed value.

9.3 The Pauli exclusion principle

From the analysis of atomic spectra, W. Pauli (1900-1960) derived in 1925 his exclusion principle, stating that:

no two electrons in an atom can exist in the same energy state.

In other words, if the energy and momentum-state of an orbital electron in an atom is defined by a certain set of values of the four quantum numbers $n$, $l_q$, $m_q$ and $s_q$, then there cannot be in the same atom another orbital electron having the same values of these quantum numbers.

The exclusion principle plays an important role in physics. This includes not only atomic spectroscopy but also atomic structure, the determination of electron configurations in atoms and the periodic classification of the elements and their isotopes. The exclusion principle was also generalized to include electrons confined to proximity in space as, e.g., the "free" electrons in metals, as also in semimetals and "degenerated" semiconductors, in which the average distance between the electrons is ~1 nm or less (Section 10.4). As such, the exclusion principle became a pillar of the electron theory of metals and of the solid state physics as a whole.

As was the case with other important principles of quantum mechanics, the exclusion principle was introduced without explanation. Being a mathematical science, not based on a physical model, quantum mechanics is unable to provide physical explanations for its principles and findings. This is not a misdoing, just a pity. If the founders of this science were to accept their inability to provide explanations as a pity and would express their sorrow about it, together with a hope for a better future, then this would certainly increase our gratitude to them. After all, if you ask for a product and get in reply: "Sorry, we are temporarily out of it, please check again by the end of the century", then you can only thank the salesman; but continue to shop around.

However, Bohr, Pauli and Heisenberg presented their inability to explain as an achievement to be proud of, as a sign of the divinity of their postulations. They insisted that there are no explanations and cannot be, because all there is are their principles and equations. They denied physical causality and the reality of atomic orbits and considered anyone seeking an understanding as a nuisance and ignoramus. In the slightly lower income group, such is the attitude of the merchant in Section 3.10, with his "there is no such thing", "you will never find it" and finally, also, "you do not know what you are talking about".

9.4 Epola explanation of the exclusion principle

In the epola model, the Pauli exclusion principle has a quite simple explanation. The energy state of an atomic orbital electron, defined by the set of values of the four quantum numbers, actually specifies the average velocity of the electron, the way its velocity is periodically changing along the elliptic orbit, the rotations of the electron and the positioning of the orbital plane. All these motions are accompanied by the accordant changes of the accompanying wave of the electron, so that the energy of the system "orbital electron – its accompanying wave" remains constant, as defined by the set of values of the four quantum numbers.

As seen in the Table of Section 3.9, the radii of the half-wave deformation clusters of the accompanying wave of an electron on a ground-state orbit are comparable to the radius of the atom. Therefore, the accompanying wave of each orbital electron involves most of the epola particles in the peripheral shell of the atom. The same epola particles must therefore simultaneously participate in the accompanying waves of many, if not of all orbital electrons. As long as these waves differ from each other by an energy, characterized by at least one quantum number, they do not interfere with each other in their superposition.

The numerous orbital electrons in the atom may approach or even cross the orbits of each other but very seldom do they come close to one another. Even if such an event occurs, it lasts for an extremely short time, because of the very high orbital speeds, and because the orbits diverge. Hence, if the two electrons are exposed to their mutual Coulombic repulsion, or the part of it which is not balanced by the attraction to the nucleus, then, due to the shortness of the interaction-time, no drastic changes in their motions should occur.

If there were two orbital electrons with exactly the same accompanying waves, then these waves would soon become superimposed on each other. Being carried by the same epola particles, as explained before, these two identical waves would turn into just one wave motion. This means that the two electrons would be moving on the same orbit, thus being all the time exposed to their mutual Coulombic repulsion, or the part of it which is not balanced by the attraction to the nucleus, Obviously, long before the orbits coincide, the Coulombic repulsion becomes sufficiently strong to change at least one of the quantum numbers; for example, to flip over the spin of one electron or to force the plane of its orbit into a different allowed position. Therefore, no two orbits may be superimposed in an atom, or no two epola waves accompanying two orbital electrons in an atom may be identical. This is the derived epola formulation of Pauli's exclusion principle.

9.5 Adjustment of atomic orbits in moving atoms

When an atom rests in the epola, there is a deformation cluster around its nucleus at rest (disregarding small motions due to the interaction of the nucleus with the orbiting electrons) and epola waves accompanying the motion of each orbital electron. When the atom moves in the epola, an epola wave accompanies the motion of the nucleus, and accompanying waves corresponding to this motion are added to the accompanying waves of each orbital electron. The additions must be performed in such a way that the circumferences of the orbits, adjusted to the motion, contain integral numbers of wavelengths of the resultant accompanying waves. Also, these numbers must be the same as in the resting atom, otherwise there would be a change in the main quantum number. (This can happen, though, at high velocities.) If the velocity is not too high, the other quantum numbers defining the orbits also remain the same after the adjustment of the orbits to the motion. Clearly, all atoms on Earth are adjusted and have all elements of their orbits adjusted to the motions of Earth in the epola.

The energy $E$ needed for an orbit adjustment in an atom moving with velocity $v$ relative to Earth can be roughly approximated as equal to the kinetic energy of an electron moving with this velocity. Thus,

With the energy $E$ expressed in electron-volts, we have an approximate formula, binding the velocity of the atom relative to Earth, with the energy required to adjust an atomic orbit to this motion,

After the adjustment, epola units open and close coherently for the motion of the nucleus and of each of the orbital electrons. The epola is then vacuum-transparent to the motion of the atom.

When an atomic body is moving relative to Earth, all and every atom of the body must have its orbits adjusted to the motion. After the adjustment, the accompanying waves of all nuclei and of all orbital electrons pre-form the epola for the motion of each particle, and the epola is vacuum-transparent for the motion of the body as a whole.

9.6 Velocity limits in the motion of bodies in the epola

We have seen that the velocity of an atom relative to Earth is connected with the energy $E$ necessary for the adjustment of an atomic orbit and can be roughly approximated as:

This means that, in order to bring the velocity of an atom to 600 km/s relative to Earth, one must supply to the atom 1 eV of energy per orbital electron for the adjustment of its orbit to the motion. Raising the velocity to 2.4 Mm/s increases the necessary adjustment energy to 16 eV per orbital electron. This energy is above the first ionization energy of atoms on Earth. Bringing an atom to such a velocity would confront its outermost electron with the option to free itself from the atom. (Similarly, when the kinetic energy of a wheel in a moving car is equal to the "binding energy" of the wheel to its axis, the wheel faces an option to go off the car). Therefore, the velocity of 2.4 Mm/s relative to Earth should be considered as the stability limit for atoms on Earth.

In atoms moving with speeds at or above the 2.4 Mm/s limit, there would be a probability of ionization by motion, due to their interaction with the epola (atop of ionization due to other reasons, e.g., to interatomic collisions). The probability of such ionization would rapidly grow with increasing speed. At each velocity, there would be a dynamic equilibrium between the numbers of atoms and ions. At sufficiently high speeds there may be, practically, ions only, and no atoms left. At certain higher velocities, the binding of the second, third, etc., orbital electron in the ions would become endangered, and the probability would appear of the second, third, etc., ionization by motion.

At velocities of 240 Mm/s the required orbit-adjustment energy is, most roughly, 160 keV. This is above the energies of the innermost $K$-electrons of the uranium atom. One might therefore expect that at such velocities there will be no atoms or ions of all but the heaviest elements, because they would have lost, or freed even the innermost orbital electrons. Atoms and ions, these these particles of atomic matter would disintegrate into bare nuclei and free electrons, which are particles of the nuclear matter, or dense particles. In other words, at velocities well below the 300 Mm/s lumic limit, there would not be a reminiscence left of particles of atomic matter, not alone of extended bodies of such matter.

There is presently no way to obtain directly atomic or molecular beams of velocities higher than thermal (i.e., in the tens of km/s) range. However, ions can be accelerated to much higher speeds. Remarkable are experiments in which hydrogen atoms are negatively charged (in a Van de Graaf machine) becoming $\mathrm{H^-}$ ions, which are then accelerated to 50 Mm/s and more. The $\mathrm{H^-}$ ions then pass through a positively charged screen, at which they loose the extra electron and become neutral atoms again. They then continue by inertia to move with speeds, exceeding the 2.4 Mm/s threshold by factors of 10 - 40.

The fast motion of the $\mathrm{H^-}$ ions and, subsequently, of the $\mathrm{H}$ atoms is actually enabled by the hundreds of free electrons, which are accelerated simultaneously with each ion (but to much higher velocities) and then pass the screen and move by inertia ahead of the $\mathrm{H}$ atoms. The fast-moving electrons modify the epola in front of the ion or atom, with their bodies and with the accompanying waves, forming 'near breakdown' channels (see Section 8.16) in which the proton and orbital electron of the $\mathrm{H}$ atom can move with much less resistance than in the unmodified epola. Hence, much less energy is needed for the orbit adjustment at these high velocities which, therefore, do not endanger the binding of the orbital electron to the atom.

Molecules are usually bonded by the outermost electrons of the constituent atoms and the energy of the bonds is usually lower than the ionization energy of the atoms. Thus, at velocities abovc 1.8 Mm/s, at which the orbit adjustment energy is above 9 eV, diatomic molecules should disintegrate. More complicated molecules usually have lower binding energies and should disintegrate at correspondingly lower velocities.

The velocities at which solids would sublimate can be approximated from their binding energies. Metals would lose their qualifying properties at velocities of 1300 km/s, requiring adjustment energies of 5 eV. This energy is equal to the "work function" for emission of electrons from the metal. Thus, 1300 km/s should be considered as the maximum velocity relative to Earth of an unmanned space mission of a metallic spaceship. The danger level to people can be approximated at an electron energy corresponding to a body temperature of 40°C (or 105°F). The electron energy of a non-degenerate electron gas in a solid (or liquid) body is given by:

Here, $k = 86.25 \ \mathrm{\mu eV / K}$ is the Boltzmann constant, and $T$ is the absolute temperature of the body in Kelvin (or K). This formula yields for 40°C or 313K an energy $E = 27 \ \mathrm{meV}$. Allowing this energy as the maximum orbit-adjustment energy in atoms of the human body, we obtain for the maximum allowed velocity for human travel the value of 100 km/s. This is more than twice the velocity achieved by the unmanned Voyager, and ten times the velocity reached by the manned Apollo mission. This velocity limit allows to reach the moon in slightly more than an hour or to reach Mars in six days. However, it absolutely excludes the completion of even the shortest interstellar tour during a human lifetime, however lengthened by science and technology. The velocity limits are presented graphically in Figure 5.

9.7 Orbit adjustment redshifts in the emission spectra or moving bodies

We have seen that when an atom moves in the epola, its orbits are adjusted to this motion. Thus, all atoms of Earth have orbits adjusted to the motions relative to the epola of the particular spot of Earth to which they belong, as well as to the instantaneous velocity of their thermal and other motions relative to the epola. Considering, however, that the orbital velocities of electrons on the outer orbits of atoms are in the order of $10^6 \ \mathrm{m/s}$, most of the other mentioned motions do not have significant effects on the atoms.

In the zero approximation, the energy $E$ needed for the adjustment of an orbit to any motion of the atom relative to the epola (with velocity $v$ much smaller than the velocity of light) seems to be identical for all possible orbits of the atom. Therefore, the energies of electron transitions between these orbits and the corresponding spectral lines should not be affected by the motion. However, a more exact treatment of the adjustment energy, and especialJy some higher order restrictions imposed by the epola on the orbits (as, e.g., those expressed in the orbital, magnetic and spin quantum numbers) may result in a dependence of the electron transitions on the velocity of the atom relative to the epola.

The general trend in physical processes is toward the reduction of involved energies, whenever possible. Thus, the changes in electron transition energies between orbits, due to the motion of the atoms, will most probably tend to reduce the transition energies. We therefore expect that:

in the emission spectra of atoms moving relative to the epola, a shift of spectral lines should be observed, mostly toward the longer wavelength (red) ends of the spectra.

This redshift depends on the velocity of the atoms relative to the epola and not on whether the emitting atoms move toward usor away from us. Hence, it is not a Doppler shift. Nor is it a gravitational or Einstein redshift. We shall refer to it as to the "orbit adjustment redshift" or, apparently very seldom, as to an "orbit adjustment blueshift".

The orbit adjustment redshift may explain why so many more celestial bodies exhibit redshifts than blueshifts. The few observed blueshifts are apparently due mostly to the Doppler effect in the radiation of bodies moving toward us. Believing in some symmetry in our world, we cannot accept the idea that everything is running away from us. Therefore, we think that the number of bodies which really move away from us should be comparable to the number of bodies moving toward us. In other words, the number of bodies exhibiting a Doppler redshift should be nearly equal to the number of bodies exhibiting the blueshift. The remaining number of observed redshifts is then due to the gravitational redshift (Section 2.10) and to bodies which just move in the epola and exhibit the orbit-adjustment redshift.

Obviously, if there is an orbit-adjustment shift in spectral lines, then it should be taken into account in the spectra of bodies exhibiting the Doppler shifts. The derived velocity values for the motions of such bodies should therefore be corrected for the orbit-adjustment (red)shifts. This may turn out to be an important factor in the spectroscopy of fast-moving atoms.

9.8 Detection of motion in the epola and frames or reference

The adjustment of atomic orbits to the motion of the atomic body and the resulting 'orbit-adjustment redshift' may serve as an indicator of motion. If it were possible to observe such a redshift (or blueshift) in the radiation spectrum of a body, then we might be able to determine from it the velocity (or rather, the speed) of the body relative to the epola around it. If the conditions in the epola there were identical with the conditions in the epola around us, then the epola could also serve as a frame of reference for the motion of bodies.

In the epola model, each uniform epola region may serve as a 'local' frame of reference for the motion of bodies. With the exclusion of some slightly distorted areas around the Sun and the closest 'regular' stars, our uniform epola region extends apparently much further than we might ever reach, considering the speed limits for manned or even unmanned space missions. In view of the vast variety of epola conditions, forms and motions in the universe, the existence of a Newtonian universal frame of reference is denied in the epola model of space, as it is in relativity, though not in a postulatory way (Section 2.1). On the other hand, just because of this vast variety, one can always select an epola frame, which will fit his practical needs and might be even more 'universal' than Newton's imagined frame.