Chapter 8. Motion of Dense Free Particles in the Epola

8.1 Motion of a free electron and the accompanying epola wave

In the Compton wave, of wavelength $\lambda_c = 2426 \ \mathrm{fm}$ , the photon energy is $m_e c^2$ . This means that in the wave, each epola particle which 'holds' the photon has during the 'holding time' the energy $m_e c^2$ . This energy is equal to the binding energy of the particle in the epola. Hence, during the holding time of the photon, the particle has a detectable mass $m_e$ and should act as a free electron or positron. We may therefore conclude that the deformation in the epola around a photon-holding particle in the Compton wave should be identical to the deformation of the epola around a free electron or positron. Vice versa,

around a free electron or positron in a sphere of radius $\lambda_c / 4$ the deformation of the epola should be identical to the deformation in the half-wave cluster, centered around a photon-holding epola particle in the Compton wave motion.

Consider a free electron (or positron) moving with a velocity $v$ which is much smaller than the vacuum light velocity $c$ ${(v << c)}$ . The electron causes an expansion of any epola region into which it enters, so that the $\lambda_c / 2$ deformation cluster is formed there around the electron. The expansion is followed by a contraction when the electron leaves. Therefore,

the epola around a moving free electron is vibrating with a period $T_e$ equal to the time which it takes the electron to travel the length of two $\lambda_c / 2$ clusters

T_e = \frac{\lambda_c}{v}

The frequency $f_e$ of this vibration. $f_e \equiv 1 / T_e$ or

\begin{align} f_e = \frac{v}{\lambda_c} \end{align}

is directly proportional to the velocity of the particle. Thus, the faster the motion of the electron, the higher must be the vibration frequency of the epola around it.

Due to the binding between epola particles, their vibrations spread in the epola with the velocity $v_d$ of bulk deform'ation waves,

v_d = \left( \frac{{}_b E}{m_e} \right)^{1/2}

which is equal to the vacuum light velocity $c$ (Sections 6.3, 6.5). This results in an epola wave, accompanying the motion of the electron. The wavelength $\lambda_e$ of the wave is

\lambda_e = \frac{v_d}{f_e} = \frac{c}{f_e}

Substituting $f_e = v / \lambda_c$ , (Equation 1) we have

\lambda_e = \frac{c \lambda_c}{v}

We have seen (Sections 7.4, 7.5) that

\lambda_c = \frac{h}{m_e c}

so that

\lambda_e = \frac{h}{m_e v} \equiv \lambda_B

This expression for the wavelength of the epola wave, accompanying the motion of the electron, is identical to the formula for the de Broglie wavelength $\lambda_B$ of an electron moving with velocity $v$ . We derived it, based solely on the epola concept, without ascribing non-particle properties to the electron and non-wave properties to the waves, as was done by de Broglie (Section 3.5).

8.2 Energy of the accompanying wave

The epola wave, accompanying the motion of a free electron, propagates with the same velocity $v_d = c$ , as any epola defonnation wave. As any epola wave, it also has energy, the more, the higher its frequency $f_e$ , i.e., the larger the velocity $v$ of the free electron. We found that this frequency is proportional to the velocity of the electron, $f_e = v / \lambda_c$ , so that

the photon energy $h f_e$ of an epola wave accompanying the motion of a free electron is proportional to the velocity $v$ of the electron,

\begin{align*} h f_e &= \frac{hv}{\lambda_c} \\ &= \frac{hv}{h / m_e c} \\ &= m_e c \cdot v \end{align*}

The energy of the accompanying wave is supplied during its build-up, i.e., during the acceleration of the free electron from rest to its final velocity $v$ . During this time, the accelerated electron is transferring energy into the epola. This energy spreads as energy of epola deformation waves. It reveals itself as electromagnetic radiation energy in its action on free electrical charges and charges in material bodies (see Section 6.6). Thus,

the energy transferred by the accelerated electron to the epola is the experimentally observed electromagnetic radiation of the accelerated electron.

When the velocity of the electron becomes constant, the accompanying wave is coherent with the motion, shifting epola particles apart in front of the free electron and converging them behind it. This action of the accompanying wave does not consume energy. Once the wave is established, both the energy of the wave and the energy of the moving electron remain constant.

8.3 Effects of the accompanying wave on the motion

The epola wave, accompanying the motion of a free electron with velocity $v$ much smaller than the velocity of light $c$ , ${v << c}$ , propagates with the velocity $c$ . Therefore, it moves much faster than the electron and modifies or pre-forms the epola for the motion in front of the moving particle. Hence, the epola around the electron vibrates coherently with the motion of the particle. Any epola region approached by the free electron is appropriately expanded by the accompanying wave, while the region behind the particle contracts.

The accompanying wave is forced into being by the motion of the free electron, so that the forming of the wave can be considered as a hindrance to the motion. But when the motion and the wave are established, then the accompanying wave assists the motion by preparing or pre-forming the epola for it. Therefore, the epola does not resist this motion. We may say that the epola becomes vacuum-transparent for the motion.

Due to the pre-forming action of the accompanying wave, the free electron is moving, always surrounded by a half-wave deformation cluster of the Compton wave. Thus, relative to the moving electron, the accompanying wave is a standing wave, with no energy transfer along the wave and no energy exchange between the electron and the wave. Relative to the epola, the accompanying wave has a standing wave pattern, which moves with the velocity $v$ of the electron.

The accompanying wave has a stabilizing effect on the motion of the accompanied particle. For example, if the particle would suddenly "decide" to reduce speed, then the contracting epola behind it would push it forward, and the expanding epola in front of it would pull the particle forward. If the particle were to increase speed, it would face epola units which did not expand for it. Hence, the accompanying wave resists any changes in the velocity of the particle and so it is in part responsible for the inertia of the particle. Rather, we should think of a system composed of the moving free particle and its epola accompanying wave.

To successfully increase the velocity of the free particle, one must pump energy to the particle and to the accompanying wave, because an increase in the particle's velocity forces a proportional increase in the frequency $f_e$ of the accompanying wave and in its photon energy $h f_e$ . When the velocity of the particle is reduced, the frequency and photon energy of the accompanying wave are proportionally reduced. The energy difference is partially returned to the decelerating particle, revealing itself in the inertial opposition of the system to the reduction in velocity.

8.4 The physical character of the accompanying wave

The accompanying wave is a special kind of wave. Its frequency is as many times smaller than the frequency of the Compton wave, as the velocity of the accompanied particle is smaller than the velocity of light. Hence, due to its low frequency, it should be a compressibility wave. However, the moving free particle in the wave is surrounded by a half-wave deformation cluster of the Compton wave, which is an impact wave. Then, the wave is guided by the moving free particle and the particle is guided by the wave. They both form a closed system, in which their energy is preserved in a narrow channel along the direction of motion, as if there were a wave-guide around them, similar to the case of the impact waves (Section 7.8). This holds as long as the velocity remains constant. With any change in velocity of the particle, the directionality of the energy is lost, as if the wave-guide were broken, and corresponding compressibility waves are radiated in the epola. This is observed as the electromagnetic radiation of the accelerated (or decelerated) electron.

As we have also seen, the accompanying wave is a standing wave in the frame of reference of the moving particle. In this frame, the particle and its accompanying wave are a closed system. But in the epola frame, the accompanying wave is a traveling wave, in which a standing wave pattern of half-wave deformation clusters of the Compton wave (or such 'wave packet') is moving with the velocity $v$ of the particle, while the propagation velocity of the accompanying wave is the velocity of light $c$ . This corresponds to the "group" and "phase" velocities in quantum mechanics.

8.5 Correspondence between accompanying waves and quantum-mechanical waves of matter

It is seen that the accompanying wave may well be the physical background for the quantum-mechanical "waves of matter" or "wave packets", in their association with the moving particle, and for the "group velocity" in its association with the velocity of the particle. However, in quantum mechanics these mathematically derived associatives became replacements for the particles. This led to the denial of the existence of the particles ("with such good waves, who needs particles?"), to the denial of physical causality and the reality of the world, all according to the philosophical tastes of the particular mathematician or the interpreter.

The accompanying wave is a real wave in the electron-positron lattice. It accompanies the motion of a real particle, which moves, surrounded by a wave packet, which is, in turn, a real wave-pattern of half-wave deformation clusters in the lattice. The particle and the wave or wave pattern each exist and do not serve as replacements of each other. The wave is connected with the quantum-mechanical probability of finding the particle only because the wave accompanies the particle. Thus, where this wave pattern is, there is the particle.

The correspondence between the accompanying waves and the de Broglie waves of matter is disturbed by the freedom of extrapolation, taken in quantum mechanics. Both our formula for the wavelength of the accompanying wave and the de Broglie formula were derived only for electrons. However, in quantum mechanics, as well as in relativity, it is customary to extend results obtained for electrons onto any ponderable mass. Therefore, what is right for an electron becomes right for transatlantic cruisers, jetliners, twin brothers and clocks, just by saying that it "obviously" should be so (recall Aristotle's 'four-horse rule', Section 2.4).

For the motion in the electron-positron lattice, there is a profound distinction between 'dense' particles, i.e., "elementary" particles and atomic nuclei, as opposed to "planetary" systems of dense particles, i.e., atoms and bodies of atoms. The dimensions of the dense particles (1 to 5 fm) allow them to move in channels between the particles of the epola. Also, the internal binding of a subparticle in the dense particles usually exceeds the 511 keV binding energy of an epola particle. Therefore, dense particles would not disintegrate during fast motion in the epola.

By virtue of the above, the formula for wavelength of accompanying waves, as well as other formulas derived for the electrons, can be shown to be valid when extended to dense particles. They cannot be applied to atoms and atomic bodies as wholes. Any such application leads to useless, if not ridiculous results (Section 9.6).

8.6 Explanation of the particle-wave duality

The experimentally observed corpuscular properties of electromagnetic waves and undular properties of particles cannot be explained by any existing theory. Therefore, as is customary, a postulate was introduced, stating, without any explanation, that in some phenomena, waves may exhibit particle properties, while in other phenomena they exhibit wave properties; similarly, particles may in some phenomena exhibit wave properties, while in other phenomena they exhibit corpuscular properties. This postulate is the "particle- wave duality principle".

It is easy to see that this principle, like so many other principles of our science, does not explain a thing; it only formulates the experimental or other observation, as if saying to the student or reader: "This is it; do not worry; this is the way it should be." The strange thing is that a lesson or page later, the teacher or author discussing a related phenomenon merely mentions the principle, and this reference is considered as an explanation of the phenomenon. Obviously, an unexplained principle cannot explain a thing.

We shall first explain why electromagnetic waves exhibit corpuscular properties. We have seen that the physical basis for the electromagnetic wave is an epola wave. The epola wave is composed of vibrating epola electrons and positrons, transferring energy and momentum among them. When there is a free particle on the path of such a transfer process or, using the photon concept for the transfer process, on the path of the photon, the free particle may be literally knocked by the last epola particle in the row. The free particle may then absorb the energy and momentum, transferred to it by the knocking epola particle. Then the whole energy and momentum transfer process along this row is finished. Using the photon concept, one would say that the photon was absorbed and disappeared. The last epola particle in the row (or on the path of the photon) which did the knocking of the free particle in his way, has revealed the corpuscular properties of the wave. Therefore,

the corpuscular properties of the electromagnttic wave are due to the fact that these waves result from vibrations of electrons and positrons of the epola; phenomena, in which the energy and momentum of an individual epola electron or positron can be detected, reveal the corpuscular properties of the electromagnetic wave.

The undular properties of elementary particles and nuclides are due to the deformation clusters which they cause in the epola, and to the epola waves which accompany their motion. An epola wave, thus, an electromagnetic wave, which approaches the particle, does not "see" the particle. It interacts with the deformation cluster around the particle or with the accompanying wave. This interaction results in the appropriate diffraction and interference phenomena. Therefore,

the wave properties of particles are actually the properties of the deformation clusters and the accompanying waves formed around the particles in the epola.

8.7 The epola basis of Heisenberg's uncertainty principle and its derivation

The uncertainty principle was first formulated by W. Heisenberg in 1927. The principle states that the uncertainty in the position of a particle, multiplied by the uncertainty in the momentum of the particle, is approximately equal to Planck's constant $h / 2 \pi \equiv \hbar$ . This means that the more exactly we know the position of the particle, the less exactly we may know the momentum, and vice versa.

As the other principles of quantum mechanics, this principle too was introduced without any explanation and without any physical model or proof. Considered as proof is that the principle works well in quantum mechanics. This is, therefore, like the proof of a pudding - in eating. Moreover, the principle is considered as a kind of divine revelation, which should be accepted as is, with no interpretations. Even the mention of possible measurement effects is forbidden.

We shall now prove that what is true and right in the uncertainty principle is merely a result of the electron-positron lattice structure of the vacuum space. As we know, an epola wave consists of half-wave epola deformation clusters, the radius of which is $\lambda / 4$ , where $\lambda$ is the wavelength of the epola wave. If we would search for a 'dense' particle with an electromagnetic wave of this wavelength, then the uncertainty in locating the particle would be equal to the radius of the half-wave epola cluster of this wave. Actually, the uncertainty $\Delta x$ should be slightly smaller, because the probability of 'catching' the particle at the boundary of the cluster is smaller. This can be corrected, according to a probability formula, by dividing $\lambda$ in the relation for the radius of the spherical cluster by $2 \pi$ , instead of by 4. Therefore, the uncertainty in position $\Delta x$ is

\Delta x = \frac{\lambda}{2 \pi}

The searching epola wave has photon energy $hf$ and momentum $hf / c = h / \lambda$ . When the searching wave catches the particle, the photon momentum may be transferred to the particle. Therefore, the uncertainty in the momcntum $\Delta M$ of the particle is the momentum of the searching photon,

\Delta M = \frac{h}{\lambda}

Thus,

\begin{align*} \Delta x \cdot \Delta M &= \frac{\lambda}{2 \pi} \cdot \frac{h}{\lambda} \\ &= \frac{h}{2 \pi} \\ & \equiv \hbar \end{align*}

This is Heisenberg's uncertainty principle, derived with the purest of physical causality.

The uncertainty principle can also be expressed by the uncertainties in time and in energy. The uncertainty $\Delta t$ in the time of detection of the particle is equal to the time it takes the epola wave or its photon to travel the distance $\Delta x$ , thus,

\begin{align*} \Delta t = \frac{\Delta x}{c} = \frac{\lambda}{2 \pi \cdot c} \end{align*}

The uncertainty $\Delta E$ in the energy of the particle is the energy of the searching photon, which might be transferred to the particle. Thus,

\Delta E = hf = \frac{hc}{\lambda}

Therefore,

\begin{align*} \Delta E \cdot \Delta t &= \frac{hc}{\lambda} \cdot \frac{\lambda}{2 \pi c} \\ &= \frac{h}{2 \pi} \\ & \equiv \hbar \end{align*}

This means that the more exactly we know the energy of the particle, the less exactly we may know the time, at which the particle has this exact energy.

As we see, the uncertainty principle is a direct result of the epola structure of the vacuum space. It can be derived and understood on the basis of the epola concepts and pure physical causality.

8.8 Epola release from quantum and relativity postulations

We have seen that the Heisenberg uncertainty principle can be derived directly by accounting for the epola structure of space. With epola considerations we have also explained the particle-wave duality, derived the mass-energy relations, etc. We will show that accounting for the epola structure of space enables the explanation and derivation of all working principles of quantum mechanics and relativity. This proves that the epola model is the right model for phenomena, dealt with by these two branches of our science.

When analyzing the position, timing and motion of a particle, we may either deny the existence of the epola and then we have to consider the quantum-mechanical and relativistic postulatory principles, forbiddals and exclusions to account for the then otherwise unpredictable and uncalculable behavior of the particle. However, when one accepts the existence of the epola, one is able to account for the quantum-mechanical and relativistic principles as effects of the electron-positron lattice. Making sure that in his analysis he took into consideration all epola effects, he automatically and intrinsically accounted for all the working postulatory principles of quantum mechanics and relativity. Therefore, it would be wrong to consider them in addition to the epola considerations, and one must not do so.

This means, particularly, that we may proceed with a micro-micro analysis of interactions in the epola on the basis of regular physical considerations, without worrying about the postulatory principles of quantum mechanics and relativity, provided that all possible effects of the epola are accounted for.

8.9 Mass and inertia of epola waves and frequency invariance

Based on the relation between the binding energy and freed or appearing mass (Chapter 4), we may say that the mass of particles, positioned exactly on their lattice sites, is bound to the lattice and cannot be detected individually. Thus, for an observer inside the lattice, the mass of all lattice particles positioned exactly on their lattice sites is zero.

In a lattice wave-motion, the lattice particles vibrate around their lattice sites. When the vibrating particle is on the lattice site, its appearing mass is zero. The energy of the particle is maximal (including also the binding energy). However, the particle cannot transfer any of this energy to other particles, as long as it is on its lattice site. When the particle leaves the lattice site, its mass appears, contributing to the mass of the half-wave deformation cluster to which it belongs, hence, to the inertial properties of this cluster. The particle is now able to transfer energy and momentum to other particles of the lattice or to a free particle. We may, therefore, state that:

At any instant, the mass, momentum and transferable energy of a half-wave deformation cluster in a lattice wave-motion belong to those vibrating particles of the cluster which are at that particular instant out of their lattice sites.

Because the epola waves –particularly, the half-wave deformation clusters– have mass, they should have inertial and gravitational properties. We shall discuss here only some inertial properties. In Section 8.3 we mentioned the inertial properties of the epola wave, which accompanies the motion of a free electron. The frequency of an accompanying wave is directly proportional to the velocity of the accompanied particle. Any change in this velocity would require a change in the frequency of the wave. Keeping its frequency constant, the accompanying wave "smooths" possible small deviations in the particle's velocity and so it takes part in the inertia of the system "moving particle – accompanying wave".

Keeping constant the frequency of a wave means keeping constant both the energy of the half-wave deformation clusters and their mass. This means that the principle of frequency invariance of electromagnetic waves is actually a corollary of the conservation of mass and energy in the epola wave. The frequency invariance is an additional evidence of a massive presence and participation of particles in the electromagnetic wave.

We may now add another understanding of the mass-energy relation, $E = m c^2$ , which we derived for the epola in Section 6.8. Now, in this formula, $E$ is the energy of a half-wave deformation cluster in an epola wave (or the energy of the resulting electromagnetic wave), and $m$ is the mass of electrons and positrons which are out of their lattice sites in this half-wave deformation cluster. This formula is therefore identical with the formula $E_{cl} = \Delta N \cdot m_e c^2$ of Section 7.6.

8.10 Mass, momentum and energy of photons

The transition from the half-wave cluster formulas to formulas for photons can also be made on the basis of our understanding that the corpuscular action of the electromagnetic-wave-photon is the action of the last epola particle on the path of the photon.

Therefore,

the energy and momentum of the photon, revealed in any experiment, are actually the transferred energy and momentum of an epola electron or positron, which was the last in the path of the photon.

Hence, the mass-energy relation is valid for the photon, when in the formula $E = m c^2$ , $E$ is the energy of the photon,

E = hf = \frac{h c}{\lambda}

and $m$ is the quasi-mass of the photon, or the part of mass of the last epola electron or positron on the path of the photon, as it appeared in the energy and momentum transfer process.

The momentum $M$ of the photon is then, as usual, its energy divided by its velocity,

M = \frac{E}{c} = \frac{hf}{c} = \frac{h}{\lambda}

From the mass-energy relation, the mass of the photon is

m = \frac{E}{c^2} = \frac{h}{\lambda c}

We may also write the momentum of the photon as its appearing or quasi mass $m$ , multiplied by the velocity of the photon,

M = m \cdot c

In the accompanying wave, the wavelength is $\lambda_e = h / m_e v$ . Therefore, the mass $m$ of the photon of the accompanying wave is

m = \frac{h}{\lambda_e c} = \frac{m_e v}{c}

The mass of this photon is as many times smaller than the mass of the accompanied free electron, as the velocity of the electron is smaller than the velocity of light. However, the momentum of this photon,

M = mc = \frac{m_e v}{c} \cdot c = m_e v

is equal to the momentum of the accompanied free electron.

8.11 Epola resistance to sublumic motion of dense particles

We found that the motion of a dense particle causes in the epola an accompanying wave, which propagates with the velocity of light $c$ and pre-forms the epola for the motion. The accompanying wave shifts epola particles apart in front of the moving free particle and converges them behind the particle. Therefore, the motion of the particle with a velocity $v$ much smaller than $c$ proceeds in the epola without any resistance. The coherence of the accompanying wave with the established motion of the particle makes the epola vacuum-transparent for the motion.

When the velocity of the dense particle is sublumic, i.e., close to the velocity of light, the accompanying wave may not have sufficient time to pre-form the epola in front of the particle. The particle must then shift apart the epola particles on its way, or "open the gates" to the epola units "with its own body". The motion faces resistance, which sharply increases as the velocity of the free particle becomes closer to the velocity of light.

A mathematical analysis of the increase of the resistance of the epola to the sublumic motion of a free particle would be very cumbersome. It could possibly be approached with the methods developed for the treatment of the "many-body problem" in solid-state physics.

8.12 Effective mass of dense particles

In the case of the effects of a solid-state lattice on the motion of "free" electrons in it, a comparatively simple solution was found by replacing the mass of the electron with an appropriately fitted "effective" mass. In various solid lattices the effective mass of the electron may be, depending on physical conditions, many times larger or smaller than the mass $m_e$ of the electron. The effective mass of the electron may even be negative. This means that under an applied electric field in the lattice, the electron would be accelerated in a direction opposite to the direction in which it would be accelerated by such a field outside the lattice. The spot in a solid lattice, out of which a negative-mass electron has escaped is "the hole". The hole has therefore a positive effective mass and a positive electric charge, and acts as a particle, analogous to the positron or to Dirac's hole (Section 4.1.).

Let us consider a free electron, moving in the epola with a velocity $v$ close to the velocity of light $c$ . We shall try to account for the resistance of the epola to this "sublumic" motion by replacing the mass $m_e$ of the electron by an effective mass $m$ , which is increasing with the velocity of the particle. However, such is exactly the mass of the system "moving free electron – its accompanying wave". We have seen in Section 8.10 that for a free electron moving with a velocity ${v << c}$ , the quasi-mass of the photon of the accompanying wave is $m_e v / c$ , so that for a dense particle with an effective mass $m$ , the mass of the photon of the accompanying wave is $m v / c$ . Hence, the effective mass of a sublumic particle may be obtained by appropriately combining the mass $m v / c$ of its photon with the 'rest' mass $m_0$ of the resting or slowly moving particle.

8.13 Derivation of the dependence of mass on velocity

We shall present the effective mass $m$ of the sublumic electron as consisting of the mass $m v / c$ of the photon of its accompanying wave and the mass $m_e$ of a resting or slowly moving electron. Thus,

m = \frac{m v}{c} \sim \!\!\!\!\! \vert \ \ m_e

The wavy plus sign $\sim \!\!\!\!\! \vert \$ indicates that we do not know how to sum the stable mass $m_e$ with the mass of a photon, transferred among the epola particles with the velocity of light. Multiplying by $c$ , we obtain

m c = m v \sim \!\!\!\!\! \vert \ \ m_e c

Here, $mc$ is the momentum of a particle of effective mass $m$ moving with velocity $c$ ; $mv$ is the momentum of the photon of the epola wave, accompanying the motion of a particle with effective mass $m$ , the velocity $v$ of which is close to $c \;\!$ ; $m_e c$ is the hypothetical momentum of an electron of mass $m_e$ , moving with velocity $c$ , in a space where there is no epola, thus no resistance to this motion. When this electron arrives at a "gate" to an epola unit, it pushes apart the epola particles there in a direction perpendicular to its momentum $\overrightarrow{m_e c}$ , thus creating the photon of the accompanying wave. Therefore, at this instant and at this instant only, the momentum $\overrightarrow{mv}$ of the photon is perpendicular to $\overrightarrow{m_e c}$ . Now we can add these two vectors according to the law of vector addition. We have

(mc)^2 = (mv)^2 + (m_e c)^2

which yields

m = m_e \left( 1 - \frac{v^2}{c^2} \right)^{-1/2}

This is the relativistic expression for the dependence of mass on velocity, introduced by A. Einstein in 1905 (Section 2.2). It is derived (as also by Einstein) for electrons only; however, it can be extended to fit dense particles. The sizes of the dense particles (1 to 5 fm) allow them to move in channels between epola particles. Also, the internal bonds per subparticle of the dense particles exceed 511 keV; therefore, the dense particles do not disintegrate during fast motion in the epola. Application of this formula to any ponderable mass, i.e., also to atomic bodies, twin brothers and clocks, is senseless because atomic bodies disintegrate at velocities a hundred times smaller than the velocity of light, and people would die at velocities above 100 km/s relative to Earth (Section 9.6).

8.14 The possibility of superlumic motion

Superlumic is a motion in the epola with velocity $v$ exceeding the vacuum light velocity $c$ , $v>c$ . We have already pointed out and shall prove in Section 9.6, that atoms and atomic bodies would disintegrate at velocities a hundred times lower than $c$ . Hence, we shall consider here only 'dense' particles or avotons, as we named them in Section 5.10. We divide the avotons into two groups: small avotons, with diameters $D$ smaller than the epola constant $l_0$ , $D < l_0$ or ${D < 4 \ \mathrm{fm}}$ , and large avotons, with $D \ge l_0$ . More precisely, the division should be at $D = l_0 \sqrt{2} - 2 R_e$ , where $R_e$ is the radius of the electron (see Figure 3, Section 5.10).

As in motion with velocities close to $c$ , in superlumic motion there is also no support from an accompanying wave, which would open the gates to epola units. Hence, the superlumic avoton must push away the epola particles on its way 'with its own body'. However, superlumic motion not only lacks the support of an accompanying wave, but is hindered by the opposite of such wave – by a Cherenkov-type wave, which it causes in the epola behind. It can be shown that both the epola wave which accompanies sublumic motions and, on the other hand, the Cherenkov-type wave, which 'accompanies' the superlumic motion, are two extremes of the epola reaction to the motion of avotons in it.

The superlumic avoton loses energy not only to collisions with epola particles on its way but also to the Cherenkov-type epola wave and to stray vibrations and waves of quadrillions of epola particles behind it. The superlumic motion lasts for as long as the energy of the avoton can afford it. When the velocity of the avoton falls below the velocity of light, its motion starts to be supported by an accompanying wave, though the avoton continues to lose energy in collisions; the less, the lower its velocity. This continues until such a velocity that the accompanying wave has sufficient time to prepare the epola ahead for an undisturbed passage of the impoverished avoton. The epola is then vacuum-transparent for its motion and the avoton can move happily forever or until disturbed by another avoton; whichever happens first.

8.15 Superlumic motion of small avotons

Small avotons can move in channels between epola particles, colliding only with those of them which happen to be sufficiently far out of their lattice sites. Hence, small avotons may have between collisions a free path of hundreds or thousands of lattice units; the longer, the smaller the avoton. The free path also depends on the conditions in the epola and is the longer, the lower the epola temperature and the smaller the concentration of defects and impurities in the epola (Sections 5.8, 11.3, 11.4).

In each collision, the superlumic avoton pays a 'road-toll energy' to the epola particle on its way. This energy is then turned into the random motion of epola particles, or radiated as electromagnetic waves. The energy-tolls vary from collision to collision and are usually smaller than $m_e c^2$ ; the smaller and slower the avoton the smaller the energy toll. Only in the rare case of a head-on collision, the epola particle would not give way until accelerated to the velocity of the avoton. The gained energy will then be radiated as a $\gamma$ -photon. However, head-on collisions have low probability; again, the smaller the avoton. the lower the probability.

The electromagnetic radiation spectrum of superlumic small avotons is more or less continuous, due to the variety of energies dissipated in various collisions. However, some spectral regions may be missing, especially if the superlumic motion did not last long enough to fill all possible energies. But even if it did, the energy distribution among the various wavelengths would not be close to that in the blackbody radiation. Hence, the blackbody radiation laws should not be automatically applied in calculations involving observable continuous cosmic spectra, if the thermal character of the source is not assured.

8.16 Superlumic motion of large avotons

Avotons of sizes equal to or larger than the gate to an epola unit cube (Fig 3, Section 5.10) have a 'free path' of one lattice unit. This means that they have to collide with every epola particle in each entered epola unit. Moreover, most of the collisions will be of the 'head-on' type, so that these epola particles are pushed by the avoton until they reach its velocity. This may mean a transfer of quite a few MeV to each epola particle. In some avotons, e.g., in the uranium nucleus, this may exceed the binding energies. Thus, they may disintegrate into smaller avotons. In other words, forcing very heavy nuclei into a superlumic motion in the epola may initiate a nuclear fission process. In any case, the superlumic motion of single large avotons must cease within very short distances, due to either a reduction in their velocities below the sublumic range or due to their disintegration into small avotons.

Consider a powerful nuclear reaction, which creates streams of large super lumic avotons in a particular direction. These avotons tear the epola off along this direction, creating a breakdown channel, analogous to electrical breakdown channels in insulating crystals or to the channel in air caused by a bolt of lightning. In such a channel, the leading avotons push the epola electrons and positrons in front of them with superlumic velocities, creating $\gamma$ -rays of mostly a single energy. The streams of avotons behind them replenish the energy resources of the leading avotons. Otherwise, they do not lose energy to radiation, because they move in the empty channel and have very little contact wilh the epola to the sides. However, they transfer energy backwards to the Cherenkov wave and to the stray vibrations and waves left behind. Such superlumic motion can continue as long as the avatons are streaming into the channel, but stops very soon after they stop.

The energy acquired by an avoton, e.g., in a cosmic nuclear reaction, may well exceed the $10^{25} \ \mathrm{eV}$ range. In the epola, the avoton loses energy on creating many $\gamma$ -photons and other electromagnetic radiation. Suppose that our spectroscopist did observe these bursts of radiation. Adding the observed energies, he obtains, say, $10^{24} \ \mathrm{eV}$ . Then, in the customary way, he ascribes this energy to a single maternal $\gamma$ -photon. His paper will be published, although nobody has ever observed $\gamma$ -photons of energies exceeding 140 MeV in a direct way. Such energies could only be obtained by summing up the much lower secondary and tertiary radiation quanta. In the epola space, the highest (cutoff) energy of a photon is 140 MeV (Section 7.10). A higher energy can only be the energy of an avoton and, if the energy is in the $10^{20}$ eV range, then it most probably is the energy of a superlumic avoton.

Chapter 7: Epola Waves and Photons Chapter 9: Motion of Orbital Electrons, Atoms and Bodies of Atoms in the Epola