Chapter 7. Epola Waves and Photons

7.1 Stabilization of epola wave motions and frequency invariance

Let us follow the beginning of a harmonic oscillation with frequency $f$ or period $T \equiv 1 / f$ , of a single epola particle in an otherwise undisturbed epola. At the moment $t = 0$ the particle, marked $O$ , is in equilibrium position, having velocity ${v_0 << c}$ . During the first quarter period, or the time $T/4$ of the amplitudinal deflection of particle $O$ , the disturbance spreads throughout with the velocity $v_d$ of epola bulk deformation waves, $v_d = \left( {}_b E / m_e \right)^{1/2}$ , which is equal to the vacuum light velocity $c$ (Sections 6.3. 6.5). At $t = T/4$ , the surrounding particles, swept gradually into motion, form a primary spherical deformation cluster with a radius $R_{cl} = c \cdot T/4$ . During the second quarter period, or time from $t = T/4$ to $t = T/2$ the particle $O$ returns to the equilibrium position, and a return signal travels through the cluster to move its particles in the opposite direction, while the first signal of the initial deflection is transferred with the velocity $c$ further out to form the neighboring clusters, and so on.

The general principle of wave motion, discovered by Huygens and used ever since in all wave constructions, is that

every point in space reached by a wave motion can be considered as a center of propagation of a new spherical wave.

To make Huygens's principle applicable for the epola we shall rephrase it as follows:

every lattice particle reached by an epola wave is forced to vibrate with the frequency of the wave; therefore, it acts as a center of a new spherical epola wave.

Part of the energy of the new spherical waves, formed by each and every particle in the epola wave, returns to the 'initiating' particle $O$ and to the 'primary' half-wave deformation cluster. Therefore, every epola particle in the wave is subject to an infinite number of signals or 'move!' orders, interfering with each other. An epola wave motion can be established if as a result of this interference in each particle, the vibrations of the primary cluster, as well as of any other half-wave deformation cluster of the wave motion, do come into compliance with the returning signals from clusters formed further and further away. Otherwise the primary vibration is damped and its energy is transferred to the random vibrations of epola particles around their lattice sites. Obviously, the interference conditions become much more complicated and severe in the presence of guest particles, atoms and bodies of atoms in the epola. This points out that not every or any vibration of epola particles or even systems of epola particles do establish a wave motion in the epola.

Once a wave motion is established in the epola, then the vibration of every epola particle in the wave is linked with quadrillions of particles, interconnected with each other and vibrating with the same compliant frequency. Therefore, if some factor would try to change the frequency of a particle or of a group of particles, the quadrillions of the other particles in the wave-motion will restore the compliant frequency. This explains the observed stability of the frequency or the principle of "frequency invariance". The frequency of wave motions in the epola is preserved even when the waves pass through regions containing bodies of atoms or through otherwise distorted zones.

7.2 Half-wave clusters in epola compressibility waves

Epola bulk deformation waves can be compressibility waves, which are waves of long and medium wavelengths, and shock or impact waves, of short and very short wavelengths. In compressibility waves the half-wave deformation clusters have instantaneous spherical symmetry, so that the volume of the cluster and the equilibrium number $N_{cl}$ of epola particles in it is proportional to the third power of the wavelength, $N_{cl} \propto {\lambda}^3$ .

The deformation in the cluster is due to the excess epola particles, which entered the cluster or left it because of their vibrations in the wave motion. As long as the lattice remains intact, the excess particles, pushed into the half-wave cluster or pulled out of it can only be particles of the boundary layer of the cluster. Therefore, the number $\Delta N$ of the excess particles is proportional to the outer area of the half-wave cluster, or to the square of the wavelengt, $\Delta N \propto {\lambda}^2$ .

Every excess epola particle entering the cluster brings in an amount of energy equal to its binding energy $m_e c^2$ , and every leaving particle takes this energy out. Therefore, the deformation energy $E_{cl}$ of a half-wave epola deformation cluster is proportional to the number $\Delta N$ of excess particles in it, thus to the square of the wavelength, $E_{cl} \propto \Delta N \propto {\lambda}^2$ .

The average deformation energy $E_p$ per particle in the cluster, i.e., the energy of the cluster divided by the number of particles in it, is, therefore, inversely proportional to the wavelength

E_p = \frac{\Delta N}{N_{cl}} \propto \frac{1}{\lambda}

7.3 Definition of photons in the epola and phonons in solids

The transfer of energy in the epola wave-motion can be represented by the motion of half-wave deformation clusters, or also by the transfer of the average per-particle energy $E_p$ in the half-wave cluster from particle to particle along any direction in the wave motion. This transfer of energy $E_p$ can be considered as the motion of a quasi-particle having this energy and moving with the velocity $c$ of the wave motion. Our definition of the photon is, hence:

The photon is an imaginary quasi-particle depicting the transfer of energy in an electromagnetic wave. The photon energy or the quantum of energy in this wave is the average energy transferred from one epola particle to the next in line in the epola wave motion.

The first sentence in this definition agrees with the original definition given by Einstein in 1905. The second part expresses the meaning of the photon in the epola. This meaning is in full agreement with all observed phenomena in which photons are involved. It releases the obscurity of the photon cancept in quantum theory (Section 3.3).

In our definition the photon is not a real particle and does not represent a real particle. It does not even represent the vibrational energy of a real particle in the epola wave motion, because this energy is proportional to the square of the amplitude of the particle. It only represents the per-particle part of the energy, transferred in the wave motion.

Our definition of the photon can also be used to define the phonon in solids (or liquids) by inserting the appropriate terms. Thus:

The phonon is an imaginary quasi-particle, depicting the transfer of energy in an elastic wave or sound wave in a solid (or liquid). The phonon energy or the quantum of energy in this wave is the average energy, transferred from one ion, atom or molecule of the body to the next in line in the wave motion.

What was said about the nature of the photon relates also to the phonon. However, as in the case of the epola $-\mathrm{NaCl}$ lattice analogy (Section 5.1), the photon-phonon analogy cannot be driven too far. Material bodies are much more complex than the epola, so that the phonon concept is also more complex and diverse than the photon concept. However, our derivation of the photon energy, of Planck's law and of Planck's constant can be carried out on phonons in crystalline solids as well.

7.4 Derivation of Planck's Law

We found that the energy $E_p$ transferred from particle to particle, thus the energy of the photon, is inversely proportional to the wavelength of the epola wave motion, $E_p \propto 1 / \lambda$ . We may write this proportionality in the form $E_p \cdot \lambda = \mathrm{const}$ , meaning that the photon energy in any epola wave motion, multiplied by the wavelength of this motion, is a constant. This can also be written in the form

E_p \cdot \lambda = E' \cdot \lambda '

where $E' \cdot \lambda '$ is the outcome of any experiment, in which the photon energy $E'$ is measured together with the corresponding wavelength $\lambda '$ . Such can be, e.g., the photon energy $E_c = m_e c^2$ of the electromagnetic wave with the Compton wavelength $\lambda_c = 2426 \ \mathrm{fm}$ . Therefore,

\begin{align*} E_p \cdot \lambda &= E' \cdot \lambda ' \\ &= E_c \cdot \lambda_c \\ &= 511 \ \mathrm{keV} \times 2426 \ \mathrm{fm} \\ &= 1.24 \ \mathrm{eV} \cdot \mathrm{\mu m} \end{align*}

Replacing $\lambda$ by $c / f$ , we have

\begin{align*} E_p & = \left( \frac{E' \cdot \lambda'}{c} \right) \cdot f \\ & = \left( \frac{1.24 \ \mathrm{eV} \cdot \mathrm{\mu m}}{300 \ \mathrm{Mm / s}} \right) \cdot f \end{align*}

E_p = hf

which is Planck's Law. Here,

\begin{align*} h & = \frac{E_c \lambda_c}{c} \\ & = \frac{1.24 \ \mathrm{eV} \cdot \mathrm{\mu m}}{300 \ \mathrm{Mm / s}} \\ & = 4.14 \ \mathrm{feV} \cdot s \\ & = 6.63 \cdot 10^{-34} \ \mathrm{J \cdot s} \end{align*}

is Planck's constant.

7.5 The Compton Wave in the epola

According to our description of epola compressibility waves, the shortest wave of this kind would be a wave with a single excess particle in the half-wave deformation cluster. The deformation energy of this cluster is then equal to the binding energy of this single particle, or to $m_e c^2$ . However, the transfer of the energy of a single excess particle to the next particle of the cluster can only be described by a single photon having the energy of the excess particle. This means that the single excess particle which entered the cluster transfers the $m_e c^2$ energy to the nearest epola particle, this particle transfers it to the next in line and so on. Thus, for $\Delta N = 1$ , the number of photons $N_p$ in the half-wave cluster is $N_p = 1$ and the energy of the photon is equal to the energy of the cluster,

\begin{align*} E_p &= E_{cl} \\ & = \Delta N \cdot m_e c^2 \\ & = m_ec^2 \end{align*}

It was experimentally established and it also follows from Planck's law, that an electromagnetic wave with photon energy $m_e c^2$ has the Compton wavelength $\lambda_c = h / m_e c = 2426 \ \mathrm{fm}$ .

We shall refer to such wave as the Compton wave.

The diameter of the half-wave deformation cluster of the Compton wave is $1213 \ \mathrm{fm}$ and the number $N_d$ of epola particles across this diameter is

\begin{align*} N_d &= \frac{\lambda_c}{2 l_0} \\ & = \frac{1213 \ \mathrm{fm}}{4.4 \ \mathrm{fm}} \\ & \approx 280 \end{align*}

If spherical symmetry can still be assumed, then the number of epola particles in this cluster is

\begin{align*} N_{cl} &= \left( \frac{4 \pi}{3} \right) \cdot \left( \frac{\lambda_c}{4 l_0} \right) \\ & = 1.1 \cdot 10^7 \end{align*}

7.6 Characteristic relations for epola compressibility waves

The characteristic values found for the Compton wave can be used as proportionality factors in the expressions for the half-wave deformation clusters of epola waves. We did it already in our derivation of Planck's law (Section 7.4) when we substituted for $E' \lambda'$ the known values of the photon energy $E_c = m_e c^2$ and the wavelength $\lambda_c = 2426 \ \mathrm{fm}$ of the Compton wave. The derived standard expression for Planck's law may thus be replaced by

\begin{align*} E_p &= \frac{E_c \cdot \lambda_c}{\lambda} \\ & = \frac{m_e c^2 \cdot \lambda_c}{\lambda} \\ & = \frac{1.24 \ \mathrm{eV} \cdot \mathrm{\mu m}}{\lambda} \end{align*}

The proportionality expression for the equilibrium number $N_{cl}$ of epola particles in a half-wave cluster, $N_{cl} \propto \lambda^3$ can now be replaced by $N_{cl} / 1.1 \cdot 10^7 = \lambda^3 / {\lambda_c}^3$ , leading to

N_{cl} = 1.1 \cdot 10^7 \left( \frac{\lambda}{\lambda_c} \right)^3

The number $\Delta N$ of excess particles, $\Delta N \propto \lambda^2$ is correct for $\lambda > \lambda_c$ . With the Compton-wave value of $\Delta N = 1$ we may rewrite the proportionality as $\Delta N / 1 = \lambda^2 / {\lambda_c}^2$ , so that

\Delta N = \left( \frac{\lambda}{\lambda_c} \right)^2

The number $N_d$ of epola particles along the diameter of the cluster is proportional to the wavelength, so that $N_d / 280 = \lambda / \lambda_c$ , thus

N_d = 280 \frac{\lambda}{\lambda_c} = \frac{\lambda}{2 l_0}

The energy $E_{cl}$ of the cluster, $E_{cl} \propto \Delta N$ , or $E_{cl} / E_c = \Delta N / 1$ , is thus

\begin{align*} E_{cl} &= E_c \cdot \Delta N \\ & = m_e c^2 \left( \frac{\lambda}{\lambda_c} \right)^2 \end{align*}

The number $N_p$ of photons in the cluster, multiplied by the photon energy $E_p$ must be equal to the energy $E_{cl}$ of the cluster at any instant, because of energy conservation:

E_{cl} = N_p \cdot E_p

Therefore,

\begin{align*} N_p &= \frac{E_{cl}}{E_p} \\ & = \frac {\displaystyle m_e c^2 \left(\displaystyle \frac{\lambda}{\displaystyle \lambda_c} \right)^2} {\displaystyle m_e c^2 \left( \frac{\displaystyle \lambda_c}{\displaystyle \lambda} \right)} \\ & = \left( \frac{\lambda}{\lambda_c} \right)^3 \end{align*}

It is seen, that $N_p$ is $1.1 \cdot 10^7$ times smaller than $N_{cl}$ . This means that in any half-wave deformation cluster of an epola compressibility wave there is at any instant one photon per every 11 million epola particles. The derived expressions for $N_{cl}$ , $\Delta N$ , $N_d$ and $N_p$ are plotted against the photon energies, the frequency and wavelength of the epola (electromagnetic) waves in Figure 4, to the left of $\lambda = \lambda_c$

Figure 4. Spectrum of epola waves (of electromagnetic waves)

Abscissas are the photon energy, in units of $m_e c^2$ (lowest axis) and in $\mathrm{eV}$ , the frequency $f$ in $\mathrm{Hz}$ , the wavelength in $\mathrm{m}$ and in units of the Compton $\lambda_c$ . $N_d$ is the number of epola particles along the diameter of the $\lambda / 2$ deformation cluster, $N_p$ is the number of photons, $\Delta N$ - the number of excess particles, and $N_{cl}$ - of equilibrium particles in the $\lambda / 2$ cluster. The vertical dotted line connecting points $E = m_e c^2$ , $\lambda = \lambda_c$ , $\Delta N = N_p = 1$ , divides between compressibility and impact waves. The right edge vertical line corresponds to the shortest epola wave, $\lambda = 2 l_0 = 8.8 \ \mathrm{fm} = \lambda_c / 280$ , $f = 3.4 \cdot 10^{22} \ \mathrm{Hz}$ , and $E_p = 280 \ m_e c^2 = 140 \ \mathrm{MeV}$ . (Section 7.10)

7.7 Epola impact waves

The half-wave deformation clusters in epola waves with wavelengths shorter than the Compton wavelength, $\lambda < \lambda_c$ , are always invaded by one excess particle. The energy of the cluster, thus also the energy of the single photon, is $E_c = m_e c^2$ , plus the kinetic energy ${}_k E$ of the invading (excess) particle. This energy defines the depth and the duration of the invasion, hence the subsequent deflection of the receiving host particle and, consequently, the energy transferred in the wave-motion, or the energy of the photon. The energy-transfer has the character of an impact and the waves in this case are epola impact waves. The characteristic relations for the impact waves are:

\begin{align*} \Delta N & = N_p = 1 \ , \\ E_{cl} & = E_p = m_e c^2 + {}_k E \ , \\ E_p & = m_e c^2 \frac{\lambda_c}{\lambda} = hf \ . \end{align*}

These relations are depicted in Figure 4 to the right of $\lambda = \lambda_c$ .

As the wavelength becomes shorter and shorter, the impact or shock character of the wave strengthens. The half-wave clusters lose their spherical shape, becoming more and more prolate ellipsoids. The long axes of the elipsoids aim in the direction of propagation of the single photon. Their length is $\lambda / 2$ and the number $N_d$ of epola particles along these axes is

N_d = 280 \ \frac{\lambda}{\lambda_c} = \frac{\lambda}{2 \ l_0}

as in compressibility waves.

We assumed that in the Compton wave with $\lambda = \lambda_c$ , the half-wave cluster is still spherical, which allowed us to find the number $N_{cl}$ of particles in this cluster as $1.1 \cdot 10^7$ . This number was then entered into the $N_{cl}$ expression for compressibility waves. In impact waves we do not know how the short axis of the ellipsoidal half-wave clusters shortens with decreasing wavelength. For the shortest possible wavelength, where $N_{cl} = 1, \ \lambda = 2l_0$ , the short axis should be considered as just the effective diameter of the electron or the positron.

7.8 Directionality of epola impact waves

It is experimentally known that the impact or shock waves in elastic media do not spread energy evenly in all directions allowed by interference, as do compressibility waves. The energy of shock waves spreads mostly in the direction of the shock or the impact which created them, the more, the higher the power of the impact. Part of the energy of the impact waves is always dissipated throughout the medium by elastic waves, but this part is small. the smaller, the larger the impact power.

Electromagnetic radiation of wavelengths much longer than the Compton wavelength, identified by us as representing epola compressibility waves, spreads evenly in all directions (allowed by interference). In order to direct the radiation, i.e., to 'squeeze', say, 90 percent of it into a solid angle as small as possible, one has to build complicated systems based on reflection, refraction, as in wave guides, on interference and on increased coherence of the radiation, as in lasers. With all this, the result is still partial, the better, the shorter the wavelength of the radiation.

The ease with which long-wave electromagnetic radiation spreads in space is due to the ease with which epola particles can be deflected from their equilibrium positions at their lattice sites. This is shown by the flatness of the particle's energy minimum at $l = l_0$ in Figure 1 (Section 5.2). Therefore, in spite of the very high binding energy of the particle in the epola, even an infinitesimal energy is able to deflect the particle from its equilibrium position at $l = l_0$ .

On the other hand, even the smallest deflection of an epola particle is resisted by the surrounding quadrillions of particles and the terrific binding forces between them. But if energy is supplied to an epola particle slowly enough, then the slowly rising deflection of the particle is not resisted by its nearest neighbors, which have the needed time to deflect, too. Similarly, their deflections are not resisted by their neighbors, and so on. Hence, the energy is absorbed 'softly' and a compressibility wave is formed (see Section 7.1). If the energy is 'thrown' at the epola particle during a very short time, then the elastic deflections of connected epola particles have no time to develop. The deflecting power has against itself a super-solid wall of epola particles resisting the deflection. The energy cannot be accepted, the deflection cannot occur and the compressibility wave is not formed.

Electromagnetic radiation of wavelengths equal to and shorter than the Compton wavelength $\lambda_c = 2426 \ \mathrm{fm}$ , or $\gamma$ -rays, identified by us as epola impact waves, are known to have an explicit directionality. Gamma-rays propagate in rectilinear channels without significantly dissipating energy to the sides, just as if they were moving in wave-guides. created by themselves in space.

The wave-guide directionality of $\gamma$ -rays can be explained, considering that they represent epola impact waves. The half-wave cluster of these waves contains a single excess particle. The energy of this particle is transferred as a single photon along the diameter of the cluster in the direction of motion of the excess particle. This direction is preserved when the single excess particle forms clusters further and further away, which is the physical basis for the propagation of the $\gamma$ -ray photon.

The energy of the impact-wave cannot be spread sidewise when the frequency is high. Suppose that an epola particle, slightly outside a half-wave cluster, i.e., off the main direction of Ihe impact wave, received from the wave a stray signal, ordering the particle to deflect in a certain sidewise direction. The signal persists only during one quarter-period of the wave, or a time $t = T/4$ . Due to the high frequency of the impact wave, this time is too short for the particle to deflect against the opposing super-solid wall of neighboring epola particles, which did not receive the signal. During the next quarter-period, the signal commands the particle to deflect in the opposite direction, which the particle cannot do because of the wall of epola particles, opposing this deflection, and so on. The energy of the stray sidewise signal cannot therefore be accepted by this particle or by any other particles of the epola, adjacent to the propagating half-wave clusters of the impact waves. The super-solid walls of their surrounding quadrillions of epola particles serve as the walls of the "wave guide" of the impact wave, i.e., of the $\gamma$ -ray.

7.9 Electrical polarity of epola impact waves

The electrical polarity of the half-wave clusters of impact epola waves is the stronger the shorter the wavelength. With only one excess particle in each cluster it is obvious that if in one cluster the excess particle is an electron, then in the next cluster the excess particle must be a positron. In the half-wave cluster of the Compton wave this means one electron or positron charge, $-e$ or $+e$ , per 11 million epola particles. Though the wave as a whole is electrically neutral, on distances shorter than the wavelength strong electric fields should be experienced.

The electric field of a half-wave cluster in epola impact waves grows very fast with decreasing wavelength. Even if the cluster were spherical, its volume and the number of epola particles in it would be reduced proportionally to $\lambda ^{-3}$ . However, with the decrease of wavelength below $\lambda_c$ , the half-wave clusters become more and more prolate ellipsoids. This is an additional factor reducing the volume of the cluster and the number $N_{cl}$ of epola particles in it, thus the number of particles per the excess $+e$ or $-e$ charge of the cluster.

7.10 The shortest epola wave, the cutoff frequency and energy

The shortest possible wavelength $\lambda_{min}$ of epola bulk deformation waves is twice the lattice constant $l_0$ ,

\lambda_{min} = 2 \cdot l_0

Such also are the shortest wavelengths of bulk deformation waves in any lattice. In the epola, the wave motion with the shortest wavelength has frequency $f_{max}$ , sometimes referred to as the 'cutoff' frequency,

\begin{align*} f_{max} & = \frac{c}{\lambda_{min}} \\ & = 3.4 \cdot 10^{22} \ \mathrm{s^{-1}} \end{align*}

The photon energy $E_{max}$ of this wave motion is

\begin{align*} E_{max} & = h f_{max} \\ & = 140 \ \mathrm{MeV} \end{align*}

In the epola wave motion with the shortest possible wavelength the space-domain of each epola particle along the path of the photon is invaded by an excess particle during the entire period: half of the period by an electron, the other half by a positron. This is somehow equivalent to tearing the lattice apart. Actually, as in the case of solid lattices, 'sublimation' in the epola, i.e., free electron-positron pair production, should occur, and occurs at wavelengths much longer than these.

It should be clear from the above that a 140 MeV photon is exactly a 140 MeV electron (or positron) and there is no way to distinguish between them. What is reported in the literature as $\gamma$ -photons with energies exceeding 140 MeV could be electrons or positrons or other 'dense' particles (avotons, Section 8. 14) which have these high energies. They create in the epola secondary, tertiary, etc., $\gamma$ -rays, out of which the high energy of the primary particle is derived.

7.11 Action of photons, holding time and Planck's constant

The action (or action-function) was introduced to describe a dynamic system of particles. If the system passes from a position it had at time $t_1$ to a position reached at time $t_2$ , then the action $A$ is expressed by the integral

A = 2 \int_{\displaystyle t_1}^{\displaystyle t_2} \! {}_k E \cdot \mathrm{d}t

where ${}_k E$ is the total kinetic energy of the system.

Keeping the spirit of this definition, we introduce the action of a photon as twice its energy, multiplied by the time $t_h$ which it takes to transfer this energy from one epola particle to the next in line. Obviously,

\begin{align*} t_h &= \frac{l_0}{c} \\ &= \frac{4.4 \ \mathrm{fm}}{300 \ \mathrm{Mm / s}} \\ &= 1.5 \cdot 10^{-23} \ \mathrm{s} \end{align*}

We may also consider $t_h$ as the time, during which an epola particle 'holds" the photon energy. Thus, $t_h$ is the 'holding time' of the photon by an epola particle.

In our presentation, the action $A_p$ of a photon is given by

\begin{align*} A_p &= 2 \, hf \, \frac{l_0}{c} \\ &= 2 \ \frac{h l_0}{2 \ l_0} \\ &= h \end{align*}

For the photon with the shortest possible wavelength $\lambda_{min} = 2 \cdot l_0$ we have

\begin{align*} A_p &= 2 \ \frac{h l_0}{2 \ l_0} \\ &= h \end{align*}

Hence,

Planck's constant $h$ is the action of the most energetic photon (or the cutoff-frequency photon).

If $h$ is to be considered as Planck's 'quantum of physical action' then the actions of photons are proportional to the frequencies of the epola waves, just as the photon energies or energy quanta. Thus, they are smaller than $h$ as many times, as the frequency of the photons is smaller than the cutoff frequency.

Chapter 6: Epola Waves and Electromagnetic Radiation Chapter 8: Motion of Dense Free Particles in the Epola