Chapter 5. Structure of the Electron-Positron Lattice

5.1 Analogy with the NaCl crystal lattice

Our acceptance of the $\mathrm{NaCl}$ crystal lattice as the closest solid-state analog of the electron-positron lattice is based on several reasons. First, the epola analog must be an alkali-halide crystal because of the equality of the electrical charges of their positive ions (e.g., $\mathrm{Na^+}$) to the charge $+e$ of the positron, and of their negative ions (e.g., $\mathrm{Cl^-}$) to the charge $-e$ of the electron. Also, in terms of the types of crystal bonds, the epola bonds correspond to 'fully ionic'. Therefore, the crystal bonding of the analog must have the highest possible rate of ionicity, which again leads to the alkali halides.

Second, the equality of the masses $m_e$ of the electron and the positron requires that the ion masses of the analog be as close as possible to one another. Therefore, the best choice would be $\mathrm{KCl}$, with $m_{\mathrm{K}} = 39 \ \mathrm{AMU}$ and $m_{\mathrm{Cl}} = 35 \ \mathrm{AMU}$. However, the symmetry in the inner structure of the electron and positron suggests that the electronic shell structure of the alkali and the halide atoms should be similar, i.e., they should belong to the same period in the periodic table of elements. Such are the $\mathrm{Na}$ and $\mathrm{Cl}$ atoms, $\left( m_{\mathrm{Na}} = 23 \ \mathrm{AMU},\ m_{\mathrm{Cl}} = 35 \ \mathrm{AMU} \right)$. Both belong to the third period, have the same filled inner shells (two electrons on the $s$-shell and eight electrons on the $d$-shell), and the same unfilled outer shell (one $p$-electron in $\mathrm{Na}$, seven $p$-electrons in $\mathrm{Cl}$). The atomic masses of other alkali and halide atom pairs from a common period have more diverse masses $\left( m_{\mathrm{Li}} = 7, m_{\mathrm{F}} = 19; \ m_{\mathrm{K}} = 39, m_{\mathrm{Br}} = 80 \ \mathrm{AMU} \right)$.

Third, the binding energy in the $\mathrm{NaCl}$ crystal is the largest among the alkali-halide crystals, which fits the strength of the bond between electrons and positrons. Then, the face-centered cubic (fcc) lattice structure of the $\mathrm{NaCl}$ crystal is most common: it is represented more or less exactly in 17 out of the 20 alkali halide crystals. It should be noted., however, that our calculations based on the analogy with the $\mathrm{NaCl}$ crystal would not be significantly affected by the use of some other alkali halide crystal as analog.

The analogy with the $\mathrm{NaCl}$ crystal cannot be drawn too far. One should bear in mind that electrons and positrons are particles of dense matter, with density comparable to that of the nucleons and nuclei. The internal binding of the subparticles, of which the electrons and positrons consist, by far exceeds the GeV range. Ions, on the other hand, are planetary systems, in which only a $10^{-15}$ part of the volume is occupied by dense particles, the electrons and the nucleus; the rest of the volume is as empty as the vacuum space. The supposed spherical symmetry of the ions is very problematic and very easily affected by split eV energies, thus very susceptible to the presence, the location, the rotations, vibrations and what not of the neighbors. Therefore, in crystals we observe directional effects, strong effects of the internal and surface boundaries and other effects. Contrarily, the epola is expected to be incomparably uniform and homogeneous, with probable directional effects observable only in the micrometer range.

5.2 Formation and stability of a NaCl molecule

The ability of electrons and positrons to form a lattice can be derived from their ability to form an $e^-e^+$ pair. The types of forces or interactions needed to form the $e^-e^+$ lattice or its analog – the $\mathrm{NaCl}$ lattice, are the same as needed to form the $e^-e^+$ pair or its analog – the $\mathrm{NaCl}$ molecule. We shall therefore start with the formation of an $\mathrm{NaCl}$ molecule out of the $\mathrm{Na^+}$ and $\mathrm{Cl^-}$ ions.

The fact is that as long as the distance $l$ between the $\mathrm{Na^+}$ and the $\mathrm{Cl^-}$ ions is larger than twice their distance $l_0$ in the $\mathrm{Na^+ Cl^-}$ molecule $\left(l_0 \approx 300 \ \mathrm{pm} \right)$, their interaction is an electrostatic attraction between two point charges of $+e$ and $-e$, described by Coulomb's Law,

Here, $F$ is the attractive force acting on each of the charges, and $k = 9 \ \mathrm{Gm/F}$ is an SI units factor.

According to the laws of electrostatics, a system of charges can act as a point charge, if the charges of the system are distributed homogeneously on concentric spherical surfaces, or in an equivalent way. The electrostatic field outside the outermost surface is then identical to the field, which would be created there by a point charge, positioned in the common center of the spheres and equal to the algebraic sum of their charges. Therefore, the $\mathrm{Na^+}$ can act as a $+e$ point charge, if the $+11 \ e$ charge of its nucleus and the $-10 \ e$ charge of its orbital electrons are homogeneously distributed on concentric spheres. Similarly, the $\mathrm{Cl^-}$ ion can act as a $-e$ point charge if the $+17 \ e$ charge of its nucleus and the $-18 \ e$ charge of its orbital electrons are distributed homogeneously on concentric spheres.

Further facts are that when the distance $l$ between the ions decreases below $2 l_0$, their attraction weakens, reaching zero at $l = l_0$, i.e., when the molecule is formed. If the ions are then forced to a distance smaller than $l_0$ (e.g., by outside pressure on the substance) they repel each other, and the repulsion steeply increases with decreasing distance. This behavior is depicted by the plots of the attractive and repulsive energies as functions of distance in Figure 1.

The repulsive interaction appears in addition to the electrostatic attraction when the distance between the ions is smaller than $2 l_0$. This means that while the electrostatic interaction has an infinite range, the repulsive interaction has a range shorter than $2 l_0$. It is therefore named 'the short-range repulsion'. The attraction between the ions at $l = l_0$ being zero means that in the molecule the short-range repulsion and the electrostatic attraction forces are equal to each other and the total energy of the two interactions has a minimum. (See Figure 1.) Thus, the two interactions together provide the stability of the molecule.

$l_0$ is the equilibrium distance between the ions in the $\mathrm{NaCl}$ molecule or lattice, also between the electron and positron in the $e^- e^+$ pair or lattice. ${}_{es}E_0$ is the electrostatic attraction energy ${}_{es}E$ at $l = l_0$. ${}_{sr}E$ is the short-range repulsion energy, and ${}_tE$ is the total energy, attractive for $l > l_0$ and repulsive for $l < l_0$ ; at $l = l_0$, ${}_tE$ has a minimum value, which is the binding energy ${}_bE$. In the lattices, ${}_bE = 0.9 \ {}_{es}E$ (Sections 5.4, 5.6).

5.3 The short-range repulsion

The origin of the short-range repulsion cannot be electrostatic, because, as textbooks teach, electrostatic forces alone cannot keep a system of electrical charges in equilibrium. Such a system, whether in the form of a particle, atom, molecule or lattice, would either explode or collapse. Hence, the repulsion cannot be due, e.g., to charge displacements in the ions at proximity, which would expose the electrostatic repulsion between the filled electron shells of the two ions or between their bare nuclei. (Unfortunately, such explanations still persist in teaching.) The gravitational interaction between the $\mathrm{Na^+}$ and $\mathrm{Cl^-}$ ions is 30 orders of magnitude weaker than their electrostatic interaction, so that the short-range repulsion cannot be gravitational in nature. Neither can it be nuclear, because this fundamenlal interaction has a range of 1fm.

The only fundamental interaction left is the electromagnetic one, therefore, the short-range repulsion must be electromagnetic in nature. (Such were also found to be viscosity, adhesion, friction and lately, also the weak nuclear force).

The electromagnetic but non-electrostatic nature of the short-range repulsion can be explained by considering that each of the ions is a complicated system of many charged particles, which are involved in very fast spinning or also in fast orbiting on non-simple orbits, the planes of which rotate around axes which are vibrating or/and precessing. When the distance between the ions exceeds $2 l_0$ all the complicated motions remain an 'internal concern' in each of them and they balance, allowing the ion to appear to the outside world as an analytically treatable coulombic point-charge. At distances smaller than $2 l_0$, the orbiting electrons of each ion interact with the orbiting electrons of the other ion by involving their orbital magnetic moments and the spin magnetic moments. This electrodynamic interaction of mostly magnetic nature results in the short-range repulsion. With decreasing distance between the ions the repulsion strengthens; first, due to the growing interaction between their outermost orbiting electrons, then, due to the increasing number of outer orbiting electrons involved. Finally, inner shell orbiting electrons may be forced into the interaction, resulting in the steep growth of the repulsion at distances below $l_0$.

5.4 Formation and stability of the NaCl lattice

In the formation and stability of the $\mathrm{NaCl}$ crystal lattice, the electrostatic attraction and the short-range repulsion between the $\mathrm{Na^+}$ and $\mathrm{Cl^-}$ ions play the same role as in the $\mathrm{NaCl}$ molecule. The only adjustment to the electroslatic energy is the introduction of the Madelung constant $\alpha$ to Coulomb's expression for the electrostatic force ${}_{es}F_0$,

and for the electrostatic energy ${}_{es}E_0$ of the ion,

In $\mathrm{NaCl}$, $\alpha = 1.75$ (for $l$ close to $l_0$), meaning that the electrostatic energy of an ion, say $\mathrm{Na^+}$, in the lattice is 1.75 times larger than its energy would be at a distance $l_0$ from the other ion, say $\mathrm{Cl^-}$, when both were outside the lattice. In other alkali halides, Madelung's constant can differ by 0.5 percent.

An exact expression for the short-range repulsion cannot be given, because exact analytical solutions are possible for the dynamic interactions of only two point masses or two point charges. The short-range repulsion energy ${}_{sr}E$ was therefore approximated either by a power function of the type ${}_{sr}E = B \cdot l^{-n}$ (Lenard-Jones potentials, 1917) or by an exponential function of the type

Here, $A$ and $r$ (or $B$ and $n$) are factors, chosen to fit experimental data. These early approaches yield acceptable results without obscuring the physics (as do the mathematically advanced approaches, which, disappointingly enough, do not always fit experimental data any better). In the $\mathrm{NaCl}$ lattice, $r = 0.1 \ l_0$; in other alkali halides $r$ is in the range from $0.08$ to $0.14 \ l_0$.

5.5 The short-range repulsion in the epola

When introducing the epola, we made a daring assumption that this lattice is analogous in structure to the $\mathrm{NaCl}$ crystal lattice. This means, particularly, that for the epola we may use the ${}_{es}E$ and ${}_{sr}E$ expressions, with the values of $\alpha$ and $r$ which fit the $\mathrm{NaCl}$ lattice. The use of the ${}_{es}E$ expression (Eq. 2) should not raise objections, because the epola lattice-constant $l_0$ is within the validity range of Coulomb's Law, and the charges ($-e$ and $+e$) are the same. Also, at distances above $2 \ l_0$ (or 10 fm), free electrons and positrons are known to interact as $-e$ and $+e$ point charges. Again, such behavior is possible only if the charges of the electron and of the positron are somehow homogeneously distributed over spherical surfaces in them.

As to the use of the ${}_{sr}E$ expression (equation 3), the very existence of a non-electrostatic short-range repulsion between electrons and positrons at distances below $2 \ l_0$ must be agreed upon, if a lattice built of these particles is to exist. As in $\mathrm{NaCl}$, the repulsion in the epola is still out of the range of nuclear forces and cannot be gravitational; therefore, it has to be electrodynamic, mostly magnetic in nature. In the epola, the explanation then is, that at distances below $2 \ l_0$ the complexity of the inner structure of electrons and positrons enters the play. The subparticles (quarks?) of each one of these particles begin to act on the subparticles of the other, affecting the moving charge distributions, their magnetic moments and spins, similar to what was described to occur in the $\mathrm{Na^+}$ and $\mathrm{Cl^-}$ ions. The $2 \ l_0$ range of the short-range repulsion in both lattices makes the calculation of the lattice constant with equations 2 and 3 independent of $A$ and insensitive to the specific small value chosen for $r$.

5.6 Calculation of the e^–e⁺ lattice constant

Adding (algebraically) the energy ${}_{sr}E$ of the short-range repulsion, given by equation 3, to the energy ${}_{es}E$ of the electrostatic attraction (with a minus sign for attraction) yields the total interaction energy ${}_t E$ per particle (or per ion pair in the $\mathrm{NaCl}$ lattice). The graphs of these energies as functions of the distance $l$ between the particles (or ions) in the lattice are shown in Figure 1. It is seen that at $l = l_0$ (here $l_0$ is the lattice constant) the total energy has a minimum value, which is the binding energy ${}_b E$ per particle (or per ion pair in the $\mathrm{NaCl}$ lattice). Therefore, at $l = l_0$ the resultant of the two interaction forces, $\left( \delta / \delta l \right) \ {}_{es}E + \left( \delta / \delta l \right) \ {}_{sr}E$ is zero. Hence,

In the $\mathrm{NaCl}$ crystal, thus also in the epola, $r = 0.1 \ l_0$, so that $A = 2200 \ {}_{es}E_0$ and

Therefore,

and

Substituting the epola binding energy ${}_bE = m_e c^2 = 0.511 \ \mathrm{MeV}$, we find $l_0 = 4.44 \ \mathrm{fm}$. This value is in fact, not sensitive to the specific small value chosen for $r$. The variation of $r$ from $0$ to $0.20 \ l_0$ (i.e., in a much wider range than that noted earlier for various alkali halides) results in values of $l_0$ between $4.92 \ \mathrm{fm}$ and $3.94 \ \mathrm{fm}$. We therefore propose for $l_0$ the value

5.7 The face-centered cubic lattice

The face-centered cubic (fcc) lattice of the $\mathrm{NaCl}$ crystal and of the epola is shown in Figure 2. Here the corners of the elementary cubes, marked alternately black and white (for positive and negative occupation) are the 'lattice sites'. The distance between two nearest lattice sites is the 'lattice constant'. (The lines in the figure were drawn to ease presentation and do not exist in the epola or in crystals.)

The $e^-$ and $e^+$ particles of the epola or the $\mathrm{Na^+}$ and $\mathrm{Cl^-}$ ions of the $\mathrm{NaCl}$ crystal are vibrating randomly in thermal vibrations around their lattice sites. In the $\mathrm{NaCl}$ crystal the lattice structure and the lattice constant $l_0 = 280 \ \mathrm{pm}$ were revealed using X-ray diffraction techniques (Laue, 1912).

The lattice constant $l_0$ of the epola, calculated in Section 5.6, is $l_0 = \left( 4.4 \pm 0.5 \right) \ \mathrm{fm}$. Neutron diffraction techniques may become able to reveal the epola structure when (and if) resolutions of $1 \ \mathrm{fm}$ are reached. Until then, we shall hold on to the analogy with the $\mathrm{NaCl}$ crystal lattice, which enables the calculations of the epola structure.

The central particle is marked $1$. Its six nearest neighbors, belonging only halfwise to the unit, are marked $1 / 2$. Fractions marking other particles show their affiliation with this fcc unit.

5.8 The electromagnetic radius of the electron

There is currently no way to measure the radius of the electron, also not in the foreseeable future. A common approximation is the 'classical' or 'electromagnetic' radius. This value, which is 2.8 fm, was obtained using two wrong assumptions. The first is the 'classical' assumption that the $-e$ charge of the electron is distributed evenly on the outermost surface of the electron, so that there is no charge in its volume. In such case the electrostatic energy of the electron can be calculated as the energy ${}_{es}E$ of a conducting sphere of radius $R$ and charge $e$, which is

Here, $k = 9 \ \mathrm{Gm / F}$ is and SI units factor.

Considering the electron as a conducting sphere means that the charge of the electron is not intrinsically connected with the particle itself. This and the assumed absence of charge in the volume of the electron contradict our knowledge of the structure of the electron. Hence, the classical contribution to the electron radius is incorrect.

The second incorrect assumption is the relativistic contribution, equating the electrostatic energy of the conducting-sphere electron to $m_e c^2$, the energy-equivalent of the electron mass $m_e$. Then,

which yields

Substituting all the known values, one obtains $R = 2.82 \ \mathrm{fm}$.

However, $m_e c^2$ is the radiation energy, appearing in the electromagnetic field when an electron 'disappears' in it. Radiation energy is definitely known to be connected with the vibrational motion of electrons. Therefore, it is also definitely connected with the electrostatic energy of the electron, but this cannot explain why these two energies should be equal. As shown in Section 2.6., equating the two energies is illegal. Therefore, the justly derived part of the expression for R means that if the charge e on a conducting sphere causes it to have an electrostatic energy $m_e c^2 = 511 \ \mathrm{keV}$, then the sphere should have a radius of 2.8 fm.

5.9 Radii of atomic nuclei

The radii of nuclei are not precisely determinable quantities, and each type of experiment serving to determine them yields slightly different values. Bombardment with fast neutrons yields the 'neutron collision' radius of the nucleus. The rate of disintegration of $\alpha$-particles and the cross-sections of nuclear reactions involving other nuclides yield the Coulomb or Gamow 'barrier radius' of the nucleus. The 'electrostatic radius' is obtained from the analysis of the binding energy of the nucleus, and the 'electron scattering' radius is obtained from the scattering of fast electrons from the nucleus (R. Hofstadter).

In the experiments, especially in the scattering experiments, a point can be located, at which the charge-density of the nucleus has a maximum. The radius R of the nucleus is defined as the distance from this point to a point, where the positive charge-density of the nucleus is decreased to 50 percent of the maximum. Then all experiments show that the radii of nuclei are proportional to the cube-root of their mass-number $A$,

The factor $R_1$ which is the radius of a nucleus having $A = 1$, would then be the radius of a proton or of a neutron. These radii were not yet measured, and in different experiments the derived values of $R_1$ vary from 1.1 to 1.3 fm. Thus,

The so-defined radii of the nuclei vary, therefore, from 1.2 fm in hydrogen $(A = 1)$ to 7.4 fm in uranium $(A = 238)$.

These results lead to important conclusions. First, that all nuclei have about the same mass density. Second, if $R_1$ is the radius of protons and neutrons and $A$ is their number in the nucleus, then $R$ can be proportional to $A^{1/3}$ only if the protons and neutrons are most closely packed in the nucleus. Therefore, the electrical charge of the nuclei must be quite evenly distributed in their volumes.

5.10 The 'nuclear' radius of the electron

Just how unfit the electromagnetic radius of the electron is can be seen from the fact that 2.82 fm is more than twice the radius of the proton, or equal to the size of the nucleus of nitrogen. Hence, the charge density in the proton would be 13 times higher than in the electron. This disagrees with the uniqueness of the electron charge-to-mass ratio, which is about 1840 times that of a proton.

Scattering experiments of fast electrons lead to values in the order of 0.1 fm for the radius $R_e$ of the electron. This value can also be obtained from the formula for nuclear radii, $R = (1.2 \ \mathrm{fm}) \cdot A^{1/3}$, by substituting for $A$ the 'atomic mass number' $A_e$ of the electron, $A_e = 1 / 1840$. The obtained $R_e = 0.1 \ \mathrm{fm}$ value may be named as the 'nuclear' radius of the electron.

The 0.1 fm electron radius means that the volumetric charge density is many times higher in the electron (or positron) than in the proton, as the volume of the electron is smaller than that of the proton. On the other hand, the mass-density in the electron or positron is similar to the mass-density in nucleons and nuclei, and is, therefore, $10^{15}$ times larger than in atomic bodies.

The proportionality of the radii of these 'dense' particles to the cubic root of their masses shows that the subparticles of which they consist should be closely packed in them. Hence, the structure of the dense particles is opposed to the planetary structure in atomic bodies. The dense particles, and nuclear matter in general, represent therefore a distinct form of matter, which may not obey laws, established for atomic matter (and vice versa).

The actual value of the radius of the electron is not important for the epola model, if only it is less than half the lattice constant $l_0$, i.e., less than 2.2 fm. Figure 3 shows a 'gate' to an ${l_0}^3$ unit cube of the epola, with electron and positron radii of 1 fm. The circle in the center depicts the size of a proton (or neutron). Black dots represent the 'nuclear' 0.1 fm radius-size electrons and positrons, all in one scale.

The closeness of the densities of nuclei, electrons, positrons and other elementary particles (of non·zero rest mass) justifies their consideration as 'dense' particles, opposed to the many orders of magnitude lower density of atomic bodies. As much as the densities of the dense particles may differ from each other, they still represent a close community, compared with the so much smaller density of atomic matter. Whenever appropriate, we shall use for 'dense' particles the name 'avotons', from the Hebrew 'avot', meaning dense, densely interwoven (The Prophet Yehezkiel, 6:13). Thus, avotons (or 'densons'?) are elementary particles (of non-zero rest mass) and nuclides.

5.11 Epola random vibrations and temperature

We have seen in Section 5.6 that when the ions of the $\mathrm{NaCl}$ crystal or the particles of the epola are on their lattice sites $(l = l_0)$, then the sum of the attractive and repulsive forces acting on them is zero. Therefore, even an infinitesimally small force $F$ exerted on such ion or epola particle by the slightest disturbance in the lattice is able to deflect the ion or the particle out of the lattice site. The deflection $\Delta l$ is then proportional to the force $F$, $\Delta l \propto F$.

When the disturbance disappears, the energy bestowed by the disturbance remains with the ion or the epola particle. They then perform harmonic oscillations around their lattice sites, with amplitude $\Delta l$. The energy of these oscillations is proportional to the square of the amplitude, $E \propto \Delta l ^2$.

In any lattice, under any circumstances, there is a tremendous number of distorting factors, so that all its particles are always vibrating around their lattice sites. In the lattice by itself the vibrations are chaotic and the vectorial sum of the displacements of all lattice particles at any moment is zero. Therefore, the lattice particles and the lattice as a whole are in dynamic or thermal equilibrium. The random vibrations of the constituent particles in the lattice represent the thermal motion in the lattice, and the overall vibrational energy defines the lattice temperature. Actually, the lattice temperature itself is defined as the average per-particle energy of the lattice random vibrations. This applies to the $\mathrm{NaCl}$ lattice as well as to the epola.

5.12 The Brownian movement

A direct experimental indication of random motions of molecules in liquids and gases is the Brownian movement. This is a chaotic jumping motion of a spore or other tiny speck of matter suspended in a fluid (liquid or gas). The diameters of the molecules of the host fluid are hundreds of times smaller than the diameter of the tiniest spore. Hence, there are millions of molecules around the spore. Due to their thermal motion, they bombard the spore from all directions, equally, most of the time. However, at unpredictable short moments, there comes an unbalanced punch in an unpredictable direction, resulting in a jump of the spore in this direction.

The Brownian movement was first observed by Robert Brown in 1827. Being a botanist, Brown ascribed the movement to a mysterious vivid energy of the spores themselves. The physical explanation of the Brownian movement was given 80 years later, in 1908, by J. Perrin, and the theory was developed by M. Smoluchowski and, finally, by A. Einstein. This even made possible the calculation of the temperature of the host fluid with data from the Brownian movement.

In solids, the analog of the Brownian movement is the motion of guest atoms, especially light atoms of, e.g., hydrogen or lithium. Being from time to lime unequally hit by the thermally vibrating atoms of the solid, the guest atoms wander in a jumping or hopping motion all over the solid. In metals, the wandering hydrogen atoms can accumulate on microscopic cracks, reducing the strength of the metal. In semiconductors, the chaotically moving atoms of lithium can reach defect sites caused, e.g., by irradiation. By 'settling' there, they may 'heal' the defect. Therefore, lithium-doped solid radiation detectors have the property of 'self-healing'.

5.13 Zero-point motion as epola analog of Brownian movement

The Brownian-like movement of light atoms, occuring in solids, may also be taking place in the epola. The known experimental facts which might be ascribed to such movement are, first of all, the zero-point energy of helium atoms, measured at temperatures close to the absolute zero, and the zero-point motion of the helium atoms which results in this energy. Similar, though much smaller zero-point motions and energies were also measured in other light atoms.

The zero-point motion cannot be understood on the basis of existing physical theories and is therefore a mysterious effect. With the use of special approaches, quantum theory provides a sometimes satisfactory mathematical description of this and other mysterious phenomena occurring in liquid helium close to zero Kelvin. They are ascribed to the special quantum character of helium atoms and to liquid helium as a 'quantum liquid'.

In the epola model of space, the zero-point energy and motion of helium and other light atoms is explained in a similar way as the Brownian movement. The billions of randomly vibrating epola particles around and inside the light atom exert most of the time an equal pressure in all directions. However, at unpredictable short moments, there comes an unbalanced pressure-flash in an unpredictable direction, causing a jump of the atom in this direction. Therefore, the zero-point motion depicts the random vibrations of epola particles around their lattice points. This is analogous to the way in which the Brownian movement depicts the thermal motion of the molecules of the host fluid, or in which the Brownian-like hopping movement of light atoms in solids depicts the thermal vibrations of the atoms of the solid around their lattice sites.

5.14 Approximate value of the epola temperature

The epola interpretation of the zero-point motion of light atoms as due to unbalanced actions of the randomly vibrating epola particles removes the mystery from this phenomenon, yielding a rational physical understanding of it. This interpretation also opens up new possibilities for a mathematical presentation of this and many other 'mysterious' quantum phenomena observed at temperatures close to zero Kelvin, as, e.g., the zero-field splitting of spin energy levels, electron and nuclear cooling, etc.

Let us first apply the thermodynamic rules to the energy-flow between the epola and an atomic body. Clearly,

a bulk of hot atomic matter, left in the epola without incoming energy, must cool down to the epola temperature,

and,

a bulk of atomic matter, initially cooled below the epola temperature and left alone in the epola, must heat up to the epola temperature.

Also,

when the temperature of the atomic body is equal to the epola temperature, then they are in thermal equilibrium, i.e., the overall energy flow from the epola to the atomic body is equal to the energy flow from this body to the epola.

The zero-point motion may also occur in thermal equilibrium between the epola and the bulk of atomic matter. Due to the randomness of the vibrations, a small epola cluster may still accumulate more vibrational energy (reach a higher 'temperature') than the surrounding epola. Interacting with a light atom, such cluster may force it to perform a Brownian-like jump. Such events are also possible when the temperature of the atomic body is even slightly higher than the epola temperature, but it is obvious that the zero-point motion and energy are much larger at temperatures below the epola temperature, and should increase rapidly with a further decrease of the temperature of the atomic body.

An analysis of the temperature-dependence of the zero-point energy of helium atoms, as well as of the much smaller zero-point energies of other atoms, might well reveal the epola temperature as a singular point or as a turning point in this dependence. A preliminary observation suggests that the epola temperature is below 4.2 K, at which helium liquefies.

The epola temperature may probably be obtained from the analysis of other zero-point and low temperature effects or from the analysis of the efforts needed to reach any particular temperature below 4.2 K. For this, one should work out an appropriate physical measure to evaluate these efforts, then plot the temperature of a body of atoms against this measure. The epola temperature should appear on such a plot as a turning point. However, astrophysics provides us readily with a 3 K value of the epola temperature, if the observed 'background' blackbody radiation, corresponding to this temperature, is interpreted as the radiation of the surrounding epola (Section 11.2).

To conclude, there is one more analogy between the Brownian movement and the zero-point motion. Perrin's physical approach to the Brownian movement freed it from the cloud of botanical mystery and made it possible to develop a complete mathematical presentation of this phenomenon. Analogously, the physical epola approach to the zero-point motion frees it from the cloud of quantum mystery and points out ways to develop a complete mathematical presentation of this phenomenon, as well as of other phenomena at temperatures close to zero Kelvin.