# Substantial electron model

The substantial electron model is a theoretical model, which is alternative to the concept of electrons’ origin as a result of the Big bang and to the electron model in quantum mechanics and the theory of elementary particles. To prove the substantial electron model such theories are used as the theory of Infinite Hierarchical Nesting of Matter, the theory of similarity of matter levels, SPФ symmetry, strong gravitation, as well as the concept of dynamic spin.

## The origin of electrons

It is known that the Big Bang theory has a number of unsolved problems, [1] moreover, the phenomena, on which it is based, can be explained by other reasons. [2] In the theory of Infinite Hierarchical Nesting of Matter, as opposed to atomism, there are no absolutely elementary particles – every natural object is made up of smaller objects. Therefore, the origin of electrons and other particles is considered from the standpoint of matter evolution in space, which occurs by the same laws at different levels of matter. The basis of this approach is SPФ symmetry, according to which the laws of motion of matter and fields under similarity transformations between the matter levels do not change their form and remain invariant.

The distribution of material objects in the Universe is described with the help of scale dimension, which extends over all levels of matter. The basic levels of matter are: the level of graons – the level of praons – the level of nucleons – the level of stars – the level of supermetagalaxies. [3] The ordinary matter, planets and stars consist of atoms or atomic nuclei, that is, of a mixture of neutrons, protons and electrons, in which the main contribution to mass is made by nucleons. Similarly, the objects at the level of nucleons, i.e., nucleons themselves, electrons, and other elementary particles consist mostly of neutral and positively charged praons and negatively charged praelectrons. Each basic level of matter consists of the objects of the underlying basic level of matter. This is the way the principle of nesting of matter is realized.

The characteristic and complementary processes in the global evolution of objects in the Universe are: 1) formation of particles and material bodies in the opposite processes of clustering and fragmentation. 2) formation of static fields, associated with the matter, and of moving fields in the form of emission from particles and bodies. It is assumed that much more massive bodies are formed under the action of graviton fluxes in the framework of the Le Sage's theory of gravitation. At the same time sufficiently massive objects at different levels of matter, such as compact stars, are the emitting sources of photons, neutrinos, relativistic particles, which are the basis of the field and graviton fluxes. Thus, the matter of the lower levels generates the field quanta, and the field produces individual bodies from scattered matter at the higher levels of matter. Based on the described picture, nucleons are similar to neutron stars and electrons in the atom correspond to the disks, discovered near the X-ray pulsars, which are the main candidates for magnetars. [4]

Formation of neutron stars is well known from the theory of stellar evolution, which allows us to understand the origin of nucleons as the analogues of neutron stars. Nucleons are divided to neutrons and protons, the first can be compared to ordinary neutron stars and the second can be compared to magnetars with strong magnetic fields and potentially positive electrical charge.

The matter of disks near magnetars has the form of broad rings with a characteristic radius close to the Roche limit, at which disintegration of planets near stars occurs due to the strong gravitational field. If a massive star had a planetary system, then over time this star could become a magnetar (after the stage of a neutron star), and the matter with domination of the heaviest iron-group chemical elements would remain after planets. The ratio of the Roche radius to the neutron star radius is equal to the ratio of the Bohr radius to the proton radius, so that the disks discovered near pulsars can be considered the analogues of an electron cloud in the hydrogen atom. Calculation shows that the matter of the object, which is the analogue of the electron around a magnetar, is quantitatively almost neutral – per each matter unit, containing 1016 nucleons in, there is only one additional electron. But due to rather large mass of the disk the total negative charge is great. Hence, by analogy, it follows that the matter of electrons in atoms is in the form of clouds due to the action of strong gravitation from the nucleus, and the considerable part of the clouds’ matter is electrically neutral. Besides, the electron’s matter differs from the nucleon’s matter just as the planetary matter differs from the matter of neutron stars. Due to the low density of the electron’s matter with respect to nucleons, the experiments with electron scattering do not reveal the internal substructure of electrons – it remains beyond the experimental accuracy.

According to the theory of similarity of matter levels, strong gravitation is acting at the scale level of elementary particles instead of the ordinary gravitation, ensuring the integrity of particles and their interactions with each other. Assuming that an electron has a certain radius ${\displaystyle ~R}$, the electron’s integrity is possible when the attracting gravitational force per any matter unit with the mass ${\displaystyle ~\Delta m}$ exceeds the force of electric repulsion of this unit with the charge ${\displaystyle ~\Delta e}$ of the total electron charge ${\displaystyle ~e}$:

${\displaystyle ~{\frac {\Gamma m\Delta m}{R^{2}}}>{\frac {e\Delta e}{4\pi \varepsilon _{0}R^{2}}},}$

where ${\displaystyle ~\Gamma }$ is the strong gravitational constant, ${\displaystyle ~m}$ is the electron mass, ${\displaystyle ~\varepsilon _{0}}$ is the electric constant. Without loss of generality we can assume that following the relation is satisfied:

${\displaystyle ~{\frac {\Delta m}{\Delta e}}={\frac {m}{e}}.}$

In view of this relation the electron’s stability condition takes the form:

${\displaystyle ~\Gamma >{\frac {e^{2}}{4\pi \varepsilon _{0}m^{2}}}.}$

However, after substituting all the quantities this condition is not satisfied, because the electron charge is so large that the proper gravitational force of the electron matter cannot counteract its electrical force. This means that an electron is not an independent particle and does not have a definite radius. Therefore, an electron in an atom can only be in the form of a rotating electron cloud similar to a disc, which is held by the gravitational field of the nucleus. The electron’s equilibrium requires the presence of the gravitational force, especially in the case of a hydrogen atom, because here the electrical force of the electron’s attraction to the nucleus is to a great extent compensated by the electrical force of repulsion of the electron’s charged matter.

In view of the above, the equality by the absolute value of the charges of proton and electron in the hydrogen atom is explained by the need to fulfill the condition of general electroneutrality and by adjustment of the charge (and mass) of the electron cloud due to the attraction or repulsion of the respective electric charges of the rarified matter, surrounding the atom. The existence of electrons in atoms in the entire observed cosmic matter and in the most remote, even unrelated points is the evidence of uniform and natural evolution of matter in the Universe. This applies not only to electron, but to all elementary particles, as well as to other objects, for the integrity of which both gravitational and electromagnetic forces are required.

## Production in reactions with elementary particles

Given the lack of independence of an electron as a particle, it becomes clear why in all reactions an electron appears only in connection with decay of more massive particles (such as neutrons and muons) or with obligatory participation of hadrons, just as in photoproduction of electron-positron pairs.

As an example we can consider the reaction of neutron beta decay, leading to the production of electron, proton and antineutrino. There is a neutron model, in which its interior is positively charged and the shell is negatively charged. This model has been created in an attempt to explain the experiments on scattering of electron beams of medium-energy (up to 20 GeV) on the liquid hydrogen and in deuterium, with calculation of the charge and magnetic form factors of the neutron. [5] [6] [7] The neutron’s analogue at the stellar level is a neutron star, in the production process of which we can also assume volumetric separation of the electric charge. For example, during supernova explosion with production of a neutron star part of electrons from the center can be displaced to the shell, giving a negative charge to it. Due to the spherical symmetry, such separation of charges can be stable enough, since electrons in the shell are not only attracted to the positively charged nucleus, but also repel each other. Rotation of a star with a separated volume charge leads to a characteristic pattern of the magnetic field – inside the field is directed along the rotation axis, but near the surface it is fully compensated by the oppositely directed magnetic field of the shell. Therefore, the total magnetic moment of the star is opposite to its rotation, as is the case with the neutron. The magnetic field is supported not only by rotation of the charges, but also by the stellar matter directed in the magnetic field, consisting mainly of neutrons.

Due to the reactions of weak interaction, transformations of neutrons into protons and electrons take place in the matter and the gradual change of the internal magnetic configuration occurs in the star’s volume (in the decay of each neutron at its constant rotation a rapid polarity reversal of the magnetic moment’s direction takes place, since the magnetic moments of a neutron and a proton with the same spin are opposite). The result of the stellar matter’s transformation is that at some point the central magnetic field, shielded by the shell’s field, breaks out near the poles. This leads to catastrophic restructuring of the entire magnetic field with release of energy, which is sufficient to discharge part of the outer shell. The star turns into a magnetar, while the discharged matter is negatively charged and magnetized, since it mainly consists of such elements as iron, located at the surface of the star. The matter ejected from the star corresponds to the electron in the neutron beta-decay.

Hence we can make the following conclusions:

1) The electron’s matter is not only charged, but also can be magnetized.

2) The reactions of weak interaction with elementary particles can be explained by transformation of the matter of these particles in the processes of weak interaction, that take place at the lower scale level of matter (if we compare the evolution of a neutron star with the evolution of matter of one neutron, then similarly we should compare a neutron itself with a praon as a corresponding small particle of a neutron).

3) The universal electroneutrality of matter in the Universe is associated with production of the same number of charged elementary particles (protons and electrons) in neutrons’ beta decay. In turn, neutrons emerge as a result of natural evolution of matter, almost like neutron stars.

4) The charge value of a proton and the equality by the absolute value of the charges of proton and electron are due to the evolution of neutrons and beta decay processes, repeating in all free neutrons approximately by the same law. For a neutron to transform into a proton it is necessary that in the neutron shell due to the weak interaction a sufficient amount of electric charges were accumulated and restructuring of the magnetic field took place. This would fix the charge value that emerges in the proton and electron.

This model of beta decay can be found in Sergey Fedosin’s works, [8] it is also described in the substantial neutron model.

Electron (positron) can also be found as a result of the decay chain of a charged pion according to the following scheme:

${\displaystyle ~\pi ^{-}\rightarrow \mu ^{-}+{\tilde {\nu }}_{\mu },\qquad \mu ^{-}\rightarrow e^{-}+{\tilde {\nu }}_{e}+\nu _{\mu }.}$
${\displaystyle ~\pi ^{+}\rightarrow \mu ^{+}+\nu _{\mu },\qquad \mu ^{+}\rightarrow e^{+}+\nu _{e}+{\tilde {\nu }}_{\mu }.}$

In this case, initially a charged muon ${\displaystyle ~\mu }$ and a muon antineutrino (neutrino) are formed, then the muon decays into an electron (positron), an electron antineutrino (neutrino) and a muon neutrino (antineutrino) ${\displaystyle ~\nu _{\mu }}$. In view of the theory of similarity of matter levels, a pion by its mass corresponds to a neutron star of the order of 0.2 Solar masses. Neutron stars of such small masses are unstable, because the force of gravitational contraction becomes insufficient to maintain the state of matter in the form of neutron liquid. [9] After the matter transformation as a result of the reactions of weak interaction during the time period of the order of 105 years, a charged and magnetized neutron star, which is the analogue of a pion, explosively transforms into a stellar object, the analogue of a muon. The emerging neutrino and antineutrino emission is equivalent to the emission of a muon neutrino in the reaction of pion decay. A muon corresponds to a stellar object with a mass of the order of 0.16 Solar masses. It turns out that such a mass exactly coincides with the Chandrasekhar mass for a white dwarf with the chemical composition made of the lightest chemical elements – from hydrogen nuclei to helium nuclei. [10] The evolution of the charged stellar object, the analogue of muon, lasts for up to 107 years. During this time thermonuclear fusion reactions begin in it, like in normal stars, leading eventually to the discharge of the charged shell. This process is equivalent to production of electron (positron), recorded in muon decay.

Another example is the reaction of an electron antineutrino’s interaction with a proton, resulting in production of a neutron and a positron. In contrast to an electron, the positron matter has a positive charge. This reaction, according to the substantial proton model, can be conveniently analyzed using the stellar models, assuming that stellar electron antineutrinos, consisting of the fluxes of electron neutrinos and antineutrinos of the corresponding helicity, fall on the magnetar, the proton’s analogue. A magnetar consists of nucleons, directed by the magnetic field. To transform a magnetar into a neutron star, the analogue of a neutron, it is necessary to transform the protons of the magnetar matter into positrons and neutrons with the help of electron antineutrinos. This is possible in case of the appropriate distribution of antineutrinos with respect to the directed magnetar matter. At the same time, the electron neutrino transforms the neutrons of the magnetar matter into protons and electrons. Part of the emerging electrons and positrons annihilate, releasing energy and heating the magnetar matter. After accumulating enough positrons in the shell, due to their repulsion from the central part of the magnetar, which is positively charged, the heated matter is discharged and a stellar object similar to positron is produced. The magnetar itself becomes a neutron star, the neutron’s analogue, because the nucleons in the magnetar’s shell reverse the direction of their magnetic moment under the action of the fluxes of neutrinos and antineutrinos, and the charge gradient appears in the matter due to the produced electrons. This leads to compensation of part of the magnetic field of the magnetar’s core by the magnetic field of the shell and to the change of the magnetic field’s configuration and of the sign of the star’s magnetic moment, as well as to the release of considerable energy, which contributes to the discharge of matter. Discharge of the positively charged matter from the magnetar’s shell means the loss of charge by the magnetar and transforming it into a neutral neutron star.

The given typical examples show how electrons and positrons are formed in the substantial model. If they are produced in reactions with other particles, then first we need to consider the respective matter transformations of these particles.

## Annihilation

As it is known, electron and positron can annihilate with emission of two photons (from the parapositronium state, the spins of particles are antiparallel) and with emission of three photons for orthopositronium. In the latter case, the spins of electron and positron are parallel, during the rotation of particles in a flat orbit there is additional magnetic repulsion, and the bound state of leptons exists longer – up to 1.4•10–7 sec. For parapositronium each emitted photon has an energy close to the rest energy of electron. This allows us to speak about the “conversion of mass into energy”, destruction of matter and complete conversion of its energy into the field energy, etc.

The pattern of electron-positron annihilation at the stellar level can be represented as follows. [8] Let the role of an electron be played by a negatively charged cloud of matter with the mass M = 1.5∙1027 kg (0.78 Jupiter masses) and the charge Q = 5.5∙1018 C, surrounding the magnetar. The mass and charge of the cloud are obtained from the mass and charge of the electron by multiplying by the respective similarity coefficients (see the similarity of matter levels). The role of a positron at the stellar level is played by a corresponding positively charged cloud. In a collision of these clouds recombination of their electric charges must take place, and the clouds’ matter becomes neutral. Under the action of the gravitational field strength and gravitational torsion field the neutral matter could fall onto the star. The clouds’ masses are large enough and the fallen matter could heat up enormously while reaching the surface. The high temperature of matter leads to a peculiar thermonuclear explosion on the star surface. Part of the energy released during the fall is converted into the emission of X-ray and gamma quanta, as well in the form of two jets that are typical of neutron stars. The main part of this energy comes from the gravitational energy of matter in the star’s strong field. For the two clouds the absolute value of the energy will be:

${\displaystyle \mid W_{g}\mid ={\frac {2GM_{s}M}{R_{s}}}=4.5\cdot 10^{43}}$ J,

where for the mass and radius of the neutron star the values are used: Ms = 2.7•1030 kg and Rs = 12 km, respectively; ${\displaystyle G}$ is the gravitational constant.

Dividing this energy by the coefficient of similarity in energy ${\displaystyle ~{\mathcal {2}}'=\Phi 'S'^{2}=8.6\cdot 10^{55}}$, we can find the corresponding energy at the level of elementary particles: 3.3 MeV. At the same time, in the annihilation of electron and positron two gamma quanta are usually recorded with energies of about 0.511 MeV each. Thus, release of the gravitational energy due to the nucleon’s strong gravitation of 3.3 MeV is enough to form two gamma quanta, corresponding to electron and positron annihilation. In this case, we do not observe full conversion of the mass-energy of lepton and antilepton matter into electromagnetic emission, but conversion of their gravitational energy near the nucleon into the electromagnetic form of energy.

## Electron capture

An electron cloud, that lost its orbital angular momentum for some reason, can fall on the nucleus and be combined with one of the nucleus’ protons in the following reaction:

${\displaystyle p^{+}+e^{-}\rightarrow n^{0}+\nu _{e},}$

forming a neutron and a neutrino.

In electron capture, in addition to the gravitational energy, electrical energy is released in the recombination of the negative charges in the electron cloud matter and the positive charges in the proton matter. In the process of recombination, the negative charges penetrate into the proton shell, significantly changing its state. During the proton rotation the shell acquires the negative charge from the fallen electron, the configuration of the proton’s magnetic field is replaced by the configuration of the neutron’s magnetic field. Becoming a neutron, the proton effectively loses its charge (compensation of part of the charge takes place, the proton center is positively charged, and the shell becomes negatively charged), its total magnetic moment is reversed and an electron neutrino is emitted. A similar event in a stellar model looks as follows: stellar electron cloud falls on the magnetar, the excess electrons of the clouds’ matter combine with the protons in the magnetar’s shell with production of neutrons and emission of fluxes of neutrinos, which are generally regarded as stellar neutrinos. It follows that electron neutrinos in the electron capture represent directed fluxes of very small particles which are similar to neutrinos by their properties and are emitted by the matter in the proton shell. [8]

## Cosmic rays

Based on similarity with the proton, it is assumed that the magnetar has a positive charge Q = 5.5∙1018 C. Calculation of the electrostatic energy per one electron or proton of the disk near the magnetar gives 6.7∙1019 eV at the stellar Bohr radius (with the Roche limit), and 4.1∙1024 eV on the magnetar surface. If we divide the charge Q by the volume of the disk near the magnetar, we can estimate the electron concentration in the disk as 108 m–3. [8] This is proved by measuring the electron concentration in the magnetosphere of the magnetar. [11] Due to interactions of the charged particles the positively charged protons and nuclei can accelerate away from the star at great energies up to 6.7∙1019 eV or more. The maximum energy of cosmic rays just fall within this range. [12] A small share of electrons in cosmic rays, up to 1%, can be explained by instability of an electron as a particle, and a large share of the nuclei of iron series can be explained by their significant presence in magnetar disks and on the surface of these stars.

## The Pauli principle

By analogy with the discovered disks in the form of rings around the neutron stars, electron clouds in the atom are considered as disks. Due to their negative charge, separate electron disks repel each other, and rotation of the charged matter in the disks creates a magnetic field penetrating the entire atom. The Pauli exclusion principle is explained as a consequence of electromagnetic induction in adjacent electron clouds – while two electron disks approach each other their matter starts rotating in the opposite direction to satisfy the Lenz's law. Two paired electron clouds will have the total angular momentum and spin equal to zero, this also applies to the filled atom shells, in which the number of electrons is even.

## Stationary states

According to the first postulate of the original version of quantum mechanics, the theory of the Bohr atom, there are stationary states of atom, in which electron travels along a stationary orbit and does not emit energy. Discreteness of such states implies that the angular momentum and energy of the electron are quantized in them.

Estimation of the minimal dimensions of the electron cloud in the form of a flat disk for the hydrogen atom gives the value of the inner disk’s edge about ${\displaystyle ~0.5R_{b}}$, the value of the outer disk’s edge about ${\displaystyle ~1.5R_{b}}$, and the disk’s height less than ${\displaystyle ~0.33R_{b}}$, where ${\displaystyle ~R_{b}}$ is the Bohr radius. The analysis of the forces acting in the electron matter in its steady rotation around the nucleus leads to the law of conservation of energy and momentum of the matter, electromagnetic field and strong gravitational field. [13]

As it is proved in the book, [8] the stationary states of the atom are characterized by the fact that in these states the kinetic energy flux of the electron cloud matter is equal to the total energy flux of the strong gravitation field and the electromagnetic field that passes through the matter, due to the redistribution of energy fluxes of general field components. In this case, the field energy flux does not exert pressure on the matter, the rotation is relatively steady, and due to the axial symmetry of the electron cloud the emission from the atom tends to zero. In multielectron atoms such principles are added as the minimum energy principle, the Pauli principle, as well as the principle of stability of some spatial electron configurations in the atom due to the compensation of the emergent electromagnetic emission from different parts of the cloud and reduction of the total emission to zero because of the geometry of the cloud. All this is the reason for quantization of the energy levels as well as of the angular momentum of the electrons in the atom.

## Spin as a dynamic property

Most of the phenomena associated with the electron spin in the atom occur at the moment of the electron’s transition from one energy state to another, when absorption or emission of the light quantum takes place. All these phenomena are studied by measuring the electron spectra. This refers to the emission spectra of chemical elements (atoms and ions), the fine structure and atomic spectra multiplicity, the Zeeman effect, etc. In the concept of the dynamic spin it is assumed that the spin as part of the electron’s angular momentum emerges when the electron cloud center is shifted relative to the atomic nucleus and revolves around it. This is possible if the electron cloud underwent interaction, for example with a photon or another electron. In the general case, the electron cloud matter rotates around its center and rotates around the nucleus together with the cloud. The difference of gyromagnetic ratios in the formulas, connecting the spin and orbital angular momenta and the corresponding magnetic moments, results from the difference of rotational motion of the matter, which is responsible for the dynamic spin and orbital rotation.

To explain the Stern–Gerlach experiment, in which atoms in the ground state are deflected by the magnetic field, the magnetic field is calculated, which is created by the orbital rotation of a single electron disk’s matter around the nucleus. We assume the same distribution of the disk’s matter as it follows from quantum-mechanical calculation of the probability of the electron’s presence in the atom in solution of the Schrödinger equation. For the ground state we obtain the value of the disk’s magnetic moment equal to the Bohr magneton. This means that in the atom’s ground state, as well as in the s-states, the magnetic moment of the electron in the atom is not connected with the spin (in these states the dynamic spin is zero, the electron cloud’s center does not rotate relative to the nucleus) but with the orbital rotation of the electron cloud’s matter. In this case, the calculation using the model of a charged disk-the electron’s analogue near the magnetar shows that the intrinsic magnetic moment of the disk due to its magnetization is small and can be neglected in comparison with the magnetic moment due to the orbital rotation. The described pattern of the dynamic spin differs fundamentally from the postulates of quantum mechanics, according to which in the ground state of the electron in the atom there is no orbital angular momentum (here the problem of quantum mechanics arises: what keeps the electron near the nucleus in the presence of a constant force of attraction to the nucleus?), there is only spin, besides the spin and spin magnetic moment as the electron’s internal characteristics never disappear. In the substantial model, the electron is not an independent particle with the spin and spin magnetic moment, they emerge only after interaction of the bound electron in the atom with other particles. The electron matter can also be rotated by the external electromagnetic field from the incident photon or due to the electromagnetic induction effect.

## Emission and absorption of quanta in the atom

The shift of the center of mass of the electron cloud relative to the nucleus and its rotation in the excited atom lead to the electromagnetic emission from the atom, which is observed in the form of electromagnetic quantum. The energy of the emitted or absorbed quantum is equal to the difference in the energy of stationary levels, between which the electron transition takes place (the second postulate of the Bohr theory of atom). Meanwhile, the frequency of the quantum is very close, but not exactly equal to the frequency ${\displaystyle ~\omega _{e}}$ of the electron cloud rotation at the energy level, where the electron moves to after emission. This is explained as follows. If we take a hydrogen atom, the electron in it can make about 107 revolutions around the nucleus, before it moves from the excited state to the ground state with the emission of a quantum. The analysis of the course of emission shows that the highest energy emission rate is achieved at the increased frequency of rotation of the electron’s center of mass relative to the nucleus, which occurs near the lowest energy level. Therefore, the main quantum energy is emitted at a rate almost equal to the frequency of rotation of the electron cloud at the lowest energy level. During the transition between the levels the electron loses the angular momentum equal to the Dirac constant ${\displaystyle ~{\hbar }}$, and the emitted quantum acquires this momentum. The formula for the quantum energy during emission from the atom has the form:

${\displaystyle ~E=\hbar \omega ,}$

where ${\displaystyle ~\omega }$ is the angular frequency of the quantum wave, measured by spectral instruments. The meaning of this formula is that the quantum energy is proportional to the rate of change of the angular momentum, which depends on the angular frequency ${\displaystyle ~\omega _{e}}$ of rotation of the electron cloud in the atom. Along with the electromagnetic emission, the gravitational emission of the electron takes place in the atom with the quanta’s energy [8]

${\displaystyle ~E_{g}={\frac {m}{m_{p}}}\hbar \omega ,}$

where ${\displaystyle ~{\frac {m}{m_{p}}}}$ is the ratio of the electron mass to the proton mass. Since after the change of the electron’s angular momentum the atomic nucleus and the electron rotate around the common center of mass in opposite directions, the gravitational emission of the nucleus compensates the gravitational emission of the electron. Consequently, the dipole gravitational emission of the atom in general is close to zero and the quadrupole emission component becomes the main component.

The substantial model of an electron in the form of a disk rotating around the atomic nucleus allows us to describe the process of a photon’s formation during transition of an electron to the lower energy levels. A rotating electron creates at the disk’s axis an alternating electromagnetic field acting on the praons – relativistic particles of a dynamic vacuum field as the contents of electrogravitational vacuum. Under the action of the field the praons form spirals and get connected with each other by means of strong gravitation forming a photon. The model allows us to calculate for praons the charge-to-mass ratio and other parameters. [14] With this in mind, the substantial model of a photon is constructed, its internal structure is found, the internal electromagnetic fields are determined, as well as the total rest mass of all the photon’s particles and the magnetic moment of the photon, depending on the charge of the nucleus of the hydrogen-like atom and the energy level of the emitting electron. [15] It is shown that the angular frequency ${\displaystyle ~\omega }$ in the formula for the photon energy is the angular frequency ${\displaystyle ~\omega _{e}}$ of the electron cloud’s rotation in the atom averaged over the time of photon emission.

## Multiplicity

To explain the multiplicity of atomic spectra it should be noted that the center of each electron cloud can rotate around the nucleus in two opposite directions, giving two different projections of the spin and the spin magnetic moment on the preferred direction. For a single electron the multiplicity is 2 and the spectrum of a hydrogen-like atom consists of doublets – each main energy level, except the energy levels in s-states, is split due to the additional low magnetic energies from the interaction of the magnetic moment of the electron cloud with the magnetic moment of the positively charged nucleus, moving in one or another direction around the cloud’s center of mass. If there are two electrons, there are six possible combinations between the directions of the matter rotation in electron disks and the directions of rotation of the disks’ centers of mass relative to the nucleus, which defines the maximum possible multiplicity of the spectra in this case. In the approximation of the spin-orbit coupling of two electrons the multiplicity calculated by the standard formula is found to be 1 and 3, that is, in the spectrum of an atom with two electrons we can expect up to 4 close lines. The same is obtained from six possible combinations described above, if we neglect in them the magnetic interaction of the electrons with each other. With the help of similar combinations we consider the multiplicity and the fine splitting of the spectra for the cases of three or more electrons.

## The Zeeman effect

In the Zeeman effect doublet splitting of the spectra in the case of s-states is observed. The standard explanation for this is the interaction of the spin magnetic moment of the electron with the external magnetic field. However, in the concept of dynamic spin there is no spin in the s-states and the magnetic moment of the order of Bohr magneton arises due to the orbital rotation of the matter in the electron cloud. Then in order to explain this effect the electron spin is not required. If the approximation of the spin-orbit coupling holds for the atom, then it also holds in the weak external magnetic field. Since all electrons in the atom interact with each other, then for each combination of magnetic states of electrons there is its own total angular momentum of the atom with the quantum number ${\displaystyle ~J}$ and the corresponding magnetic moment. The number of orientations of this magnetic moment in the magnetic field is ${\displaystyle ~2J+1}$ that specifies the number of sub-levels to which this atom’s energy level is split in the Zeeman effect.

## The Lamb experiment

In 1947, the experiment carried out by Willis Eugene Lamb and Robert Retherford discovered the so-called Lamb shift of energies in three close states of the electron in the hydrogen atom, which is located on the second energy level. This shift is usually explained by relativistic corrections, as well as with the help of the hypothesis of quantum-mechanical vacuum fluctuations in the form of radiation effects of emission and absorption of virtual photons and creation and annihilation of virtual electron-positron pairs. Using the idea of the dynamic spin the result of the experiment can be seen from a different perspective. On the second energy level in the hydrogen atom the electron can be in s-state (singlet state) with the orbital rotation, as well as in two p-states, in which the electron has both the orbital angular momentum and the spin with the corresponding sign with respect to the orbital angular momentum. The dynamic spin causes the magnetic energy of interaction of the electron’s magnetic moment and the magnetic moment of motion of the nuclear charge. As a result, the difference in the three electron states arises from both the relativistic corrections, that depend on the spin direction and the shape of electron clouds, and from the corresponding magnetic corrections to the level’s energy. [8] The use of the dynamic spin of the electron, considered as a cloud and not as a point particle, allows us to eliminate the hypothesis of virtual particles and vacuum corrections, which are required in the quantum mechanics.

## Magnetomechanical phenomena

It is believed that one of the proofs of the presence of the electron’s spin in the s-states of atoms are the results of experiments carried out by Barnett and Einstein–de Haas with ferromagnetic samples. In Barnett's experiment on rotating the sample he observed magnetization and the magnetic field, and in Einstein–de Haas’ experiment the sample, when placed in the magnetic field, starts to rotate. When analyzing the Barnett effect two moments of force are equated, one of which is associated with rotation of the electron’s angular momentum around the sample’s rotation axis at a certain angular velocity, and the other is associated with the influence of the magnetic field on the electron’s magnetic moment. As a result the g-factor of the electron in the atom is equal to 2, as is the case for the spin. On the other hand, it would be more correct not to compare the moments of force but to use the law of conservation of energy. The latter leads to the fact that the work of the moment of force, when the sample is rotated, is expended not only to rotate the electrons’ angular momenta towards the rotation axis, but also on the work of the emerging magnetic field on rotation of the electrons’ magnetic moments in this field. This leads to the fact that for the g-value we obtain the value equal to 1. [8] Consequently, in the atom’s ground state the magnetic moment appears not due to the electron spin (the dynamic spin is zero in the stationary states), but due to the orbital rotation of the electron clouds’ matter.

## The spin and chemical properties

In the atom’s ground state the electrons’ dynamic spin is zero, but the distribution of electrons by shells is subject to the Pauli principle (due to the electric and magnetic interactions of the adjacent electron disks) and to the principle of minimum energy in the system. There are also geometric limitations – on each shell at a certain distance from the nucleus the possible number of electrons in the form of disks must be even and at a certain amount of matter for each electron this number is limited by the shell’s area. The total number of electrons on the shell with the quantum number n is 2 n2, where due to the Pauli principle electrons are combined into closely-connected pairs, and the quadratic dependence on the number of the shell is associated with the squared distance to the shell, which is proportional to the shell area. Emerging of a new electron in the atom is due to the necessity for the electroneutrality of the atom as a whole. The described conditions for the presence of electrons in the atom lead to the changes in the chemical properties of atoms (they are determined by the outer shell electrons) in the periodic table of chemical elements. This approach does not need to involve the dynamic spin to explain the atom’s structure and the chemical laws.

## The hydrogen and hydrogen-like atoms

In comparison with a free electron with indefinite radius, the electron in the atom is influenced by additional gravitational and electric forces from the nucleus, so that the electron has the ability to be held by these forces, staying in the form of a cloud around the nucleus. In the hydrogen-like atom, the electron’s radius, velocity and orbital angular momentum on the energy level with the principal quantum number ${\displaystyle ~n}$ and at the circular orbits in the s-states equal:

${\displaystyle ~r_{n}={\frac {4\pi \varepsilon _{0}\hbar ^{2}n^{2}(A+Z-\zeta )}{mZ^{2}e^{2}}},}$
${\displaystyle ~u_{n}={\frac {Ze^{2}}{4\pi \varepsilon _{0}\hbar n}},}$
${\displaystyle ~L_{n}={\frac {n\hbar (A+Z-\zeta )}{Z}},}$

where ${\displaystyle ~A}$ and ${\displaystyle ~Z}$ are the mass number and charge of the nucleus, ${\displaystyle ~\zeta }$ is the coefficient that shows the repulsion of the charge elements in the electron’s matter from each other (${\displaystyle ~\zeta =1}$ for one electron).

In contrast to the standard expressions, the electron’s radius and angular momentum depend not only on the nuclear charge ${\displaystyle ~Z}$ of the atom, but also on its mass number (as a result of taking into account the strong gravitation). In the presence of the dynamic spin as the motion of the electron cloud’s center of mass relative to the nucleus and in the general case, containing elliptical orbits, the formulas for the electron’s energy, angular momentum and magnetic moment are much more complicated and contain elliptic integrals. Into the magnetic moment contribution is made by the proper motion of the matter in the electron cloud (the orbital component), as well as by the spin component. For each electron state, described by a set of quantum numbers, we can calculate the parameters of the corresponding elliptic orbit, the shift of the electron’s center of mass relative to the nucleus, the velocity of matter and other quantities. [8]

## Spatial quantization

The spatial quantization occurs as quantization of the projections of the magnetic moment and the total mechanical moment of the electron on an arbitrary axis, specified by the magnetic field. The number of possible projections is determined by the number of values of the quantum numbers μj = –j, –j+1,…j–1, j , where j is the quantum number of the electron’s total angular momentum, which is determined using the quantum numbers of the orbital and spin angular momentum, so that j can be half-integer. The physical reason for the quantization of projections of the electron cloud’s total angular momentum on the preferred axis is the need to change the angular momentum by a multiple of ${\displaystyle {\frac {\hbar (A+Z-\zeta )}{Z}}}$ in the processes of excitation or emission of an electromagnetic quantum. In this case the main contribution into emission is made by the longitudinal component of the magnetic moment. This is why in the mechanical moment quantization the major role is played by its projection on the axis, specified by the magnetic field.

## Helium

The calculation of the ground state of the helium atom is performed on the assumption that the nucleus is located between two parallel electron disks in the form of rings, the matter of which rotates in opposite directions. The stability of the electrons’ matter is determined by the electrical forces between the nucleus and the rings, while the force of action of one charged ring on the other is determined by the elliptic integral, depending on the radius and width of the rings and the distances between them. In addition, we should take into account the strong gravitation from the nucleus and the centrifugal forces of the matter’s rotation in the disks. By solving the equations for the forces and the binding energy of the electrons we can find the main parameters of the helium atom, including the sizes of the disks and the speed of the matter’s rotation in them and the distance between the disks. [8]

## Electron diffraction

In the well-known experiment with diffraction of light at two close slits on the screen, we can see interference pattern appear on the observation plane of the screen. The similar pattern is observed for electrons. Thus, experiments were conducted to prove the existence of diffraction of electrons flying almost one by one. [16] If an electron is regarded as a point object, it is very difficult to understand its diffraction, because the theoretical analysis of the diffraction experiment predicts simultaneous passage of the electron through both slits. [17]

From the point of view of the substantial model, electrons in atoms represent the objects similar to charged clouds. At the level of stars, the similar clouds consist of the matter, which contains negative charges − electrons. The external electromagnetic field interacts with these charges, creating electrical currents as well as charge-density fluctuations in the clouds’ matter. Therefore, a moving cloud in an excited state is accompanied by its intrinsic electromagnetic oscillations, which are converted for an external observer to the corresponding de Broglie wavelength. When the cloud is detached from star attracting it (as in case of electron’s detachment from an atom or a piece of matter), the cloud may break into pieces, which then will move in different directions. This is due to the fact that the gravitational forces of the cloud are unable to keep its matter from the electrical forces of repulsion. Therefore, it can be expected that some electrons during the diffraction experiment will fall into pieces, which in this case can be coherent with respect to de Broglie waves. If such pieces of electrons fall into different slits of the screen, then after diffraction, at an appropriate path difference, there should be places on the observation plane, where these pieces come to either in phase or in antiphase of oscillations. This leads to either addition of oscillation energies or to their subtraction. As a result, on the observation plane we periodically see the recurrent places, where the incident electrons are strongly excited and have increased energy, and the places, where the electrons have almost no excitation energy. This gives the possibility of emergence of the interference pattern.

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