The substantial proton model describes the origin, structure, state of
matter, and other properties of the proton based on the idea of two fundamental
force fields, acting in the nature. According to the theory of Infinite Hierarchical Nesting of Matter,
these fields include the electromagnetic field and gravitational field. In the
presented model, the de Broglie wavelength
of moving protons is explained as a consequence of conversion of the wavelength
of internal oscillations of potentials of the fundamental fields of these
particles into the laboratory reference frame.
Fig.1. The structure of the magnetic field of a magnetar. a) The magnetic
lines with induction
.
and
are the magnetic moment and the
star’s spin. b) The orientation of the magnetic moments and spins of neutrons
and protons in the equatorial plane;
and
and
are the
magnetic moments and spins of neutrons and protons, respectively.
As an effective tool for the study of the proton’s
properties we can use the theory of similarity
of matter levels. In this theory, one-to-one correspondence is established
between the objects of basic levels of matter, and SPФ symmetry predicts similarity in the
progress of similar processes. [1] In
particular, at the level of stars a strongly magnetized neutron star — magnetar
corresponds to the proton. Despite the huge difference in the masses and sizes,
there is a great similarity between these objects. If the average matter
density of a neutron star is about
kg/m³, the average density of a proton is about
kg/m³. The magnetic induction on the surface of a magnetar is more than 1011
T. [2] At a tabular value of the proton magnetic
moment of
J/T, the magnetic induction at the proton pole must equal the value of
T. It is assumed that the magnetic field of a quarter of neutron stars
exceeds the value of 1010 T, and their properties are similar to
those of magnetars.
The magnetic field at the center of a neutron star is
created mainly by the neutron phase of matter. Here, under conditions of high
matter density and strong pressure of about 1033 Pa, the magnetic
moments of neutrons are arranged in a parallel way, increasing the general
magnetic field. If the neutron spins are opposite to the star spin, then this
magnetic field will be close by direction to the star spin. The neutron star
mass must be permeated by the magnetic filaments, just as it has been already
discovered in ordinary superconductors on the Earth. [3]
However, at the assumed threads’ thickness of 10−13 m the magnetic
field induction in the filaments is very high — of the order of 1011
T.
To estimate the magnetic moment
of
the magnetar, according to the theory of dimensions of physical quantities and
the theory of similarity, we must multiply the magnetic moment of proton by the
corresponding similarity coefficients:
J/T.
Here
and
are the coefficients of similarity in size, mass
and velocities, respectively, [4] as
it follows from the similarity of matter levels.
On the other hand, if the magnetic moments
of
all nucleons (mainly neutrons) that make up the magnetar, have the same
direction, then the magnetic moment equals
J/T, which
is an order of magnitude greater than
. Hence it follows that in the formation
of the magnetic moment of the magnetar practically all the particles are
involved, of which it consists. But then the proton, similarly to the magnetar,
is an object with the maximum possible magnetization of its matter. The
experimental dependences of the charge density and the density of the magnetic
moment of the proton are close to each other. Then it can be concluded that the
contribution to total magnetic moment of the proton is made by individual
magnetic moments of the proton matter, just as it happens in magnetars.
From the standpoint of classical electrodynamics, the
magnetic moment of the proton is anomalous – it is 2.79 times larger than the
nuclear magneton, that is the magnetic moment of a particle with the mass and
charge of the proton, which has the quantum spin of the proton equal to ħ/2 (ħ
is the Dirac constant).
The maximum magnetic moment of the proton can be
expressed in terms of its spin using the formula for a rotating charged ball:
![]()
where
is
the elementary charge,
is the proton mass, and the maximum spin is determined by the formula:
![]()
Here
and
are the moment of inertia and the proton
radius,
is the maximum velocity at the
proton’s equator,
is
the maximum angular velocity of rotation. The formula for the magnetic moment
is obtained by integrating over the proton
volume and is based on the condition that the electric charge of the proton is
uniformly distributed over its volume, and during the proton’s rotation this
charge creates the magnetic moment. The quantity
can be found from the equality of the
centripetal force and the gravitational force at the equator:

where
is
the strong gravitational constant.
For the magnetic moment of the proton
we obtain the expression, which gives almost exact tabular value of the
magnetic moment:
![]()
If we take into account that the magnetic moment of the
proton is determined by the standard formula:
![]()
then from comparison with the expression for
it
follows that the maximum spin of the proton is equal to
.
As we can see, there is a close relation between the
magnetic moment and the rotation of the positive volume charge of the proton
with limiting angular velocity. As a result, the magnetic moments of the proton
matter particles are oriented by the general magnetic field and support this
field, even during the subsequent deceleration of the proton’s rotation. This
situation explains the anomalous magnetic moment of the proton in comparison
with the nuclear magneton and corresponds to the structure of the magnetic
field of the magnetar, shown in Figure 1.
The attempts to calculate the electric charge of the
proton only through the rotation of its magnetic moment or through the internal
currents show that the proton charge is created mainly by the internal volume
electric charge. An additional contribution to the effective value of the
proton charge can be made both by the internal currents and the magnetic moment
of the proton matter, taking into account its spin rotation. [4]
The magnetic field induction outside the proton is
determined by the formula for the magnetic dipole field:
![]()
where
is
the vacuum permeability,
is the proton’s magnetic moment vector,
is the radius-vector from the center of the proton to the point, at which
the magnetic field is determined.
If we assume that the proton is a uniformly charged ball,
then rotation of such a ball at the angular velocity
generates inside it the magnetic field
induction for a non-rotating observer, which is found by the formula: [4]
![]()
where
is
the ball’s magnetic moment vector,
is the ball’s charge.
There are several methods to estimate the effective
radius of the proton. In literature we can find the charge and magnetic radii
as well as the radius found from the cross-sections of particles’ interaction.
All these radii can differ from the actual proton radius
. Thus, in experiments on electrons
scattering by protons the root-mean-square charge radius
m was found. [5]
The cross-section of nucleons’ interaction with each other, established at
energies greater than 10 GeV, equals 38 mbn. [6] In the classical limit we can assume that this
cross-section is close to the total geometrical cross-section of colliding
particles, that is, to the value
. Since mb = 10−31 m², then we
obtain
m.
Theoretical calculations of the proton radius were
performed by Sergey Fedosin using several methods: by examining the standing
electromagnetic waves inside the proton; by equating the difference between the
binding energies of the proton and neutron to the mass-energy of the electromagnetic
field of the proton; [1] and using the
limiting angular momentum of the gravitational field of the proton. [7] These methods provide the value of the proton radius
(6.7 ± 0,1)∙10−16 m. If we calculate the
proton radius from relation (1), we obtain the value 7.7 ∙10−16 m.
The mass and radius of the proton can be estimated by
analogy with the way, in which the masses and radii of neutron stars were
found, based on the quantum state of their matter and the relation between the
gravitational energy and quantum-mechanical energy. [8]
Just as in case of neutron stars, the proton mass is determined by the
properties of its matter, as well as by the strong gravitational constant,
ensuring the nucleon integrity. Hence, it follows that in every gravitational
field of any basic level of matter the objects have only one mass-radius ratio,
at which the greatest gravitational energy density is achieved. Besides, the
values of the mass and radius are fixed by the laws of quantum mechanics, which
points to the significant matter degeneracy. For the relation between the
radius and mass of the proton we obtain the formula: [9]

where
is
the constant that depends on the properties of the proton matter.
The self-consistent proton model takes into account the
non-uniform matter distribution inside the proton (increase in density at the
center), the formulas for the binding energy and magnetic moment at maximum
rotation. It allows us to determine the central density
kg/m³ of the proton matter and to
estimate the rate of change of the density with the change of the radius. The
proton radius equals
m and the maximum angular velocity
of its rotation reaches 6.17∙1023 Hz. [9] [10] For
comparison, the website of Particle data group [11]
gives the value of the charge radius of the proton
m.
Taking into account the law of redistribution of energy
fluxes, for the proton we find the angular velocity of its steady rotation,
equal to 2.98∙1023 Hz, at which the equality of the total energy
flux of the gravitational field and the kinetic energy flux of the rotating
matter is achieved in it. In case if the magnetic moment of the proton and its
angular momentum fully coincide in direction, the electromagnetic emission from
the proton is zero and it can be in the state of long-term, steady rotation at
the constant velocity. [9]
A neutron star contains about
nucleons, and it is assumed that a proton
contains the same number of minute quantum particles — praons. This helps us explain why in
collision of high-energy gold ions we do not find the gas of quarks and gluons,
as is expected in quantum chromodynamics, but jets of almost ideally liquid
hadronic matter. [12] At such energies of
collisions the hadronic matter cannot be in the form of gas, because it is
pulled together by strong gravitation
into self-gravitating objects, which over time take a spherical form. [9]
The analysis of the electromagnetic energy and the energy
of strong gravitational field in the proton shows that the ratio of the proton
mass to its charge is associated with the balance of energies of the field
quanta and of the proton matter particles during its formation. The proton
charge is close to the limiting value, at which the action of the
electromagnetic field begins to destroy the minute particles of the hadronic
matter, so that at a greater charge the proton could not exist.
According to the substantial
neutron model, the charge and magnetic configurations of the neutron are
gradually changing as a result of the weak interaction reactions that occur in
its matter. Then a rapid restructuring of the general magnetic field takes
place, the negatively charged shell is ejected, turning into an electron. At
the same time an antineutrino is emitted, and the rest part of the neutron,
positively charged in general, becomes a proton. This process is called
-decay of the neutron, and it shows why
the proton charge has a discrete value and is the same practically for all
protons – this is a consequence of mass discreteness of neutrons and of the
properties of their matter in the strong gravitational field.
The relation between the average pressure
and the average density
of the proton matter has the form:
![]()
where
in SI units is the coefficient,
which is found through the proton radius, its mass and the strong gravitational
constant. [9]
Assuming that the characteristic
speed of the matter inside the proton is the speed of light, for the rest
energy and the total energy of the proton, in view of the matter energy in the
strong gravitational field and the virial theorem, the following relation
holds:

where
is
the coefficient that depends on the matter distribution in the proton.
Relation (2) reflects the equivalence of mass and energy
as a consequence of the principle of proportionality of the mass and the
binding energy of the proton. It also means that in all processes with nucleons
the change in their total energy should be taken into account.
Taking into account the expression for the strong
gravitational constant, another estimate of the proton radius follows from
relation (2):
fm,
where
is
the electric constant,
is
the electron mass.
The last-mentioned equation can be interpreted as
follows. If we put a positron into a neutron and mix the entire matter and
charge of the positron over the volume of the neutron, we will obtain a
particle close to a proton. Any matter in the proton has the characteristic
speed of the order of the speed of light, and the energy equal to the rest
energy. On the other hand, the electric energy of the positron during its
compression into the nucleon’s volume increases to a maximum and is determined
by the proton radius. From expression for the energy (2) it follows that the
energy of strong gravitation depends on the proton mass and is equal to the
doubled rest energy of all the matter. Similarly, the electric energy depends
on the proton charge and is equal to the doubled rest energy of the matter of
the positron as an effective charge carrier inside the proton.
Unlike neutrons, protons are practically stable
particles, which ensures their maximum prevalence in the nature as part of the
hydrogen atom and in atomic nuclei. The stability of the proton in the strong
gravitational field is due to the balance of gravitational forces and repulsive
forces between the particles of matter inside the proton. On the other hand,
the proton matter is stable with respect to the weak interaction reactions, and
decays of free protons are not observed. The proton structure is similar to the
structure of a magnetar, in which the magnetic moments of neutrons are aligned
along the magnetic field of the star, the neutron spins are oriented along the gravitational torsion field of the star,
and as a result the stellar energy is minimal.
The proton is the basis of the matter of atoms and it
forms a number of compounds with other particles. The coupling between a
neutron and a proton by means of strong interaction can lead to formation of a
deuteron. The compounds of two protons (diproton) and two neutrons (dineutron)
have low binding energy, they are unstable and decay immediately.
In the gravitational
model of strong interaction, the strong interaction appears as a result of
summation of electromagnetic forces, strong gravitation and forces from the
gravitational torsion field. The main components are the gravitational
attraction force and the spin-spin repulsion force. When the distances between
the particles are smaller than the nucleon radius, the balance of forces and
formation of such composite objects, as atomic nuclei, are possible. [4]
Another example is strange particles, many of which are
assumed to be the compounds of nucleons and pions. So, Λ-hyperon can consist of
fast-rotating near each other and along one axis proton and pion, which are
held by strong gravitation and spin torsion fields, [9]
and Σ-hyperon is a compound of neutron and pion. The strange Ξ-baryons contain
two pions in addition to a proton, and Ω-baryon contains three or four pions,
which gives a baryon strangeness, equal to 3. Pions can combine with each other
even in the absence of nucleons. An example is K-mesons, consisting of three
pions in various combinations.
It is known that at high hadron
collision energies, the transverse momenta of arising charged pions with the
invariant energy
GeV have a value from
GeV to
GeV, here c
is the speed of light. These pion momenta are considered to be a
fundamental quantity for the interaction of hadrons – they depend little on the
type and energy of the colliding particles, on the multiplicity of particle
production, etc. To explain the origin of such pion momenta, one can use the
idea of strong gravitation. If the pion rotates near the surface of the proton
in a circular orbit at a speed of
, then from the equality

where
is the Lorentz factor, the quantities
and
are determined. So
the pion momentum will be
GeV.
If the pion moves away from the proton
to infinity with a minimum energy, there is the equality for the energy

It gives the Lorentz factor
, the initial pion speed
,
the momentum
GeV.
In contrast to interactions between protons and neutrons,
for annihilation of nucleons as a rule some antinucleons are required. At low
energies an antiproton annihilates with a proton with production of 4–5 pions
on the average, one of which is neutral and decays into two photons. Additionally,
production of K-mesons and less often of certain gamma-ray photons is possible.
An antiproton can also annihilate with a neutron.
According to the substantial model, proton-antiproton
annihilation is most effective, when they encounter along the line, which is
the axis of rotation of both particles. At the same time the spins of these
particles must be opposite, and the magnetic moments must have the same
direction. Then all the forces, acting from the spins, magnetic moments,
charges and masses, are the forces of attraction. During collision, the energy
of the opposite rotation of nucleons can fully go into the internal energy of
nucleons and heat the matter up to a high temperature. The nucleon matter
decays into large parts flying away in all directions, which turn into pions
under action of strong gravitation. The rotation energy also can go to
electromagnetic quanta. Sometimes in annihilation of nucleons, gamma-ray
photons are observed with energies up to 180 MeV, which is about 19 % of
the rest energy of nucleon. It is close to the maximum possible rotation energy
of one nucleon, reaching almost 20% of the rest energy of nucleon. [4]
Similarly we can consider interaction of antiproton
and neutron. The strong gravitational field releases enough energy in order to,
under rapid opposite rotation, divide all the matter into parts and scatter it
in space. If we calculate the mutual gravitational energy of two nucleons at
the moment of their contact, it will be almost equal to the proper
gravitational energy of one nucleon. Release of this energy in a collision is
not enough to fully scatter the entire nucleon matter, but is enough to divide it
into several large fragments with the masses of the order of meson masses.
As a rule, protons emerge in the nature in beta-decay of
free neutrons in the reaction:
![]()
in this process electrons
and
electron antineutrinos
are also emitted. This reaction at the level of transformations in the
neutron matter is analyzed in the substantial
neutron model.
The reaction of electron capture by proton has the
following form:
![]()
When the negatively charged electron matter falls on the
surface of the proton, its electric energy of attraction to the proton and the
strong gravitational energy are converted into the kinetic energy, which, upon
falling, goes into the thermal energy and heats up the matter. Simultaneously,
the negative charge of the electron flows into the proton shell and compensates
its charge. Since the negative charge cannot penetrate into the central part of
a proton due to the high pressure in the matter, the proton core remains
positively charged. The electric charge configuration emerges, which is typical
for the neutron. Since the proton spin practically does not change, rotation of
the negative charge in its shell leads to reversal of the magnetic moment. As a
result, a proton is converted into a neutron, with zero total charge and the
magnetic moment opposite to the spin.
Transformation of
nucleons in interaction with neutrinos takes place with emission of leptons.
The following reactions are the examples:
These reactions take
place with very low probability. For example, the cross-section in reaction 1
is equal to
m²
with the energy of antineutrino of 4 MeV, while the strong interaction
cross-section during pion-proton scattering has the value of the order of
m² with the energy of colliding particles
of about 1200 MeV in the center-of-inertia system. The probability of reactions
with neutrinos and the cross-sections of these reactions are directly
proportional to the neutrino energies. Obviously, all this is due to the fact
that neutrinos consist of beams of minute particles moving at relativistic
velocities.
In reaction 1 the electron antineutrino with right-handed
helicity transforms the proton into a neutron and a positron. Analysis of this
reaction can be conveniently performed at the level of stars, assuming that the
stellar electron antineutrino
falls on the magnetar, which is the analogue
of the proton. This stellar electron antineutrino consists of two parts,
including the fluxes of ordinary electron antineutrinos and neutrinos:
![]()
The magnetar consists of nucleons oriented by the
magnetic field, as is shown in Figure 1. To transform the magnetar into the
neutron star, the analogue of neutron, we need to transform the protons of the
magnetar’s matter into positrons and neutrons with the help of electron
antineutrino. This is possible only when the antineutrino
is
propagating in the direction from the south to the north magnetic pole of the
magnetar. In this case, the right-handed helicity of the fluxes
and the proper right-handed helicity of
antineutrino
in
(3) would coincide with the direction of the magnetar’s spin and the spins of
protons of the magnetar’s matter. At the same time, the fluxes
of left-helicity neutrinos
reach the magnetar’s neutrons from the side, in which the spins of neutrons
are directed, and produce electrons and protons. Part of the emerging electrons
and positrons annihilate, releasing energy and heating up the magnetar’s
matter. After accumulating a sufficient number of positrons in the shell, due
to their repulsion from the central part of the magnetar, which is positively
charged, ejection of the heated matter takes place with formation of a stellar
object of positron-type. The magnetar itself is transformed into a neutron
star, the neutron’s analogue, since the nucleons in the magnetar’s shell under
the action of the fluxes of neutrinos and antineutrinos reverse the direction
of their magnetic moment, and the charge gradient appears in the matter due to
the electrons produced. This leads to compensation of part of the magnetic
field of the magnetar’s core by the magnetic field of the shell, to changing
the magnetic field configuration and the sign of the star’s magnetic moment,
and to releasing a considerable amount of energy, which contributes to the
matter ejection. Ejection of the positively charged matter from the shell of
the magnetar means the loss of charge by the magnetar and its transforming into
a neutral neutron star. Similarly, an electron antineutrino interacts with the
matter of the proton, transforming it into a neutron in reaction 1.
In reaction 2 an electron neutrino with left-handed
helicity transforms a neutron into a proton and an electron. Consequently, the
stellar neutrino
should also transform a neutron star, the
neutron’s analogue, into a magnetar with ejection of part of the shell as an
object, which is the analogue of an electron. For this it is necessary to turn
the protons and neutrons in the stellar shell into the neutrons and protons in
reactions 1 and 2, respectively, with reversal of their magnetic moment. For
this to happen, the stellar neutrino
should reach the neutron star from the south
pole of its magnetic moment in the direction opposite to the spin of the star.
The stellar neutrino has left-handed helicity and consists of the fluxes of
electron neutrinos and antineutrinos:
![]()
Then the left-handed helicity of
, as is shown in (4) in square brackets,
will correspond to the direction of the star’s spin. In this case, the left
neutrinos
will fall onto neutrons, and the right
antineutrinos
onto the protons of the star from the
corresponding south magnetic pole of each nucleon. In reactions 1 and 2
electrons and positrons are produced, which partially annihilate with release
of energy. The reversal of the magnetic moments of nucleons in the shell of the
star takes place, which leads eventually to transformation of the magnetic
field configuration of the neutron star into the configuration of the magnetar,
which is shown in Figure 1. In the shell of a neutron star, the neutron’s
analogue, there is excess of the negative charge. Rapid changes in the magnetic
configuration of the star lead to changes in the magnetic pressure, resulting
in ejection of part of the shell’s matter that bears the negative charge. This
is equivalent to formation of a new stellar object of electron-type. The star
itself becomes a magnetar. From this stellar model we can see, what can happen
with the neutron in interaction with the neutrino and how weak interaction
reactions occur in the neutron matter. The interactions of nucleons with muons
and muon neutrinos are considered in a similar way. As a result, weak
interaction in the objects of the same level of matter is reduced to weak
interaction reactions at lower levels of matter. Therefore, weak interaction
should not be considered a special kind of force, and the use of it for
description of gauge W and Z bosons should be considered just as a convenient
way of mathematical assessment of phenomena. [4]
According to the substantial
neutron model, the first result of the matter evolution at the level of
elementary particles is production of neutrons and nuons from the more massive objects, the
analogues of which at the level of stars are the main sequence stars. These
objects emerge under the action of strong
gravitation as a result of gravitational bunching of scattered matter and
produce neutrons, just as neutron stars are formed during collapse of massive
stars. Then, in the minute particles of the matter of emerging neutrons weak
interaction reactions take place, leading eventually to
-decay of neutrons into protons, electrons
and electron antineutrinos. Similarly, weak interaction reactions in the matter
of ordinary neutron stars in a cosmologically long period of time of about 2•1015
years should end with
-decay of these stars with formation of
magnetars.
Thus the origin of protons is explained
without the use of the idea of quarks in quantum chromodynamics and the Big
Bang concept, according to which at the time of explosion gluons and quarks
should be produced, which then gather in mesons and baryons. In the model of quark quasiparticles, the quarks
are considered as quasi-particles, the properties of which are associated with
the properties of hadronic matter. The primary particles are considered to be
not quarks but nucleons themselves, which in the theory of infinite nesting of
matter represent the main objects at the level of elementary particles.
On the other hand,
the basic levels of matter are the following: the level of graons – the level
of praons – the level of nucleons – the level of stars – the level of supermetagalaxies. [13] Due to
the similarity of matter levels, each
basic level of matter consists of the objects of the underlying basic level of
matter. Hence it follows that protons, neutrons, electrons, and all elementary
particles consist of neutral and positively charged praons and negatively charged
praelectrons. Fluxes of relativistic
praons and graons form the main content of electrogravitational
vacuum, generating electromagnetic and gravitational forces between bodies.
1.
1.0 1.1 Fedosin S.G. (1999), written at Perm, pages
544, Fizika i filosofiia podobiia
ot preonov do metagalaktik, ISBN 5-8131-0012-1.
2.
Heyl J. S. Magnetars.
— arXiv: astro-ph 0504077
v1, 4 Apr 2005.
3.
Ruderman Malvin. A
Biography of the Magnetic Field of a Neutron Star. — arXiv:
astro-ph / 0410607 v2, 2004.
4.
4.0 4.1 4.2
4.3 4.4 4.5 Sergey Fedosin, The
physical theories and infinite hierarchical nesting of matter, Volume 1,
LAP LAMBERT Academic Publishing, pages: 580, ISBN-13: 978-3-659-57301-9.
(2014).
5.
Хофштадтер
Р.// Сб.: Физика атомного ядра. — М.: ГИФМЛ, 1962. — С.72‒86.
6.
Барашенков
В. С. Сечения взаимодействия элементарных частиц. — М.: Наука, 1966, 531 с.
7.
Fedosin S.G. Sovremennye
problemy fiziki: v poiskakh novykh printsipov. Moskva: Editorial URSS, 2002, 192 pages.
ISBN 5-8360-0435-8.
8.
Ландау Л. Д. On the theory of stars. — Phys. Z. Sowjetunion,
1932, Vol. 1, P. 285.
9.
9.0 9.1
9.2 9.3 9.4 9.5 Comments to
the book: Fedosin S.G. Fizicheskie teorii i beskonechnaia
vlozhennost’ materii. –
Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in
Russian).
10.
Fedosin S.G. The radius of the
proton in the self-consistent model. Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012). http://dx.doi.org/10.5281/zenodo.889451.
11.
J. Beringer et al. (Particle Data Group), Phys. Rev. D86, 010001 (2012). [1]
12.
"'Perfect'
Liquid Hot Enough to be Quark Soup". Brookhaven National Laboratory
News. 2010.
Retrieved 2010-02-26.
13. Fedosin S.G. The
graviton field as the source of mass and gravitational force in the modernized
Le Sage’s model. Physical Science International Journal, ISSN: 2348‒0130, Vol. 8, Issue 4, pp. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197.
Source: http://sergf.ru/smpen.htm