On the question of the radius of elementary particles

. This article belongs to the field of theoretical physics, specifically the study of elementary particles. We discuss the parameters of elementary particles, their interactions with gravitational potential and magnetic fields, stability with respect to their lifetimes, as well as interactions between particles with different parameters, including different masses. We clarify the concept of electric charge for particles. We use methods and formulas from classical physics that are accessible to a wide range of interested individuals. Key words: photon


Introduction
This article explores the motion of microcosmic objects through the lens of a planetary model. This model is particularly useful for understanding the gravitational field that characterizes the microcosm. The specificity of this model lies in the extremely small size of the interacting objects, allowing for closed systems and the motion of elementary particles occurring in a vacuum. The reasoning is based on the fundamental laws of energy and momentum conservation. All calculations are performed using the SI unit system.

Gravitational potential field in the microcosm
In our three-dimensional space, the gravitational potential field represents a system of potentials or energy levels that are associated with the choice of reference frame. The majority of material objects in the microcosm have a paired nature. To determine the parameters of such objects, such as distance, velocity, acceleration, and energy, at least two fixed points of center of mass are required. All elementary particles in the microcosm have their counterparts in the form of antiparticles. Many stable objects consist of such pairs. Let us consider a system consisting of two particles located in a common reference frame. This reference frame exists objectively. The reference point is defined as the point in space that has the same potential energy level with respect to the center of mass of both particles, or in other words, the point of potential balance. The potential of the center of mass of the particles relative to the point of potential balance is calculated using the following formula [9]: P1 = -G*M/R; where P1 is the potential of the center of mass of the particle; G is the gravitational constant, assumed to be equal to 6,67*10^(-11); M is the mass of the particle; R is the radius-vector of the center of mass of the particle in the reference frame of the point of potential balance.
The kinetic energy of a particle is expressed by the following formula: where P2 is the kinetic energy of the particle; V is the tangential component of the particle's velocity relative to the potential balance point; R is the radius vector of the particle's center of mass relative to the potential balance point.
Taking into account the potential of the particle's center of mass relative to the potential balance point, its energy with respect to the potential balance point is equal to: where V^2*R^2 = constant, according to the law of conservation of angular momentum.
The normal acceleration of the force acting on a particle relative to the point of potential balance can be calculated using the following formula [4]: Both particles rotate around the point of potential balance, for which the normal acceleration of the force acting on the centers of mass of the particles is zero in the case of stable periodic motion. This point corresponds to the following relationships according to formula (1): G*M/Rb^2 = Vb^2/Rb; G*M/Rb = Vb^2; (2) where M is the mass of the particle; Rb is the radius vector of the particle relative to the point of potential balance; Vb is the linear velocity of the particle relative to the point of potential balance, which is the same for both particles.
In accordance with formula (2), at the equilibrium point, the potential of the center of mass of the particle is equal to twice the value of its kinetic energy. This is a periodic motion in a closed system of two objects without energy loss. An example of such a system is a photon.

Photon
A photon is a paired construct consisting of a particle and its antiparticle, whose centers of mass are symmetrically located on a single axis. Photons have a wide range of energies. The concept of the particle's kinetic energy is relative and is autonomously determined in three-dimensional space for each of the three Cartesian axes. The total energy of a photon in a gravitational field relative to each of the axes can be expressed by the following where h is the Planck constant, which is taken to be equal to 6,626*10^(-34); Vb is the linear velocity of the centers of mass of the particles in a photon or in any pair of particles relative to the point of potential balance; c is the speed of light in vacuum, which is taken to be equal to 2,987*10^8; n is the number of autonomous small particles in half of the photon. In commonly accepted terms, this is the frequency of the photon.
Let us consider a photon with the smallest energy, consisting of one small elementary particle and its antiparticle. The energy of motion of such a photon is given by: Eo = h*n = 2*Mo*n*c^2; n = 1; where Eo is the total energy of the photon according to Einstein's formula, Mo is the mass of one of the particles that make up the elementary photon, and n is the number of autonomous small particles in half of the photon, commonly known as the frequency of the photon.
The mass Mo can be calculated using the following formula: Mo = h*n/2*n*c^2; Mo = 0,368*10^(-50); The energy of motion along the axis of the small particle in the elementary photon is: The internal rotational energy around its center of mass is: Hence, the small particle represents an elementary loop with a linear velocity equal to the speed of light in a vacuum. When the speed of light in vacuum is reached, the particle gains stability in that point.
The small particle and its antiparticle in the elementary photon rotate relative to an axis perpendicular to the plane of their common motion, opposite to each other. This difference in the particles' construction is perceived as electric charges of the particles. An approximate calculation scheme for a photon is shown in the following figure. The gravitational radius of a small particle is calculated by formula (2): Photons of the entire energy spectrum consist of paired particles, each of which has a mass calculated by the following formula: A photon can be split into two autonomous particles with equal masses but opposite charges, meaning the charge here is applied conditionally, as there are no special charges, it is just an indication of the structural differences between elementary particles. Each particle gains independence and can form paired structures with other particles, some of which are stable. One example of a stable paired structure consisting of particles with different masses is the hydrogen atom with one proton in the nucleus.

Magnetic field of a paired particle
An elementary loop with a linear velocity equal to the speed of light in a vacuum relative to its center of mass, which acts as an electric current, creates a magnetic field according to the formula of circular electric current. The magnetic induction in the loop is calculated by the following formula: where Io is the electric current of the loop, and Ro is the radius of the loop.
The electric current of the elementary loop is determined by the following formula: where e is the elementary electric charge of any loop with a linear velocity equal to the speed of light in a vacuum taken as 1.6*10^ (-19), and N is the number of turns of the loop corresponding to its linear velocity given by the following formula [3]: For an elementary circuit, the magnetic induction is equal to: Therefore, the magnetic flux in the elementary circuit is: where So is the area of the elementary circuit, equal to 3,14*Ro^2 = 25,4*10^(-156); The magnetic flux in a pair of elementary particles in a photon is closed to each other. The particle and antiparticle are attracted to each other according to Ampere's law as two circular currents. For an elementary photon, the attraction force is calculated by the following formula: where lo is the perimeter of the elementary circuit, equal to 17,1*10^(-78);

Electron
An electron is a stable elementary particle formed during the splitting of a photon into an electron and a positron, with a mass equal to [2]: where n is the number of smaller particles composing the electron and is equal to 2,48*10^20.
Structurally, an electron is represented by a circuit around its center of mass, along which elementary photons move at the speed of light in a vacuum. The radius of a free electron is given by formula (2) and is equal to the gravitational radius: where Me is the mass of the electron, taken to be 9,11*10^(-31); While remaining in a free state, an electron maintains its stability and infinite lifespan, obeying the law of conservation of momentum, and its connection with the positron in space remains. However, this connection is smoothed out by numerous contacts with other objects. Formula (2) can be expressed as follows: where Rbe is the radius vector of the center of mass of the electron-positron system and Vbe is the linear velocity of the centers of mass of the electron and positron relative to the potential balance point. According to the law of conservation of angular momentum, the right-hand side of equation (4)  Then, the special radius vector of the centers of mass of particles according to equation (4) is: Rbe = 2*c^2/n; n = 2,48*10^20; The relation (4) is maintained for any value of the radius vector of the centers of mass of particles relative to the point of potential balance. If the particles approach each other, a photon is restored, and if the distance is forcibly increased, the angular momentum that each particle had before the photon split is preserved. From equation (4) at the point of the special radius vector, the following unique relations are obtained for the electron-positron pair: Me*Rbe = h; G*h = Vbe^2*Rbe^2; An electron can form a pair with other particles, such as an electron antineutrino. In this case, the antineutrino plays the same role as the positron, differing from the positron in mass and trajectory.

Proton
The proton is a stable elementary particle formed during the splitting of a photon into a proton and an antiproton, and has a mass equal to: where n is the number of small particles comprising the proton, and is equal to 4,5475*10^23.
Structurally, the proton is represented by a contour around the point of its center of mass, along which photons consisting of electrons and positrons move at the speed of light in vacuum. The radius of a free proton is equal to the gravitational radius of the proton, given by the formula (2): Rp = G*Mp/c^2 = G*h*n/2*c^4 = 1,24*10^(-54); where Mp is the mass of the proton, taken to be 1,6735*10^(-27).
Writing equation (2) in the following form: As the radius-vector of the centers of mass decreases and the particles approach each other, the photon is restored. When the radius-vector increases, each particle retains the angular momentum it had when the photon split.

Electron antineutrino
The electron antineutrino is a stable elementary particle that is formed when a photon is split into a neutrino and an antineutrino. Its mass can be calculated using the following formula:

Hydrogen atom with one proton in the nucleus
The hydrogen atom with one proton in the nucleus is a paired particle consisting of two particles with different masses, a proton and an electron. The energy of this atom is calculated using formula (3): E = n*h*(Vbp^2/c^2 + 1)/4; where n is the number of small particles in the proton, and Vbp is the linear velocity of the center of mass of the proton and electron in the hydrogen atom relative to the balance point potential. This velocity for stable periodic motion of the closed system of the two given particles is determined by formula (2) where Rbp is the radius-vector of the center of mass of the proton, and Rbe is the radiusvector of the center of mass of the electron, with the balance point potential taken as the reference point.
The energy in formula (3) includes the energy of the particle's motion around the balance point potential and the energy of rotation of the particle around its center of mass. The rotational energy of the proton around the balance point potential is: The rotational energy of the proton around its center of mass is: The rotational energy of the electron around the balance point potential is: The rotational energy of the electron around its center of mass is: During the motion of particles in a hydrogen atom, the law of conservation of angular momentum applies, because a closed system is considered. For the proton and the electron, the angular momentum relative to the potential balance point is expressed as follows: Lp = Mp*Rbp*Vbp = constant; Le = Me*Rbe*Vbe = constant; From equation (2), it follows that: G*Mp*Rbp = Vbp^2*Rbp^2; Mp = n*h/2*c^2; (6) Equation (6) for the hydrogen atom indicates that there exists only one numerical value of the radius vector of the center of mass of the proton relative to the potential balance point, at which the force balance is not violated in the system, and the atom becomes a stable structure. This value is given by: Then, the following equality holds true: which, according to equation (6), leads to the following expression: Thus, for the pair of elementary particles, the proton and the electron, there is only one E3S Web of Conferences 389, 01059 (2023) https://doi.org/10.1051/e3sconf/202338901059 UESF-2023 potential balance point for the creation of a stable hydrogen atom. It is achieved when the electron approaches the proton. At this point, the normal accelerations of the forces acting on the centers of mass of the particles relative to the potential balance point are equal to zero. The following relationship is satisfied: This relationship means that in a hydrogen atom with one proton in the nucleus, the distances to the potential balance point of the proton and electron are proportional to their masses.

Stability of elementary particles
Considering the parameters of stable elementary particles, it can be observed that they are all composite parts of photons and attain individuality upon separation from them. However, the particle does not completely lose its connection with its antiparticle. The dependence remains through the potential balance point, given by the following formula: M*Rb = h; M = n*Mo; Thus, both particles exist in a common reference system of the potential balance point. In this common reference system, the conservation of particle angular momentum is maintained. When particles were part of a photon, the unit angular momentum was equal to L^2 = c^2*R^2. Upon separation of the photon into two particles, the unit angular momentum does not change due to the law of conservation. When particles interact with particles that have different parameters, such as different masses, it is necessary to enter a common reference system to form a stable structure with them. For example, the hydrogen atom has a single common reference point for the stable state of the proton and electron, with equal linear velocities of their centers of mass relative to this point. Upon approach of the electron to the proton, the balance of forces occurs relative to this point, resulting in the formation of a hydrogen atom. Upon approach of a particle and antiparticle, the balance of forces occurs after reaching their gravitational radii, resulting in the restoration of the photon.
Thus, elementary particles create groups of two types of particles: stable and unstable, depending on the distances to the potential balance point, which are proportional to their masses. This can be seen from formula (2), which can be rewritten as follows: G*M = Vb^2*Rb^2/Rb; (8) The stability point for paired structures is found from formulas (7) and (8) according to the following relationship: Rb = 2*c^2/n; A unit particle is stable at its gravitational radius when formula (8) takes the following form: G*M/R = c^2; M/R = c^2/G; Here, a decrease in particle radius is impossible, while an increase includes a positive

Neutron
A neutron in free form undergoes decay [8]. However, it can exist as a part of the atomic nucleus structure in a chain with other particles. As a part of the atomic nucleus, a neutron is also unstable, like in its free form, but inside the nucleus, it has a probability of interacting with random particles equal to zero. The mass of a neutron is: Mn = n*Mo = 1,6749*10^(-27); where n is the number of small particles in the composition of a neutron.n = 4,5513*10^23. The gravitational radius of a free neutron is: Rn = G*Mn/c^2 = 1,2413*10^(-54); The radius vector of the balance point of neutron potentials is: Rbn = 2*c^2/n = 3,965*10^(-7).
One of the important features of the neutron is its instability and limited lifetime in free state. The reason for this lies in the structure of the neutron. The neutron is not an elementary particle born from the division of a photon. It is a composite structure consisting of three elementary particles, which is revealed upon its decay. It consists of an ordinary proton in combination with an electron and an electronic antineutrino. If the radius-vector of the balance point of potentials does not satisfy the following relationship: Rbp = 2*c^2/n; (9) then, in the case of a smaller value, the acceleration of the force acting on the centers of mass of the particles is positive, and the force brings them closer to the limit, i.e., to the gravitational radius of the proton and electron. The electron carries the antineutrino with it, which, in order to be stable in combination with the electron, needs to obtain a distance from the center of mass of the electron to the balance point of potentials in accordance with formula (9) equal to 7,28*10^(-4).
In free form, the neutron structure consists of three elementary particles at gravitational radii from each other. These distances do not satisfy the equality (9) for paired particles and therefore the neutron is an unstable construction. Any deviation under the action of external forces destroys it. The boundary between particles and photons is not clear-cut. Mutual transitions are carried out in accordance with the laws of conservation.

Carbon Atom
Under favorable conditions, chains of interacting elementary particles can be formed. To ensure stability, the relationship (7) must be maintained. As an example, consider the carbon atom. All bonds satisfy the relationship (8). Thus, in the nucleus of the carbon atom, there are not only protons but also electrons bound in neutrons, as well as antineutrinos. The carbon atom consists of three rings of particles. In the center are six neutrons, bound to each other through their electrons and to electronic antineutrinos in a ring. The distance between particles is equal to their gravitational radius. Next, in the following circle, are protons bound to neutron electrons. Attached to them are shell electrons similar to those in a hydrogen atom. As such, the nucleus of the atom is absent. Simply put, distances between particles are much smaller closer to the center than at the periphery. Other atoms form spatial structures in the same way.

Muon
Muon is an unstable particle [2]. Its mass is accepted to be Mm = 1,8883*10^(-28). The number of subatomic particles in muon is: n = Mm/Mo = 0,5117*10^23; The gravitational radius of a free muon is: Rm = 1,39694*10^(-55); The radius vector of the potential balance point of the muon is: Rb = 2*c^2/n = 37,13*10^(-7); Muon is not an elementary particle born from the decay of a photon. It consists of chains of three elementary particles: electron, electron antineutrino, and muon neutrino. The chains of particles can be long to maintain the conservation laws. Like the neutron, the constituent particles of the muon do not correspond to formulas (7), (8), and (9), therefore the muon cannot be stable in a free state.