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- Sladkov P

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**Received Date**: June 09, 2016; **Accepted Date:** August 19, 2016; **Published Date**: August 23, 2016

**Citation: **Sladkov P (2016) Solitonic Model of the Electron, Proton and Neutron. J Phys Math 7: 193. doi: 10.4172/2090-0902.1000193

**Copyright:** © 2016 Sladkov P. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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In paper, which is submitted, electron, proton and neutron are considered as spherical areas, inside which monochromatic electromagnetic wave of corresponding frequency spread along parallels, at that along each parallel exactly half of wave length for electron and proton and exactly one wave length for neutron is kept within, thus this is rotating soliton. This is caused by presence of spatial dispersion and anisotropy of strictly defined type inside the particles. Electric field has only radial component, and magnetic field - only meridional component. By solution of corresponding edge task, functions of distribution of electromagnetic field inside the particles and on their boundary surfaces were obtained. Integration of distribution functions of electromagnetic field through volume of the particles lead to system of algebraic equations, solution of which give all basic parameters of particles: charge, rest energy, mass, radius, magnetic moment and spin.

Structure of elementary particles; Structure of matter; Theory of elementary particles; Electron; Proton; Neutron; Nuclei; Electromagnetic field; Atom; Microcosm; Elementary particles; Fundamental interactions; New theory; New physical theory

In present article alternative (to Standard Model) hypothesis of structure of electron, proton and neutron is suggested. The others elementary particles (except photon and neutrino) are not stable and they are considered as unsteady soliton-similar formations. In series of experiments indirect confirmations of existence of quarks were obtained, for instance in experiments by scattering of **electrons** at nuclei, performed at Stanford linear accelerator by R. Hofshtadter, look for instance [1]. At that, experiments by elastic and deeply inelastic scattering gave quite different results: in first case take place pattern of scattering at lengthy object, in second case is pattern of scattering at "point" centers, that is interpreted as confirmations of existence of quarks. However what "point" formations appear only in deeply inelastic scattering don’t may be an evidence of quarks existence, because to above-mentioned fact may be given and another explanations: in moment of birth of new particles, which take place in deeply inelastic scattering, structure of nucleon change, it sharply diminish in volume, but after appearance of new particles nucleon return to initial state. Or process of birth of new particles occur in "point" volume inside nucleon and these energy "point" centers disappear after completion of process particles birth. And fact that experiments by elastic scattering gave pattern of scattering at lengthy object prove inexistence of quarks in nucleus. In theory of Standard (quarkual) Model come into at least 20 parameters artificially introduced from outside, such as "colour" of particles, "aroma" etc., that is its fundamental demerit. Theoretical work, which is present here, has no demerits of Standard Model, it completely describe structure of elementary particles therefore it can help in discovery new ways of making energy, elaboration perfectly new devices for its production and to achieve progress in such fields as nuclear power engineering, nanotechnology, high-powerful lasers, clean energy and others.

Let us write down Maxwell’s equations in spherical coordinates supposing that:

1) There are no losses;

2) Only are not equal to zero.

(1)

(2)

(3)

(4)

(5)

(6)

Here r,θ, ? spherical coordinates of the observation point; - components of the electromagnetic field, - density of electric current, - volume charge density; ω-circular frequency of field alteration i - imaginary unit ε-dielectric permittivity - magnetic permeability **Figure 1**.

Substituting the expression for Hθ ? from (2) in (4), we obtain:

(7)

(7‘)

This is Helmholtz homogeneous equation. Let us designate.

(7‘‘)

Wave number - General solution of Helmholtz equation:

(8)

This expression describes two waves, moving to meet one another by circular trajectories, along the parallels. Pointing’s vector in each point is directed at tangent to the corresponding parallel [2,3].

Let us consider a wave, moving in positive direction ?.

(9)

Here

Wave phase;

K_{1} - Dimensionless analog of the wave number. If to introduce a wave number of traditional dimension

The wave phase will be written down as

Where

Arc length along the corresponding parallel. In the considered case the wave number is a function of coordinates and frequency. Thus, the wave, which is described, can exist only at availability of spatial and frequency dispersion [4-6]. Dispersion equations will be obtained below, apart from the already found expression (7′′).

From expression (2), taking into account (7′′) and (9), we have:

(9‘)

For actual amplitudes:

(10)

(10‘)

Here

Means characteristic impedance.

The last expressions describe an electromagnetic wave, rotating around axis Z in positive direction ?.Conditions of self-consistency:

1) z=constant

2) Along each parallel on the circle length, the integer number of half-waves must be kept within.

(11)

Here

Wave length, v - phase velocity of wave, f - frequency, n=1,2,3…

Let us consider the case when n=1,

(11‘)

Along each parallel, exactly half of wave length is kept within.

Phase velocity of wave is the function of frequency and distance up to the axis of rotation.

(11‘)

We are substituting in (11′′):

(12)

(12‘)

We are substituting in (12′).

(12‘‘)

(12‘‘‘)

Taking into account (8) and (11′′)

Then

(13)

(13‘)

Function sin is on valued in angles interval 0 ≤φ ≤ 2π

This situation can be interpreted as rotation of spherical coordinate system around axis z in positive direction ? with **angular velocity **

Let us find it from the condition:

Having differentiated this expression on t, we receive,

At the same time the electromagnetic field, about spherical coordinate system, is determined by expressions (13) and (13′). Further from (3): as H_{?}=0

(14)

From equation (6)

Follows

(14‘)

To receive field dependence from , let us find solution of three-dimensional Helmholtz equation in spherical coordinates.

(15)

Er does not depend from θ, look (14), therefore three-dimensional Helmholtz equation transfers into two-dimensional one.

(15‘)

Let us suppose that

now

(15‘‘)

This equation can be satisfied, if

(16),(17)

Thus, initial Helmholtz equation has split into the system of two equations. We substitute in these equations instead of (i.e. we are searching the solution as the product of two functions) and divide the first equation byf(r), and the second – by g(?). We receive

(18), (19)

Equations (16) and (18) are equivalent to equations (7) ? (7′), which were received earlier from Maxwell’s equations, and

The solution of equation (18) was found earlier, look (13).

Let us copy (19) as:

(19′)

Where

Where

19’ - centrally **symmetric** Helmholtz equation. Let us suggest, k_{4}=k_{3} r

Where vr- phase velocity of electromagnetic wave in radial direction. As in the central symmetric equation angular dependence is absent, it is logical to assume that

v_{r}=v=2ωrsinθ

at i.e.

v=2ωr

(21)

(21′)

Instead of (19′), we are having

(19′′)

This is Euler equation, it has the solution

(22)

Let us converse expression (22).

Here a-value of radius r, at which the rotating monochromatic electromagnetic wave ceases to exist, and E_{r}=E_{0}; f=1 hence

(22′′)

In view of this,

Let us designate C=p now

Thus, for E_{r} we are having

Really

So that at alteration of r within the interval from 0 to a, Er would not change its sign, observance of the following requirement is necessary P≤0

Basing on results of the previous section, let us write down expressions for electromagnetic field inside the electron, assuming that it is concentrated inside the orb of radius a

(23′)

(23′′)

Here a is electron radius, E0 amplitude of electric field intensity at r=a; z=const characteristic impedance inside the electron, P- unknown coefficient and P≤0

At that the internal electron medium possesses frequent and spatial dispersion, as well as **anisotropy**. Dispersion equations have the following appearance [7-11].

(24)

(24′)

(24′′)

Here v_{r},v_{θ},v_{?} - phase velocity of rotating monochromatic **electromagnetic wave **in corresponding direction. In viewed case, the electromagnetic wave is being spread only in the direction ?, and we shall need expressions v_{r} and v_{θ} for searching the formulas of dielectric and magnetic **permeability**, as well as wave numbers of corresponding directions; z_{r}, z_{θ}, z_{?}, z characteristic impedances inside the electron; ε_{φ}, ?, μ_{φ} were found before, see (12′) (12′′)

In view of (24)(24′)(24′′), let us write down expressions for εr, εθ,μr,μθ

(24′′′)

From considerations and formulas adduced, it follows that **dielectric** and magnetic permeability are tensor values.

Let us find dimensionless wave numbers.

Thus

Let us remind that in the viewed case, the electromagnetic wave is spread only in the direction of ?

At r=0 we are having a special point:

Despite of this, all basic electrons’ parameters – charge q rest energy W, magnetic moment M-expressed through integrals by volume from the functions specified above, prove to be finite quantities [12-15]. Look further.

From (5), we find volume charge **density** inside electron

Integrating ρ on electron’s volume, we shall receive this expression for its charge q.

(26)

On the other hand, from the third integral Maxwell’s equation, it is possible to find electron’s charge as a stream of vector electric induction D through the surface of the orb of radius a

(26′)

As we can see, expressions (26) ? (26′) are equivalent to each other.

From (1), we obtain expression for current density j_{?}

(27)

From expressions (25), (27) it is visible that in the interval of change of r from 0 to a,ρ and j? once change the sign. It can be explained by the fact that in the viewed structure, the substantial role is played by the rotating monochromatic electromagnetic wave, and the space charge density and electric current density – are auxiliary or even fictitious quantities in the sense that inside the particle there is neither any charged substance nor its motion [16]. Inside the electron, it is not the charge that is the source of electric field, but electric field is the source of the charge. In its turn, it is not the electric current that is the source of magnetic field, but magnetic field is the source of the electric current [17-22]. Thus, a deduction about vector nature of elementary charge can be made.

Now we shall determine electron’s rest **energy** as electromagnetic wave energy inside a particle.

Here w- is volume density of electromagnetic wave energy,

? – Pointing vector,

vφ -phase velocity of electromagnetic wave in direction of v?.

v?=2ωrsinθ

(28)

(28′)

Here is Planck’s constant.

We shall be searching electron’s magnetic moment in the form of a sum. M=M_{m}+M_{L}

M=M_{m}+M_{L}

Where M_{m}- is magnetic moment, created by volumetric current; M_{L} -magnetic moment, attributed to impulse moment, i.e. to rotation.

M_{L}=γL

Where γ-gyromagnetic ratio; L-impulse moment of electron.

Basing on Barnett effect, we are making a supposition, that the impulse moment, attributed to rotation, creates additional magnetic moment [21,23].

Being aware of the fact that electron’s impulse moment is equal from (28′) we find expression for L.

Let us calculate M_{m} as electric current magnetic moment in volume V, relating to axis z by the formula:

See for instance [3], page 111, where rz - distance to axis z,

r_{z}=rsinθ

(29)

(29′)

Or

(29′′)

Thus, we have received the system of algebraic equations for electron.

Here e - charge of electron, m- it’s mass.

Three equations contain five unknown quantities: E_{0}, a, z, p, γ Let us add this system with equations, which we shall receive from boundary conditions.

At r=a, R=a

(33)

In the exterior area, the same as and in the interior area, electric field intensity possesses only radial component. Here R - distance from electron’s center to the observation point in the exterior area, ε0- vacuum dielectric permeability [24].

Further. (34)

In the exterior area, the same as and in the interior area, magnetic field intensity possesses only meridional component.

It is obvious that

(33′)

then from (33) follows:

(33′′)

On the other hand it is known that the electric field, having passed through dielectric layer, cannot increase, therefore

(33′′′)

In other words, correlations (33′) (33′′) (33′′′) will be simultaneously executed only in one case, if

ε_{r}=ε_{0};(35)

(36)

Now under Biot-Savart’s law, we are finding magnetic field in the exterior area.

In last expression we substitute (12′′) and (27).

(37)

(38)

At r=a R=a

(39)

On the other hand, from (24′′′)

At r=a

We substitute in (39).

(40)

Thus, at r=a

(41)

Here c - velocity of light, ω=7,7634421*1020 Hz- Compton circular frequency of electron.

(42)

As it is known, atom’s radius approximately equals to 10^{-10} m, volume of atom - 4,18879*10-30 m^{3}. We found, that radius of electron equals to 1,930796*10^{-13} m, volume of electron –3,0150724*10^{-38} m^{3}. That is one electron occupies 0,7197955*10^{-8} from atom’s volume and, for example, 100 electrons (as in atoms located at the end of the periodic system) occupy 0,7197955*10^{-6} from atom’s volume [8,25].

We substitute (42) ? (39).

(43)

Let us solve the system (30), (31), (32), taking into account (42) and (43).

(30′)

(31′)

(32′)

We substitute (30′) in (32′).

P must be negative, therefore we select

We substitute (30′) in (31′)

We substitute p meaning in (31′′) and find γ

From solution of equation (31), it is visible that two components of magnetic moment of electron M_{m} ? M_{L} are directed to opposite sides and

Let us also calculate numerical value of E_{0} by formula (30′)

"Dimensions" of electron for the present are not discovered by experimental way, though precision of measuring is led to 10-18 m. Within the framework of the model considered it may be explained by the next way: electron is not hard particle with this quantity of vector E, which exist inside it, unlike from proton and neutron, quantity of vector E inside which approximately 10^{7} times as much [26-31].

For positron, the system of equations will take a somewhat different view.

Boundary conditions are the same as for electron. Hence

The system of equations (44), (45), (46) with exactness to a sign, has the same solutions, as the system (30), (31), (32).

By applying reasoning and mathematical calculations of the previous section in relation to proton, we shall receive the relevant system of equations.

Here corresponding letters mean parameters of proton.

Boundary conditions: at r=a

Here:ω=1,425486*10^{24} Hz- Compton circular frequency of proton [32,33].

Solving the system (47), (48), (49), we shall receive

From the solution of equation (48) it is visible that two components of proton’s magnetic moment M_{m} ? M_{L} have identical direction, and

Let us write down the system of equations for antiproton explained in [34].

(50)

(51)

(52)

Boundary conditions: at r=a

Hence

System of equations (50), (51), (52) with exactness to a sign has the same solutions, as system (47), (48), (49).

**System of equations for neutron**

(53)

(53′)

Along each parallel, exactly one wave length is kept within. In this case:

(54)

(54′)

(54′′)

In other words, anisotropy is taking place, ε and μ are tensor quantities.

Here and further, corresponding letters mean parameters of neutron.

Let us find rest energy of neutron.

Further. Charge of neutron is equal to zero.

.

Really,

It is obvious that

It is logical to assume that

Then

(56)

Magnetic moment for neutron will be searched as the sum:

M=M_{m} + M_{L}

Where M_{m} - magnetic moment created by volume current; M_{L} - magnetic moment, attributed to impulse moment, i.e. to rotation.

as

(57)

Now we shall write down the system of equations for neutron.

Boundary conditions: at r=a

Hence

From (54) ? (54′) follows that

and from (54) ? (54′′)that

So

Here ω=1,4274508*1024 Hz- Compton circular frequency of neutron.

Let us solve system (56)(55′)(57′)

(56′′)

We substitute (56′′) B (55′)

P must be negative, therefore we select

From (57′) we find γ

Let us write down the system of equations for antineutron.

Boundary conditions are the same, as at neutron, hence

The last system with exactness to a sign has the same solutions, as system (56)(55′)(57)

Within the framework of the model, which is considered, electron, proton and neutron represent a monochromatic electromagnetic wave of corresponding frequency spread along parallels inside the spherical area, i.e. a wave, rotating around some axis. At that along each parallel, exactly half of wave length for electron and proton and exactly one wave length for neutron, is kept within, thus this is rotating soliton. This is caused by presence of spatial dispersion and anisotropy of a strictly defined type inside the particles. In electron vector E is directed to center of particle, that correspond to negative charge, and in proton vector E is directed from center of particle, that correspond to positive charge. Thus, by natural way, all basic parameters of particles are obtained: charge, rest energy, mass, radius, magnetic moment and spin, that is confirmed by mathematical expressions, which are discovered.

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