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Articles and Publication  Physics Theoretical physics SOLITONIC MODEL OF ELECTRON, PROTON AND NEUTRON
SOLITONIC MODEL
OF ELECTRON, PROTON AND NEUTRON
© P. Sladkov
Contact to
the author: sladkovpi@mail.ru
Introduction
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 and others.
Abstract
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 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.
1. Rotating monochromatic electromagnetic wave.
Let us write down Maxwell’s
equations in spherical coordinates supposing that:
- there are no losses;
2) only  , , 
are not equal to zero.
(1)
(2)
  
(3)
(4)
 ;
(5)
(6)
Here 
- spherical coordinates of the observation point; 
è 
- components of the electromagnetic field, 
- density of electric current, 
- volume charge density; 
- circular frequency of field alteration
- imaginary unit dielectric
permittivity
magnetic permeability.
Fig.1
Substituting the expression for 
from (2) in (4), we obtain:
(7)
This is Helmholtz homogeneous equation. Let us
designate
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.
Let us consider a wave, moving in positive
direction 
(9)
Here
wave phase;
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. Dispersion equations will be obtained
below, apart from the already found expression 
From expression (2), taking into account (7″
) and (9), we have:
. 
For actual amplitudes:
 ;
(10)
 .
Here
means characteristic impedance.
The last expressions describe an
electromagnetic wave, rotating around axis Z in positive direction  .Conditions
of self-consistency:
 - 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,
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.
 ;
we are substituting in  :
(12)
 .
From 
we are substituting in  .
 .
Taking into account 
and
 .
Then
 ;
(13)
 .
Function 
is onevalued in angles interval  .
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 
and .
Further from (3): as 
(14)
From equation (6)
follows
To receive field dependence from  ,
let us find solution of three-dimensional Helmholtz equation in spherical
coordinates.
(15)
 does
not depend from  ,
look (14), therefore three-dimensional Helmholtz equation transfers into
two-dimensional one.
Let us suppose that
 
now
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 by  ,
and the second - by  .
We receive
(18), (19)
Equations (16) and (18) are equivalent to
equations (7) è  ,
which were received earlier from Maxwell’s equations, and
The solution of equation (18) was found earlier,
look (13).
(20)
Let us copy (19) as:
where
centrally symmetric Helmholtz equation. Let us suggest,
where 
phase velocity of electromagnetic wave in radial direction. As in the central
symmetric equation angular dependence is absent, it is logical to assume that
at 
i.e.
(21)
Instead of  ,
we are having
This is Euler equation, it has the solution
(22)
Let us converse expression (22).
(22′ )
Here 
- value of radius  ,
at which the rotating monochromatic electromagnetic wave ceases to exist, and 
hence
(22″ )
In view of this,
Let us designate  now
Thus, for 
we are having
(23)
At 
Really
So that at alteration of
within the interval from 0 to would
not change its sign, observance of the following requirement is necessary: 
At 
At 
2. System of
equations for electron.
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 
Here 
is electron radius, 
- amplitude of electric field intensity at 
- characteristic impedance inside the electron, 
- unknown coefficient and  .
At that the internal electron medium possesses
frequent and spatial dispersion, as well as anisotropy. Dispersion equations
have the following appearance.
(24)
Here 
- 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 
and 
for searching the formulas of dielectric and magnetic permeability, as well as
wave numbers of corresponding directions;  -
characteristic impedances inside the electron; 
è 
were found before, see 
In view of  ,
let us write down expressions for 
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 
we are having a special point:
Despite of this, all basic electron’s
parameters - charge 
rest energy 
magnetic moment  -
expressed through integrals by volume from the functions specified above, prove
to be finite quantities. Look further.
From (5), we find volume charge density inside electron 
 .
(25)
Integrating 
on electron’s volume, we shall receive this expression for its charge  .
(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
As we can see, expressions (26) è  are
equivalent to each other.
From (1), we obtain expression for current
density 
 .
(27)
From expressions (25), (27) it is visible
that in the interval of change of
from 0 to 
and 
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. 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. 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 
- is volume density of electromagnetic wave energy,
 where
Ï – Pointing vector,
- phase velocity of electromagnetic wave in direction of
.
(28)
here 
is Planck’s constant.
We shall be searching electron’s magnetic
moment in the form of a sum.
where  is
magnetic moment, created by volumetric current;  magnetic
moment, attributed to impulse moment, i.e. to rotation.
where  gyromagnetic
ratio; 
impulse moment of electron.
Basing on Barnett effect, we are making a
supposition, that the impulse moment, attributed to rotation, creates additional
magnetic moment.
Being aware of the fact that electron’s
impulse moment is equal  ,
from 
we find expression for L.
or 
Let us calculate 
as electric current magnetic moment in volume V, relating to axis z by the
formula:
See for instance  ,
page 111, where 
- distance to axis z,
(29)
Or
Thus, we have received the system of algebraic
equations for electron.
Here 
- charge of electron, 
- its mass.
Three equations contain five unknown
quantities: 
Let us add this system with equations, which we shall receive from boundary
conditions.
At 
(33)
In the exterior area, the same as and in
the interior area, electric field intensity possesses only radial component.
Here 
- distance from electron’s center to the observation point in the exterior
area, 
- vacuum dielectric permeability.
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
then from (33) follows:
On the other hand it is known that the electric
field, having passed through dielectric layer, cannot increase, therefore
In other words, correlations 
will be simultaneously executed only in one case, if
 ;
(35)
(36)
Now under Biot-Savart’s law, we are finding
magnetic field in the exterior area.
In last expression we substitute 
and (27).
(37)
(38)
At 
(39)
On the other hand, from
At 
We substitute in (39).
(40)
Thus, at 
(41)
Here 
- velocity of light,  -
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 m3. We found,
that radius of electron equals to 1,930796*10-13 m, volume of
electron –3,0150724*10-38 m3. That is one electron
occupies 
from atom’s volume and, for example, 100 electrons (as in atoms located at the
end of the periodic system) occupy 
from atom’s volume.
We substitute (42) â (39).
(43)
Let us solve the system (30), (31), (32), taking
into account (42) and (43).
We substitute (30′) in (32′).
must be negative, therefore we select
We substitute 
in 
We substitute meaning
in 
and find 
From solution of equation (31), it is
visible that two components of magnetic moment of electron  è
are directed to opposite sides and 
Let us also calculate numerical value of
by formula 
"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 107 times as much. Look below.
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).
3. System of equations for proton.
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 
hence
 m.
Here  -
Compton circular frequency of proton.
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
è
have identical direction, and
Let us write down the system of equations for
antiproton.
Boundary conditions: at 
hence
System of equations (50), (51), (52) with
exactness to a sign has the same solutions, as system (47), (48), (49).
4. System of equations
for neutron.
(53)
Along each parallel, exactly one wave length is
kept within. In this case:
(54)
In other words, anisotropy is taking place,
and 
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:
where  -
magnetic moment created by volume current; 
- 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
hence
From (54) è  follows
that
and from (54) è  that
So
Here  -
Compton circular frequency of neutron.
Let us solve system 
We substitute 
â 
must be negative, therefore we select
From 
we find 
Let us write down the system of equations for
antineutron.
Boundary condidions are the same, as at neutron,
hence
The last system with exactness to a sign has the
same solutions, as system
Conclusion
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 dispersion and anisotropy of a strictly defined type
inside the particles. In electron vector E is directed to centre of particle,
that correspond to negative charge, and in proton vector E is directed from
centre 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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_____________________
P. S.
Further researchs on the basis of results, which were obtained, intend solution
of following tasks:
1. Elaboration of physic-mathematical model of
photon and neutrino structure.
2. Elaboration of physic-mathematical model of
atomic nuclei structure for all chemical elements.
It is my firm belief that solution of this tasks
will assist to achieve great leap in following fields: discovery new ways of
making energy; elaboration perfectly new devices for its production; nuclear
power engineering; nanotechnology, high-powerful lasers and others.
Publishing date: December 5, 2010
Source: SciTecLibrary.ru
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