SciFed Journal of Nuclear Science

#### The Electromagnetic Model of Neutron and Nature of Nuclear Forces

#### Research Article

**Received on:** October 22 2017

** Accepted on:** October 27 2017

**Published on:** November 9 2017

**Boris Vasiliev V**

*Corresponding author:
Independent Researcher

Russia

#### Abstract

The main principle of natural Sciences was formed in the middle ages. Scientists of the time realized that
all theoretical constructs that claim to be scientific must be verified and confirmed experimentally. In our time it
is impossible to find a scientist who would not agree with this principle. Nevertheless, there are some theories in
modern physics that do not satisfy to that. In nuclear physics there are the model of the quark structure of nucleons
and related concept of the strong interaction. These theories seem plausible, although experimental data do not
confirm them, because they do not allow to calculate the main parameters of nucleons and the binding energy
between them. This article presents the model that gives an ability to calculate basic properties of neutron and
describe the quantum-mechanical nature of nuclear forces.

#### FullText

**Introduction
1 **

**The Main Principle of Natural Sciences**

**1.1 The Gilbert's Principle**

The central principle of natural science was
formulated more than 400 years ago by William Gilbert
(1544-1603).

One might think that this idea, as they say, was in the air
among the educated people of the time. But formulation
of this postulate has come down to us due to W. Gilbert's
book [1].

It formulated simply: all theoretical ideas claiming to
be scientific concepts must be verified and confirmed
experimentally. Without this principle, theoretical thought
has no boundaries for the imagination.

Galileo Galilei (1564-1642) lived a little later W.Gilbert
had developed this doctrine and formulated three phase of
testing of theoretical propositions

(1) to postulate a hypothesis about the nature of the
phenomenon, which is free from logical contradictions;
(2) on the base of this postulate, using the standard
mathematical procedures, to conclude laws of the
phenomenon;

(3) by means of empirical method to ensure, that the nature
obeys these laws (or not) in reality, and to confirm (or
not) the basic hypothesis. The use of this method gives a
possibility to reject false postulates and theories.

**1.2 The Characteristic Properties of PseudoTheories**

It is wrong to think that fantastic pseudo-theory is completely devoid of experimental evidence. They are characterized by the replacement of experimental evidence. All nuclear objects have fundamental properties that can be called primary. For example, primary properties of atomic nuclei are their binding energies, for elementary particles - their masses and magnetic moments (we do not consider the electrical charge and spin, as their value can simply be determined).

Quasi-theories are not able to predict the individual
paramount properties of considered objects. They replace
the study of physical mechanisms of the formation of these primary parameters on a describing of general
characteristics of the physics of the phenomenon and on
the systematization of objects using tables.
These tables give the impression that there is an agreement
between this theory and experimental data.
Let consider some specific pseudo-theory by theoretical
physics in the twentieth century.

**2. Neutron**

**2.1 The Quark Model and Neutron**

It is assumed that the basis of modern elementary
particle physics is the quark model.

The formation of this theory seems quite natural
in the chain of sciences on the structure of matter: all
substances consist of atoms and molecules. The central
element of atom is nucleus. Nucleus consists of protons
and neutrons, which in turn are composed of quarks.

The quark model assumes that majority of
elementary particles are composed of quarks. In order
to describe all their diversity, the quarks must have a
fractional electric charge (equal to 1/3 e or 2/3 e) and other
discrete properties, referred to as ﬂavor, color, etc.

In the 60 years after the formulation of the
foundations of the quark model, many experimenters
sought to fnd particles with fractional charge. But to no
avail.

After that was coined by the confinement, i.e
property of quarks, prohibiting them in any way to express
themselves in a free state.

The confinement was introduced to reconcile
the quark theory with data of observation (or rather non observation), but at the same time it removes the quarks
from subordination of the Gilbert's postulate. As such,
the quark model claims to scientific validity without its
confirmation by the measurement data.

Really, the quark model aptly describes some
experiments on the scattering of particles at high energies,
for example, the formation of jets or a feature of the high
energy particles scattering of without their breaking.

However, that is not very strong argument.

The basic quarks of the first generation (u and d)
are introduced in such a way that their combinations could
explain the charge parameters of protons and neutrons.
Naturally, the neutron is considered at that as an elementary
particle in the sense that it consists of a different set of
quark than a proton. In the 30s of the XX-th century,
theoretical physicists have come to the conclusion that a
neutron must be an elementary particle without relying on
the measurement data, which was not at that time.

The neutron mass, its magnetic moment and the
energy of its beta-decay were precisely measured. The
quark model does not allow to calculate these parameters,
but they can be calculated in the electromagnetic model
of neutron. Suppose that the neutron is not an elementary
particle, and as well as Bohr's hydrogen atom consists of
proton and electron which rotates round proton on a very
small distance. Near proton the electron motion must be
relativistic. Consider the electromagnetic model of the
neutron [2] in more detail.

**2.2 Main Properties of Proton and Neutron**

The main physical properties of proton and neutron was scrutinized. There are measuring of their mass, charge, spin, etc. Since the measured values of the masses of the proton and neutron are:

(1)

Their magnetic moments are measured with very high
accuracy too. In units of the nuclear magneton they are [3]

(2)

**2.3 The Electromagnetic Model of Neutron**

For the first time after the discovery of the neutron,
physicists was discussing whether or not to consider it as
an elementary particle. Experimental data, which could
help to solve this problem, was not exist then. And soon
the opinion was formed that the neutron is an elementary
particle alike proton [4]. However, the fact that the
neutron is unstable and decays into proton and electron
(+ antineutrino) gives a reason to consider it as a non
elementary composite particle.

Is it possible to now on the basis of experimentally
studied properties of the neutron to conclude that it is
elementary particle or it is not?

**2.3.1 Equilibrium in the System of Relativistic Electron + Proton**

Let's consider the composite corpuscle, in which
electron with the rest mass me and charge -e is spinning on a circle of radius
R

_{e }with speed around proton.(The presence of the intrinsic magnetic moment
of the rotating particle does not matter because of the particularities of the resulting solutions (25).)

Since we initially assume that the motion of
electron can be relativistic, it is necessary to take into
account the relativistic effect of the growth of its mass:

(3)

Where

(4)

is the relativistic factor and

The rotation of heavy electron does not allow
simplistically consider proton as resting. Proton will also
move, revolving around a common center of mass .let's introduce the parameter characterizing the ratio of
mass of relativistic electron to proton mass:

(5)

From the condition of equality of pulses, it follows that and therefore the radii of the orbits of the
electron and proton can be written in the form:

(6)

The relativistic factor of electron is then equal to

(7)

**Figure 1:**Composite Particle Consisting of Proton and Heavy (relativistic) Electrons Orbiting a Common Center of Mass

The Larmor theorem To describe the characteristic
feature of the proton rotation, we can use the Larmor
theorem [5]. According to this theorem, in a reference
frame which rotates together with proton with frequency
Ω , a magnetic feld is applied to it. This magnetic feld is
determinated by the gyromagnetic ratio of the particle.

(8)

As a result of the action of this field, the magnetic
moment of the proton turns out to be oriented perpendicular
to the plane of rotation. In other words, we can say that
due to interaction with this field, the electron rotation must
occur in the plane of the proton's "equator". The energy of
interaction of proton and Larmor's field is equal to:

(9)

It should be noted that almost the same magnitude
has energy of the magnetic field created by rotation of
electron. Because this field tends to break current e-ring,
the energy of this field has the opposite sign

(10)

Because this field tends to break current e-ring, the
energy of this field has the
positive sign.

Due to the fact that the motion of electron in orbit
must be quantized, also the magnetic ﬂux penetrating
e-ring of radius R

_{e}needs to be quantized too, and we have the magnetic ﬂux in e-ring equals to quantum of magnetic ﬂux (11)

The main part of electron current is equal to Beside
it we need to consider a small addition due to the Coulomb
and magnetic effects of protons on the electronic orbit. So
energy

(12)

where is the fine structure constant. Thus, we obtain that the energy associated with magnetic
ﬂux is almost exactly compensates the spin-orbit interaction
described by the Larmor field:

(13)

The balance of forces in the proton-electron
system In a stable bound state, the Coulomb attraction
between electron and proton and the Lorentz force acting
from the proton magnetic moment on moving electron
should be differently oriented that the total energy of their
interaction was less.

In equilibrium state, these forces are compensated by the
centrifugal force:

(14)

After simple transformations we obtain the equation

(15)

Where is the Compton radius and

(16)

From these equations we obtain the solution

(17)

And

(18)

**2.3.2 Spin of Neutron**

The spin of neutron is the sum of spin of proton,
the generalized moment of momentum of the e-current
ring and the generalized moment of momentum of proton.
Moment of the generalized electron momentum can be
written as

(19)

or

(20)

The generalized moment of current ring of proton is equal to

(21)

or

(22)

The total angular momentum of current rings

(23)

Due to the fact that the expression in brackets of this
equation coincides with the left part of Equation (15), we obtain

(24)

Thus, spin of neutron is equal to spin of proton.

**2.3.3 The Magnetic Moment of Neutron**

The magnetic moment of neutron is composed of
proton magnetic moment and the magnetic moments of
currents of electron and proton.
The total magnetic moment generated by circulating
currents

(25)

If to express this moment in Bohr magnetons we get

(26)

Given the values of we have

(27)

The summation of this quantity with the magnetic
moment of the proton (Equation (2)) gives

(28)

This value is in good agreement with the measured
value of the magnetic moment of neutron (Equation (2)):

(29)

**2.3.4 The Mass of Neutron**

It is important that the measured value of the
neutron mass

(30)

At first glance it seems that this fact creates an
obstacle for the electromagnetic model of neutron with a
binding energy between proton and electron.

It should lead to the opposite inequality: the mass
of a neutron, it would seem, must be less than the combined
mass of proton and electron on energy of their connection
(there must exists a defect of mass).

For this reason, it is necessary to conduct a detailed
examination of these energies.

The electron energy To clarify this question, first
let us write the energy of electron. It consists of kinetic
energy and potential energy of interaction with the proton.
In addition we need to consider the energy of the magnetic field of a current ring, which creates a rotating electron

(31)

or

(32)

Taking into account Equation (17), we obtain

(33)

Since the total energy of electron is negative,
that indicates on the existence of a stable bound state of
electron in the field of proton.

The kinetic energy of proton The positive
contribution to neutron mass give kinetic and magnetic
energies of the proton, which carries out the movement on a circle of radius .

The kinetic energy of the proton, taking into account the
relativistic supplements

(34)

Additionally, due to its rotation in a circle with radius for proton generates a magnetic field with energy and has energy of spin-orbital interaction

(35)

It is equal to the Larmor energy

Summing up these additional contributions to the energy
of proton and electron we obtained

(36)

Thus the mass of neutron is equal to

(37)

This is in qualitative agreement with the measured
value of the mass of the neutron Equation (1)

This excess energy needs to limit the spectrum of
β-electrons produced by the neutron decay and it also
agrees satisfactorily with the measured data.

Thus the electron energy in fields of proton is
negative Equation(33). It means that the bound state exists
between proton and electron. However, the additional
contribution to neutron mass makes the energy of proton
movement. The kinetic energy of proton slightly overlaps
the negative binding energy of electron. As a result of
addition of these energies, the neutron mass becames
slightly greater than the sum of the masses of free proton
and electron what explains the existence of inequality (30).

**2.4 Discussion**

This consent of estimates and measured data
indicates that the neutron is not an elementary particle. It
should be seen as a some relativistic analog of the Bohr
hydrogen atom. With the difference: a non-relativistic
electron in the Bohr atom forms a shell by means of
Coulomb forces and in a neutron the relativistic electron is
held by the magnetic interaction [2].

This must change our approach to the problem
of nucleon-nucleon scattering. The nuclear part of an
amplitude of the nucleon-nucleon scattering should be the
same at all cases, because in fact it is always proton-proton
scattering (the only difference is the presence or absence
of the Coulomb scattering). It creates the justifcation for
hypothesis of charge independence of the nucleon-nucleon
interaction.

The above considered electromagnetic model of
neuron is the only theory that predicts the basic properties
of the neutron. For this reason, all other models (and in
particular the quark model of neutron) that cannot describe
properties of neutron, according to Gilbert's postulate can be regarded as erroneous. The measurement confirmation
for the discussed above electromagnetic model of neutron
is the most important, required and completely sufficient
argument of its credibility.

Nevertheless, it is important for the understanding
of the model to use the standard theoretical apparatus at its
construction. It should be noted that for the scientists who
are accustomed to the language of relativistic quantum
physics, the methodology used for the above estimates
does not contribute to the perception of the results at a
superficial glance. It is commonly thought that for the
reliability, a consideration of an affection of relativism
on the electron behavior in the Coulomb field should
be carried out within the Dirac theory. However that is
not necessary in the case of calculating of the magnetic
moment of the neutron and its decay energy. In this case, all
relativistic effects described by the terms with coefficients compensate each other and completely fall out. The neutron considered in our model is the quantum
object. Its radius is proportional to the Planck constant
But it can not be considered as relativistic particle, because coefficient is not included in the definition of In the particular case of the calculation of
the magnetic moment of the neutron and the energy of its
decay, it allows to find an equilibrium of the system from
the balance of forces, as it can be made in the case of non relativistic objects.

The another situation arises on the way of an
evaluation of the neutron lifetime. A correct estimation of
this time even in order of its value do not obtained at that.

**3 About Nature of Nuclear Forces**

**3.1 The One-Electron Bond between Two Protons**

Let us consider a quantum system consisting of
two protons and one electron. If protons are separated by a
large distance, this system consists of a hydrogen atom and
the proton. If the hydrogen atom is at the origin, then the
operator of energy and wave function of the ground state
have the form:

(38)

If hydrogen is at point R, then respectively

(39)

In the assumption of fixed protons, the Hamiltonian
of the total system has the form:

(40)

At that if one proton removed on infinity, then the
energy of the system is equal to the energy of the ground
state E

_{ 0}, and the wave function satisfies the stationary Schrodinger equation: (41)

We seek a zero-approximation solution in the form
of a linear combination of basis functions:

(42)

where coeffcients a

_{1}(t) and a_{2}(t) are functions of time, and the desired function satisfies to the energy-dependent Schrodinger equation (43)

where V

_{1;2 }is the Coulomb energy of the system in one of two cases. Hence, using the standard procedure of transformation, we obtain the system of equations (44)

where we have introduced the notation of the
overlap integral of the wave functions

(45)

and notations of matrix elements

(46)

Given the symmetry

(47)

after the adding and the subtracting of equations of the
system (44), we obtain the system of equations

(48)

where

(49)

As a result, we get two solutions

(50)

where

(51)

From here

(52)

And

(53)

As

(54)

with the initial conditions

(55)

and

(56)

or

(57)

we obtain the oscillating probability of placing of
electron near one or other proton:

(58)

Thus, electron jumps into degenerate system
(hydrogen + proton) with frequency from one proton to
another.

n terms of energy, the frequency corresponds to
the energy of the tunnel splitting arising due to electron
jumping.

**Figure 2:**The Schemes Representation of Potential Well with Two Symmetric States.In Ground State, Electron can be either in the Right or in the Left Hole. In the Unperturbed State, its Wave Funtionare either ϕ

_{1}and ϕ

_{2}with the energy E

_{0}.The Quantum Tunneling Transition from One State to another Leads to the Splitting of Energy Level and to the Lowering of the Sublevel on ∆

As a result, due to the electron exchange , the
mutual attraction arises between protons. It decreases their
energy on

(59)

The arising attraction between protons is a purely
quantum effect, it does not exist in classical physics.
The tunnel splitting (and the energy of mutual attraction
between protons) depends on two parameters:

(60)

where
E

_{0}is energy of the unperturbed state of the system (ie, the electron energy at its association with one of proton, when the second proton removed on infinity), and function of the mutual distance between the protons∧ .
This dependence according to Equation (54) has the form:

(61)

The graphic estimation of the exchange splitting
∆ε indicates that this effect decreases exponentially
with increasing a distance between the protons in full
compliance with the laws of the particles passing through
the tunnel barrier.

**3.2 The Molecular Hydrogen Ion**

The quantum-mechanical model of simplest
molecule - the molecular hydrogen ion - was first
formulated and solved by Walter Heitler and Fritz London
in 1927 [6,7].
At that, they calculate the Coulomb integral:

(62)

the integral of exchange

(63)

and the overlap integral

(64)

Where is the dimensionless distance between the protons.

The potential energy of hydrogen atom

(65)

and with taking into account Equation (62)-Equation (64)

(66)

At varying the function we find that the
energy of the system has a minimum at where As a result of permutations of these values
we find that in this minimum energy the mutual attraction
of protons reaches a maximum value

(67)

This result agrees with measurements of only the
order of magnitude. The measurements indicate that the
equilibrium distance between the protons in the molecular
hydrogen ion and its breaking energy on proton and
hydrogen atom is close to

The remarkable manifestation of an attraction
arising between the nuclei at electron exchange is showing
himself in the molecular ion of helium. The molecule
He
2 does not exist. But a neutral helium atom together
with a singly ionized atom can form a stable structure -
the molecular ion. The above obtained computational
evaluation is in accordance with measurement as for
both - hydrogen atom and helium atom - the radius of
s-shells is equal to a

_{B}, the distance between the nuclei in the molecular ion of helium, as in case of the hydrogen molecular ion, must be near and its breaking energy near . In order to achieve a better agreement between calculated results with measured data, researchers usually produce variation of the Schrodinger equation in the additional parameter- the charge of the electron cloud. At that, one can obtain the quite well consent of the calculations with experiment. But that is beyond the scope of our interest as we was needing the simple consideration of the effect.

**3.3 Deutron and Other Light Nuclei**

**3.3.1 Deutron**

The electromagnetic model of a neutron, discussed above, gives possibility on a new look on the mechanism of the proton-neutron interaction. According to this model a neutron is a proton surrounded by a relativistic electron cloud. Therefore a deuteron consists of the same particles as the molecular ion of hydrogen. There is a difference. In the case of a deuteron, the relativistic electron cloud has the linear dimension Equation (19). One might think that a feature occurs at such a small size of the electron cloud. When an electron jumps from one proton to another, a spatial overlap of the wave functions will not arise and therefore the overlap integral S (Eq.(65)) can be set equal to zero.

In accordance with the virial theorem and Equation (66), the potential energy of this system at the unperturbed state is

(68)

The function (Equation (61)) at S = 0 and taking into account Equation (64) obtains the form

(69)

(where is a dimensionless distance between the protons.)

When varying this expression we find its maximum value at x=1.618

After substituting these values, we fnd that at the minimum energy of the system due to exchange of relativistic electron, two protons reduce their energy on

(70)

To compare this binding energy with the measurement data, let us calculate the mass defect of the three particles forming the deuteron

(71)

where M

_{d}is mass of deuteron. This mass defect corresponds to the binding energy (72)

Using the relativistic electron mass in Equation (71) does not seem obvious. However, this is confrmed by the fact that at the fusion reaction proton and neutron to form a deuteron

(73)

Thus the quantum mechanical estimation of the bonding energy of deuteron Equation (70), as in the case of the hydrogen molecular ion, consistent with the experimentally measured value Equation 72), but their match is not very accurate.

**Figure 3:**Schematic Representation of the Structure of Light Niclei. Dotted Lines Schematically Indicate the Possibility of a Relativistic Electron Hopping between Protons

**3.3.2 Nucleus**

As can be seen from the schematic structure of this nucleus (Figure 3), its binding energy is composed by three pairwise interacting protons. Therefore it can be assumed that it equals to the tripled energy of deuteron:

(74)

The mass defect of this nucleus

(75)

Thus mass defect corresponds to the binding energy

(76)

Consent energies and can be considered as very good.

**3.3.3 Nucleus**

As can be seen from the schematic structure of this nucleus (Figure 3), its binding energy is composed by six pairwise interacting protons which are realised by two electrons. On this reason its binding energy can be considered as:

(77)

The mass defect of this nucleus

(78)

Thus mass defect corresponds to the binding energy

(79)

Consent of these energies can be considered as alright.

**3.3.4 Nucleus**

The binding energy of Li-6 can be represented by the sum of binding energy of He-4 and deuteron. The last placed on next shell and has a weak bounding with He - 4:

(80)

The mass defect of this nucleus

(81)

and corresponding binding energy

(82)

That really confirms the weak link between the protons in different shells. It should be noted that the situation with the other light nuclei are not so simple. The nucleus consists of three protons and two communicating electrons between them. Jumps of two electrons in this system should obey to the Pauli exclusion principle. Apparently this is the reason that the binding energy of tritium is not very much greater than the binding energy of He-3. Nuclear binding energy of Li-7 can be represented as

But it is quite a rough estimate. At that the binding energy of unstable nucleus Be-8 very precisely equal to twice binding energy of He-4

**4 Conclusion**

The good agreement between the calculated binding energy of some light nuclei with measured data suggests that nuclear forces (at least in the case of these nuclei) have the above-described exchange character. These forces arise as a result of a purely quantum effect of exchange relativistic electrons.

For the first time the attention on the possibility of explaining the nuclear forces based on the effect of electron exchange apparently drew I.E.Tamm [10] back in the 30s of the last century. However, later the model of the pi-meson and gluon exchange becomes the dominant in nuclear physics. The reason for that is clear. To explain the magnitude and range of the nuclear forces need particle with a small wavelength. Non-relativistic electrons does not ft it. However, on the other hand, the model pimeson or gluon exchange was not productive: it gives not possibility to calculate the binding energy of even light nuclei.

Therefore, the simple assessment of the binding energy given above and consistent with measurements is the clear proof that the so-called strong interaction (in the case of light nuclei) is a manifestation of the quantum mechanical effect of attraction between protons produced by the relativistic electron exchange.

The Gilbert's postulate is the main tool to distinguish between theoretical models that reliably describe the object under study, from speculative quasitheories that seek to do the same, but use the wrong approach.

In physics of the 20th century, some of these farfetched theory became commonly accepted [11].

The reason for this is probably that a theory can not be constructed on on arbitrary reasonings. Since the formation of a table representing the quark structure of elementary particles illustrates the ability to some systematization but it is not proof of the correctness of their description with fractional charged quarks.

The main attribute of a quasi-theories is that they can not give an explanation of the individual primary characteristics of the objects and try to explain the general characteristics of the phenomenon as a whole.

The fact that the electromagnetic model allows us to predict most important characteristics of neutron forces us recognize that an use of presentation of structure of elementary particles based on quarks with fractional charge appears to be erroneous.

The force of attraction between the protons arising
at the relativistic electron exchange allow us to explain the
mechanism of occurrence of nuclear forces (in the case of
light nuclei). This gives possibility do not use gluons for it
and to simplify this theory.

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