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: 
  f1
                                 f1                     (1)
    Their magnetic moments are measured with very high accuracy too. In units of the nuclear magneton they are [3]
f2                        (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 Re  with speed f 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:
f3              (3)
Where
f4            (4)
is the relativistic factor and f
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
f1

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 Re 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 a1(t) and a2(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 ϕ1 and ϕ2 with the energy E0 .The Quantum Tunneling Transition from One State to another Leads to the Splitting of Energy Level and to the Lowering of the Sublevel on ∆
f2

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 E0 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 aB, 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 Md 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)
γ quantum takes energy equal to 3.563.10-6 erg [8, 9].
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.

References

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