Two types of proton-electron atoms in a vacuum and an extremely strong magnetic field
R. Fedaruk Institute of Physics, University of Szczecin, 70-451, Szczecin, Poland
The Rutherford planetary model of a proton-electron atom is modified. Besides the Coulomb interaction of the point elec- tron with the proton, its strong Coulomb interaction with the physical vacuum as well as the magnetic interaction between moving charges are taken into account. The vacuum interaction leads to the motion of the electron with the velocity of light c in the circle with the radius being equal to rthe so-called classical electron radius re. Therefore,r the velocity of the electron consists of two components: the velocity υ of the mechanical motion and the velocity c of the photon-like mo- r v tion. We postulate that υ ⊥ c , and υ < c. Hence, the electron inside the atom moves with the resulting faster-than-light velocity. The existence of two types of proton-electron atoms, the hydrogen atom and the neutron, is interpreted by the different motion and interaction of particles at large ( r >> re) and short ( r < re) distances. In the first atom, the effect of photon-like motion is small, and the electron moves around the proton with the velocity υ << c in an orbit of the radius >> r re . In the second atom, the photon-like motion is the determining factor, and the electron moves around the proton < with the faster-than-light velocity in an orbit of the radius r re . The calculated ground-state properties of the free hy- drogen atom and the free neutron are in good agreement with the experimental data. The properties of these atoms in ex- tremely strong magnetic fields ( B > 10 8 T) are discussed.
PACS numbers: 32.10.-f; 14.60.Cd; 03.50.Kk; 14.20.Dh
1. Introduction Difficulties in describing nucleons as the strongly bound three-quark systems are usually related to computational The hydrogen atom is the simplest two-body bound problems. system consisting of a proton and an electron. The hy- Note that two alternative approaches, constructing pothesis of the second allotropic type of proton-electron nucleons either from the stable particles observed in the atom, the strongly bound proton-electron atom called nucleon decay or from quarks nonexistent as free parti- neutron, was suggested by Rutherford in 1920 [1]. The cles, confront with the same problem. That problem is the existence of the neutron was confirmed in 1932 by strong interaction of elementary constituents. Unlike the Chadwick, however the proton-electron model of the quark model, the strongly bound proton-electron model neutron was left because of its contradictions with quan- was definitely rejected. Nevertheless, some new attempts tum mechanics [2]. (a) According to quantum mechanics, to improve of Rutherford’s model of neutron were made the total angular momentum of the proton-electron neu- [6–8]. In these attempts the problem of strong interaction tron must be 0, while the neutron reveals as a particle between the electron and the proton was concealed by with spin 1/2. (b) The binding energy of the proton- assuming the unusual structure of these particles. Re- electron system must be negative, but the measured mass cently, claims that so-called hydrino states exist have of the neutron is 0.782 MeV/c 2 larger than the sum of the been discussed [9–12]. In such states the electron is masses of the proton and the electron. (c) Via the use of strongly bound and is unusually close to the proton. It is the spin magnetic moments of the proton and the electron obvious that, if strongly bound proton-electron states it is impossible to obtain the neutron magnetic moment. really exist as free atoms, their description could not be (d) There is no possibility to reach the small neutron done in the frame of conventional approach. radius (of about 10 –15 m) since the smallest radius of the On the other hand, quantum mechanics predicts proton-electron atom predicted by quantum mechanics strongly bound states of the hydrogen atom in strong and the Schrödinger equation is of 10 –10 m. Although magnetic fields ( B >> 10 5 T). Such states have been thor- these contradictions are based on the theory describing oughly studied theoretically using the Schrödinger and only weakly bound atomic systems, they are generally Dirac equations (see, e.g. Refs. [13–18]). In this case the accepted arguments against the possibility of existence of Coulomb force is treated as a small perturbation com- a strongly bound proton-electron atom. pared to the magnetic force. Because of the strong mag- Today, there are no experimental data related to the netic confinement of the electron, the hydrogen atom hydrogen atom which could give trouble to quantum attains a cylindrical structure elongated along the mag- electrodynamics (QED) [3]. At the same time, the preci- netic field and a much greater binding energy compared sion of the measured proton and neutron properties, in to the zero-field case. For example, in a field of about 1.2 particular their magnetic moments and the mass differ- × 10 10 T, the hydrogen atom becomes a long cylinder 200 ence, is appreciably higher than the precision of their times narrower than its normal diameter [17]. The highest quantum chromodynamics (QCD) calculations [4,5]. magnetic fields (of about 10 2 T) produced in laboratories
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are too small for the verification of theoretical results. electron has no any additional motion, its average posi- The appropriate magnetic fields (up to 10 13 T) can exist tion does not change and it is observed as the resting on the surfaces of neutron stars [18]. However, adequate electron. experimental data confirmed the predicted properties of It is obvious that in our model the radius re of the cir- the hydrogen atom in such strong-field conditions were cular motion of a point electron is not related to its struc- not found until now. ture. Historically, the term “classical electron radius” for In the present paper, we discuss an opportunity of ex- the value re was introduced in classical electrodynamics istence of strongly bound proton-electron states based on of an extended electron using the idea of its electromag- the modification of the Rutherford planetary model. netic momentum and its electromagnetic mass [20–22]. Apart from the Coulomb interaction between the electron The photon-like motion of an electron differs from the and the proton, an additional strong Coulomb interaction Zitterbewegung. According to the Dirac equation [23], a of charged particles with the physical vacuum as well as point electron, besides its slow motion, also possesses the the magnetic interaction between moving charges should oscillatory motion with the velocity of light (the Zitter- be taken into account. The atomic model presented pre- bewegung [24]). This oscillatory motion occurs with the dicts existence of two allotropic types of proton-electron ω = 2 doubled Compton frequency C e cm / h and the bound states, the hydrogen atom and the neutron. The amplitude of the order of the Compton wavelength ground-state properties of the hydrogen atom and the r = h / cm , where h is the reduced Planck constant. neutron in a vacuum and in extremely strong magnetic C e fields are analyzed. It is important to point out that, in quantum theory, the radius of the Zitterbewegung is determined by the quantization of mechanical properties of an electron by 2. Allotropy of a proton-electron atom means of the Planck constant. In our model, the radius re of the photon-like motion of an electron is due to the In an attempt to search the origin of possible exis- quantization of an electrical charge and is much smaller tence of two types of proton-electron atoms we accent the than the Compton wavelength. fact of duality of radiation and matter. It was experimen- We now consider the motion of two point charged tally proved that photons having energies equal or larger particles in the proton-electron atom. In general, the elec- 2 tromagnetic forces between two moving charged particles than 2 ecm can create electron-positron pairs or these satisfy neither the condition that the forces are along the pairs could annihilate into photons. Here, me is the elec- line joining them nor that these forces are equal in magni- tron rest mass, and с is the speed of light in vacuum. The tude and oppositely directed [20–22]. There is a problem duality of radiation and matter testifies that an electron in the relation between Newton’s third law and electro- and a photon are different manifestations of the same magnetism. However, the ground state of the proton- entity. Therefore, we postulate that, like a photon, a point electron atom represents the extraordinary case of this electron possesses not only the rest energy E = cm 2 , relation. Due to the atom’s stability, the atom is an isolate r e r and closed system, and the emission of any electromag- = = but also a corresponding momentum p E / c ecm netic radiation by the atom is absent. Therefore, we sup- resulting from the electron motion with the velocity of pose that there is the full validity of Newton’s third law light (photon-like motion). for the interaction of moving charges inside the atom. As Let us assume that the photon-like motion of the elec- a result, at any instant, the forces of action and reaction tron is caused by its strong interaction with the environ- between the electron and the proton are equal in magni- ment (physical vacuum). This interaction can be ex- tude and opposite in direction, and Newton’s second law pressed by means of the Coulomb force between elemen- can be used to describe the motion of particles. Then, the tary charges. The law of photon-like motion of the elec- centripetal force is determined by instantaneous values of tron can be written in the following form: velocities of the particles and is balanced by the Coulomb and Lorentz forces acting along the line joining the parti- 2 2 cles. Note that, owing to the exact validity of Newton’s e cm e = , (1) third law, the process of propagation of interaction be- r πε 2 e 4 0 re tween the particles inside the nonradiative atom can be excluded from the consideration. Consequently, Newto- ε where e is the elementary charge, 0 is the electric con- nian dynamics can be used in describing motions of the stant. According to equation (1), the photon-like motion particles inside the atom and relativistic effects could be of the electron occurs in the circle with the radius being taken into account only in terms of the Lorentz force equal to the so-called classical electron radius instead of special relativity. In the atom, apart from the Coulomb and Lorentz in- r = e 2 4πε m c 2 = 2.817940325(28) × 10 –15 m [19]. e 0 e teraction between the electron and the proton, the parti- Due to its strong interaction with physical vacuum, an cles strongly interact with the physical vacuum, too. electron is never at rest and moves at any time with the Therefore, the electron performs two types of motions velocity of light. Since the angular frequency of the mo- and its instantaneousr velocity consists of two compo- tion is so high and its radius is so small, the photon-like nents: the velocity υ of the mechanical motion, and the motion of an electron cannot be directly measured. If the
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r velocity c of the photon-like motion. We postulate that atom in the nonradiative state, Newtonian dynamics can r v these velocities are perpendicular to each other υ ⊥ c , be used. and υ < c. Hence, inside an atom, the mechanical motion In the hydrogen atom, neglecting the photon-like mo- of the electron inevitably results in a faster-than-light tion, only the orbital motion of the electron with the ve- υ с locity around the proton will be considered. This mo- velocity which does not exceed 2 . The same as the tion is balanced by the force of the electromagnetic inter- photon-like motion, the faster-than-light motion of the action between the electrical charges. The electromag- electron inside the atom cannot be directly measured, and netic force has two components. The first component, the it is necessary to recognize its manifestation in the world r of conventional (much less than c) velocities. This mo- Coulomb force FC , describes the electrostatic interac- tion leads to a modification of the classical planetary tion. Since each moving charge produces a magnetic model and reveals in a different way in weakly and field, therer exists also the second component, the Lorentz strongly bound atoms. force FL , describing a magnetic interaction of the In a weakly bound (hydrogen) atom, the distance be- r charges. The Lorentz force F is along the line connect- tween the electron and the proton is large, r >> r , and L e ing the charges and coincides in the direction with the the quick motion of the electron with the velocity c in the r Coulomb force F . The magnetic field, created by an circle of the radius re is superimposed on its slow motion C around the proton with the velocity υ << c in an orbit of orbital motion of the proton in the position of the elec- the radius r. Therefore, the electron moves around the tron, is given by Biot-Savarte’s law: proton on a helical path. The velocity of the electron changes continuously its direction relative to the radius of µ 2 π i µ 2 π ef 1 e υ B = 0 = 0 = , (2) the orbit. In result, only the instantaneous velocity of the π π πε 2 2 4 rH 4 rH 4 0 r c electron is faster-than-light, but its orbital velocity is H much less than c. Because of the averaging over the or- µ υ= ( π ) bital period, the effect of photon-like motion on the prop- where 0 is the magnetic constant, i = ef , f 2/ rH , erties of the hydrogen atom is small. rH is the distance between the electron and the proton. The faster-than-light velocity of the electron mani- Using formula (2), the Lorentz force can be written as fests quite differently when the distance between the electron and the proton is small, r < r . In such strongly r r r e 2 υ 2 e F = e ×υ B = . (3) L πε 2 2 bound (neutron) atom, the photon-like motion of the 4 0 r H c electron is the determining factor, and the electron circles the proton r with a r faster-than-light velocity. Since both Newton’s second law for the motion of the electron in υ velocities and c are perpendicular to the orbital ra- the nonradiative hydrogen atom is dius, its value does not change. A path of the electron lies on the surface of a sphere, but it is not flat. υ2 2 2 me e υ Hence, the strong Coulomb electron-vacuum interac- = F + F = 1+ . (4) C L πε 2 2 tion gives an additional substantial contribution to the rH 4 0 rH c electromagnetic interaction between the proton and the electron, and results in the existence of two allotropic Using equation (4), the radius of the hydrogen atom forms of proton-electron atoms. can be written as
= ( β+ 2 ) β 2 3. Features of proton-electron atoms rH re 1 , (5)
υ=β с The simultaneous existence of the mechanical and where re is the classical electron radius, / . photon-like motions of the electron gives rise to difficul- In the neutron, the distance between the electron and ties in the precise analytical description of even the sim- the proton is smaller than re. Due to the photon-like mo- plest, proton-electron atom. Therefore, only approximate tion, the electron moves around the proton with the fast- > < analysis of two cases when r re and r re will be er-than-light velocity done. In the first case, the average effect of the photon- 2 2 2 like motion of the electron is small and the slow motion υ+ = c υ+ = c 1 β+ . (6) of the electron around the proton with the velocity υ can be used as the zeroth approximation. In the second case, the photon-like motion of the electron must be taken into The centripetal force, acting on the electron, is account from the outset. 2 2 Let us consider an electron and a proton are point par- m υ + m c F = e = e 1 β+ 2 , (7) ticles. To simplify a description, the mass of the proton is r+ rn assumed to be infinite. As it was argued above, for the
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2 2 2 2 where r+ = r 1 β+ , r is the distance between the ϕ = E +E = cm 1 1 β− −(1 β+ ) . (12) n n ∞ 1 2 e proton and the electron in the neutron.
We assume that the Lorentz force, acting on the elec- υ Since the electron-vacuum interaction has been ne- tron, is determined only by the velocity of the me- glected in the law of motion (Eq. 4), the energies of me- chanical motion of the electron. However, the electro- chanical and photon-like motions of the electron in the magnetic interaction between the electron and the proton hydrogen atom could be treated separately. In the neu- > is different at r re , when the electron moves around the tron, the electron-vacuum interaction is the determining < proton with the slower-than-light velocity, and at r re , factor, and the mechanical and photon-like motions of the when the electron moves around the proton with the electron should be described simultaneously (Eq. 8). faster-than-light velocity. In the second case, the mag- Consequently, the potential and kinetic energies could netic field created by the relative motion of the proton not be represented for the electron-proton and electron- has to change its sign. Thus, the Lorentz force is de- vacuum interactions separately. Moreover, in order to scribed as usual by formula (3), but its direction is oppo- obtain the measured energies, the energies of the faster- site to the Coulomb force. Then, the law of the motion of than-light electron should be normalized by the energies the electron in the neutron can be written as: of the mechanical slower-than-light motion. Using equation (8), the potential energy of the faster- 2 than-light electron can be written as cm e 2 e 1 β+ 2 = F − F = ()1 β− 2 . (8) C L πε 2 rn 4 0 rn 2 −= e ()β− 2 −= 2 β+ 2 U 2 1 m e c 1 . (13) 4 πε r From equation (8) we obtain the radius of the neutron: 0 n
This energy includes the electron-proton and electron- = ( β− 2 ) β+ 2 rn re 1 1 . (9) vacuum interactions. We have to modify the potential − 2 energy of the electron-vacuum interaction, m e c , Binding energy . Forces, acting inside the atom, cannot taking into account the energy U and normalizing this be measured directly. Only the change of the energy of 2 energy by the total kinetic energy E of the slower-than- the atom or its total energy can be determined. The ki- 1 light motion given by equation (10). In this way, we netic and rest energies of particles can be described by obtain the measured part of the kinetic energy of the special relativity but the potential energy is not included faster-than-light electron: in this theory. Thus, it is necessary to take into account the potential energy by a different way. Additionally, the −= 2 () = 2 β− 4 Coulomb electron-vacuum interaction has to be included. E 3 m e c U 2 E1 m e c 1 . (14) In the hydrogen atom, the kinetic energy of the elec- tron consists of the energy of its mechanical motion, We also have to modify the kinetic energy of the pho- 2 β− 2 − 2 2 m e c 1 m e c , and the energy of its photon- ton-like motion, m e c , normalizing this energy by the 2 total potential energy E2 given by equation (11). Then the like motion, m e c . The total kinetic energy of the elec- measured potential energy of the faster-than-light elec- tron is tron can be written as follows:
E = m c 2 1 β− 2 . (10) = 2 ( 2 ) −= 2 ( β+ 2 ) 1 e E 4 m e c m e c E 2 m e c 1 . (15)
The electromagnetic force given by the right part of The binding energy of the neutron is given by equation (4) is along the line connecting the charges, and it is a central force. The velocity of the electron is per- pendicular to the line connecting the charges and is con- Φ = + = 2 β− 4 − ( β+ 2 ) ∞ E3 E4 ecm 1 11 . (16) stant. Therefore, using equation (4) the potential energy of the electron can be expressed as The binding energy of the neutron does not include U −= e 2 (1 β+ 2 ) 4πε r −= m υ2 . Similarly, the po- 1 0 H e the kinetic energy of the mechanical motion of the elec- tential energy of the electron-vacuum interaction is equal tron. Such energy can be described separately and is − 2 to m e c . The total potential energy of the electron is E = cm 2 1 1 β− 2 −1 . (17) m∞ e E −= m υ2 − cm 2 −= cm 2 (1 β+ 2 ). (11) 2 e e e
It gives the significant contribution to the total energy of The binding energy of the particles in the hydrogen the neutron. atom is given by
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Correction due to the finite proton mass. The distance Magnetic and angular moments of atoms. The mag- between the electron and the proton in the atom is netic moment of the electron at its orbital circular motion is given by ξ= ξ+ r e p , (18) µ −= υ * ∞ re 2 , (27) where ξe and ξp are the distances that the electron and the proton are from the center of mass, respectively. Accord- where r * = r 1 β− 2 for the hydrogen atom and ing to the definition of the center of mass, H * = = β+ 2 r r+ rn 1 for the neutron. m ξ = m ξ . (19) ee pp Similarly, the orbital angular momentum of the elec- tron is From equations (18), (19) we obtain: = υ * L∞ me r . (28) m m ξ = p r , ξ = e r . (20) e + p + m p m e mp me A correction due to the finite proton mass should be done both for the magnetic and angular moments. The The linear velocities of the proton and the electron correction for the angular momentum results in the re- are: place of me by the reduced mass. The masses of a proton and an electron have essentially different values, but the m m same signs. On the other hand, their electric charges have υ = e υ = e υ . (21) p e + the same absolute values, but the different signs. There- m p m p m e fore, instantaneous positions of the center of mass and the center of charge do not coincide. In order to correct the Equations (12), (16) and (17) corrected for the finite position of the center of charge, it is necessary to change proton mass can be written as: the position of the proton to the new position being sym- metrical relative to the center of mass. Then, the distance = ξ′ − ξ′ ϕ=ϕ − 2 [ β− 2 − ( β+ 2 )] between the electron and the proton is r e p , ∞ m p c 1 1 p 1 p , (22) ξ ′ = − ξ′ = − where e m p r /( m p m e ) , p e rm /( m p me ) . Φ=Φ − m c 2 [ 1 β− 4 − (11 β+ 2 )], (23) ∞ p p p The magnetic and angular moments of the hydrogen µ µ E = cm 2 (1 1 β− 2 −1)− m c 2 (1 1 β− 2 −1), (24) atom ( LH, H) and the neutron ( Ln, n) with the correction m e p p due to the finite proton mass can be written as
β υ= where p p / c . ( β+ 2 ) m + m µ −= ecr e 1 p e Mass of atoms . The rest energy of the hydrogen atom H , (29) β β− 2 m − m is defined as the sum of the rest energies of its constituent 2 1 p e 2 + particles and the binding energy: ecr β(1 β− ) m p me µ −= e , (30) n − 2 m p me m c 2 = m c 2 + m c 2 ϕ+ , (25) H e p ( β+ 2 ) m = crm ee 1 p LH , (31) 2 m + m where m H is the mass of the atom. For bound states the β 1 β− p e binding energy is negative, and mH is smaller than the m p sum of the masses of the proton and the electron, giving L = crm β()1 β− 2 . (32) n ee + rise to the mass defect. m p me The rest energy of the neutron consists of, in addition to the rest energies of the proton and the electron, the kinetic energy of the mechanical motion Em and the 4. Discussion binding energy Φ : Our model describes the nonradiative states of the hy- m c 2 = m c 2 + m c 2 + E + Φ , (26) drogen atom and the neutron. The velocity of the electron n e p m specifies these states. The dependencies of radii of the atoms on the mechanical velocity υ of the electron are where m n is the neutron mass. Only the binding energy shown in Fig. 1. The radius r of the hydrogen atom has negative values and ensures stable states of the neu- H increases infinitely at υ → 0, but it tends to the value 2r tron. The energy E is positive, and much larger than the e m at υ → c. On the other hand, the radius r of the neutron binding energy. Therefore, unlike the mass defect in the n at a small velocity is limited by the value r and tends to hydrogen atom, there is the mass “excess”. e
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zero at υ → c. Note that these atoms never have the same as well as the atomic radius and the Coulomb force, but radius. the Lorenz force could never exceed the Coulomb force.
10 4 1,0
0,8 10 3
0,6 E m 10 2 0,4
rH 2 c
1 e e 10 0,2 m / r/r E
0,0 10 0 ϕ -0,2 Φ
10 -1 rn -0,4
0,0 0,2 0,4 0,6 0,8 1,0 10 -2 0,0 0,2 0,4 0,6 0,8 1,0 υ/c υ/c Fig. 2. Binding energies of the hydrogen atom ( ϕ) and the Fig. 1. Radii of the hydrogen atom ( rH) and the neutron ( rn) vs. neutron ( Φ) vs. the velocity of the electron. The dash line the velocity of the electron given by formula (5) and (9). shows the kinetic energy of the mechanical motion of the electron in the neutron.
Fig. 2 shows the binding energies of the atoms as Comparison with experimental data . Measured values functions of the velocity υ of the electron. The dependen- of the binding energy, the mass or the magnetic moment cies demonstrate that the binding energy of the hydrogen can be used to calculate the ground-state properties of the υ ≤ atom is negative at 0.8 c , whereas the binding energy free hydrogen atom and the free neutron. As an illustra- υ ≥ of the neutron is negative at 0.9 c . So, these atoms tion, the results of computations, which use measured have stable states in a different range of velocity values. values of the magnetic moments, are presented. In the neutron, the kinetic energy of the mechanical mo- The measured ratio of the magnetic moment of the Φ υ ≥ tion Em∞ is much larger than the at 0.9 c , that µe electron in the hydrogen atom H to that of the free results in the mass “excess”. electron (µ = −1.00115965218111(74) µ [19]) is equal The orbital velocity of the ground-state electron in the e B × -6 µe free atom is determined by the origin of this atom. An to 1 – 17.709(13) 10 [3]. Consequently, H = – external magnetic field alters the orbital velocity of the 1.001141943(13) µB, where µB is the Bohr magneton. electron in the free atom, thus transferring the atom into Using this value and equation (29), we obtain βH = its new nonradiative state. The velocity of the electron 0.00729755563(2). The radius of the hydrogen atom increases or decreases depending on the orientation of the –10 given by equation (5) is rH = 0.52917594(3) × 10 m. electron orbital magnetic moment relative to the magnetic According to equation (22), at β = βH, the binding energy field. In strong external magnetic fields, when the change equals – 13.5985038(8) eV and is close to the experimen- of the orbital velocity of the electron cannot be treated as tal value – 13.59844 eV [25]. The difference of the ener- a perturbation, it is difficult to describe quantitatively the gies is comparable with the Lamb shift 8172840(22) kHz effect of external magnetic field. In this case the quantum (3.38 × 10 –5 eV) [26]. model supposes that the Coulomb force can be treated as Using the experimental value of the neutron magnetic a small perturbation compared to the magnetic force [13– moment [19] µn = – 1.9130427(5) µN, ( µN is the nuclear 18]. As a result, the atomic structure is changed from magneton), for the neutron, we found the following val- spherical to cylindrical in extremely strong magnetic ues: β = 0.91914955(3), r = 0.321911718(8) × 10 –15 m, fields that are typical for neutron stars. In our model, a n n Φ = – 3826.22(3) eV, ( m – m )c2 = 1.2933083(3) MeV. magnetic field does not change the symmetry of the n n p The experimental neutron-proton mass difference is (m – atom. This field alters the orbital velocity of the electron n 2 2 mp)c = 1.2933317(5) MeV [19]. The calculated value rn
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= 0.103627 fm 2 correlates with the root-mean-square numerous attempts were made. Moreover, as it is known, 2 2 projections of spin angular momenta of all particles are charge radius of a neutron rn = – 0.1161(22) fm ob- multiple of h /2 and do not depend on any of their prop- tained from elastic electron-deuteron scattering using erties (mass, charge, structure, magnetic moment, etc.). β− 2 At the same time, masses and magnetic moments have no QCD [19]. The value rn 1/ n = 0.8172379(3) fm correlates with the magnetic root-mean-square radius of multiplicity, i.e. their quanta are absent. Such strong the neutron 0.873(15) fm [27]. difference between the mechanical and magnetic proper- The calculated physical quantities characterizing the ties has no physical interpretation. hydrogen atom and the neutron are in good agreement Electric and magnetic fields inside atoms. We shall with the experimental data. However, in more precise now discuss the relation between atomic properties and description, some fine effects should be taken into con- extremely strong electromagnetisms. We found that, in sideration. In our model, only the static electron-vacuum hydrogen atom, the orbital motion of the electron is rela- interaction was taken into account. Besides, the electron tively slow (about 137 times slower than the light), the interacts with vacuum fluctuations. This interaction with magnetic field BH is of 12.5 T and its effect is small in comparison with the effect of the electrostatic field ( EH = virtual photons and electron-positron pairs has a stochas- 11 5.14 × 10 V/m). The Lorentz force FL is about of 0.53 × tic nature and is described by QED. The static and fluctu- –4 10 FC. In a neutron, the electron moves in extremely ating vacuum interactions exist simultaneously, result in 22 strong electric ( En = 1.39 × 10 V/m) and magnetic ( Bn = completely different physical effects and require two 13 different (deterministic or probabilistic) descriptions. In 4.26 × 10 T) fields exceeding appreciably the QED 9 × 18 particular, vacuum fluctuations cause “the trembling” of critical values ( Bcr = 4.4 × 10 T, E cr = 1.3 10 V/m). the electron resulting in the Lamb shift of atomic levels. In this case the influence of the magnetic field is compa- We estimate that in the hydrogen atom the effect of vac- rable to the electrostatic interaction; FL is about of 0.84 3 uum fluctuations is stronger by factor of about 10 than FC. that of the photon-like motion of the electron, and that The faster-than-light motion of the electron and the could be the main reason of the small discrepancy be- extremely strong electric and magnetic fields inside the tween our calculations and the experimental results. neutron result in its unusual properties and give rise to On the other hand, our results disclose the substantial the fundamental contradictions between the proton- difference between the inner (non-observed, hidden) and electron model of the neutron and quantum mechanics. observed mechanical properties of the proton-electron These contradictions indicate restrictions of quantum atoms. According to the planetary model, if the mass and mechanics beyond the description of free particles and the electric charge of the electron are localized in the weakly bound atomic systems. This theory has problems same point, its orbital angular momenta and magnetic in describing strongly bound systems and cannot give moments have similar dependencies on its orbital motion. arguments against the proton-electron model of the neu- The calculated orbital magnetic moments of the electron tron. in the hydrogen atom and in the neutron are close to their As it is easy to see, physics of extremely strong elec- experimental values. But, as it follows from equations tromagnetisms, existing according to our model at short (31) and (32), the respective orbital angular momenta are distances, r < re, merges with physics of strong interac- –3 tions given in a different way by QCD. The force of in- LH ≈ 0.9995 h and Ln ≈ 1.0413 ×10 h and do not co- incide with the measured value of h 2/ . teraction between the proton and the electron into the One possibility to reconcile these features is the as- neutron (Eq. (8)) represents basic properties attributed to sumption that the observed mechanical and magnetic the interaction between quarks in QCD and called the υ → properties relate to two different entities. We assume that confinement and the asymptotic freedom. When 0, the measured gyromagnetic ratio of the bound electron is the Lorentz force vanishes, only the Coulomb force acts the ratio of its orbital magnetic moment to the angular and the radius rn of the orbit of the electron tends to its υ → momentum h 2/ resulting from the process of resonant maximum value re. When c, the Lorentz force com- absorption or emission of photons. So, the orbital mag- pensates the Coulomb force, and the radius rn tends to netic moment of the electron is directly measured in zero. Most likely, even at very short distances, the objec- magnetic-resonance experiments. Quite the contrary, the tive reality is not described by one model only, but it is finite orbital angular momentum, though really exists, necessary to reconcile the deterministic and probabilistic does not change the energy of the magnetic moment in aspects. the magnetic field and is non-observable. Unlike quan- tum mechanics, in our concept the angular momentum h 2/ relates to the process of the resonant absorption of 5. Conclusions electromagnetic radiation rather than to the own proper- ties of the electron. Therefore, it does not depend on the In the planetary model of the proton-electron atom, orbital motion and it is the same for the free electron or we take into account, besides the Coulomb interaction of the electron bound in any atoms. Note that the unjustified the point electron with the proton, its strong Coulomb attribution of the angular momentum h 2/ to the own interaction with the physical vacuum as well as the mag- properties of the electron did not allow until now to pro- netic interaction between moving charges. These modifi- pose the consistent physical model of the spin though cations allow us to construct its two different bound
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states: the hydrogen atom and the neutron. Our model 13. H. Ruder, G. Wunner, H. Herold, and F. Geyer. Atoms in enables to analyze from the new point of the simplest Strong Magnetic Fields. (Springer, Berlin, 1994). bound states in atomic and nuclear physics. Such ap- 14. M. A. Liberman and B. Johansson. Phys. Usp. 38, 117 proach might be useful for better understanding static (1995). 15. D. Lai. Rev. Modern Physics. 73 , 629 (2001). properties of nucleons and low-energy hadron physics 16. A. Y. Potekhin and A. V. Turbiner. Phys. Rev. A, 63 , where QCD is confronted with difficulties. 065402 (2001). 17. B. M. Karnakov and V. S. Popov. JETP, 97, 890 (2003). I am grateful to K. Czerski and M.P. D ąbrowski for 18. A. E. Shabad and V. V. Usov. Phys. Rev. Lett. 98 , 180403 reading the manuscript and useful comments and discus- (2007). sions. 19. S. Eidelman et al. (Particle Data Group). J. Phys. G 33 , 1 (2006). 20. R. P. Feynman, R. B. Leighton, and M. Sands. The Feyn- man Lectures on Physics , Vol. II, Chap. 28 (Addison- References Wesley, Reading, 1964). 21. P. Caldirola. Riv. Nuovo Cimento. 2, 2 (1979). 1. E. Rutherford. Proc. Roy. Soc. A 97 , 374 (1920). 22. K. T. McDonald. e-print physics.class-ph/ 0003062. 2. Yu. A. Mostovoi, K. N. Mukhin, and O. O. Patarakin. 23. P. A. M. Dirac. The Principles of Quantum Mechanics , Phys. Usp. 39 , 925 (1996). (Clarendon, Oxford, 1958) p. 262. 3. S. G. Karshenboim. Phys. Rep. 422 , 1 (2005). 24. E. Schrödinger. Sitz. Preuss. Akad. Wiss. Physik-Math. 4. T. R. Hemmert and W. Weise. Eur. Phys. J. A 15 , 487 Berlin 24 , 418 (1930). (2002). 25. W. C. Martin, A. Musgrove, and S. Kotochigova. Ground 5. D. B. Leinweber, D. H. Lu, and A. W. Thomas. Phys. Rev. Levels and Ionization Energies for Neutral Atoms , D 60 , 034014 (1999).
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