The Incorporated Institute of Nuclear Researches (Dubna)



Model of Structure Electron and Muon

By the initial precondition of model of structure electron and muon we shall accept universality of known quantum of the magnetic flow which has been found out at research of macroscopic superconductor rings. More precisely, its meaning


F0 =h*c/(2*e)=2.06785E-7

also we shall assume, that the charge of a particle in regular intervals is distributed on a ring toroidal surface with large radius R and small radius of circular section of a ring r0, and r0<<R.

Essential the prospective presence constant azimuth of speed of a charge equal speed of light "c" is also.

One of arguments for the benefit of this assumption is the equality of forces electrostatic repulsion of a charge and its magnetic tightening directed lengthways perpendicular to a surface toroid. It is possible also, that following from the Dirac's theory "vibration" electrons with speed c, actually is this ordered by azimuth movement of its charge. Let's choose cylindrical system of coordinates with the beginning at the centre of a ring and axis Oz, its directed on an axis symmetry.

So, according to the first assumption the magnetic flow F, created above mentioned by azimuth a current, in electron and muon is expressed as


F = M*F0

( M =...-2,-1,0,1,2,...),

and the total electrical charge on each ring is equal "e".

Using known expressions of vector potential of a ring current Af(r, z), scalar potential both components of intensity of magnetic and electrical fields Hr, Hz, Er, Ez through complete elliptic integrals, for close to a surface toroid of a vicinity is received the following approached formulas for Af at r> = r0:



Af(r) = 2*J*(ln(R/r)+ 0.0794)/c

H(r) = E(r) =~ 2*J/(r*c)



where r - distance from a circle of radius R, located in an equatorial plane of a toroid with the centre in the beginning of coordinates;

J - azimuth a current in rings.

From electrodynamic ratio connecting the magnetic moment of a particle "m", a current J and radius R, we find


m =~ 2*pi*R*J/c,

J/c =~ e/(2*pi*R)


where from


R =~ 2*m/e,

J/c =~ e**2/(4*pi*m)


Then the magnetic flow penetrating a ring F = 2pi*R*Af(r0), will be equal


F =~ 2*e*( ln(R/r)+ 0.0794) = M*F0


From last, the expression for a spectrum of allowable discrete meanings ln(R/r) follows:


ln(R/r)=~ M* F0/(2*e) - 0.0794,


dimensionless quantum which is - F0/(2*e) =215.2557.

The contribution of an electromagnetic field to energy of rest W0 makes


W0 = 1/(8*pi)* ( (H**2+ E**2)dV.


According to /3/ it shall present as


W0= e/(2*pi*R)* (ln(Rm/r0)+ ln(Re/r0)),


As there are such meanings of size Rm and Re, dimensionality of length, which substitution in /9/ gives exact meaning W0.

To facilitate understanding of results of the further consideration, we shall take advantage of that circumstance, that the overwhelming part of integral /8/ contains in the field of small r, that gives the basis approximately to put


Rm ~ Re ~ R.






Nonquantized item in the right part /9 '/ is omitted, as its module at M> 0 makes < 4E-4 from the module of the basic member.

Calculation of the mechanical moment /of a spin/ in particles, which, obviously, is equal to multiplication of size of radius R of a ring multiplication, on which the charge is concentrated, and to a field part of a pulse of a charge P in the same place, with the regard of /5/ and /7/ gives


s=R*e*Af(r0)/c= 2*e*( ln(R/r0)+ 0.0794)/(pi*c) = M*h/(4*pi)


From here follows, that all particles having of spin h/(4pi) should be characterized by quantum number of a magnetic flow M=1. This caused by model ratio a spin to size h/(4pi) causes the certain optimism.

But the particle, having the mechanical moment, besides energy of a field /9/ or /9 '/ has also additional energy of rest W0" at the expense of rotation, equal in view of light speed of a charge


W0"= P*c = M*e*h*c/(8*pi*m)


The complete energy of rest of a particle, thus, appears equal


W0"= W0 + W0" =~ M*e*h*c/(4*pi*m);

(M =1)


And by solving /12/ regarding magnetic moments "m", we have


m = e*h/(4*pi*me*c) = e/( me*c)*s,


Whence we receive correct quantum-mechanical the relation


m/s = e/( me *c).


Is characteristic, that quantums of energy of rest W0 for all four leptons in /12/:


W0 = e*h*c/(4*pi*m)

are amazingly close to experimental meanings of their energy of rest and make 0.51041MeV for a electron and 105.533MeV for a muon. The truth, from behind the above mentioned approximation, allowed by us, in /9 '/ it is impossible to consider concurrence it while exact. The specified calculation /8/ for an electron (positron) and muon (antimuon) was done on the computer.

Part of integral on area contiguous to a surface of the ring r0 < r < R*1E-6, were calculated analytically, and outside of a toroid r = R*1E-6 is count.

Before initial to result the table of the parameters, calculated on the basis of offered model, an electron (positron) and muon (antimuon) we shall emphasize, that for it the universal physical constants h, c, e, F0 = h*c/(2*e) and two experimentally measured meanings of the magnetic moments electron and muon, equal accordingly 9.28485E-21 and 4.49062E-23. Fitting parameters in model are absent. The general structural features of particles were assumed only:

a) toroidal symmetry;

b) homogeneous distribution of a charge on a surface toroid;

c) equality of speed of azimuth movement of a charge of speed of light.

The calculation meanings M, s, W0, R, ln(R/r0), r0 and J/c - are given in the table.

The table




































For tau-lepton, known experimental energy of which rest W0 = 1784MeV, foretell by model the magnetic moment "m", is =2.65E-24erg/gauss. Besides already marked concurrence calculated the spin with their experimental meanings the concurrence of mass of rest electron (positron) and muon (antimuon) not worse 0.12% is visible also. It can form the basis for the statement, that, apparently, nature of mass of rest all lepton completely electromagnetic. The same statement about a nature of lepton muon was made Barut's [1].

The distribution of mass, charge and magnetic moment has appeared similar partoning of model hadronics.

All specified characteristics are concentrated /truth, not in points/ in an extremely small vicinity of a circle of radius R. Large radius electron Re = 3.87E-11 centimeter, on the first sight, is paradoxical is great. However unique on size the magnetic moment a electron could not be created with the help even of light speed of the ordered movement of a total charge e at the small characteristic size of a particle. From the ratio vagueness, not connected to model, also follows that characteristic the size 2R of any particle should satisfy to expression


2R > h*c/(p*c)


/p - characteristic pulse of internal movement/. Whereas known of the Dirak's "vibration" even rest an electron has mathematical expectation of the operator of speed equal "c", for it


2R > h*c/(m*c**2) =3.86E-11 centimeter

Magnetic induction in the considered rings-particles are so great /at a surface toroids ~ 1E100Gs in the electron and ~ 1E109Gs in the muon, and at the centre of rings - accordingly 3.2*10E11Gs and 1.37E16Gs /, that short-range the interaction of such rings of the magnetic moments is capable to cause strong interaction. In effect it was already shown Barut's and Kraus's [2] shown theoretically c by the help of introduction of certain effective potentials of interaction of the abnormal magnetic moment of the lepton with a field motionless the antilepton an opportunity of occurrence of resonances with the characteristic sizes the hadronic.

Probably, the nuclear forces also are caused by interaction of the magnetic moments nucleons /though the considered elementary ring model of structure for a proton has appeared not quite suitable, electromagnetic mass, designed on its basis, of rest has made only - 336 MeV/.

Last proves to be true by the following important feature of structure of nucleuses. In all nucleuses the even part the nucleons always supposes splitting into pairs: a proton - a neutron, a proton - a proton and neutron - a neutron, any of which has the antiparallel magnetic moments. Charge-independent is the mutual orientation in the specified pairs just of the magnetic moments, whereas the spin in them are both antiparallel, and parallel /as in deuteron/. It means, that the interaction of the magnetic moments can be the dynamic factor causing strong connection.

Besides there is a concurrence of dependence from distance between particles of nuclear forces and interaction mutually of antiparallel magnetic moments created by ring currents with conterminous equatorial planes of rings. It is easy to understand with the help of known expression of energy of interaction of two magnetic moments, that at distance between rings r, such, that r>2R, occurs short-range an attraction | ~ 1/r ** 3 |.

If r=0, the strong pushing away is observed. In the field of 0<r<2R with increase r the pushing away passes in an attraction.

Thus, it is clear, why just the antiparallel orientation of the magnetic moments causes their strong connection. Here outside there is not a barrier of pushing away, as would be in case of the parallel moments, which would do connection impossible, but zone of a strong attraction.

In light of these representations some is natural also non-additivity of the magnetic moments, which, for example, in the deuteron makes about 2%. Quantized in the nucleons there are magnetic flows, and the moments because of the mutual inductance change.

Would be biassed not mention the following feature of the offered model, which can be interpeted doubly. In well investigated stationary quantum-mechanical systems, for example, in atom of hydrogen or in superconductor macroscopic rings in the least a condition / n = 1 for hydrogen and = 1 for a ring /, length of a wave de Broil's accordingly for the electrons and consisting from two the electrons quasi-particle, calculated on mathematical expectation of the operators of a pulse, in accuracy is equal to length of a circle with Bohr's radius and radius of a ring accordingly. At the same time this length of a wave of a charge "e" in our model appropriate impulse - Af(r0) from /10/ makes not 2*Pi*R, but 4*Pi*R.

The skeptically adjusted reader in this connection has the right to regard it as difficulty of model. The optimist, on the contrary, can think, that in elementary particles the standing waves de Broil's are carried out.



1. Barut A.O. In: Lecture Notes in Physics. Ed. A.Bohm, Springer6 p.440.

2. Barut A.O., Kraus J. Phys. Lett., 1975, 59B, No.2, p.175.

February 4, 1981.


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