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Volume 14 (2018) PROGRESS IN PHYSICS Issue 2 (April)

Helical Solenoid Model of the Electron

Oliver Consa

Department of Physics and Nuclear Engineering, Universitat Politècnica de Catalunya

Campus Nord, C. Jordi Girona, 1-3, 08034 Barcelona, Spain

E-mail: oliver.consa@gmail.com

A new semiclassical model of the electron with helical solenoid geometry is presented.

This new model is an extension of both the Parson Ring Model and the Hestenes

Zitterbewegung Model. This model interprets the Zitterbewegung as a real motion that

generates the electron’s rotation (spin) and its magnetic moment. In this new model,

the g-factor appears as a consequence of the electron’s geometry while the quantum of

magnetic ﬂux and the quantum Hall resistance are obtained as model parameters. The

Helical Solenoid Electron Model necessarily implies that the electron has a toroidal

moment, a feature that is not predicted by Quantum Mechanics. The predicted toroidal

moment can be tested experimentally to validate or discard this proposed model.

1 Introduction

Quantum mechanics (QM) is considered the most accurate

physics theory available today. Since its conception, however,

QM has generated controversy. This controversy lies not in

the theory’s results but in its physical interpretation.

One of the most controversial interpretations of QM was

postulated by Bohr and Heisenberg. The “Copenhagen In-

terpretation” described QM as a system of probabilities that

became deﬁnite upon the act of measurement. This interpre-

tation was heavily criticized by many of the physicists who

had participated in the development of QM, most notably Al-

bert Einstein. Because of its probability features, Einstein

believed that QM was only valid for analyzing the behavior

of groups of particles and that the behavior of individual par-

ticles must be deterministic. In a famous quote from a 1926

letter to Max Born, Einstein stated, “He (God) does not play

dice with the universe”.

A major ﬂaw in QM becomes apparent when the theory

is applied to individual particles. This leads to logical con-

tradictions and paradoxical situations (e.g., the paradox of

Schrödinger’s Cat). Einstein believed that QM was incom-

plete and that there must be a deeper theory based on hidden

variables that would explain how subatomic particles behave

individually. Einstein and his followers were not able to ﬁnd

a hidden variable theory that was compatible with QM, so the

Copenhagen Interpretation was imposed as the interpretation

of reference. If we assume that Einstein was correct, and that

QM is only applicable to groups of particles, it is necessary

to develop a new deterministic theory to explain the behavior

of individual particles.

2 Spinning models of the electron

2.1 Ring Electron Model

In 1915, Parson [1] proposed a new model for the electron

with a ring-shaped geometry where a unitary charge moves

around the ring generating a magnetic ﬁeld. The electron be-

haves not only as the unit of electric charge but also as the unit

of magnetic charge or magneton. Several important physi-

cists, including Webster, Gilbert, Grondahl and Page, con-

ducted studies that supported Parson’s Ring Electron Model.

The most important of these studies was conducted by Comp-

ton [2], who wrote a series of papers showing that his new-

found Compton Eﬀect was better explained with Parson’s

Ring Electron Model than with the classical model that de-

picted the electron as a sphere. All these studies were com-

piled in 1918 by Allen [3] in “The Case for a Ring Electron”

and discussed at a meeting of the Physical Society of London.

The Ring Electron Model was not widely accepted and

was invalidated in 1923 by Schrödinger’s wave equation of

the electron. The Ring Electron Model has been unsuccess-

fully revisited several times by investigators like Iida, Carroll,

Giese, Caesar, Bergman and Wesley [4], Lucas [5], Ginzburg

or Kanarev [6]. Other researchers, such as Jennison [7], Gau-

thier [8], and Williamson and van der Mark [9], proposed

similar models, with the additional assumption that the elec-

tron is a photon trapped in a vortex.

The Ring Electron Model proposes that the electron has

an extremely thin, ring-shaped geometry that is about 2000

times larger than a proton. A unitary charge ﬂows through the

ring at the speed of light, generating an electric current and

an associated magnetic ﬁeld. This model allows us to com-

bine experimental evidence that the electron has an extremely

small size (corresponding to the thickness of the ring) as well

as a relatively large size (corresponding to the circumference

of the ring).

The Ring Electron Model postulates that the rotational

velocity of the electric charge will match the speed of light

and that the angular momentum will match the reduced

Planck constant:

vr=c,(1)

L=mRvr=~.(2)

As a consequence of (1) and (2), the radius of the ring

will match the reduced Compton wavelength and the circum-

80 Oliver Consa. Helical Solenoid Model of the Electron

Issue 2 (April) PROGRESS IN PHYSICS Volume 14 (2018)

Fig. 1: Ring Electron Model.

ference will matches the Compton wavelength

R=~

mvr

=~

mc =oc,(3)

2πR=h

mc =λc.(4)

Meanwhile, the frequency, angular frequency and rotation

time period of the ring electron are deﬁned by:

fe=vr

2πR=mc2

h,(5)

we=2πfe=mc2

~,(6)

Te=1

fe

=h

mc2.(7)

The electron’s ring acts as a circular antenna. In this type of

antenna, the resonance frequency coincides with the length

of the antenna’s circumference. In the case of the electron

ring, the resonance frequency coincides with the electron’s

Compton frequency.

Substituting the electron’s frequency (5) in the Planck

equation (E=h f ), we obtain the Einstein’s energy equation

E=h fe=hmc2

h=mc2.(8)

The moving charge generates a constant electric current. This

electric current produces a magnetic moment that is equal to

the Bohr magneton:

I=e fe=emc2

h,(9)

µe=IS =emc2

hπR2=e

2m~=µB.(10)

The relationship between the magnetic moment and the angu-

lar momentum is called the “gyromagnetic ratio” and has the

value “e/2m”. This value is consistent with the magnetic mo-

ment generated by an electric current rotating on a circular

surface of radius R. The gyromagnetic ratio of the electron

can be observed experimentally by applying external mag-

netic ﬁelds (for example, as seen in the “Zeeman eﬀect” or in

the “Stern-Gerlach experiment”):

E=e

2mB.(11)

The energy of the electron is very low, but the frequency

of oscillation is extremely large, which results in a signiﬁcant

power of about 10 gigawatts:

P=E

T=m2c4

h=1.01 ×107W.(12)

Using the same line of reasoning, the electric potential can be

calculated as the electron energy per unit of electric charge,

resulting in a value of approximately half a million volts:

V=E

e=mc2

e=5.11 ×105V.(13)

The electric current has already been calculated as 20 amps

(I=e f =19.83 A). Multiplying the voltage by the current,

the power is, again, about 10 gigawatts (P=VI ).

The Biot-Savart Law can be applied to calculate the mag-

netic ﬁeld at the center of the ring, resulting in a magnetic

ﬁeld of 30 million Tesla, equivalent to the magnetic ﬁeld of a

neutron star:

B=µ0I

2R=3.23 ×107T.(14)

For comparison, the magnetic ﬁeld of the Earth is 0.000005 T,

and the largest artiﬁcial magnetic ﬁeld created by man is only

90 T.

The electric ﬁeld in the center of the electron’s ring

matches the value of the magnetic ﬁeld multiplied by the

speed of light:

E=e

4π0R2=cB =9.61 ×1012 V/m.(15)

The Ring Electron Model implies the existence of a

centripetal force that compensates for the centrifugal force

of the electron orbiting around its center of mass:

F=mv2

r

R=m2c3

~=0.212 N.(16)

Electromagnetic ﬁelds with a Lorentz force greater than

this centripetal force should cause instabilities in the elec-

tron’s geometry. The limits of these electric and magnetic

ﬁelds are:

F=eE +evB,(17)

E=m2c3

e~=1.32 ×1018 V/m,(18)

B=m2c2

e~=4.41 ×109T.(19)

Oliver Consa. Helical Solenoid Model of the Electron 81

Volume 14 (2018) PROGRESS IN PHYSICS Issue 2 (April)

In quantum electrodynamics (QED), these two values are

known as the Schwinger Limits [10]. Above these values,

electromagnetic ﬁelds are expected to behave in a nonlinear

way. While electromagnetic ﬁelds of this strength have not

yet been achieved experimentally, current research suggests

that electromagnetic ﬁeld values above the Schwinger Limits

will cause unexpected behavior not explained by the Standard

Model of Particle Physics.

2.2 Helical Electron Model

In 1930, while analyzing possible solutions to the Dirac equa-

tion, Schrödinger identiﬁed a term called the Zitterbewegung

that represents an unexpected oscillation whose amplitude is

equal to the Compton wavelength. In 1953, Huang [11] pro-

vided a classical interpretation of the Dirac equation in which

the Zitterbewegung is the mechanism that causes the elec-

tron’s angular momentum (spin). According to Huang, this

angular momentum is the cause of the electron’s magnetic

moment. Bunge [12], Barut [13], Zhangi [14], Bhabha, Cor-

ben, Weyssenhoﬀ, Pavsic, Vaz, Rodrigues, Salesi, Recami,

Hestenes [15, 16] and Rivas [17] have published papers in-

terpreting the Zitterbewegung as a measurement of the elec-

tron’s oscillatory helical motion that is hidden in the Dirac

equation. We refer to these electron theories as the Hestenes

Zitterbewegung Model or the Helical Electron Model.

The Helical Electron Model assumes that the electron’s

charge is concentrated in a single inﬁnitesimal point called

the center of charge (CC) that rotates at the speed of light

around a point in space called the center of mass (CM).

The Helical Electron Model shares many similarities with

the Ring Electron Model, but in the case of the Helical Elec-

tron Model, the geometric static ring is replaced by a dynamic

point-like electron. In this dynamic model, the electron’s ring

has no substance or physical properties. It need not physically

exist. It is simply the path of the CC around the CM.

The CC moves constantly without any loss of energy so

that the electron acts as a superconducting ring with a persis-

tent current. Such ﬂows have been experimentally detected in

superconducting materials.

The CC has no mass, so it can have an inﬁnitesimal size

without collapsing into a black hole, and it can move at the

speed of light without violating the theory of relativity. The

electron’s mass is not a single point. Instead, it is distributed

throughout the electromagnetic ﬁeld. The electron’s mass

corresponds to the sum of the electron’s kinetic and poten-

tial energy. By symmetry, the CM corresponds to the center

of the electron’s ring.

We can demonstrate the principles of the Helical Electron

Model with an analogy to the postulates of the Bohr Atomic

Model:

•The CC always moves at the speed of light, tracing cir-

cular orbits around the CM without radiating energy.

Fig. 2: Helical Electron Model.

•The electron’s angular momentum equals the reduced

Planck constant.

•The electron emits and absorbs electromagnetic energy

that is quantized according to the formula E=h f .

•The emission or absorption of energy implies an accel-

eration of the CM.

The electron is considered to be at rest if the CM is at

rest, since in that case the electric charge has only rotational

movement without any translational movement. In contrast,

if the CM moves with a constant velocity (v), then the CC

moves in a helical motion around the CM.

The electron’s helical motion is analogous to the observed

motion of an electron in a homogeneous external magnetic

ﬁeld.

It can be parameterized as:

x(t)=Rcos(wt),

y(t)=Rsin(wt),

z(t)=vt.

(20)

The electron’s helical motion can be deconstructed into

two orthogonal components: a rotational motion and a trans-

lational motion. The velocities of rotation and translation are

not independent; they are constrained by the electron’s tan-

gential velocity that is constant and equal to the speed of light.

As discussed above, when the electron is at rest, its rotational

velocity is equal to the speed of light. As the translational

velocity increases, the rotational velocity must decrease. At

no time can the translational velocity exceed the speed of

light. Using the Pythagorean Theorem, the relationship be-

tween these three velocities is:

c2=v2

r+v2

t.(21)

Then the rotational velocity of the moving electron is:

vr=cp1−(v/c)2,(22)

vr=c/γ . (23)

82 Oliver Consa. Helical Solenoid Model of the Electron

Issue 2 (April) PROGRESS IN PHYSICS Volume 14 (2018)

Where gamma is the coeﬃcient of the Lorentz transforma-

tion, the base of the Special Relativity Theory:

γ=1

p1−(v/c)2.(24)

Multiplying the three components by the same factor (γmc)2:

(γmc)2c2=(γmc)2v2

r+(γmc)2v2

t.(25)

Substituting the value of the rotational velocity (vr=c/γ) and

linear momentum (p=γmv), results in the relativistic energy

equation:

E2=(γmc2)2=(mc2)2+(pc)2.(26)

With this new value of the rotational velocity, the frequency,

angular frequency and rotational time period of the helical

electron are deﬁned by:

fe=vr

2πR=mc2

γh,(27)

we=2πfe=mc2

γ~,(28)

Te=1

fe

=γh

mc2.(29)

The rotation time period of the electron acts as the elec-

tron’s internal clock. As a result, although there is no absolute

time in the universe, each electron is always set to its proper

time. This proper time is relative to the electron’s reference

frame and its velocity with respect to other inertial reference

frames.

The electron’s angular momentum is always equal to the

reduced Planck constant. This implies that the electron’s

mass has to increase γtimes in order to compensate for the

decrease in its rotational velocity:

L=mRvr=(γm)R(c/γ)=mRc =~.(30)

If the electron moves at a constant velocity, the particle’s

trajectory is a cylindrical helix. The geometry of the helix is

deﬁned by two constant parameters: the radius of the helix

(R) and the helical pitch (H). The helical pitch is the space

between two turns of the helix. The electron’s helical motion

can be interpreted as a wave motion with a wavelength equal

to the helical pitch and a frequency equal to the electron’s

natural frequency. Multiplying the two factors results in the

electron’s translational velocity:

λefe=v , (31)

λe=H=v

fe

=vγh

mc2=γβλc.(32)

The rest of the parameters representative of a cylindrical helix

can also be calculated, including the curvature (κ) and the

torsion (τ), where h=2πH=γβoc:

κ=R

R2+h2=1

γ2R,

τ=h

R2+h2=β

γR.

(33)

According to Lancret’s Theorem, the necessary and suf-

ﬁcient condition for a curve to be a helix is that the ratio of

curvature to torsion must be constant. This ratio is equal to

the tangent of the angle between the osculating plane with the

axis of the helix:

tan α=κ

τ=1

γβ .(34)

2.3 Toroidal Solenoid Electron Model

In 1956, Bostick, a disciple of Compton, discovered the exis-

tence of plasmoids. A plasmoid is a coherent toroidal struc-

ture made up of plasma and magnetic ﬁelds. Plasmoids are so

stable that they can behave as individual objects and interact

with one another. From Parson’s Ring Electron Model, Bo-

stick [21] proposed a new electron structure, similar to that of

the plasmoids. In his model, the electron takes the shape of

a toroidal solenoid where the electric charge circulates at the

speed of light. In the Toroidal Solenoid Electron Model, we

assume that the electric charge is a point particle and that the

toroidal solenoid represents the trajectory of that point elec-

tric charge.

In a toroidal solenoid, any magnetic ﬂux is conﬁned

within the toroid. This feature is consistent with the idea

that the mass of a particle matches the electromagnetic en-

ergy contained therein. Storage of electromagnetic energy in

a toroidal solenoid superconductor without the loss of energy

is called superconducting magnetic energy storage (SMES).

According to the Toroidal Solenoid Electron model, an elec-

tron is a microscopic version of a SMES system.

Toroidal solenoid geometry is well known in the electron-

ics ﬁeld where it is used to design inductors and antennas. A

toroidal solenoid provides two additional degrees of freedom

compared to the ring geometry. In addition to the radius (R)

of the torus, two new parameters appear: the thickness of the

torus (r) and the number of turns around the torus (N) with N

being an integer.

The toroidal solenoid can be parameterized as:

x(t)=(R+rcos Nwt) cos wt,

y(t)=(R+rcos Nwt) sin wt,

z(t)=rsin Nwt.

(35)

Where the tangential velocity is:

|r0(t)|2=(R+rcos Nwt)2w2+(rNw)2.(36)

We postulate that the tangential velocity is always equal to

the speed of light (|r0(t)|=c). For RrN , the rotational

Oliver Consa. Helical Solenoid Model of the Electron 83

Volume 14 (2018) PROGRESS IN PHYSICS Issue 2 (April)

Fig. 3: Helical Toroidal Electron Model.

velocity can be obtained as:

c2=(Rw)2+(rNw)2,(37)

c/vr=s1+rN

R2

.(38)

The second factor depends only on the geometry of electron.

We call this value the helical g-factor. If RrN, the helical

g-factor is slightly greater than 1,

g=s1+rN

R2

.(39)

As a result, the rotational velocity is dependent on the helical

g-factor and slightly lower than the speed of light:

vr=c/g. (40)

With this new value of the rotational velocity, the frequency,

angular frequency and time period are deﬁned by:

fe=vr

2πR=mc2

gh,(41)

we=2πfe=mc2

g~,(42)

Te=1

fe

=gh

mc2.(43)

The length of a turn of the toroidal solenoid is called the arc

length. To calculate the arc length, we need to perform the

integral of the toroidal solenoid over one turn:

l=Zp|r0(t)|2dt

=Zp(R+rcos Nwt)2w2+(rNw)2dt .

(44)

Approximating for RNr and replacing the helical g-factor

(39) results in:

l=Zp(Rw)2+(rNw)2dt

=ZRwp1+(rN/R)2dt =gRZwdt =2πgR.

(45)

Fig. 4: Toroidal and Poloidal currents.

This means that the arc length of a toroidal solenoid is equiv-

alent to the length of the circumference of a ring of radius

R0=gR:

l=2πgR=2πR0.(46)

In calculating the electron’s angular momentum, we must

take into consideration the helical g-factor. The value of the

rotational velocity is reduced in proportion to the equivalent

radius, so that the angular momentum remains constant:

L=mR0vr=m(gR) c

g!=~.(47)

The electric current ﬂowing through a toroidal solenoid

has two components, a toroidal component (red) and a

poloidal component (blue).

By symmetry, the magnetic moment due to the poloidal

components (red) is canceled, while the toroidal component

(blue) remains ﬁxed. No matter how large the number of turns

in the toroidal solenoid, a toroidal component generates a cor-

responding axial magnetic moment [22]. This eﬀect is well

known in the design of toroidal antennas and can be canceled

with various techniques. The exact value of the axial mag-

netic moment is:

m=IπR2"1+1

2r

R2#.(48)

A comparison of the Toroidal Solenoid Electron Model

(v=0,r>0) with the Ring Electron Model (v=0,r=0)

reveals that the radius still coincides with the reduced Comp-

ton wavelength. The electric current is slightly lower, since

the electron’s rotational velocity is also slightly lower:

IπR2=e f πR2=evrR

2=ec~

2gmc =e~

2mg=µB

g,(49)

m=µB

g"1+1

2r

R2#,(50)

m'gµB.(51)

In calculating the angular momentum, the rotational veloc-

ity decreases in the same proportion as the equivalent radius

increase, compensating for the helical g-factor. However, in

84 Oliver Consa. Helical Solenoid Model of the Electron

Issue 2 (April) PROGRESS IN PHYSICS Volume 14 (2018)

the calculation of magnetic moment, the rotational velocity

decreases by a factor of g, while the equivalent radius in-

creases by a factor approximately equal to gsquared. This

is the cause of the electron’s anomalous magnetic moment.

2.4 Helical Solenoid Model

The geometries of both the Ring Electron Model and the

Toroidal Solenoid Electron Model represent a static electron

(v=0). For a moving electron with a constant velocity

(v > 0), the ring geometry becomes a circular helix, while

the toroidal solenoid geometry becomes a helical solenoid.

On the other hand, if the thickness of the toroid is negated

(r=0), the toroidal solenoid is reduced to a ring, and the

helical solenoid is reduced to a helix.

Experimentally, the electron’s magnetic moment is

slightly larger than the Bohr magneton. In the Ring Electron

Model, it was impossible to explain the electron’s anomalous

magnetic moment. This leads us to assume that the electron

has a substructure. The Toroidal Solenoid Electron Model al-

lows us to obtain the electron’s anomalous moment as a direct

consequence of its geometry.

Geometry v=0v > 0

r=0 Ring Helix

r>0 Toroidal Solenoid Helical Solenoid

The universe generally behaves in a fractal way, so the

most natural solution assumes that the electron’s substructure

is similar to the main structure, that is, a helix in a helix.

Fig. 5: Helical Solenoid Electron Model.

The trajectory of the electron can be parameterized with

the equation of the helical solenoid:

x(t)=(R+rcos Nwt) cos wt,

y(t)=(R+rcos Nwt) sin wt,

z(t)=rsin Nwt+vt.

(52)

Like the other electron models discussed above, the Helical

Solenoid Electron Model postulates that the tangential veloc-

ity of the electric charge matches the speed of light and that

the electron’s angular momentum matches the reduced Planck

constant.

|r0(t)|2=c2=(Rw)2+(rNw)2+v2

+rw(2Rw+rwcos Nwt+2vN) cos Nwt.(53)

This equation can be obtained directly from the helical

solenoid geometry without any approximation. This equation

shows a component that oscillates at a very high frequency

with an average value of zero. Consequently, the Helical

Solenoid Electron Model implies that the electron’s g-factor

is oscillating, not ﬁxed. Since the value oscillates, there is a

maximum level of precision with which the g-factor can be

measured. This prediction is completely new to this model

and is directly opposite to previous QED predictions. For

RrN, this oscillating component can be negated, and the

equation reduces to

c2=(Rw)2+(rNw)2+v2.(54)

The rotational velocity can be obtained as a function of

the speed of light, the Lorentz factor, and the helical g-factor:

c2=(Rw)2(1 +(rN/R)2)+v2,(55)

c2=(vr)2g2+v2,(56)

gvr=cp1−v2/c2,(57)

vr=c/gγ. (58)

With this new value of the rotational velocity, the frequency,

angular frequency, rotation time period and the wavelength

(o pitch) of the helical solenoid electron are deﬁned by:

fe=vr

2πR=mc2

gγh,(59)

we=2πfe=mc2

gγ~,(60)

Te=1

fe

=gγh

mc2,(61)

λe=H=v

fe

=gγβλc.(62)

In 2005, Michel Gouanère [18] identiﬁed this wavelength in

a channeling experiment using a beam of ∼80 MeV electrons

aligned along the <110 >direction of a thick silicon crys-

tal (d=3.84 ×10−10 m). While this experiment has not

had much impact on QM, both Hestenes [19] and Rivas [20]

have indicated that the experiment provides important experi-

mental evidence consistent with the Hestenes Zitterbewegung

Model:

d=gγβλc=(γmv)gh

(mc)2=pgh

(mc)2,(63)

Oliver Consa. Helical Solenoid Model of the Electron 85

Volume 14 (2018) PROGRESS IN PHYSICS Issue 2 (April)

p=d(mc)2

gh=80.874 MeV/c.(64)

In the Helical Solenoid Electron Model, the rotational ve-

locity is reduced by both the helical g-factor and the Lorentz

factor. In contrast, the equivalent radius compensates for the

helical g-factor while the increasing mass compensates for

the Lorentz factor. The angular momentum remains equal to

the reduced Planck constant:

L=m0R0vr=(γm)(gR)(c/γg)=mRc =~.(65)

3 Consequences of the Helical Solenoid Electron Model

3.1 Chirality and helicity

In 1956, an experiment based on the beta decay of a Cobalt-60

nucleus demonstrated a clear violation of parity conservation.

In the early 1960s the parity symmetry breaking was used by

Glashow, Salam and Weinberg to develop the Electroweak

Model, unifying the weak nuclear force with the electromag-

netic force. The empirical observation that electroweak in-

teractions act diﬀerently on right-handed fermions and left-

handed fermions is one of the basic characteristics of this the-

ory.

In the Electroweak Model, chirality and helicity are es-

sential properties of subatomic particles, but these abstract

concepts are diﬃcult to visualize. In contrast, in the Helical

Solenoid Electron Model, these concepts are evident and a

direct consequence of the model’s geometry:

•Helicity is given by the helical translation motion (v >

0), which can be left-handed or right-handed. Helicity

is not an absolute value; it is relative to the speed of the

observer.

•Chirality is given by the secondary helical rotational

motion, which can also be left-handed or right-handed.

Chirality is absolute since the tangential velocity is al-

ways equal to the speed of light; it is independent of

the velocity of the observer.

3.2 Quantum Hall resistance and magnetic ﬂux

The movement of the electric charge causes an electrical cur-

rent (I=e fe) and a electric voltage (V=E/e=h fe/e). Ap-

plying Ohm’s law, we obtain a ﬁxed value for the impedance

of the electron equal to the value of the quantum Hall resis-

tance. This value is quite surprising, since it is observable at

the macroscopic level and was not discovered experimentally

until 1980:

R=Ve

Ie

=h fe/e

e fe

=h

e2.(66)

According to Faraday’s Law, voltage is the variation of the

magnetic ﬂux per unit of time. So, in a period of rotation,

we obtain a magnetic ﬂux value which coincides with the

quantum of magnetic ﬂux, another macroscopically observ-

able value. This value was expected since, in this model, the

electron behaves as a superconducting ring, and it is experi-

mentally known that the magnetic ﬂux in a superconducting

ring is quantized:

V=φe/Te,(67)

φe=VeTe=h fe

e

1

fe

=h

e.(68)

3.3 Quantum LC circuit

Both the electrical current and the voltage of the electron

are frequency dependent. This means that the electron be-

haves as a quantum LC circuit, with a Capacitance (C) and

a Self Inductance (L). We can calculate these coeﬃcients for

a electron at rest, obtaining values L=2.08 ×10−16 H and

C=3.13 ×10−25 F:

Le=φe

Ie

=h

e2fe

=gγh2

mc2e2,(69)

Ce=e

Ve

=e2

h fe

=gγe2

mc2.(70)

Applying the formulas of the LC circuit, we can obtain the

values of impedance and resonance frequency, which coin-

cide with the previously calculated values of impedance and

natural frequency of the electron:

Ze=rLe

Ce

=h

e2,(71)

fe=1

√LeCe

=mc2

gγh=fe.(72)

As the energy of the particle oscillates between electric and

magnetic energy, the average energy value is

E=LI2

2+CV 2

2=h f

2+h f

2=h f .(73)

The above calculations are valid for any elementary particle

with a unit electric charge, a natural frequency of vibration

and an energy which match the Planck equation (E=h f ).

From this result, we infer that the electron is formed by

two indivisible elements: a quantum of electric charge and a

quantum of magnetic ﬂux, the product of which is equal to

Planck’s constant. The electron’s magnetic ﬂux is simultane-

ously the cause and the consequence of the circular motion of

the electric charge:

eφ=h.(74)

3.4 Quantitative calculation of the helical G-factor

The g-factor depends on three parameters (R, r and N) but we

do not know the value of two of them. We can try to ﬁgure out

the value of the helical g-factor using this approximation [28]:

Using this expansion series:

p1+(a)2=1+1/2(a)2+. . . (75)

86 Oliver Consa. Helical Solenoid Model of the Electron

Issue 2 (April) PROGRESS IN PHYSICS Volume 14 (2018)

The helical g-factor can be expressed as:

s1+rN

R2

=1+1

2rN

R2

+. . . (76)

QED also calculates the g-factor by an expansion series where

the ﬁrst term is 1 and the second term is the Schwinger factor:

g. f actor(QE D)=1+α

2π+. . . (77)

The results of the two series are very similar. Equaling the

second term of the helical g-factor series to the Schwinger

factor, we obtain the relationship between the radius of the

torus and the thickness of the torus:

1

2rN

R2

=α

2π,(78)

rN

R=rα

π.(79)

What gives a value of helical g-factor of

g=p1+α/π . (80)

This gives us a value of the helical g-factor =1.0011607. This

result is consistent with the Schwinger factor, and it oﬀers a

value much closer to the experimental value.

3.5 Toroidal moment

In 1957, Zel’dovich [23] discussed the parity violation of ele-

mentary particles and postulated that spin-1/2 Dirac particles

must have an anapole. In the late 1960s and early 1970s,

Dubovik [24, 25] connected the quantum description of the

anapole to classical electrodynamics by introducing the polar

toroidal multipole moments. The term toroidal derives from

current distributions in the shape of a circular coil that were

ﬁrst shown to have a toroidal moment. Toroidal moments

were not acknowledged outside the Soviet Union as being

an important part of the multipole expansion until the 1990s.

Toroidal moments became known in western countries in the

late 1990s. Finally, in 1997, toroidal moment was experimen-

tally measured in the nuclei of Cesium-133 and Ytterbium-

174 [26].

In 2013, Ho and Scherrer [27] hypothesized that Dark

Matter is formed by neutral subatomic particles. These par-

ticles of cold dark matter interact with ordinary matter only

through an anapole electromagnetic moment, similar to the

toroidal magnetic moment described above. These particles

are called Majorana fermions, and they cannot have any other

electromagnetic moment apart from the toroid moment. The

model for these subatomic particles of dark matter is compat-

ible with the Helical Solenoid Electron Model.

In an electrostatic ﬁeld, all charge distributions and cur-

rents may be represented by a multipolar expansion using

Fig. 6: Electric, Magnetic and Toroidal dipole moments.

only electric and magnetic multipoles. Instead, in a multi-

polar expansion of an electrodynamic ﬁeld new terms appear.

These new terms correspond to a third family of multipoles:

the toroid moments. The toroidal lower order term is the

toroidal dipole moment. The toroidal moment can understood

as the momentum generated by a distribution of magnetic mo-

ments. The simplest case is the toroidal moment generated by

an electric current in a toroidal solenoid.

The toroidal moment is calculated with the following

equation [24]:

T=1

10 Zh(j·r)r−2r2jidV.(81)

In the case of the toroidal solenoid, the toroidal moment can

be calculated more directly as the B ﬁeld inside the toroid by

both the surface of the torus and the surface of the ring [25]:

µT=BsS =Bπr2πR2,(82)

B=µNI

2πR.(83)

Using B, the toroidal moment is obtained as [22]:

T=NI

2πRπr2πR2=

NI πr2R

2.(84)

Rearranging and using the relation (79):

T=µB

R

g2NrN

R2

=µB

oc

gNα

2π.(85)

According the Helical Solenoid Electron Model, the elec-

tron’s theoretical toroidal moment is about T'10−40 Am3.

The theoretical toroidal moment value for the neutron and the

proton should be one million times smaller. The existence

of a toroidal moment for the electron (and for any other sub-

atomic particle) is a direct consequence of this model, and

it may be validated experimentally. Notably, QM does not

predict the existence of any toroidal moments.

3.6 Nucleon model

By analogy to the theory underlying the Helical Solenoid

Electron Model, we assume that all subatomic particles have

the same structure as the electron, diﬀering mainly by their

Oliver Consa. Helical Solenoid Model of the Electron 87

Volume 14 (2018) PROGRESS IN PHYSICS Issue 2 (April)

charge and mass. Protons are thought to be composed of

other fundamental particles called quarks, but their internal

organization is beyond the scope of this work.

The radius of a nucleon is equal to its reduced Comp-

ton wavelength. The Compton wavelength is inversely pro-

portional to an object’s mass, so for subatomic particles, as

mass increases, size decreases. Both the proton and the neu-

tron have a radius that is about 2000 times smaller than the

electron. Historically, the proton radius was measured using

two independent methods that converged to a value of about

0.8768 fm. This value was challenged by a 2010 experiment

utilizing a third method, which produced a radius of about

0.8408 fm. This discrepancy remains unresolved and is the

topic of ongoing research referred to as the Proton Radius

Puzzle. The proton’s reduced Compton wavelength is 0.2103

fm. If we multiple this radius by 4, we obtain the value of

0.8412 fm. This value corresponds nicely with the most re-

cent experimental radius of the proton. This data supports our

theory that the proton’s radius is related to its reduced Comp-

ton radius and that our Helical Solenoid Electron Model is

also a valid model for the proton.

The current of a nucleon is about 2000 times the current

of an electron, and the radius is about 2000 times lower. This

results in a magnetic ﬁeld at the center of the nucleon’s ring

that is about four million times bigger than that of the elec-

tron or thousands of times bigger than a neutron star. This

magnetic ﬁeld is inversely dependent with the cube of the

distance. This implies that while the magnetic ﬁeld inside

the neutron’s ring is huge, outside the ring, the magnetic ﬁeld

decays much faster than the electric ﬁeld. The asymmetri-

cal behavior of the neutron’s magnetic ﬁeld over short and

long distances leads us to suggest that the previously identi-

ﬁed strong and weak nuclear forces are actually manifesta-

tions of this huge magnetic ﬁeld at very short distances.

3.7 Spin quantum number

In 1913, Bohr introduced the Principal Quantum Number to

explain the Rydberg Formula for the spectral emission lines

of atomic hydrogen. Sommerfeld extended the Bohr the-

ory with the Azimuthal Quantum Number to explain the ﬁne

structure of the hydrogen, and he introduced a third Magnetic

Quantum Number to explain the Zeeman eﬀect. Finally, in

1921, Landé put forth a formula (named the Landé g-factor)

that allowed him to explain the anomalous Zeeman eﬀect and

to obtain the whole spectrum of all atoms.

gJ=gL

J(J+1) −S(S+1) +L(L+1)

2J(J+1)

+gS

J(J+1) +S(S+1) −L(L+1)

2J(J+1) .

(86)

In this formula, Landé introduced a fourth Quantum Num-

ber with a half-integer number value (S =1/2). This Landé

g-factor was an empirical formula where the physical mean-

ing of the four quantum numbers and their relationship with

the motion of the electrons around the nucleus was unknown.

Heisenberg, Pauli, Sommerfeld, and Landé tried unsuccess-

fully to devise a new atomic model (named the Ersatz Model)

to explain this empirical formula. Landé proposed that his

g-factor was produced by the combination of the orbital mo-

mentum of the outer electrons with the orbital momentum

of the inner electrons. A diﬀerent solution was suggested

by Kronig, who proposed that the half-integer number was

generated by a self-rotation motion of the electron (spin), but

Pauli rejected this theory.

In 1925, Uhlenbeck and Goudsmit published a paper

proposing the same idea, namely that the spin quantum num-

ber was produced by the electron’s self-rotation. The half-

integer spin implies an anomalous magnetic moment of 2. In

1926, Thomas identiﬁed a relativistic correction of the model

with a value of 2 (named the Thomas Precession) that com-

pensated for the anomalous magnetic moment of the spin.

Despite his initial objections, Pauli formalized the theory of

spin in 1927 using the modern theory of QM as set out by

Schrödinger and Heisenberg. Pauli proposed that spin, angu-

lar moment, and magnetic moment are intrinsic properties of

the electron and that these properties are not related to any

actual spinning motion. The Pauli Exclusion Principle states

that two electrons in an atom or a molecule cannot have the

same four quantum numbers. Pauli’s ideas brought about a

radical change in QM. The Bohr-Sommerfeld Model’s ex-

plicit electron orbitals were abandoned and with them any

physical model of the electron or the atom.

We propose to return to the old quantum theory of Bohr-

Sommerfeld to search for a new Ersatz Model of the atom

where the four quantum numbers are related to electron or-

bitals. We propose that this new atomic model will be com-

patible with our Helical Solenoid Electron Model. We also

propose that the half-integer spin quantum number is not an

intrinsic property of the electron but a result of the magnetic

ﬁelds generated by orbiting inner electrons.

Submitted on January 25, 2018

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