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Helical Solenoid Model of the Electron

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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.
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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 Eect 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 eect” 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
4π0R2=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, Weyssenho, 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)
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Issue 2 (April) PROGRESS IN PHYSICS Volume 14 (2018)
Where gamma is the coecient 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) 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)=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 eect 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'B.(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=cp1v2/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
h,(59)
we=2πfe=mc2
~,(60)
Te=1
fe
=h
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 ×1010 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 dierently 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 dicult 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 coecients for
a electron at rest, obtaining values L=2.08 ×1016 H and
C=3.13 ×1025 F:
Le=φe
Ie
=h
e2fe
=h2
mc2e2,(69)
Ce=e
Ve
=e2
h fe
=e2
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
h=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 oers 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)r2r2jidV.(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'1040 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, diering 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 eect. Finally, in
1921, Landé put forth a formula (named the Landé g-factor)
that allowed him to explain the anomalous Zeeman eect 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 dierent 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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Oliver Consa. Helical Solenoid Model of the Electron 89
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