## IV. RADIUS DETERMINATION FROM SPIN

As a test of this model it is examined whether characteristics
like the spin are associated with the mass of the electron,
if the mass equivalent of the electromagnetic field rotates
with speed of light „c". The classical electron radius is
derived from the capacity of the electron as sphere condenser
with charge e:

Equation (2) after: Mohr & Taylor[7]

The equatorial peripheral speed v of this classical electron
would exceed the speed of light in several approaches, although
it is a postulate of quantum mechanics not to contradict
the classical mechanisms. The classical approach leads to
"no meaningful result" according to the own statements of
modern physics (R. Gross[8]).

The peripheral speed „v" in the current model is assumed
as speed of light „c". This approach is strictly forbidden
according to the standard model, since no „mass" can move
with speed of light. Only „mass less" particles or photons
which nevertheless are provided with an exactly defined
mass/energy equivalent can move with c. If the approach
is wrong, it should therefore lead to an unreasonable result.

The angular momentum L of a mass (e.g. m_{el})
around an axis and the radius r is defined as:

The mass of the electron m_{el} (= field energy
/c^{2}) rotates with „c" around the radius "r" in
a first approximation. With v = c and with
the angular momentum L becomes .

The electron spin is defined as .
Now the radius is the only unknown parameter and can be
calculated:

Equation (3)

which gives.

This radius corresponds to a measured value for the scattering
of an individual photon at the electron. The quantity m
is well known as Compton wavelength and also has been determined
as electron diameter by Giese[9]
and Mills[10]. What,
if the Compton wavelength were more than a historical observation
without further meaning? It should be considered that the
analysis of atomic distances with x - rays is a usual procedure
in physics and e.g. metallurgy. The measured Compton wavelength
according to this model becomes a measurement of the diameter
of the circulating wave of the electron.

The computation of the electron radius from the spin as
rotation of the mass/ energy of the electromagnetic field
with speed of light "c" around the radius "r" leads to a
meaningful result in contrast to classical computations
of the electron radius.

There are plenty of data suggesting that the diameters
of electrons, protons or neutrinos are below a certain size
of e.g. 10^{-15} m. This would certainly be correct
if looking for the small "stone with charge". It should
be regarded, however, that the diameter will decrease at
increasing energies of the particle/ c-tron. An electron
accelerated to 1 GeV total energy has a diameter of 1.9
* 10^{-16} m according to eq. 3, which perfectly
fits to the current model and to the observations concerning
measured particle diameters or effective cross sections
at high particle energies.

The radius determined before has to be compatible with
the classical quantum physics observations as the de Broglie
wavelength and the mass energy equivalence. One full phase
of the c-tron of the current model is completed after passing
the circumference "C" of the particle twice, i.e. .
With the electron diameter d = 2 r = 3,86 * 10^{-13}
m derived from the spin, the wavelength can be calculated
to =
2.425 * 10^{-12} m and with
the circulation frequency of the particle in rest to .
It is interesting that this rest mass frequency often is
cited in physics books, but its nature rarely is explained
or commented[11].

In a very clear way the natural frequency by
de Broglie for an electron in rest gets a realistic meaning
as the true frequency of the c-tron of the energy .

In the model, the c-tron wavelength of
differs from the de Broglie wavelength of an electron of
1 eV (5.92 10^{5} m/s) of ca. 10^{-9} m.
The de Broglie wavelength of a moving particle, however,
is interpreted as the group velocity of matter waves. It
is assumed that the displacement of the **E**
- field maximum of the circulating wave for a certain particle
velocity is identical with the de Broglie wavelength.

The fraction 1/X of the field energy of the c-tron that
forms the external charge to the total field energy can
be calculated. The electron is regarded as sphere capacitor
with the stored energy E = 1/2 Q^{2} /C.

.
With the capacity,
the charge Q is

Equation (4)

With the values for the de Broglie frequency ,
the electron radius
determined above and the factor 1/X = 1/137, the electron
charge is calculated correctly to 1.603 10^{-19}
C. The dimensionless fraction 1/X equals the fine structure
constant,
whose derivation may have been similar to the above approach.
The figure 1/137 could be coincidence, too. If, nevertheless,
the known formula of
is inserted into eq. 4, the product inside the root equals
to
and Q = e. The coupling constant
perfectly confirms the current model.

Above approach is regarded the quantum realistic origin
and meaning of the coupling constant as the ratio of the
electric field energy forming the charge of the electron
to the total field energy .

## V. C-TRON MODEL AND QUANTUM MECHANICS

The interpretation of the electron as a circulating wave
or "photon" leads from one elementary constant, the spin
of the electron, to the elementary charge using another
well known dimensionless factor, the fine structure or coupling
constant. It gives a realistic meaning to the spin up and
down characteristics and provides an approach for the neutrino.

The model solves the mystery of the application of the
equation
to solid matter, which led to the wave aspects and the de
Broglie wavelength of matter. At the same time it opens
a door to the equivalence of energy and matter itself, E
= m c^{2}. Electromagnetic energy and matter are
equivalent because they are of identical nature, at least
shown for leptons so far. The application to hadrons or
quarks in a similar approach is assessed elsewhere[12].

The consequences of the model are the following:

o The particle can behave as a wave, as it is a wave. Nevertheless
it has the mass of m=E/c^{2}, which can cause all
effects a solid particle can cause.

o As the field revolves with the speed of light c, a macroscopic
prediction of the results other than with the statistical
methods of quantum mechanics is hardly possible.

o The Zeeman Effect is the effect of the c-tron oscillating
around the nucleus in reality.

o There is a real sense of rotation in c-tron particles.
In collision experiments, a parity violation must be observable
due to superposition of the local field strength - The decay
of W+ or W- particles exactly shows this effect: These can
only decay into leptons with matching helix direction[13].

o The fine structure constant
is the fraction of uncompensated field energy (charge) to
total field energy.
should therefore increase, if the proportion of the total
rotating field that is effective for an interaction increases
e.g. for small distances to the rotary electromagnetic field
or for high energies. This is exactly what is found[14].

o The model describes matter as closed loops of electromagnetic
waves, which do not have a point like centre and therefore
are free of singularities. Mathematical problems in dealing
with point-like mass concentrations therefore are eliminated.

o The old question why the electron accelerated on its
path around the nucleus shows no emission of energy is solved:
no charge circles the atom but an electromagnetic wave with
some excess field on the outside.

The model presented is based solely on the definition of
the electric field itself and on classical equations of
physics and early quantum physics. The model offers a great
unification and simplification for the quantum world of
the leptons so far.

## VI. EXPERIMENTAL VERIFICATION

Experimental verification to verify or falsify a basic
concept is required for every new theory. Absolutely identical
to proposing new experiments is the requirement that a theoretical
model shall fulfill the results of experiments already performed
- if it offers simplification compared to the assumptions
required by the old theory.

The natural and realistic explanation of the spin 1/2 is
a strong aspect in favor of the current particle model.
The experiments proposed would have been to observe the
spin balance in particle collision experiments and watch
for the law of spin conservation with the postulated spin
½ for the electron.

All formulations and correlations between e.g. charge and
fine structure constant have already been found. The formula
for
had been found earlier - but with what interpretation? Here
the interpretation is given in a simple and realistic way.

An oscillating movement of the electron and the interaction
with the magnetic field of the nucleus - the Zeeman-effect
- already has been discovered. Fine and hyperfine structures
in the spectra of light emitted by atoms had been found
and interpreted in this way. In the c-tron model it can
easily be visualized that each of the small loops or ripples
on the orbit[15]
corresponds to one field rotation with the De Broglie frequency.

The Moebius ribbon has another remarkable property, which
was found experimentally: a set of eight paper ribbons was
produced, which were marked with a polarity - red or black
for positive or negative field, both pointing up and down,
all twisted either to the left or to the right before closing
the loop with paper adhesive in identical manner. Table
1 shows the results of this little experiment.

TABLE 1: All possibilities of making
a paper strip Moebius ribbon. "cwi" and "cwo" is seen clockwise
with inward, respectively outward torsion, "cci" and "cco"
are the same seen counter-clockwise.

There are four positive and four
negative combinations. Basically they contain two groups
either being the clockwise inward or clockwise outward type.
Each particle has the property "inward" torsion and "outward"
torsion at the same time. It shows an inward twist if observed
from one side and outward twist if seen from the other side
in space. This means that each beam of electrons will sort
itself into two beams in a Stern - Gerlach experiment. Each
split beam, which is allowed to loose its orientation in
space (having been aligned by the magnetic orientation),
will have - again - both properties of inward torsion and
outward torsion depending on the orientation of the individual
particle as it enters a next quantum separator. This is
exactly what is been found experimentally.

As usually a continued split up of
every filtered beam in two beams is observed, it might be
worth while looking at the results under the aspect of the
current model. The + and - spin directions classically should
be identical in magnitude and energy and differ only in
the sense of rotation. The c-trons might show different
energy levels with the twist inward or outward and stay
that way if filtered once. Any splitting of the beam from
then on would have to be attributed to a randomization of
the orientation between the filters and a sorting in the
next magnetic field depending on how they enter this field.

There are experiments to verify the
model, e.g. the existence of the two basic versions of the
electron with torsion outwards and torsion inwards to give
an additional magnetic moment. It already has been performed
- the Stern Gerlach experiment led to the description of
the properties "spin up" and "spin down".

Experimental verification of the
proposed model can be performed with a Stern - Gerlach type
of experiment, too, where first the clockwise-inward are
sorted from the clockwise outward electrons. Then one beam
can again be sorted in two, if the electrons are allowed
to randomize - e.g. by thermal movement, by interaction
with other magnetic fields or by gaps in the magnetic field
keeping the alignment.

If, however, the electrons are e.g.
cooled and the orientation in the magnetic field constantly
is kept aligned, the filtered beam should not spilt in two,
again. Extremely low energy electrons with low beam densities
should be preferable in this test, no interaction with magnetic
fields of nearby atoms or neighboring electrons can be allowed.

Proposed for verification further
is a three dimensional, time dependent field simulation
to show the statistical - like behavior of the c-tron in
collision experiments. It cannot behave other than the observed
quantum waves; however, interactions are strictly local
and causal.

Second, mathematical integration
over volume and time of the field of the wave circling the
Moebius ribbon (e.g. based on eq. 1) is proposed. It should
prove that 1/137 of the total field-energy and mass equivalent
of the electron of eq. 2 are sufficient to form the field
of the elementary charge, stipulated that E_{0}
is determined from another source.

There is no accepted formulation
for the structure of the electron, rather than effort to
avoid this crucial question by using the abstracted "black
box" with quantum properties. The current formulation of
the electron is a basic concept and is not in contradiction
to accepted theories.

## LIST OF FIGURE CAPTIONS

FIG. 1: Electrical field strength E of an electromagnetic
sine wave (left side).

Right side: Vector field of E in the x-y plane (z-axis
out of plane): the vectors give the direction and magnitude
of the acceleration of test charges

(Color online) FIG. 2: Path of the
rotating electromagnetic wave forming the electron in shape
of a Moebius ribbon

FIG. 3: Rotating vector field in
the plane of the ribbon for time t = 0 and for t = t0 +
Pi/2. The radial component has a cos2 amplitude.

(Color Online) FIG. 4: Schematic
formation of the magnetic field of the positive c-tron in
three steps with the classical notation of the electron
as tiny magnet (lower right).

FIG. 5: Comparison of natural twist
of a Moebius ribbon with the classical electron orbit.

## REFERENCES

[1]
R. Gross, *Das Wasserstoffatom*, http://www1.physik.tu-muenchen.de/~kressier/SS02/phy34/4.pdf,
Chapter 4

[2]
A. Giese, *Proc. Spring Meeting of the DPG*, Dresden,
Germany, (March, 24, 2000), cited after http://www.ag-physics.org

[3]
R. L. Mills, http://www.blacklightpower.com/theory/book.shtml

[4]
F. Grotelüschen, *Der Klang der Superstrings,* dtv,
München,(1999), p. 30

[5]
O. Höfling, P. Waloschek, *Die Welt der kleinsten Teilchen*,
8474, rororo, Reinbek bei Hamburg, (1988), p. 455

[6]
F. Grotelüschen, *Der Klang der Superstrings*, dtv,
München, (1999), p. 30ff

[7]
Mohr & Taylor *Physical Constants*, (1999) http://pdg.lbl.gov/2000/consrppbook.pdf

[8]
R. Gross, *Das Wasserstoffatom*, lectures, http://www1.physik.tu-muenchen.de/~kressier/SS02/phy34/4.pdf,
Chapter 4

[9]
A. Giese, (Spring Meeting of the DPG, Dresden, Germany March,
24, 2000), after http://www.ag-physics.org

[10]
R. L. Mills, http://www.blacklightpower.com/theory/book.shtml

[11]
W. H. Westphal, *Physik,* Springer Verlag Berlin
(1963), p. 620

[12]
C. R. Caesar, *Elektromagnetisches Teilchenmodell sowie
Verfahren zu seiner Berechnung und Simulation,* (submitted
2003.09.08, published 2005.04.14) German Patent Application
DE 103 41 341 A1

[13]
O. Höfling, P. Waloschek, *Die Welt der kleinsten Teilchen*,
8474, rororo, Rheinbek b. Hamburg 1984, p. 420

[14]
http://www.didaktik.physik.uni-erlangen.de/grundl_d_tph/lexikondf.html#Feinstrukturkonstante

[15]
R.Gross,* Das Wasserstoffatom*, http://www1.physik.tu-muenchen.de/~kressier/SS02/phy34/4.pdf,
p. 133

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