proof
145
7.5 Myon
We now have discussed all conceivable possibilities, which two elementary vortices can
form: the creation of a pair for like charge and the annihilation under emission of photons
via the formation of the positronium as an intermediate result for unequal charge. Next
another elementary vortex shall be added and all different possibilities and configurations
will be derived, which can be formed by amassing or overlapping.
The positronium can, as said, only take the in fig. 7.3 shown bound structure, if it is
stabilized from the outside. This task now a further electron shall take over. According to
the shell model the innermost elementary vortex an electron (e
-
), is overlapped by a
positron (e
+
) and that again overlapped by an electron (e
-
).
With an in
the sum single negative charge, a completely symmetric structure as well as a
half-integer spin this particle will show a behaviour corresponding to a large extent to that
of the electron. Merely the mass will be considerably larger, because every vortex each
time compresses the other two.
It therefore concerns the myon
which also is called "heavy electron". The myon
was discovered 1937 in the cosmic radiation (Anderson and others).
In fig. 7.5 are drawn above each other the shell-shaped structure of the myon and the
electric field E(x) of the three elementary vortices.
It is visible that merely in the proximity of the particle the actual course of the field
deviates from and is smaller, than the course which theoretically can be expected for a
single negatively charged body. The difference is marked by a hatching.
We now can tackle the calculation of the myon. For that the following considerations to
begin with are useful:
Mass is an auxiliary term founded in usefulness, which describes the influence of the
electromagnetic field on the speed of light and with that on the spatial extension of the
"point mass".
Without exception the local cosmic field E
o
has an effect on a free and unbound
elementary vortex, thus on an individual e
-
or e
+
, and determines so its size and its mass.
But as long as we haven't determined this field strength, the calculation of its quantum
properties won't succeed.
Instead the masses of compound particles will be compared to each other, which are so
heavy that the field strength of the neighbouring vortices is predominant over the basic
field E
0
, so that a neglect of E
o
seems to be allowed. The course of the calculation is made
for all elementary particles in the same manner, which is explained hereafter.
146
calculation of the vortex fields
Fig. 7.6: Calculation of the electric field strength E(r) of
the myon from its dependency on radius
proof
147
7.6 Calculation of the vortex fields
The tension voltage of an elementary vortex, like for a spherical capacitor, is determined
by integrating over the electric field strength from the inner radius r
i
up to the outer radius
r
a
:
(7.1)
For the electron (r
i
= 0 und r
a
= r
e
) we already have carried out the integration and
determined the tension voltage to be 511 kV (equation 6.31 *).
Doing so we further had discovered that it won't change, if the radius r varies. Even for a
shell configuration, in which electrons and positrons alternately overlap, the approach is
valid:
U
1
= U
2
= U
3
= U
4
= ... = U
n
(7.2)
At a certain radius all elementary vortices show the same density of field lines and with
that also the identical field strength, so that we can solve the integral (7.1) for the each
time neighbouring vortex shells and can compare the results:
At the radius r
1
with E(r
1
) = E
1
the agreement, according to equation 7.1* (fig. 7.6), is
valid for the innermost and the overlapped vortex shell.
At the radius r
2
with E(r
2
) = E
2
the agreement according to equation 7.1** (fig. 7.6) is
valid analogously for the 2nd and 3rd shell.
If still more shells are present, then we can arbitrarily repeat this procedure. For the radius
of each shell we always obtain relation 7.3, which, related to the innermost radius,
provides the following simple expression for the individual radii:
r
2
= 2 * r
1
; r
3
= 3 • r
1
; ... ; r
n
= n * r
1
(7.4)
From the comparison of the integration results 7.1* and 7.1** follows further that all
elementary vortices produce the same field strength:
E
1
= E
2
= E
3
= ... = E
n
(7.5)
We infer from the transformation table (fig. 6.18, eq. 6.27) that the field strengths E and H
decrease with 1/r. In fig. 7.5 the decrease of the fields with 1/r is shown. Up to the radius
r, the field of the innermost vortex E
1
has worn off to the value E
31
= - E
1
• (r
1
/r
3
).
This field is overlapped by E
32
= E
2
* (r
2
/r
3
) as well as the cosmic basic field E
o
:
E(r
3
) = E
31
+ E
32
+ E
0
= E
1
• (r
2
- r
1
)/r
3
+ E
o
(7.6)
The local basic field E
o
is not known, but it is very small with regard to the field of the
neighbouring vortex shells, so that a neglect seems to be allowed.
From equation (7.6) in this way follows with the radius relation (7.4):
(7.7)
For the shell-shaped configuration of the myon (fig. 7.5) relation (7.7) indicates, which
field the outside vortex shell is exposed to. From this can already be seen, how much it is
compressed thanks to the field dependent speed of light and how much its mass as a
consequence is increased.