14. Conclusions

The Spinning Particle model provides an analytical description of the classical electron that is determined by the evolution of a single point
r
, interpreted as the center of charge CC, that is a different point than the center of mass CM, and satisfies a system of ordinary fourth-order differential equations, in such a way that all observables can be defined in terms of this point and the subsequent time derivatives.
This particle has two distinguished points, the center of mass
q
and the center of charge
r
, where the CC velocity
u
is |
u
| = c and the CM velocity
v
satisfies |
v
| < c.
The motion of the CC is, in general, a curve with torsion what is known in the literature as the zitterbewegung motion.
For the electron and positron, the intrinsic magnitudes are in natural units:
mass m = 1
the separation between CC and CM
R
o
= 1/2
spin S = m
R
o
c = 1/2
electric charge e = ±1
electric dipole moment d = e
R
o
= 1/2
magnetic dipole moment
µ
B
= eℏ/2m = 1/2.
We have derived the equations of the dynamics and explored the interaction of the spinning Dirac particle with uniform and oscillating electric and magnetic fields. We use the minimal coupling interaction and the classical parameters m and e, are those of the electron. The analysis is a kind of microscopic analysis at the scale of the separation between the CC and the CM.The motion of the center of mass of the particle under these fields, is the expected motion, similar to the motion of the point particle with a slight zitter. The new feature has been to determine how the spin evolution is affected by the interaction. ​In the dynamics of the electron we have two different scales: The scale of the separation between the CM an CC, of order of
R
o
≈ 1
−13
0
m, to show the relative motion between these two points, and the macroscopic scale at the laboratory level. The other is the time scale. The CC is turning around the CM at the speed of light and in natural units the time taken during a turn is π γ(v) where v is the velocity of the CM
q
, and this period is of order of 1
−21
0
s, for low velocity Dirac particles and multiplied by the γ(v) factor for high energy particles.To depict in the same figure the global and the internal motion we have needed to use very high external fields. The numerical analysis has been performed with very high hypothetical fields that are difficult to produce at the laboratory. Since the dynamical equations are relativistically invariant, they can be solved for arbitrary high velocities. If we use laboratory fields we will not distinguish at that scale a difference between the CC and CM motions, but we can always analyze the spin evolution.​​The uniform and oscillating electric field seems not to affect the evolution of the CM spin
S
CM
that remains in almost constant direction during a few periods of the relative internal motion at the analyzed low velocity. ​The magnetic field produces the precession of the
S
CM
with the Larmor angular velocity, and this spin precesses backwards, in the opposite direction to the angular velocity of the CM motion, such that when the CM has given two turns the spin comes back to its initial orientation. In the interaction with an oscillating magnetic field, each time the magnetic field reverses, the angular velocity of the CM reverses and also the Larmor angular velocity of the center of mass spin. Combinations of oscillating electric and magnetic fields produce the modification of the CM spin orientation, although in these examples the CM velocity is not high and the absolute value of the spin is not modified.

15. Dynamic equations and Particle Initial Conditions

In this second article we add the influence of external electric and magnetic fields both uniform and oscillating, so the dynamic equations of the electron will be as follows.
The Center of Mass is accelerated, it moves according to equation (3):

v
t
=
2


2
t
q
=
e
mγ(v)
E
+
u
×
B
-
1
2
c
v

E
+
u
×
B

v

(
9
)
and the remaining equations are the same as in the free case :
a
=

u
t
=
2

r

2
t
=
2
c
-
v
.
u
2

q
-
r


q
-
r

(
10
)
u
=

r
t
(
11
)
v
=

q
t
(
12
)
To integrate the system of differential equations, we have to establish the appropriate boundary conditions for the positions and velocities of both points, the CC and the CM of each particle . These are the 12 boundary values for the variables
r
0
,
u
0
,
q
0
and
v
0
. These parameters are those parameters that define the relationship between the Center of Mass observer of the particle and any arbitrary inertial observer or laboratory observer who sees the CM of the particle moving at the speed
v
.
The initial or boundary conditions of the particle are calculated as a function of the parameters that describe the system (user defined parameters in the code of section 17): - The initial location of the Center of Mass of the particle is defined by the coordinates
d
=(dx,dy,dz). - The velocity
v
of the Center of Mass of the particle is defined by its module v, β the zenithal angle and λ the azimuthal angle. - The Center of Mass Spin is defined by its zenithal angle θ and azimuthal angle ϕ. - The Center of Charge initial location is defined through the initial phase ψ.These initial conditions are the same no matter if the particle is free or is in the presence of external fields, and they have been derived in Part I.Section17 ​