Download Simple Models for Classical Electron Radius and Spin

Simple Models
for
Classical Electron Radius and Spin
Vedat Tanrıverdi
Physics Department, METU
[email protected]
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Abstract
Some simple models about classical electron radius and spin are considered.
These simple models are considered for a better understanding spin and its
relation with other electron properties, i.e. charge, inertia, energy. These
models have different inconsistencies with current theories, however they are
still helpful for understanding.
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Introduction
In a previous work, history of electron spin is reviewed [1] and some problems
are underlined related with it. In physics, electron is considered as a point
particle and it has spin and magnetic moment. However, there is not any
explanation why a point particle have magnetic moment and angular momentum. It is said that spin is an intrinsic property. It is just escaping from
the question instead facing it. More importantly, to define a point particle
one could have 3 independent variable, however electron is defined with 4
quantum numbers. These are most basic conceptual problems related with
electron spin. Yet, there is not any consistent and explanatory theory on it.
These basic problems are related with how we treat electron spin. Some of
basic problems are conceptual and some of them are qualitative. In the conceptual part, most of the answers addressed to one word ”intrinsic”, there is
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no underlying answers or relations. Other than intrinsic, we have some comments on experimental results and relations. These relations are basically to
treat experimental results, not to explain spin in a causal way. Qualitative
parts are related with basically experimental observations. Experimental extractions from observations related with electron can be considered having
mass, charge, 4 quantum number, spin angular momentum, magnetic moment; and any qualitative model should be consistent with the observations
in the Stern-Gerlach experiment, having two distinct trace; in the sequential
Stern-Gerlach experiment, not depending on the previous observation if spin
is measured in a different perpendicular direction; and also should give correct experimental measurements. A successful model should explain these
experimental results in a qualitative way.
In this work, some simple models related with electron are considered
and aim of this paper is not to explain all things in a qualitative way. The
ones related with classical electron radius calculations are reconsidered and
two other simple model is also considered. These are only toy models and
they are studied to understand the problem in a better way. These models
are considered in the concepts of angular momentum, energy and inertia and
analyzed by using some simple relations.
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Models
Here, 4 different models are considered; spherical shell model, spherical
model, two point particle model and three point particle model. In the
first one, the electron is considered as a spherical shell, this is a model that
is considered in some quantum mechanics books[2, 3]. This model is also
considered in the calculations of classical electron radius[4]. In this model,
electron is considered as a spherical shell with radius a carrying charge e
and having mass m̃. Total charge will be always equal to the electron
charge e = −1.602 × 10−19 C, and mass m̃ will be equal to electron mass
m = 9.109 × 10−31 kg except rotational energy consideration. For the spherical model, electron is considered as a sphere instead of shell in a similar way.
For the two and three particle model, electron is considered as constituent
either from two or three particle, having charge e/2 or e/3 and mass m̃/2 or
m̃/3, Figure 1. Total of these constituent particles will be equal to mass of
the electron, m, except rotational energy consideration.
The radius a will be calculated by using angular momentum and energy.
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We will mainly consider that angular momentum of the shell is the source
of spin angular momentum of the electron, and the energy of the shell in
different ways corresponds to electron mass. In a case, vr /c speed at rim
to the speed of the light ratio will also be calculated. For these four model
we will apply three calculation scenario; the first one is the view of angular
momentum as a source of spin, the second one is similar to classical electron
radius calculations and the final one is a consideration of mass as a result
of total of rest energy of the system and its rotational kinetic energy. In
the last case, differently from classical electron radius calculations, source of
the mass or the inertia is considered as rotational kinetic energy plus rest
energy. While doing this, we defined m̃ as the mass of shell, sphere or total
rest masses of sub-particles. Since we have an extra parameter, instead of
calculating vr /c ratio we take it as 1, as a limit value, and calculated electron
radius due to these considerations. Due to changes in moment of inertia or
electrostatic energy, there will be some changes among the model results.
• Angular momentum
In this consideration, spin angular momentum of the electron is taken simply
equal to the angular momentum of rotating shell. Rotating shell has moment
L = Iw and spin angular
of inertia I = 23 ma2 and angular momentum
√
3
~
momentum has a magnitude S = |S| = 2 h̄. In the vr /c = 1 case, here
vr = wa speed at the rim, if L = S one can find a = 5.018 × 10−13 m.
For spherical model only change comes from the difference in moment of
inertia I = 52 ma2 , then in the vr = c case a = 8.364 × 10−13 m.
For the two or three particle model, we consider 2 or 3 particle with
masses m/2 or m/3 rotating in a plane around the center of mass, also in a
circular path with constant radius a. Then moment of inertia of the systems
will be same, I = ma2 . Then in the vr /c = 1 case, a = 3.345 × 10−13 m for
both case.
• Electrostatic energy
In the classical electron radius calculations, electron is considered as a spherical shell or sphere, electron’s charge and mass distributed on or within either
of these uniformly. For such cases, electrostatic potential energy is calculated
for a radius a, and this energy is considered as the source of inertia. After
equating the electrostatic energy to the rest energy of the electron, electron’s
radius are calculated and this radius is named as classical electron radius.
For uniformly distributed charge, the electrostatic energy for the spherical
1 e2
shell is UES = 21 4π
. If one considers this energy as the source of inertia,
0 a
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(a) 2 point particle model
(b) 3 point particle model
Figure 1: 2 and 3 point particle models
2
1
e
then it should be equal to E = mc2 . Then one can obtain a = 12 4π
2 and
0 mc
−15
its numerical value is a = 1.414 × 10 m. In this case, by using L = S √speed
h̄
at the rim to the speed of the light ratio can be calculated as vr /c = 3 4 3 mca
and its numerical value is vr /c = 354.9.
If similar assumptions are considered, one can obtain a = 1.696 × 10−15 m
and vr /c = 493.1 for spherical model; a = 3.534 × 10−16 m and vr /c =
946.6 for two particle model; and a = 5.441 × 10−16 m and vr /c = 614.9 for
three particle model. For spherical model charge is considered as uniformly
distributed within the sphere, for two and three particle model charge is
equally split amongst the particles.
• Rotational motion and energy
If one considers total of rest energy of the shell and its rotational kinetic
energy as the source of inertia, then E = m̃c2 + 2I1 L2 should be equal to
E = mc2 , here m̃ is the mass of the shell and m is the mass of the electron.
Since total of rest energy of the shell and its rotational kinetic energy is
considered as the source of the inertia, m̃ is different than mass of electron
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m. Then one can obtain the relation mc2 = m̃c2 (1 + 13 vc2r ). If vr /c is taken
as 1, m̃ = 34 m; so m̃ > 34 m. If vr /c is taken as 0 (very close to zero),
m̃ = m;
so m̃ < m. Hence m > m̃ > 43 m. If one considers L = S, then
√
m̃ = 3 4 3 vh̄r a and if one considers vr /c as 1, then from previous inequality
√
√ h̄
h̄
one can obtain 3 4 3 mc
< a < 3 mc
. From this, one can reach the numerical
−13
values 5.018 × 10 m < a < 6.691 × 10−13 m.
If one follow similar steps, then one can obtain 8.364 × 10−13 m < a <
10.04 × 10−13 m for spherical model, 3.345 × 10−13 m < a < 5.018 × 10−13 m
for two and three particle models.
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Comparison
Four different model is resulted in different numerical values due to their considerations. Smallest radius is obtained in two point particle model from the
electrostatic energy consideration as a = 3.534×10−16 m. Biggest radius is obtained spherical model rotational energy consideration as a = 10.04×10−13 m
as an upper limit. The other results are between these values. However all
these are much bigger than previous limiting studies[5], even most results are
bigger than charge radius of the proton, 8.751 × 10−16 m [6].
Smallest values for radius are obtained from the electrostatic energy considerations, however these considerations are resulted in very high vr /c ratio.
For the smallest radius value a = 3.534 × 10−16 m, the biggest vr /c ratio
obtained as 947. This is really huge number for vr /c ratio, and none of our
current physical theories support such a thing.
If one eliminates results with vr /c > 1, we left with results at the order
10−13 m. These are really big radius values for electron. From these results,
one can say that in this format these models are away from the experimental
measurements.
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Conclusion and discussion
The above models are simple ones that are considered to understand spin
problem in a better way. These classical models are only dealt with radius
and rotation velocity in terms of angular momentum and energy. Magnetic
moment of the electron and its relation with the magnetic field should also
be considered for such cases for better understanding.
Now let us discuss the assumptions that have been done in this work.
One of the assumption is that electrostatic energy is considered as source
1 e2
, where c̃ is a constant depending on the
of inertia as saying mc2 = c̃ 4π
0 r
model, r is the radius of the electron in the model. Is it a valid assumption?
We have theories on some bound systems like nuclei and mesons. In the
nuclei, mass comes from masses of nucleons and mass of formed nuclei is
always smaller than total masses of protons and neutrons in the nuclei. The
energy difference is considered as binding energy and this is one of the cases
that a bound system has less mass than the constituted particles total mass.
In the nuclear theory we do not have any contribution of the charge to the
mass. So if we want to use knowledge of nuclear theory we can write a theory
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whose constituent particles total mass bigger than the constituent particle.
Also we should not include charge as the contributor to the mass. Also in
meson theory, total mass of the constituent particles can be bigger than the
meson[7]. In that theory charge does not contribute to the mass. So, the
1 e2
relation mc2 = c̃ 4π
is not supported by nuclear theory and meson theory.
0 r
Hence, we can say that classical electron radius calculations are not true if
we consider the successes of the meson and nuclear theory.
In meson theory, rest energies of constituent quarks and kinetic energy
of them contribute to mass of the meson. The interaction energy, i.e., color,
spin orbit and spin-spin, in general reduces the mass of the meson. Similar
reduction is also valid for the nuclear theory. So mc2 = m̃c2 + L2 /2I is not
valid always. In the mc2 = m̃c2 + L2 /2I assumption, the biggest value that
m̃ can take is m. So we can modify this equation by adding some negative
constants. This negative constant can not be found in a deterministic way,
since we do not have a theory that describes such interactions. One of the
possibility is preon models to explain such interactions, however they are
not advanced enough to determine this negative constant. These toy models
can be improved in different ways and these improvements can be helpful to
understand spin in a better way.
References
[1] V. Tanrıverdi, vixra: 1504.0181
[2] J. L. Powell and B. Crasemann, Quantum Mechanics. Addison-Wesley,
(1961).
[3] W. M. Wilcox, Qunatum Principles and Particles. CRC Press, (2012).
[4] Wikipedia, Classical electron radius. Web. 30 Sep. 2015.
[5] H. Dehmelt, Physica Scripta. T22, 102 (1988).
[6] ”CODATA Internationally recommended 2014 values of the Fundamental Physical Constants”, NIST, Web. 30 Sep. 2015.
[7] S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
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