On the existence of natural self-oscillation of a free electron

0.1 MB·English

by  Zhixian Yu

On the existence of natural self-oscillation of a free
electron
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0 Zhixian Yua,b, Liang Yuc,∗
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r aDepartment of Physics and Astronomy, University of New Mexico, Albuquerque NM
a
87131 USA
M
bCollege of Physics Science, Qingdao University, Qingdao 266071 China
cLaboratory of Physics, Jiaming Energy Research, 36 Yushan Rd, Qingdao 266003 China.
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]
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p
- Abstract
s
s
The possibility of the existence of natural self-oscillation of a free electron is
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l suggested. This oscillation depends on the interaction of the electron with
c
. its own electromagnetic fields. Suitable standing wave solutions of the elec-
s
c tromagnetic fields are chosen. A kind of displacement dependent electric po-
i
s tential and mechanism of energy exchange between velocity and acceleration
y
h dependent electromagnetic fields are analyzed. Conditions for the existence
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of natural self-oscillation are given.
[

2 Keywords:
v natural self-oscillation, standing wave solution, displacement dependent
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potential
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2 PACS: 03.50.De
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.
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0 1. Introduction
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Thesubtle andbeautifulexperiment ofDehmelt et alshowed thatasingle
:
v
isolated electron might be driven by static electric field in a Penning trap to
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oscillate as a “mono-election oscillator”[1]. We want to show here that a free
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electron may also possess natural self-oscillation by itself under some selected
a
conditions.
The property of a free electron has been discussed by many authors
through both classical and quantum theories[2, 3]. There are still difficulties
∗
Corresponding author. Tel.: [phone]. Fax: [phone].
Email addresses: [email protected](Zhixian Yu), [email protected](Liang Yu)
Preprint submitted to Annals of Physics March 13, 2012
and paradoxes about the model of electron. Here we take a simple model of
electron used by Konopinsky[4] and Dehmelt[5], in which a free electron is a
system consisting of a particle with mass m, charge e and distributed electro-
magnetic fields. These fields may have their own energies, momentums and
angular momentums. The more detailed internal structures of the electron
are neglected at first, thus many difficulties about the model of electron are
avoided.
2. Standing wave solutions of electromagnetic fields of an electron
in harmonic oscillation
The electromagnetic fields of an electron in harmonic oscillation may be
derived from the vector potential or Hertz vector of the oscillating electron[6,
7, 8]. Following the method of Bateman[9] or Adler, Chu and Fano[10], we
may also solve the wave equation of the electromagnetic fields on the basis of
Maxwellequationsinsphericalcoordinatesfirst, andthenmatchthesolutions
with the motion of the electron. From both methods we may have the dipole
term of the electron’s electromagnetic fields as[6]
1 ik
E = 2eZ [ − ]cosθeiωte∓ikrrˆ (1)
r e0
r3 r2
1 ik k2
E = eZ [ − − ]sinθeiωte∓ikrθˆ (2)
θ e0 r3 r2 r
ik k2
H = eZ [− − ]sinθeiωte∓ikrφˆ (3)
φ e0 r2 r
where E and E are electric fields in r and θ directions, H is magnetic
r θ φ
field in φ direction, rˆ, θˆ and φˆ are corresponding unit vectors (Besides (1),
(2) and (3), there are also a monopole Coulomb field of the electron and a
dipole field at the origin[7], but they are not useful in following analysis),
r is the distance from the electron to a field point, e is charge of electron,
Z is amplitude of its oscillation, terms of 1/r3, 1/r2 and 1/r correspond
e0
to oscillating static fields, velocity dependent near fields and acceleration
dependent fields respectively.
The ∓ signs in the space phase factor represent that retarded and ad-
vancedtimesolutionsareusedrespectively. These twoarecomplexconjugate
solutions which represent progressive divergent wave and regressive conver-
gent wave, and any combination of them is also an allowed solution. Usually
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the solution of advanced time is rejected by the requirement of causality,
but here it is used as one of the conjugate waves, there is no problem about
causality. According to Adler, Chu and Fano the two kinds of space factors
may add together to form complete standing wave modes[10] of sinkr and
coskr. We choose the sum of retarded time and advanced time solutions to
form standing wave solutions. Then their space and time phase factors are
formed as
ei(ωt−kr) +ei(ωt+kr) = 2(coskrcosωt+icoskrsinωt) (4)
thus for the static and acceleration dependent fields, the phase factors are
2coskrcosωt. For the velocity dependent electromagnetic fields, they are
2coskrsinωt, since
−i2(coskrcosωt+icoskrsinωt) = 2(coskrsinωt−icoskrcosωt) (5)
In order to match the motion of the oscillating electron, we have to choose
the static electric fields E , E and acceleration dependent electromagnetic
sr sθ
fields E , H with their phase factors as
aθ aφ
4eZ 2eZ
eo eo
E = cosθcoskrcosωt E = sinθcoskrcosωt (6)
sr sθ
r3 r3
2k2eZ 2k2eZ
eo eo
E = sinθcoskrcosωt H = sinθcoskrcosωt (7)
aθ aφ
r r
The velocity dependent electromagnetic fields E , E and H with their
vr vθ vφ
standing wave phase factors are chosen as
4keZ 2keZ
eo eo
E = cosθcoskrsinωt E = sinθcoskrsinωt (8)
vr vθ
r2 r2
2keZ
eo
H = sinθcoskrsinωt (9)
vφ
r2
3. Displacement dependent electric potential and force for self-
oscillation of a free electron
It is known that for any electric field E at a point in free space, there is
a corresponding energy density E , which is
d
1
E = E2 (10)
d
8π
3
and the total energy E of the electric field is the volume integration of its
energy density through whole space, which is
1
E = E2dv (11)
8π Z
Thus for the electrical fields E and E of the static zone, there is the
sr sθ
corresponding energy E . As the time factor is suppressed at first,
s
1
E = (E2 +E2 )dv
s 8π Z sr sθ
1 rmax 4eZ 2eZ
= [( eo cosθcoskr)2 +( eo sinθcoskr)2]2πr2drsinθdθ
8π Z r3 r3
rmin
(12)
where r and r are upper and lower limits of the integration. It’s
max min
reasonable to take the upper limit r as infinite or the total length of
max
certain number of standing wave lengths, and the lower limit r as M r ,
min s 0
where r is taken as the classical radius of electron which is defined as[11]
0
e2
r = = 2.82×10−13cm (13)
0 mc2
and M is an undetermined numerical constant. Taking r = ∞ and
s max
r = M r we get the integration of E as
min s 0 s
1+cos(2kM r ) ksin(2kM r )
E = 4e2Z2 [ s 0 − s 0
s e0 6(M r )3 6(M r )2
s 0 s 0
k2cos(2kM r ) 2 ∞ sin2kr
− s 0 + k3 dr] (14)
3M r 3 Z r
s 0 Msr0
Since r and M r are comparatively small, we may use the approximations
0 s 0
∞ sin2kr π
cos(2kM r )≈1 sin(2kM r )≈2kM r dr≈ (15)
s 0 s 0 s 0 Z r 2
Msr0
Then we have
1 2k2 π
E = 4e2Z2 [ − + k3] (16)
s e0 3(M r )3 3M r 3
s 0 s 0
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Since Z is the amplitude of the harmonic oscillation of the electric fields,
e0
E is a displacement dependent energy which is proportional to the square of
s
Z and may has a kind of restore force with maximum value f , which is
e0 s0
∂E 1 2k2 π
f = − s = −8e2[ − + k3]Z (17)
s0 ∂Z 3(M r )3 3M r 3 e0
e0 s 0 s 0
This restore force will drive the electron to take harmonic oscillation and we
may take energy E equal the maximum of the electron’s kinetic energy E ,
s k
which is
1 1
E = mv2 = mc2k2Z2 (18)
k 2 0 2 e0
since v = ckZ is the amplitude of the electron’s velocity. Then we have
0 e0
E = E (19)
s k
1 2k2 π 1
4e2Z2 [ − + k3] = mc2k2Z2 (20)
e0 3(M r )3 3M r 3 2 e0
s 0 s 0
π 2 1mc2 1
k3 −( + 8 )k2 + = 0 (21)
3 3M r e2 3(M r )3
s 0 s 0
then by (13) we get
π 2 1 1
k3 −( + )k2 + = 0 (22)
3 s 3M r 8r s 3(M r )3
s 0 0 s 0
where k is labeled as k . (22) gives the numerical relation of k with M
s s s
and r , and is one of the selection conditions for the natural self-oscillation
0
of the free electron. This equation may be modified by some factors such as
the range of the upper limit r , the relativistic variation of the electron’s
max
kinetic energy, but the main feature of (22) is not affected.
4. The energy storage and exchange of the velocity and accelera-
tion dependent electromagnetic fields
Using the electromagnetic fields E , E and H of (8) and (9), we may
vr vθ vφ
calculate the energy of the velocity dependent field E as
v
1
E = (E2 +E2 +H2 )dv
v 8π Z vr vθ vφ
(23)
4k2e2Z2 rmax cos2kr
= e0 (4cos2θ+2sin2θ)2πr2drsinθdθ
8π Z r4
rmin
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Taking r = ∞ and r = M r , where M is also an undetermined
max min v 0 v
constant as the M in (14), we get
s
8k2e2Z2 1+cos(2kM r ) ∞ sin2kr
E = e0[ v 0 −2k dr] (24)
v 3 M r Z r
v 0 Mvr0
Since r and M r are comparatively small, we may use the approximations
0 v 0
∞ sin2kr π
cos(2kM r )≈1 dr≈ (25)
v 0 Z r 2
Mvr0
then we have
8k2e2Z2 2
E = e0( −kπ) (26)
v
3 M r
v 0
The acceleration dependent energy E may be calculated from the acceler-
a
ation dependent electromagnetic fields E and H in θ and φ directions.
aθ aφ
According to (2) and (3), E is
a
1
E = (E2 +H2 )dv
a 8π Z aθ aφ
4k4e2Z2 rmax 2sin2θ
= e0 cos2kr2πr2drsinθdθ (27)
8π Z r2
rmin
4k4e2Z2 sin2kr rmax
= e0(r + )
3 2k (cid:12)
(cid:12)rmin
(cid:12)
(cid:12)
where k = 2π/λ, λ is wave length of the standing wave. The acceleration
dependent energy E contains a number of energy bands which are standing
a
wave spherical shells with equal energy. We take the number of the bands
each with width λ/2 as N. The central band is bisected by the electron at
its center, half of its width is 1 · λ, thus the upper limit is r = (N + 1)λ.
2 2 max 2 2
The lower limit here may be taken as r = 0, since M r is small and here
min s 0
r is not in the denominator as above. Thus
min
4k4e2Z2 sin2kr rmax=(N+12)λ2
E = e0(r + )
a 3 2k (cid:12)
(cid:12)rmin=0
(cid:12)
4k4e2Z2 1 λ (cid:12)
= e0(N + ) (28)
3 2 2
4k3e2Z2 1
= e0(N + )π
3 2
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The electric and magnetic fields of E or E respectively have equal time
v a
phase and they can not exchange energy within E or E alone, but the
v a
electromagnetic fields of E and E are out of phase π/2 in time, thus E and
v a v
E can exchange their stored energy between each other. The exchange of
a
energy between velocity and acceleration dependent fields of an oscillating
electric dipole had been discussed by Mandel[12] and Booker[13] and they
also showed that there was energy flow between these two fields. This kind
of energy flow is also discussed in antenna theory. The condition for the
complete energy exchange between E and E is
v a
E = E (29)
v a
From (26) and (28), we get
8k2e2Z2 2 4k3e2Z2 1
e0[ −kπ] = e0(N + )π (30)
3 M r 3 2
v 0
then we have
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k = (31)
v M r [(N + 1)π +2π]
v 0 2
here k is labeled as k . This is a selection condition of k with M and r .
v v v 0
For the free electron to have any kind of persisted natural self-oscillation,
both conditions of k in (22) and k in (31) should be satisfied at the same
s v
time, that is
k = k (32)
s v
which is the combined relation of possible natural self-oscillation.
5. Short discussion
Natural self-oscillation is popular in electric circuits, microwave structure
or mechanical system. Standing wave solutions of electromagnetic waves are
also used in antenna theory. Above energy analyses show that a free electron
may also possess natural self-oscillation through the interaction with its own
electromagnetic fields under certain conditions. These conditions are deter-
mined by the electron’s standing wave modes, the number N of its standing
wave bands in acceleration dependent fields and two numerical constants in
its energy integrations. For an electron oscillating in a Penning trap as that
in the experiment of Dehmelt et al for the “mono-electron oscillator”, there
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is a strong static magnetic field along the oscillating direction of the elec-
tron to keep the oscillating electron localized and in stable movement. For
the self-oscillation of a free electron this kind of external auxiliary is unnec-
essary, since localization is not a problem for a free electron. The natural
self-oscillation will have some effects on the interaction of the electron with
its environment and thus could be measured by suitable experiments. We
will discuss these in connection with the stability of natural self-oscillation
of a free electron. Here we only give an energy analysis and suggest that its
existence is possible.
6. Acknowledgements
TheauthorLiangYuthankstheteaching andkindadviceoflateprofessor
Leonard Mandel.
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