Periodic electron motion in the field of intensive plane light wave
Results of integration of the Lorentz-Dirac equation for electron motion in the field of intensive light wave are given in this work. Diagrams and parameters, which characterize the periodic motion of a relativistic electron, are shown. Formulas of the spectral-angular distribution of the electromag...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2001 |
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| Мова: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | Periodic electron motion in the field of intensive plane light wave / Yu.N. Grigor’ev, O.D. Zvonaryova // Вопросы атомной науки и техники. — 2001. — № 1. — С. 92-93. — Бібліогр.: 2 назв. — англ. |
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Grigor’ev, Yu.N. Zvonaryova, O.D. 2015-03-18T17:34:37Z 2015-03-18T17:34:37Z 2001 Periodic electron motion in the field of intensive plane light wave / Yu.N. Grigor’ev, O.D. Zvonaryova // Вопросы атомной науки и техники. — 2001. — № 1. — С. 92-93. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 41.60.-m, 41.85.-p., 29.27.-a https://nasplib.isofts.kiev.ua/handle/123456789/78520 Results of integration of the Lorentz-Dirac equation for electron motion in the field of intensive light wave are given in this work. Diagrams and parameters, which characterize the periodic motion of a relativistic electron, are shown. Formulas of the spectral-angular distribution of the electromagnetic field irradiated by a relativistic electron, moving towards electromagnetic wave, were obtained. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Theory and technics of particle acceleration Periodic electron motion in the field of intensive plane light wave Периодическое движение электрона в поле интенсивной плоской световой волны Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Periodic electron motion in the field of intensive plane light wave |
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Periodic electron motion in the field of intensive plane light wave Grigor’ev, Yu.N. Zvonaryova, O.D. Theory and technics of particle acceleration |
| title_short |
Periodic electron motion in the field of intensive plane light wave |
| title_full |
Periodic electron motion in the field of intensive plane light wave |
| title_fullStr |
Periodic electron motion in the field of intensive plane light wave |
| title_full_unstemmed |
Periodic electron motion in the field of intensive plane light wave |
| title_sort |
periodic electron motion in the field of intensive plane light wave |
| author |
Grigor’ev, Yu.N. Zvonaryova, O.D. |
| author_facet |
Grigor’ev, Yu.N. Zvonaryova, O.D. |
| topic |
Theory and technics of particle acceleration |
| topic_facet |
Theory and technics of particle acceleration |
| publishDate |
2001 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| title_alt |
Периодическое движение электрона в поле интенсивной плоской световой волны |
| description |
Results of integration of the Lorentz-Dirac equation for electron motion in the field of intensive light wave are given in this work. Diagrams and parameters, which characterize the periodic motion of a relativistic electron, are shown. Formulas of the spectral-angular distribution of the electromagnetic field irradiated by a relativistic electron, moving towards electromagnetic wave, were obtained.
|
| issn |
1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/78520 |
| citation_txt |
Periodic electron motion in the field of intensive plane light wave / Yu.N. Grigor’ev, O.D. Zvonaryova // Вопросы атомной науки и техники. — 2001. — № 1. — С. 92-93. — Бібліогр.: 2 назв. — англ. |
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2025-11-25T22:51:41Z |
| last_indexed |
2025-11-25T22:51:41Z |
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1850575169172537344 |
| fulltext |
PERIODIC ELECTRON MOTION
IN THE FIELD OF INTENSIVE PLANE LIGHT WAVE
Yu.N. Grigor’ev, O.D. Zvonaryova
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
Results of integration of the Lorentz-Dirac equation for electron motion in the field of intensive light wave are
given in this work. Diagrams and parameters, which characterize the periodic motion of a relativistic electron, are
shown. Formulas of the spectral-angular distribution of the electromagnetic field irradiated by a relativistic electron,
moving towards electromagnetic wave, were obtained.
PACS: 41.60.-m, 41.85.-p., 29.27.-a
1. In [1] the Dirac-Lorentz equation for motion of a
relativistic electron in the field of travelling linearly
polarized plane light wave was integrated. The
bremsstrahlung force produced by the electron radiation
was taking into account. It was shown, that the
bremsstrahlung force results in an appearance of
decrement. However, under some conditions the
periodic motion of a radiating electron is possible.
In the present work approximate conditions of the
existence of periodic motion of a radiating electron are
analyzed. Diagrams of electron periodic motion were
calculated without taking into account the radiation.
Formulas of spectral-angular distribution of
electromagnetic radiation intensity for an electron,
which is moving periodically in a wave field, were
obtained.
2. The Dirac-Lorentz equation of electron motion in
the electromagnetic theory [2] can be written as
RLm
dt
d FFv += , (1)
( ) 0
2
1
2
0 ,v,1 m
c
mm =−= ββ
is the rest mass of an
electron, c is the light velocity, v is the vector of
electron velocity, FL is the Lorentz force, FR is the
bremsstrahlung force.
It could be shown [1] that when FR = 0 in the field of
plane electromagnetic wave the integral of motion takes
place:
( )( ) ( )( ) ( )( ) ( )( ) Btt xx =−+=−+
212212 010111 ββββ , (2)
where cxх v=β , vx is the velocity projection onto the
x-axes, "0" designates a value at the initial time instant.
By proceeding to new variable “S” (instead of “t”)
Eq. (1) could be reduced to the form:
( ) ( )( )
( )( ) +
−
−±++
+
+=
c
dS
dvSEr
dS
d
c
ee
dS
d
x
ae
ri
rEnEr
01
012cos1
3
2
2
2
222
0
00
2
2
β
βδπ
µ
µµ
( ) ( )( )
( )
×
β−
β
λ
δ+ππ
µ
+
01
012sin21
3
2
20
xae vSeErr
,ik
±−× c
dS
dz
(3)
where 2
0
2 cmere = is the radius of electron, νλ c=
is the light wavelength, Bm00 =µ , k and i are unit
vectors of the z-axes and x-axes, respectively.
+
−= δπ ν
c
taEzE rn2cos , [ ]EnH
= , (4)
where n is the normal to the wave front, r is a radius-
vector, δ is the phase initial value. E, H are vectors of
electric and magnetic fields, respectively.
21;1; ββ −=−=−= B
dt
dS
dt
dS
c
tS x
nr
. (5)
3. It could be shown that Eq. (3) under some
conditions has a periodic solution. Approximate
conditions for electron periodic motion without taking
into account bremsstrahlung force (FR ≡0) are reduced to
following expressions
( ) ( ) ( )π νµ
δ
2
;sin0;00
0
aeE
ppzy ==′=′ ; (6)
( ) ( )
[ ]caB
dNSMSL
1
1
0
2
1
22cos22cos
1
+
=++++
−
∫
−
ν
ξδπ νδπ ν
ν
(7)
[ ]
1
2
;1
2
1;2
22
2
2
1
1
2224244
+
+=
+−==
−
−−−
BcpaN
caBcpMpBcL
( ) δ2cos
4
10
2
1 c
pxa +′= ; (8)
( ) ;
cos1
cos
0;
cos1
sin
)0(
αβ
αβ
αβ
αβ
−
==′
−
==′ c
dS
dxx
c
dS
dzz
where α is the angle between initial directions of v and
i.
Eq. (6) is the condition of electron periodic motion
with the period ν--1 on z. Eq. (7) is the condition of
periodicity on time. It was obtained from the
requirement that, when electron passed along x-axis the
distance equal to 1
1
−νa , it came to the same wave phase.
The numerical calculations have shown, that Eq. (7) is
the result of Eq. (6). To obtain the spectral - angular
distribution of radiation intensity of electron moving in
the xz plane, we calculate electric field components in
the wave zone [2]:
92 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 1.
Series: Nuclear Physics Investigations (37), p. 92-93.
( ) ( ) ( ) ( )dSezx
T
ilw
Rc
ellw SilwF
c
R
ilw
x ∫
−
′+′−−=
1
0
2
2 sinsin
ν
ϕϕε , (9)
( ) ( ) ( ) ( ) dSexz
T
ilw
Rc
el
lw SilwF
c
Rilw
z ∫
−
′+′−−=
1
0
2
2
coscos
ν
ϕϕε ,
(10)
( ) 0=lwyε ,
yzx εεε ,, are the projections of a radiating vector of
electrical component.
( )ϕϕ sincos1)( zx
cc
xSSF +−+= , (11)
( ) ( ) ;2sin;22cos
4 1
2
δπ νδπ ν +=′++=′ SpzaS
c
px
( ) ( )
( ) ( )
( ) ( ) ( ) ;cos
2
0;2sin
44
0
2cos
2
;22sin
44
2
2
2
2
21
2
δ
νπ
δ
νπ
δνπ
νπ
δνπ
νπ
pzc
c
Rxa
cSpz
aSaS
c
px
+=+=
++−=
+++−=
R is the distance from an electron to the point of
observation, ϕ is the angle between unit vector of x-axis
i, and vector, directed from the coordinate origin to the
point of observation.
+= −
c
aT 11 1ν is the period of
electron motion on the trajectory, Tw π2= .
Using the given formulas, we can obtain fundamental
frequencies of the electromagnetic field radiation in the
point of observation:
( ) 12
1
1 2;cos11 ff
c
a
f =
−+= ϕν ; (12)
( )
−+
+= ϕcos111 11
c
a
c
a
l ;
l is the number of radiation harmonic.
The formula for l (whole number) puts a
requirement on a relation between parameters of wave
and particle, and ensures a linear spectrum of radiation,
typical under a periodic electron motion on time.
During the calculation Eq. (12) the small oscillations
x and z were not taken into account. Small x and z will
give higher radiation harmonics but very small in
comparison with fundamental harmonics of radiation.
The known formula of maximum frequency of
Compton scattering for εz follows from Eq. (12):
νγ 2
1 4≈f (13)
when 2
0,, cmW=== γπϕπα , W is the electron
energy.
The formula (13) is obtained at limitation on intensity
of a plane wave Ea:
( ) 22
1
0
−
< <
=
γ
νπµ
aeE
c
p
. (14)
For example, at W = 50 MeV, = 3*1014 Hz Ea
should be less than magnitude 8102 ∗π V/m.
From these formulas the important result follows:
with increasing of the incident wave intensity Ea the
frequency of electron radiation decreases and tends to
frequency of the incident wave at very large values. It is
connected with decreasing of velocity of electron, which
oscillates in a wave field towards the observer. So, at
a1=0 the electron stops to move along the x-axis and
Doppler effect disappears. The electron reradiates the
incident wave.
Figures 1-3 show the trajectories of an electron with
energy 50 MeV in the field of a travelling wave for
typical cases: ν = 3*1014 Hz, ϕ = π. 1) Ea = 108 V/m, α=
π-3*10-7, δ=1,3; f1=1,2*1019 Hz; 2) Ea = 5*1014 V/m, α=
π-10-2, δ=0,0064, f1=9,93*1014 Hz; 3) Ea = 9,1*1014
V/m, α=π-10-2, δ=0,0035, f1=3*1014 Hz.
-410-6 -310-6 -210-6 -110-6 X, m
Z, m
3 10 -14
2 10 -14
1 10 -14
n
Fig. 1
10-6 10-6 -510-7 X, m
Z, m
2.510-7
2 10-7
1.5 10-7
1 10-7
5 10-8
-1.5-2.5
n
Fig. 2
-310 -7 -110 -7 110 -7 310 -7
110 -6
210 -6
310 -6
410 6 -
Z, m
X, m
n
Fig. 3
REFERENCES
1. Yu.N. Grigor’ev, O.D. Zvonaryova, I.M. Karnau-
khov. Electron motion in the field of intensive plane
light wave // VANT, 2000, N2, p. 72-75.
2. L.D. Landau and E.M. Lifshits A field theory. 1960,
Moscow: “Nauka”, p. 416.
2
PERIODIC Electron Motion
in the Field of Intensive Plane Light Wave
Yu.N. Grigor’ev, O.D. Zvonaryova
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
REFERENCES
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