Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave
Plane monochromatic electromagnetic wave scattering by charged particles is investigated theoretically taking into account the influence of scattered incoherent electromagnetic radiation on particles motion. The spread in the particles-velocity due to interaction of particles with each other via the...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave / V.V. Ognivenko // Вопросы атомной науки и техники. — 2007. — № 1. — С. 130-132. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1105242017-01-05T03:03:33Z Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave Ognivenko, V.V. Plasma electronics Plane monochromatic electromagnetic wave scattering by charged particles is investigated theoretically taking into account the influence of scattered incoherent electromagnetic radiation on particles motion. The spread in the particles-velocity due to interaction of particles with each other via the fields of spontaneous incoherent radiation is shown to appear in the process of scattering of incident wave. Теоретично досліджено розсіяння плоскої монохроматичної електромагнітної хвилі зарядженими частинками з урахуванням впливу розсіяного некогерентного електромагнітного випромінювання на рух частинок. Показано, що в процесі розсіяння падаючої хвилі з'являється розкид по імпульсах частинок, обумовлений їх взаємодією одна з другою через поля спонтанного некогерентного випромінювання. Теоретически исследовано рассеяние плоской монохроматической электромагнитной волны заряженными частицами с учетом влияния рассеянного некогерентного электромагнитного излучения на движение частиц. Показано, что в процессе рассеяния падающей волны появляется разброс по импульсам частиц, обусловленный их взаимодействием друг с другом через поля спонтанного некогерентного излучения. 2007 Article Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave / V.V. Ognivenko // Вопросы атомной науки и техники. — 2007. — № 1. — С. 130-132. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 52.25.Gj; 52.50.Jm; 52.59.Rz; 41.60.Cr http://dspace.nbuv.gov.ua/handle/123456789/110524 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Plasma electronics Plasma electronics |
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Plasma electronics Plasma electronics Ognivenko, V.V. Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave Вопросы атомной науки и техники |
| description |
Plane monochromatic electromagnetic wave scattering by charged particles is investigated theoretically taking into account the influence of scattered incoherent electromagnetic radiation on particles motion. The spread in the particles-velocity due to interaction of particles with each other via the fields of spontaneous incoherent radiation is shown to appear in the process of scattering of incident wave. |
| format |
Article |
| author |
Ognivenko, V.V. |
| author_facet |
Ognivenko, V.V. |
| author_sort |
Ognivenko, V.V. |
| title |
Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave |
| title_short |
Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave |
| title_full |
Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave |
| title_fullStr |
Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave |
| title_full_unstemmed |
Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave |
| title_sort |
heating of charged particles at thomson scattering of a monochromatic electromagnetic wave |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2007 |
| topic_facet |
Plasma electronics |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110524 |
| citation_txt |
Heating of charged particles at Thomson scattering of a monochromatic electromagnetic wave / V.V. Ognivenko // Вопросы атомной науки и техники. — 2007. — № 1. — С. 130-132. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT ognivenkovv heatingofchargedparticlesatthomsonscatteringofamonochromaticelectromagneticwave |
| first_indexed |
2025-07-08T00:42:14Z |
| last_indexed |
2025-07-08T00:42:14Z |
| _version_ |
1837037345918943232 |
| fulltext |
130 Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 130-132
HEATING OF CHARGED PARTICLES AT THOMSON SCATTERING
OF A MONOCHROMATIC ELECTROMAGNETIC WAVE
V.V. Ognivenko
NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine,
e-mail: ognivenko@kipt.kharkov.ua
Plane monochromatic electromagnetic wave scattering by charged particles is investigated theoretically taking into
account the influence of scattered incoherent electromagnetic radiation on particles motion. The spread in the particles-
velocity due to interaction of particles with each other via the fields of spontaneous incoherent radiation is shown to
appear in the process of scattering of incident wave.
PACS: 52.25.Gj; 52.50.Jm; 52.59.Rz; 41.60.Cr
1. INTRODUCTION
Investigation of processes of electromagnetic waves
interaction with charged particles is of considerable
interest for many applications of plasma physics and
charged particles beams, such as heating of plasma by the
external fields [1], stability of plasma in the external
electromagnetic fields [2], its diagnostics [3],
accelerations of electrons in the fields of laser radiation
[4], increase of frequency in plasma [5]. The scattering of
external electromagnetic waves by the charged particles is
also concerned with generation of narrow-band ultrashort
wavelength radiation at motion of relativistic electron
beams in the external periodic fields [6]. The peculiar
significance for these applications has finding out effects
of radiation reaction on particles motion.
In the given paper the results of theoretical research of
the plane monochromatic electromagnetic wave Thomson
scattering process by charged particles taking into account
the influence of scattering radiation on particles motion
are presented, the incoherent scattering of external wave
by the charged particles being discussed.
The charged particles motion in the total
electromagnetic field in the limit case of a small value of
unit-less wave strength parameter is considered. The
spread in particles momentum is shown to appears in the
process of incident wave scattering. The dependences of a
mean-square value of particles longitudinal velocity on
time are found and investigated by analytically and
numeral methods.
2. FORMULATION OF THE PROBLEM
Let us consider the ensemble of the identical charged
particles with the charge of q, mass of m and
homogeneous average density n0, moving in a field of a
plane monochromatic electromagnetic wave (EMW)
( ) ( )krEE −= text ωcos0 , (1)
where E0 is the amplitude of the wave electric field; ω, k
are its frequency and wave vector.
Assume that the E0 and k vectors are directed along
axes 0X and 0Z accordingly Cartesian coordinates.
Let us find the spread in the longitudinal velocity of
particles, due to the interaction of particles via
electromagnetic fields, produced by these particles at
motion in the external field (1). Let us call the direction
parallel to that of scattered wave propagation a
longitudinal one (viz. along the OZ axis).
In order to find solution in explicit analytical form let
us consider ωmceEa 0= parameter as a small one
which characterises a relative size of transverse
oscillatory velocity of particle in the external field.
Solution of a problem will be searched by the
following method: let us consider the ensemble of charges
as large number of the individual charged particles.
Having found the field produced by the individual
charged particle, moving in the external field, in the limit
case of a small value of a parameter let us find the total
longitudinal force, acting on the individual (test) particle,
the mean-square longitudinal velocity being expressed via
the ensemble average of the product of pair interaction
forces of particles.
3. RESULTS AND DISCUSSION
In the external field the charged particles oscillate in
the direction parallel to the E0 vector. The equations for
coordinate and velocity of particles can be written as
follows:
( ) ( ) ( ) ( )tkztcat izxii ∆+−−= eerr ωω cos0 ,
( ) ( ) zzxi kztcat vsin eev +−= ω ,
where { }iiioi zyx 000 ,,=r is the coordinate of i-th particle
in the initial moment of time t0 (let t0=0),
( ) ( )tztz iii ∆+= 0 be the longitudinal displacement of
particle trajectory relative to the initial position.
Let us express electric and magnetic fields produced
by the individual particle via Lienard-Wiechert potentials
[7]. In the nonrelativistically motion of particles and small
a value the expression for the electric field, produced by
the i-th particle in r coordinate in t time is:
( )
−−−
+
−−=
ϕ
ϕ
cos3111
sin31,
2
2
2222
2
2
3
i
xi
ii
ii
x
i
x
i
i
i
R
R
RkRk
kRR
R
R
kaq
R
qt eRrE
, (2)
where ( ) ( ) ( )[ ]tzzyyxx iziyixi ′−+−+−= eeeR 00 ,
( )[ ]tzRkt ii −−= ωϕ .
A total field in the observation point of r in the time t
will be determined by charges initial coordinates of which
satisfy condition:
sct 0rr −> (3)
Let us consider the motion of some individual (test)
particle both in the field (1) and in the fields, produced by
mailto:ognivenko@kipt.kharkov.ua
131
all other particles. Equations of longitudinal motion of
such particle (let it be the i-th particle) in these fields can
be written as follows:
( )[ ] ( ) ( )[ ]∑==
s
si
s
ziz
zi xttFtt
dt
d
;,,F
p
rr , (4)
( ) ( ) ( ) ( )[ ]
+= zszss
s
z t
c
tEextF ,1,;, rHvrr , (5)
where ri , pi are the position and momentum of the i-th
particle at time t; ( ) ( )tt ss ,,, rHrE are the strength of
electric and magnetic fields, produced by the s-th particle
at time t in r coordinate.
Let’s the considered system consists of N charged
particles the coordinate and momentum of which at the
initial time are random values. We introduce the
distribution function of system states DN(x01,…, x0N; t0) at
time t0. This function is normalized as:
∫DN(x01,…, x0N; t0)dx01…dx0N=1, where x={r, p},
x0s=xs(0). By integrating of Eq. (4) we can obtain the
deviation of longitudinal momentum from the mean value
for test particle
( )[ ]∫ ′′′=∆
t
izzi tdttFp
0
,rδ , (6)
and expression for rate of change of mean-square spread
in the longitudinal momentum of particles
( ) ( )[ ] ( )[ ]∫ ′′′=∆
t
izizzi tdttFttFp
dt
d
0
2 ,,2 rr δδ , (7)
where ∆pzi=pzi-<pzi>, zzz FFF −=δ , angular brackets
indicate average values,
( ) ( ) ( )[ ] ( )∫= 0010
1 ;;,, dxxfxtxtt rFrF ,
( ) ( )∫= NNN dxdxtxxxDNxf 0020002001 ...;...,, ,
f1 is the single-particle distribution function.
The right-hand side of Eq. (7) is founded in neglecting
of influence of fields, produced by particles on its motion.
Thus in the right-hand side of Eq. (7) ri (t) can be
replaced by unperturbed trajectory ( )( ) ( )0;0 =∆= tt ii rr of
particle in the external field (1). Taking into account that
particles are identical and neglecting initial correlation
between them Eq. (7) can be expressed as
( ) ( )
( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]ossi
s
zossi
s
z
t
V
ososzi
xtxtFxtxtF
xfdxtdp
dt
d
,;,;
2
00
0
1
2
′′×
×′=∆ ∫ ∫∫
′
ξξ
, (8)
where ( )( ) ( )( ){ }ttrt ii ,00 =ξ
Analytical expression for the force, acting on an
individual particle in the electromagnetic field, produced
by the other particle will be derived by substituting
expression for field (2) in Eq. (5):
GkqFzs
22= ,
ρ
ψ
+ψ−
ρρ
ρ
=
ttt
z aG cossin1 2
02
, (9)
where zt ρρψ −= , ( ) 2122
zt ρρρ += ⊥ , ( )siz zzk −=ρ ,
( ) ( )2
00
2
00 sisi yyxxk −+−=⊥ρ , 20 aa = .
Neglecting initial momentum spread the Eq. (8) for
rate of change of root mean square longitudinal particles
momentum is:
( ) ( ) ( )[ ]
( )
∫∫ ⊥′=∆
'V
0
2
0
4
0
2
0
,4
t
sszi dGtdqknp
dt
d rρρπ
τ
, (10)
where V0 is the integration region determined by
condition (3).
Having integrated Eq. (10) taking into account the fact
that particles are at some finite distance one from other,
we find:
( ) 222
QRz σσβ +=∆ , (11)
where ( ) ttNa
r
R ωω
λ
σ 212
0
0
3
= , tNr
Q ω
λ
π
σ 320
33
= ,
3
0λnN = , 22
0 mcqr = , cv zz =β .
As a minimum distance between particles the value of
31
0min
−= nr was taken.
The first term in right hand side Eq. (11), that depends
on parameter a, is the longitudinal particle velocity
spread, due to interaction of particles via the
electromagnetic waves. The second term in Eq. (11), that
is not dependent on parameter a, describes the spread, due
to the coulomb interaction of particles.
On the ( )4231 KNt π>>ω times the particle velocity
spread will be determined by interaction of particles with
each other via electromagnetic waves. The expression for
the momentum spread in this case may be presented as
follows:
( )[ ] ( )( ) ( )tt
eff
R
z N
2 0
1212
ε
ε
β
∆
=∆ , (12)
where 22πωtNNeff = , ( ) 32 2
0
21 tkaqR ωε =∆ is the energy
losses of an individual charge on radiation, 2
0 mc=ε .
Now let us consider the EMW scattering on the
clusters of the charged particles, having a form of a
circular cylinder with the height of lb and radius of rb. Let
the wave vector k be parallel to the axis of a cylinder
(cluster). The spread in the longitudinal particle velocity
will be described by Eq. (11) for particles present in the
initial moment of time in the center of cluster on the
cylinder axis and at the distance of lb/2 from its base.
Thus the first term in this equation, corresponding to the
radiative interaction, is:
21
2
0
0)( 2
2
Φ
λ
π
τ
λ
=σ=σ
b
bbc
RR l
rl
Na
r , (13)
where Φ(x)=x arcctg(x).
When Eq. (13) was derived a considered test particle
was assumed to be located in the influence region of
fields of all other particles of the cluster. Eq. (13) shows,
that for the sufficiently wide charged particles cluster
(2rb>>lb) longitudinal velocity spread due to the radiative
interaction of particles will be proportional to the root
square from the number of particles in the effective
volume bl
2
effV πλ= .
The dynamics of the change of particles longitudinal
velocity spread was investigated by particle simulation of
the EMW scattering process on the charged particles
cluster. The cluster has the form of a cylinder, whose axis
coincides with the direction of external wave propagation.
The radius of a cluster is equal to 5λ, and its length equals
10λ. A cluster consists of 5000 particles, a0=0.7,
132
λ=10µm. The motion of particles was described by
equations (4), (5), (9).
The dependence on the dimensionless time τ=ωtρ
normalized value of the spread in the longitudinal velocity
of N1 particles, located in the center of a cluster at the
initial time t=0, is shown in the figure, where
( ) λπρ /32 21
00 Nlra b= . Namely the initial coordinates
of these particles (test particles) were in the region,
limited by the surface of a cylinder with the radius of
r1=0.15rb and the height of l1=0.4lb . Axes of cylinders
and planes of symmetry (a plane perpendicular to the axis
of a cylinder and dividing its in two) coincide.
0 5 1 0 1 5 2 0 2 5
0 ,0
0 ,2
0 ,4
0 ,6
0 ,8
σ z /ρ
1
2
τ
In this figure the curve 1 corresponds to the particle
simulation, and curve 2 - the dependence of the
longitudinal velocity spread, determined by Eq. (11), (13).
Here ( )222
zzz β−β=σ , ∑
=
=
1
11
1A
N
i
iA
N
- summation is
carried on over all test particles, N1 is their total number.
The figure shows that the spread in velocity is
proportional to the t on the initial stage of scattering. The
results of numeral simulation agree with the analytical
estimations by Eqs. (11), (13). On the times τ>5 the rate
of change in longitudinal velocity spread decreases.
Thus, the EMW scattering on the charged particles
leads to the increase of the velocity spread of particles.
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