Stochastic heating of electrons by electromagnetic waves
In the present paper the nonlinear dynamics of electrons motion in the fields of two combinational waves or in the fields of one combinational wave and a longitudinal wave is considered. The mechanism of electrons energy grows under an overlap of two nonlinear resonances is investigated and upper li...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Stochastic heating of electrons by electromagnetic waves / V.V. Ognivenko // Вопросы атомной науки и техники. — 2000. — № 6. — С. 131-133. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859795143747436544 |
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| author | Ognivenko, V.V. |
| author_facet | Ognivenko, V.V. |
| citation_txt | Stochastic heating of electrons by electromagnetic waves / V.V. Ognivenko // Вопросы атомной науки и техники. — 2000. — № 6. — С. 131-133. — Бібліогр.: 6 назв. — англ. |
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| description | In the present paper the nonlinear dynamics of electrons motion in the fields of two combinational waves or in the fields of one combinational wave and a longitudinal wave is considered. The mechanism of electrons energy grows under an overlap of two nonlinear resonances is investigated and upper limit of the electron energy is determined.
|
| first_indexed | 2025-12-02T13:36:06Z |
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Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 131-133 131
UDC 533.9
STOCHASTIC HEATING OF ELECTRONS BY ELECTROMAGNETIC
WAVES
V.V.Ognivenko
NSC, Kharkov Institute of Physics & Technology, Kharkov, Ukraine,
e–mail: ognivenko@kipt.kharkov.ua
In the present paper the nonlinear dynamics of electrons motion in the fields of two combinational waves or in
the fields of one combinational wave and a longitudinal wave is considered. The mechanism of electrons energy
grows under an overlap of two nonlinear resonances is investigated and upper limit of the electron energy is
determined.
1. Introduction
The investigation of electromagnetic waves
interaction with charge particles is of interest for number
applications in particular for plasma heating and a
generation of electromagnetic radiation [1–4]. In the
paper [4] the method of stochastic plasma heating in the
field of several (more then three) external
electromagnetic waves is considered. A gain in energy
of electrons in this method of heating is due to a
stochastic instability of electron motion under the
overlap of nonlinear resonances of combinational
waves, which are formed by nonlinear interaction of
pairs electromagnetic waves with electrons of plasma.
The stochastic instability of relativistic electron motion
in the field of several electromagnetic waves and
periodic transverse magnetic field is studied in [3]. The
purpose of the present paper is the determination of the
upper limit of electron energy, which they can acquire in
stochastic heating, when two nonlinear resonances
overlap one other. The nonlinear dynamics of electrons
in the fields of two combinational waves or in the fields
of one combinational wave and a longitudinal wave is
considered. It is shown, when two nonlinear resonances
are overlap, the maximal energy of electrons is equal to
the energy corresponding to the sum of those two
nonlinear resonances.
2. The equations of motion
Let us consider a motion of electrons in the fields of
several monochromatic electromagnetic waves:
( )E e e= − +∑ x n
t
n n
n
z lA k z t Acos sinω ϕ! , (1)
where A An
t , ! – are the amplitudes of waves; upper
indexes "t" and " ! " denote the external transverse
electromagnetic waves and a longitudinal one,
respectively; ϕ ω! ! != −k z t ; ω,k are the frequency
and wavenumber; e ex z, are the unit vectors in the x and
z directions.
The energy of electrons, which we shall represent by
its relativistic factor ( )γ β= −
−
1 2 1 2
(where β = v c ),
evolves according to the equations:
( )d
dt
e
mc
γ
= 2 vE , d
dt
r v= . (2)
A motion of electrons in the fields (1) can be
presented as the fast oscillations in the fields of the
transverse electromagnetic waves and slow motion in
the field of longitudinal wave or combinational wave
formed by pairs of transverse waves. Taking the
oscillating part of the electron velocity from
conservation of canonical transverse momentum in the
transverse electromagnetic waves and substituting its in
eq. (2), we obtain the following equations for a slow
motion of electrons:
( ) ( )
( ) ( )
d
dt
a a
e A
mc
m n
m n
m n mn
m n mn
z
γ
γ
ω ω θ
ω ω θ β ϕ
= − −
− +
+
∑ −
+
1
2 ,
sin
sin sin .
!
!
(3)
dz
dt
vz= , (4)
where ( ) ( ) ( )θ ω ωmn m n m nk k z t± = ± − ± , a eA
mcn
n
t
n
=
ω
.
If A! = 0 the eq. (3) becomes the same as in [3,4].
The first terms on the right–hand side of eq. (3),
proportional to the square of the wave amplitudes,
describe a motion of electrons under conditions of
combinational resonance of electrons with external
transverse electromagnetic waves:
( )ω ωm n z m nv k k± = ± .
The last term on the right–hand side eq. (3) is
responsible for a Cherenkov resonance interaction of the
electrons with a longitudinal wave. ω! != k vz .
We consider below two cases. In the first case (a)
there are only three external electromagnetic waves: the
wave E1 propagates in positive z direction, and the both
waves E2 and E3 propagate in the opposite direction:
( )ω ω1 2 1 2− = +v k kz , ( )ω ω1 3 1 3− = +v k kz .
In the second case (b) there are two electromagnetic
waves E1, E2 (the same as in the case (a)) and the
longitudinal wave E ! :
( )ω ω1 2 1 2− = +v k kz , ω! != k v .
For those two cases the equations of motion (3), (4)
in dimensionless variables become:
( )dw
dτ
ε ϕ ε κϕ τ= + +1 2 2sin sin ∆ , (5a)
132
d
d
w
ϕ
τ
= + ∆1 . (5b)
The initial condition for the eqs. (5a), (5b) is:
( )w 0 0= .
Where w = −γ γ0 , ϕ ω= −k z tc c , ( )τ β γ= ck tc 0 0
3 ,
ε
ω β γ
1
1 2 0 0
2
2
=
a a
k c
c
c
, ∆1
0
0
2
0
31= −
ω
β γc
cv k
ω ω ωc = −1 2 ,
k k kc = +1 2 .
In the case (a):
( )
( )ε β γ
ω ω
2 1 3 0 0
2 1 3
1 32
=
−
+
a a
c k k
,
κ =
+k k
kc
1 3 ,
( )
∆2
1 3
1 3
0 0
3
1 3= −
−
+
+ω ω ω β γc
c ck k k
k k
k c
.
In the case (b): ε
β γ
2
0
2
0
3
2=
eA
mc kc
! , κ =
k
kc
! ,
∆2
0 0
3
= −
ω ω β γc
c ck k
k
k c
!
!
! .
The nonlinear equations (5a), (5b) have general form
and describe a stochastic instability of the charge
particles motion in the different electromagnetic waves
[3–5].
3. The overlap of resonances
Let us obtain the analytical estimations for the rate of
electron energy increase and the upper limit of electrons
energy. For the electron moving in the field of the first
combinational wave (ε1) the half–width of the nonlinear
resonance in energy (δw)1 and in wave number (δκ)1 (in
dimensionless variable) are:
( ) ( )δκ δ ε1 1 12= =w . (6a)
In the field of second wave (ε2) the half–width of
nonlinear resonance is equal to:
( ) ( )δκ κ δ κ ε2 2 22= =w . (6b)
The total width of nonlinear resonance in the fields
ε1 and ε2 is:
( ) ( )δκ δκ δκ= +1 2 . (7а)
The distance between isolated resonance of those
waves is
( )∆κ ∆ ∆= − −1 1 2κ . (7b)
The stochastic instability arises when separatrices of
nonlinear resonances touch each other [6]:
δκ > ∆κ .
Using (7а) and (7b) the condition for the overlap of
nonlinear resonance takes the form:
( )2 2 11 2 1 2ε κ ε κ+ > − −∆ ∆ . (8)
When the overlap of nonlinear resonance takes place
the motion of electrons becomes random. Assuming that
condition (8) is satisfied and phases of electrons are
random one can obtain the estimation for the rate of
electron energy growth [4]:
w a a t e
mc
A t
c
2 1 2
0
2
0
2
2
=
+
π
γ
ω
β
ω
!
!
.
For the determination of maximal energy of
electrons, which they can acquire under the overlap of
resonances, we take into account that electrons can
move only from one resonance to other. Therefore the
energy of electron grows up to the value corresponding
to the total width of two nonlinear resonances in energy
units:
( )wmax = +2 2 21 2ε ε κ . (9)
Thus the saturation of electron energy grows in the
fields of two wave ε1, ε2 takes place as a result of the
electron trapping in a potential well formed by a total
nonlinear resonance of these two waves.
4. Numerical results and discussion
The nonlinear dynamics of individual particle
motion, in according to eqs. (5a), (5b), has been carry
out numerically for the following value of parameters: ε1
= 0.2, ε2=0.06, κ =0.2, ∆1 =0.02, ∆2 =0.3, and several
initial phases ( )ϕ π0 2 =0.3 (fig.1); 0.7 (fig.2); 0.8
(fig.3), 1.1 (fig.4). It is consider the case when criterion
of stochastic motion (8) is satisfied and nonlinear
resonances are overlap.
In the figures 1–4 the square of normalized energy
w2 as a function of dimensionless time τ is presented.
0 100 200 300 400 500 600
0
2
4
6
8
τ
w2
Fig. 1
0 100 200 300 400 500 600
0
2
4
6
8
τ
w2
Fig.2
133
0 100 200 300 400 500 600
0
2
4
6
8
τ
w2
Fig.3
0 100 200 300 400 500 600
0
2
4
6
8
τ
w2
Fig.4
It is shown that motion of particles is substantially
irregular. At first the energy of electrons grows slightly
that is due to the oscillation of electrons in potential well
of first wave ε1 and a weak influence of the second wave
ε2.
When the energy of electron increases it approaches to
the separatrix and the influence of the second wave
becomes the same as the first one. Since the resonances
of these waves are overlap the electron can move from
one resonance to other and its energy grows up to the
peak value. Then the energy of electrons decreases
down to not far from initial value. For the some value of
initial phases (fig.2, fig.3) this process is repeated. The
results of the numerical calculation depend strongly
from initial parameters, as the motion of electrons is
stochastic. A gain in energy of electrons in considering
case is due to a stochastic motion of electrons in
potential well formed by sum of nonlinear resonances of
the waves ε1 and ε2. Substituting the value of parameters
into eq. (9) we obtaine wmax .2 7 92= . The maximal value
of energy obtained by numerical calculation well agrees
with analytical result.
The author thanks V.A. Buts and K.N. Stepanov for
fruitful discussion. The work was partially supported by
the Science and Technology Center in Ukraine, grant
No 253.
References
1. K.N.Stepanov. Fiz. Plazmy. 1983, vol. 9, p.45 – 61.
2. V.V.Alikaev et al. Fiz. Plazmy. 1976, vol.2, p.390 –
395.
3. V.A.Buts, V.V.Ognivenko . Pis’ma v Zh. Exper.
Teor. Fiz. 1983, vol. 38, N 9, p. 434 –436.
4. V.A.Buts, K.N.Stepanov. Pis’ma v Zh. Exper. Teor.
Fiz. 1993, vol. 58, N 7, p. 520 –523.
5. G.M.Zaslavsky, N.N.Filonenko. Zh. Exper. Teor.
Fiz. 1968, vol.54, N 5, p.1590-1602.
6. B.V.Chirikov. A universal instability of many–
dimentional oscillator systems // Physics Reports.
1979, vol.52, N 5, p.263–379.
|
| id | nasplib_isofts_kiev_ua-123456789-78548 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-02T13:36:06Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Ognivenko, V.V. 2015-03-18T18:47:05Z 2015-03-18T18:47:05Z 2000 Stochastic heating of electrons by electromagnetic waves / V.V. Ognivenko // Вопросы атомной науки и техники. — 2000. — № 6. — С. 131-133. — Бібліогр.: 6 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/78548 533.9 In the present paper the nonlinear dynamics of electrons motion in the fields of two combinational waves or in the fields of one combinational wave and a longitudinal wave is considered. The mechanism of electrons energy grows under an overlap of two nonlinear resonances is investigated and upper limit of the electron energy is determined. The author thanks V.A. Buts and K.N. Stepanov for fruitful discussion. The work was partially supported by the Science and Technology Center in Ukraine, grant No 253. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Вeams and waves in plasma Stochastic heating of electrons by electromagnetic waves Article published earlier |
| spellingShingle | Stochastic heating of electrons by electromagnetic waves Ognivenko, V.V. Вeams and waves in plasma |
| title | Stochastic heating of electrons by electromagnetic waves |
| title_full | Stochastic heating of electrons by electromagnetic waves |
| title_fullStr | Stochastic heating of electrons by electromagnetic waves |
| title_full_unstemmed | Stochastic heating of electrons by electromagnetic waves |
| title_short | Stochastic heating of electrons by electromagnetic waves |
| title_sort | stochastic heating of electrons by electromagnetic waves |
| topic | Вeams and waves in plasma |
| topic_facet | Вeams and waves in plasma |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78548 |
| work_keys_str_mv | AT ognivenkovv stochasticheatingofelectronsbyelectromagneticwaves |