Stochastic heating of charged particles in absence of first order resonances
Dynamics of charged particles in external electromagnetic fields in the absence of resonances proportional to the first degree of intensity of an electromagnetic field is investigated. Two schemes are investigated. In the first existence of several regular electromagnetic waves is supposed. It is su...
Gespeichert in:
| Datum: | 2015 |
|---|---|
| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
|
| Schriftenreihe: | Вопросы атомной науки и техники |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/112194 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Stochastic heating of charged particles in absence of first order resonances / A.N. Antonov, V.A. Buts, E.A. Kornilov, V.G. Svichensky, V.V. Kuzmin, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2015. — № 4. — С. 161-166. — Бібліогр.: 6 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-112194 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1121942017-01-18T03:04:29Z Stochastic heating of charged particles in absence of first order resonances Antonov, A.N. Buts, V.A. Kornilov, E.A. Svichensky, V.G. Kuzmin, V.V. Tolstoluzhsky, A.P. Плазменно-пучковый разряд, газовый разряд и плазмохимия Dynamics of charged particles in external electromagnetic fields in the absence of resonances proportional to the first degree of intensity of an electromagnetic field is investigated. Two schemes are investigated. In the first existence of several regular electromagnetic waves is supposed. It is supposed that frequencies and wave vector of these waves are such that the phase velocity one of beating waves is close to thermal velocity of particles. It is essential that thus organized Cherenkov resonances are proportional to square of small parameter (a square of dimensionless intensity of the field). In the second scheme is supposed that the phase of the field of wave changes by jump under the random law. Parameters of the studied systems at which one or other scheme of transmission of energy the field to particles has advantage are found. Досліджується динаміка заряджених частинок у зовнішніх електромагнітних полях при відсутності резонансів, які пропорційні першому ступеню напруженості електромагнітного поля. Досліджуються дві схеми. У першій передбачається наявність декількох регулярних електромагнітних хвиль. Передбачається, що частоти і хвильові вектори цих хвиль такі, що фазова швидкість однієї з хвиль биття близька до теплових швидкостей частинок. Істотно, що таким чином організовані черенковські резонанси пропорційні квадрату малого параметра (квадрату безрозмірної напруженості поля). У другій схемі передбачається, що фаза поля хвилі змінюється стрибком за випадковим законом. Знайдено параметри досліджуваних систем, при яких має перевагу одна або інша схема передачі енергії поля частинкам. Исследуется динамика заряженных частиц во внешних электромагнитных полях в отсутствие резонансов, пропорциональных первой степени напряженности электромагнитного поля. Исследуются две схемы. В первой предполагается наличие нескольких регулярных электромагнитных волн. Предполагается, что частоты и волновые векторы этих волн таковы, что фазовая скорость волн биения близка к тепловым скоростям частиц. Существенно, что таким образом организованные черенковские резонансы пропорциональны квадрату малого параметра (квадрату безразмерной напряженности поля). Во второй схеме предполагается, что фаза поля волны изменяется скачком по случайному закону. Найдены параметры изучаемых систем, при которых имеет преимущество одна или другая схема передачи энергии поля частицам. 2015 Article Stochastic heating of charged particles in absence of first order resonances / A.N. Antonov, V.A. Buts, E.A. Kornilov, V.G. Svichensky, V.V. Kuzmin, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2015. — № 4. — С. 161-166. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 05.45.PQ, 52.35.MW, 41.20.JB, 41.75.Jv http://dspace.nbuv.gov.ua/handle/123456789/112194 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Плазменно-пучковый разряд, газовый разряд и плазмохимия Плазменно-пучковый разряд, газовый разряд и плазмохимия |
| spellingShingle |
Плазменно-пучковый разряд, газовый разряд и плазмохимия Плазменно-пучковый разряд, газовый разряд и плазмохимия Antonov, A.N. Buts, V.A. Kornilov, E.A. Svichensky, V.G. Kuzmin, V.V. Tolstoluzhsky, A.P. Stochastic heating of charged particles in absence of first order resonances Вопросы атомной науки и техники |
| description |
Dynamics of charged particles in external electromagnetic fields in the absence of resonances proportional to the first degree of intensity of an electromagnetic field is investigated. Two schemes are investigated. In the first existence of several regular electromagnetic waves is supposed. It is supposed that frequencies and wave vector of these waves are such that the phase velocity one of beating waves is close to thermal velocity of particles. It is essential that thus organized Cherenkov resonances are proportional to square of small parameter (a square of dimensionless intensity of the field). In the second scheme is supposed that the phase of the field of wave changes by jump under the random law. Parameters of the studied systems at which one or other scheme of transmission of energy the field to particles has advantage are found. |
| format |
Article |
| author |
Antonov, A.N. Buts, V.A. Kornilov, E.A. Svichensky, V.G. Kuzmin, V.V. Tolstoluzhsky, A.P. |
| author_facet |
Antonov, A.N. Buts, V.A. Kornilov, E.A. Svichensky, V.G. Kuzmin, V.V. Tolstoluzhsky, A.P. |
| author_sort |
Antonov, A.N. |
| title |
Stochastic heating of charged particles in absence of first order resonances |
| title_short |
Stochastic heating of charged particles in absence of first order resonances |
| title_full |
Stochastic heating of charged particles in absence of first order resonances |
| title_fullStr |
Stochastic heating of charged particles in absence of first order resonances |
| title_full_unstemmed |
Stochastic heating of charged particles in absence of first order resonances |
| title_sort |
stochastic heating of charged particles in absence of first order resonances |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2015 |
| topic_facet |
Плазменно-пучковый разряд, газовый разряд и плазмохимия |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112194 |
| citation_txt |
Stochastic heating of charged particles in absence of first order resonances / A.N. Antonov, V.A. Buts, E.A. Kornilov, V.G. Svichensky, V.V. Kuzmin, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2015. — № 4. — С. 161-166. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT antonovan stochasticheatingofchargedparticlesinabsenceoffirstorderresonances AT butsva stochasticheatingofchargedparticlesinabsenceoffirstorderresonances AT kornilovea stochasticheatingofchargedparticlesinabsenceoffirstorderresonances AT svichenskyvg stochasticheatingofchargedparticlesinabsenceoffirstorderresonances AT kuzminvv stochasticheatingofchargedparticlesinabsenceoffirstorderresonances AT tolstoluzhskyap stochasticheatingofchargedparticlesinabsenceoffirstorderresonances |
| first_indexed |
2025-07-08T03:31:28Z |
| last_indexed |
2025-07-08T03:31:28Z |
| _version_ |
1837047993065275392 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №4(98) 161
PLASMA-BEAM DISCHARGE, DISCHARGE
AND PLASMACHEMISTRY
STOCHASTIC HEATING OF CHARGED PARTICLES
IN ABSENCE OF FIRST ORDER RESONANCES
A.N. Antonov1, V.A. Buts1,2, E.A. Kornilov1, V.G. Svichensky1, V.V. Kuzmin1,
A.P. Tolstoluzhsky1
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2V.N. Karazin Kharkiv National University, Kharkov, Ukraine
E-mail: tolstoluzhsky@kipt.kharkov.ua
Dynamics of charged particles in external electromagnetic fields in the absence of resonances proportional to the
first degree of intensity of an electromagnetic field is investigated. Two schemes are investigated. In the first exist-
ence of several regular electromagnetic waves is supposed. It is supposed that frequencies and wave vector of these
waves are such that the phase velocity one of beating waves is close to thermal velocity of particles. It is essential
that thus organized Cherenkov resonances are proportional to square of small parameter (a square of dimensionless
intensity of the field). In the second scheme is supposed that the phase of the field of wave changes by jump under
the random law. Parameters of the studied systems at which one or other scheme of transmission of energy the field
to particles has advantage are found.
PACS: 05.45.PQ, 52.35.MW, 41.20.JB, 41.75.Jv
INTRODUCTION
Previously, it was shown that in all cases for the
same energy stored in the wave scheme with regular
fields significantly more effective than scheme heating
particles by the noise fields [1]. However, if resonances
of particles with electromagnetic fields are absent, the
only mechanisms of heating are mechanisms using the
noise fields. In particular waves with random jumping
phase [2, 3]. In all previous studies suggested that the
resonances are usual resonances of the first-order of the
field [4, 5]. It is possible to organize such resonances
only in the presence medium (Cherenkov resonances),
or in the presence of an external constant magnetic field
(cyclotron resonances, resonances on normal and anom-
alous effect of Doppler). In absence of medium and the
external magnetic field such resonances can't be orga-
nized. However, even in vacuum it is possible to organ-
ize Cherenkov resonances fields’ proportional to the
second order of the field strength.
Such resonances, in particular, are used in the FEL
schemes and in schemes of the accelerating type of in-
verse free-electron laser. It must be said that the effec-
tiveness of the interaction of charged particles with
fields with such resonances are significantly lower than
in the interaction of first-order of the field. However, as
we will see below, they can also play an essential role in
processes of transmission of energy from electromag-
netic waves to particles. Below we will investigate
mechanism of the charged particles heating in a vacuum
in the absence of resonances of the first-order in the
field. The essence of this mechanism is that the velocity
the charged particles in the field of several waves can be
in the Cherenkov resonance with the field of beating
wave (combination wave).
The essence of this mechanism is that the speed of
the charged particles in the field multiple waves can be
in the Cerenkov resonance with the wave field heartbeat
(combination wave) and, when the width of this reso-
nance is large enough so the separatrix of this resonance
touches separatrix of nonlinear resonance on other com-
bination wave, the motion of the particles becomes lo-
cally unstable. The dynamics of particles is similar to
the dynamics of particles in a random field, and there is
their stochastic heating
1. STATEMENT OF PROBLEM
AND THE BASIC EQUATIONS
Let's consider dynamics of the charged particles in a
field of several electromagnetic waves. Expressions for
electric and magnetic fields of these waves can be pre-
sented in such kind:
n
n
E E=∑
, ,n
n
H H=∑
Re( ),ni
n nE eψ=
[ ],n n n
n
cH k E
ω
=
(1)
where n n nk r tψ ω= −
.
The equations of movement in fields (1) look like:
[ ].dP eeE vH
dt c
= +
(2)
These equations are convenient rewrite in dimen-
sionless variables:
PP
mc
≡ ,
0
n
n
ωω
ω
≡ , 0tτ ω≡ , dPP
dτ
≡
, vr
c
≡
,
n
n
n
eEE
mcω
≡
, ,n
n
n
e
mcω
≡
n
n
n
k ck
ω
≡
, 0r r
c
ω
≡
(3)
Substituting fields (1) in the equations (2) and using
dimensionless variables it is possible to receive the fol-
lowing, the equations:
( ) ( ),n n n n n
n n
P E k r k rEω= − +∑ ∑
(4)
n n
n
P Eγ ω
γ
= ∑
,
where ( )Re ni
n nE eψ=
.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 162
For the further analysis it is convenient to enter also
a certain auxiliary characteristic of a particle which we
further shall name partial energy of a particle which
satisfies to the following equation:
( )n n nrEγ ω=
. (5)
From definition of this partial to energy follows, that
it determines those value of energy which the particle
would have if it moved only in the field of one n -th
electromagnetic wave. Using definition of this partial
energy, from the equations (4), (5) it is possible to re-
ceive the following integral of movement:
Re( )ni n
n n
n n n
kP i e Cψ γ
ω
− − =∑ ∑
. (6)
Generally the equations (4), (5) together with inte-
gral (6) can be investigated only by numerical methods.
For obtain of analytical results we shall consider, that
the parameter of force of each of waves working on a
particle is small. In this case all description particles (its
energy, a pulse, coordinate, speed) can be presented as
the sum slowly and quickly varying sizes:
P P P= +
n n nγ γ γ= + .
In this case it is possible to receive the following ex-
pressions and the equations which connect fast and slow
variables:
n
n
n n
kP Cγ
ω
= +∑
,
Re( ) /ni
n n n n
n n
P i e kψ γ ω= +∑ ∑
, (7)
Re( )ni
n n n n nvE v eψγ ω ω= =
,
n n nvEγ ω=
, Re( )ni
n neψγ = Γ ,
where /n n n ni vω ψΓ = −
.
The equations for fast variables can be integrated:
( )Re /ni
n n n n ni v e k vψγ ω ω = −
,
{ }Re ( ) /ni
n n n n
n
P ie k vψ ω = + ∑
.
The equations for slow variables will get the kind:
( ) ( )
,
1 Re Rem ni i
n m n
m n
P k i e eψ ψ
γ
= ∑
,
( )
( )
,
,
1 Re( ) Re( )
1 cos / 2
2
cos / 2 .
m ni i
m n n
m n
n n m m n
m n
m n
i e eψ ψγ ω
γ
ω ψ ψ π
γ
ψ ψ π
= =
= + + +
+ − +
∑
∑
(8)
The equations (8) are equivalent to the equation of a
nonlinear pendulum (a mathematical pendulum) on
which external periodic force operates. We shall show
it. Let among those waves which operate on a particle,
are available two waves (at number 1 and 2) which pal-
pation form a combinational wave which phase speed, is
close to average speed of a particle. We shall designate
a difference of phases of these waves through θ :
1 2θ ψ ψ≡ − . For this difference of phases it is possible
to receive the following differential equation.
( )d v
dt
θ χ γ= −Ω = ∆
, (9)
where 1 2k kχ ≡ −
, 1 2ω ωΩ ≡ − .
Thus we count that / vχΩ ≅ . Equation (8) we now
can rewrite as:
( )1 cosd F
d
γ θ τ
τ γ
= Ω + , (10)
where 1 2=
, ( )F τ – periodic function.
Let's consider that initial energy of a particle in ac-
curacy corresponds of Cerenkov resonance of a particle
with a combinational wave. It means, what: 0( ) 0γ∆ = .
Besides we shall take into account that as a result of
interaction of waves with particles energy of a particle
has changed not on many. In this case detune it is possi-
ble to spread out in Taylor's number:
0
0( )
γ
γ δγ
γ
∂∆
∆ = ∆ + ∂
.
Then the equations (9) and (10) will be completely
closed and will accept the following kind:
0
d
d γ
θ δγ
τ γ
∂∆
= ∂
,
0
cos ( )d F
d
δγ θ τ
τ γ
Ω
= +
. (11)
The system of the equations (11) is equivalent to the
equation of a mathematical pendulum taking place un-
der influence of external periodic force ( )F τ
( ) ( )
0 0/ ( / ) cos F
γ
θ γ γ θ τ= ∂∆ ∂ Ω + . (12)
The equation (12) correctly describes dynamics of
particles at small amplitudes of waves working on them.
And the described dynamics is less than amplitude of
these waves, the more precisely.
2. NUMERICAL RESEARCHES
OF DYNAMICS OF PARTICLES
We are interested in the dynamics, both at small and
at high field strengths. Therefore, we performed a series
of numerical studies of the initial system of equations
(4). We investigated the dynamics of the particle num-
ber in the most interesting field configuration, which is
the field 3n ≥ of propagating electromagnetic waves.
The dispersion diagram in Fig. 1 illustrates the appear-
ance of combination waves.
Fig. 1. The dispersion diagram of interacting waves
The conditions for the appearance of stochastic in-
stability are conditions of overlapping nonlinear reso-
nances of combination waves
1 1
0
0 1 02
0 0 0
( )
i i i iph ph i iv v
k v
ω ω
γ+ ++
− ≤ ∆ + ∆
, (13)
where { }0 0 0/ ( ), 1,2,.. , 1
iph i i i iv k k i nω ω ω= ∆ + = ∆ ≡ − .
ISSN 1562-6016. ВАНТ. 2015. №4(98) 163
We chose two values for the field strengths of each
of the waves 0.03i = , 0.3i = and different values of
the wave numbers. The initial velocities of the particles
were chosen to be zero.
Fig. 2 shows the change of energy versus time for a
single particle with the initial phase 0 0ψ = three waves
with strengths 0.03i = , 0.3i = and wave numbers
1 2 30.8, 1, 1.2k k k= − = − = .
Fig. 2. The energy of
single particle 0.03i =
Fig. 3. The energy of
single particle 0.3i =
From Fig. 2 and Fig. 3 it is visible that at small
strength of electromagnetic field of waves the particles
oscillate regular, being in single nonlinear resonance of
the combination wave. With increasing field strength
under the action of fields there is transition of the parti-
cle from resonance in the resonance, dynamics of the
particles motion has irregular character with significant
changes of particles energy.
To determine the laws of interaction of charged par-
ticles with electromagnetic fields we will investigate the
energy averaged over ensemble of 30 particles. Depend-
ences on time of average energy with various initial
values of phases 0π ϕ π− < ∆ < for three waves with
strength 0.3i = and various values of wave numbers
are given in plots of Figs. 4,5.
Fig. 4. Energy averaged
over ensemble of particles
1 2 30.8, 1, 1.2k k k= − = − =
Fig. 5. Energy averaged
over ensemble of particles
1 2 30.9, 1, 1.1k k k= − = − =
Dependences on time of average energy with various
initial values of phases 0π ϕ π− < ∆ < for five waves
with strength 0.3i = and various values of wave num-
bers are given in plots of Figs. 6,7.
Fig.6. Energy averaged
over ensemble of particles
1 2 30.8, 1, 1.2k k k= − = − =
4 50.6, 0.4k k= − = −
Fig.7. Energy averaged
over ensemble of particles
1 2 30.9, 1, 1.1,k k k= − = − =
4 50.7, 0.7k k= − = −
As can be seen from these Figures the growth rate of
the average energy of the ensemble of particles and its
maximum energy depends on the magnitude of the
strength of electromagnetic waves, and the number of
combination waves involved in the interaction, as well
as the distance between their nonlinear resonances. So
the maximum energy that can collect particles in the
case of overlapping of Cherenkov resonances from the
combination of waves, is the sum of the distances be-
tween resonances
1 0
1
0
i i N
N
ph ph ph ph
n
v v v v
+
−
=
− = −∑ .
The degree of nonlinear resonances overlapping, de-
pending on the amplitudes of the interacting waves,
determines the time of particle transition from reso-
nance to resonance, thereby determining the velocity
and dynamics of the particles on time. The dynamics of
growth of the average energy of the particles also de-
pends on the time spent by the particles inside the non-
linear resonance.
3. MODEL OF WAVE WITH RANDOMLY
JUMPING OF PHASE
For a base for formation of the wave with chaotically
jumping phase, the travelling harmonic wave of kind
0( , ) cos( )f t r a t krω ϕ= − +
is taken (regular wave), to
phase of which we will add stochastic function of
time ( )tξ . For a numerical analysis the scheme of the
numerical analysis which allows to vary the quantity of
an interval of phases in which there is the jump of phase
change, is realized. Also it is realized the possibility to
select the interval of time in which, during the random
moment of time, the phase jump occurs. Time of jump is
supposed considerably smaller than the wave period.
On plots Fig. 8, as example, one can see the initial
part of realization (length of 1000 period) of the wave
field strength time dependence at random jump of the
phase at each period of wave for interval of phases jump
( 0π ϕ π− < ∆ < ) and spectral density of power of this
realization.
а
b
Fig. 8. Field of wave and spectrum
From these plots it is visible that phase jump occurs
at random moment of time at each period of the regular
wave Fig. 8,а, and quantity of this jump also is random
and lies in the range of phases (-π, π). The spectrum (see
Fig. 8,b) is widened enough with a maximum near to
unity.
а
b
Fig. 9. Spectrum of wave: one at one period with jump
( 0/ 2 / 2π ϕ π− < ∆ < ) (a); one at 5-th period with
jump ( 0π ϕ π− < ∆ < ) (b)
ISSN 1562-6016. ВАНТ. 2015. №4(98) 164
With increasing of the interval of time on which
there is jump of the phase and reduction of an interval
of the phase jumps, the spectrum of the wave is consid-
erably narrowed see plots Fig. 9. Spectral bandwidth
reduction is proportional both to reduction of quantity of
jump, and increasing interval of time in which this jump
takes place.
4. DYNAMICS OF PARTICLES
IN THE FIELD OF WAVE WITH RANDOM
CHANGING PHASE
The dynamics of the particle obeys to the vector
equation (14), in which ( )tξ is a stochastic function
changing under the law described above. Numerical
modeling of the particle motion in the field of wave
with chaotically changing phase is carried out in the
absence of a magnetic field 0 0H = at various intervals
of change of the phase jump ( 0π ϕ π− < ∆ < ) and various
intervals of time in which, at random moment, there is
the phase jump.
On Fig. 10 time dependence of the energy change
for single particle with initial phase 0 0ψ = and aver-
aged on ensemble from 30 particles with initial phases
from interval ( 0π ϕ π− < ∆ < ) for case of single jump at
period and interval of the phase jumps ( 0π ϕ π− < ∆ < ),
is presented. On the same plot, for comparison with the
diffuse law of the energies growth with time, the curve
of the time dependence of the energy change is given:
( )dγ τ α τ= at value of coefficient α = 0.5. The pa-
rameter of the wave force is 1ε = .
а
b
Fig. 10. Energy particle gain at the field:
one particle (a); ensemble averaging (b)
From these graphs it is visible that the single particle
at interaction with the wave field in a random way gain-
ing and loses energy. However when averaged over
ensemble of particles (a particle with different initial
phases) a certain regularity of the particles energy
growth is observed.
The graphs of dependence of the particles energy on
time averaged over ensemble of 30 particles for interval
of jump of phases 0π ϕ π− < ∆ < and values of the force
wave parameter 0.3ε = and 1ε = are given in Fig. 11
a
b
Fig. 11. Energy particle gain at the field with random
jumping phase with one jump at period:
0.3ε = (a); 1ε = (b)
From Fig. 11 it is visible that dependence of average
energy on time has close to diffusion character – smooth
curves ( )dγ τ α τ= with α = 0.5.
For a more detailed analysis of the influence of dif-
ferent parts of the spectrum in the dynamics of energy
exchange of charged particles with the wave field with
randomly changing phase of this wave identified three
main region of the spectrum: a low, basic and high fre-
quency. In this case, the missing parts of the frequency
spectrum supplemented by zero values. With the help of
the inverse Fourier transform has been restored realiza-
tions which correspond to each parts of the spectrum.
Figs. 12-14 shows plots the spectral power parts of spec-
trum and the corresponding initial part of restored field
for regions low 3
0 010 0.5ω ω ω− < < , main 0 00.5 1.5ω ω ω< <
and the high frequency 01.5ω ω> .
а
b
Fig. 12. a – low region of spectrum; b – initial part of
restored field from low region of spectrum
а
b
Fig. 13. a – main region of spectrum; b – initial part
of restored field from main region of spectrum
а
b
Fig. 14. High frequency region of spectrum (a);
initial part of field restored main region of spectrum (b)
For each of the reconstructed field realizations has
been investigated the dynamics of particles in these
fields for different values of the field amplitude. For
small amplitudes of the field strength of the wave pa-
rameter’s 0.01,ε ≤ the main contribution to the energy
exchange between the field and particles is in the low
frequency range. Figs. 15, 16 shows graphs of the longi-
tudinal momentum (energy) of the particles from time to
time, averaged over an ensemble of 30 particles using
the restored realization from various parts of the spectral
expansion.
Graphics averaged momentum for the high frequen-
cy part of the spectrum is similar graphs for the middle
part of the spectrum. As seen from these graphs for
small field amplitude ( 0 0.1ε< < ) a major role in the
energy exchange of particles with field has low frequen-
cy.
γ
ISSN 1562-6016. ВАНТ. 2015. №4(98) 165
а
b
Fig. 15. The average pulse 0.01ε = . a – low-frequency
part of spectrum;b – main part of spectrum
а
b
Fig. 16. The average pulse 0.1ε = : low-frequency
part of spectrum (a); main part of spectrum (b)
With increase of the amplitude ( 0.1ε > ) major role
in the energy exchange of particles with field has main
part of frequency spectrum.
It should be noted that the model of wave, consid-
ered above, with jumps of the phase leads to significant
broadening of the spectrum and thus the most part of the
spectrum can be a small effective for heating. However
there are schemes with jumps of phases [6, 7] which
don't lead to significant broadening of the spectrum in
vicinity of basic frequency ω (by the statement of the
authors). In this case efficiency of heating can be high.
5. EXPERIMENTAL INVESTIGATIONS
Higher we saw that it is necessary for formation of
slow virtual waves (beating waves) that the particle in-
teracted at least with two running towards each other
waves. In the resonator filled with rare plasma such
conditions can be realized at decay of a high-frequency
wave on a high-frequency wave and on the plasmas
wave. Such process especially effectively proceeds in
conditions when distance on frequencies between the
main modes of the resonator are close to plasmas fre-
quency.
In addition, the nature of the decay process (speed of
its flow, regular or chaotic dynamics) generally depends
on the intensity of the decaying waves.
Besides, character of the decay process (speed of its
flow, regular or chaotic dynamics) depends in general
on intensity of the decaing wave. To understand these
features we put a series of experiments. The resonator
filled with rare plasma ( 2 2
pω ω<< ) was excited in these
experiments from an external source (magnetron) at a
frequency 2.7 GHz. The spectrum of the oscillations
excited in the resonator depending on the level of the
RF power entered into resonator was studied. It was
found that at the level of the input power close to 52 kW
or more in the resonator regular process of decay takes
place. As a result of such decay there is a low-frequency
wave at frequency close to pω besides, there was a new
high-frequency wave which frequency there was less
than frequency of the magnetron on close to pω . Thus
in these cases (cases rather small strength of RF wave)
in the experiment classical regular process of decay was
observed. Such processes of decay (cascade of decays)
can form the required set of slow combination wave for
effective self-consistent plasma heating. It should, how-
ever, bear in mind that with further increase in the am-
plitude of the RF wave which excites the cavity, decay
process becomes less regular. Moreover at excess mag-
netron power over some critical value the decay process
becomes irregular. Note that this occurs when the
growth rate of decay instability becomes greater of the
plasma frequency. The listed above features are illus-
trated by Figs. 17-20. In these figures the spectrum of
the excited oscillations is presented at the inputting
power of 52 kW (see Fig.18). Plasma frequency in these
figures isn’t presented. When the power of the excited
oscillations increased that the spectrum of the excited
oscillations has essentially extended. When reaching of
power level 167 kW the spectrum becomes almost con-
tinuous (see Fig. 19). Dependence of the spectrum width
on the level of input power is presented on Fig. 20.
Fig. 17. Oscillations
spectrum at P=17 kW
Fig. 18. Oscillations
spectrum at P=52 kW
0
50
100
150
200
250
300
350
400
0 50 100 150 200
P, kW
S, MHz
Fig. 19. Oscillations
spectrum at P=167 kW
Fig. 20. Dependence
of the spectrum
width on the input power P
CONCLUSIONS
To compare the effectiveness of a set of energy par-
ticles in the combination of waves (in terms of overlap-
ping nonlinear resonances of the second order in the
field), and in the wave field with a randomly jumping
phase will proceed from the natural assumption of the
equality of power in the regular waves and wave with
jumping phase reg chW W= .
Here we have:
2 2
0
~ /
N
reg i i reg
i
W N Qε ω ε
=
= ∆∑ ;
~ 1/i iQω∆ ; 2
ch chW ε ω= ∆ ; ~ 1chω∆ .
From these relations we find the following relation
between the amplitudes of the regular wave and noise
wave: 2 2~ /reg chQ Nε ε .
We will also use the natural assumption that at de-
velopment of dynamic chaos, between the regular and
the amplitudes of the noise field exist the relation:
ISSN 1562-6016. ВАНТ. 2015. №4(98) 166
2~ ~comb reg chε ε ε . In this case, one can enter the follow-
ing coefficient of efficiency ( ) /chK Q Nε= ⋅ .
If this ratio is greater than one ( 1K > ), efficiency of
the heating by the field of regular wave is higher than
the heating by the noise field.
The provided experiment show that in the cavity
filled with the rare plasma, it is possible to realize the
conditions for heating of plasma by exciting in the cavi-
ty several eigenmodes of the resonator. Note also that
the scheme described above is relevant only to the heat-
ing of the plasma at relatively low field strengths. If the
power source rf oscillations (magnetron) which excites
resonator close to 150 kW or more, in the cavity devel-
oping a modified decay. The fields excited thus are
ramdon and there is no need for realization of the de-
scribed mechanism.
This work was supported in part by special-purpose
program “Perspective study of plasma physics, con-
trolled thermonuclear fusion and plasma technologies.”
(Order of Presidium of the National Academy of Sci-
ences of Ukraine № 151 of 12 March 2014).
REFERENCES
1. V.A. Buts, K.N. Stepanov. Stoxasticheskij nagrev
plazmy polem lazernogo izlucheniya // Pisma v
ZhETF. 1993, v.58, p. 520-523 (in Russian).
1. 2. F.G. Bass, Ya.B. Fainberg, V.D. Shapiro. Kvazi-
linejnaya teoriya slaboturbulentnoj plazmy s
uchetom korrelyacii elektricheskix polej // ZhETF.
1965, v. 49, p. 329-334 (in Russian).
2. V.I. Karas’, Ya.B. Fainberg, A.F. Alisov,
A.M. Artamoshkin, R. Bingham, I.V. Gavrilenko,
V.D. Levchenko, M. Lontano. V.I. Mirny, I.F.
Potapenko, A.N. Starostin. Interaction of Microwave
Radiation Undergoing Stochastic Phase Jumps with
Plasmas or Gases // Plasma Physics Reports. 2005,
v. 31, № 5, p. 746-760.
3. V.A. Balakirev, V.A. Buts, A.P. Tolstoluzhskii,
Yu.A. Turkin. Randomization of motion of a beam
of phased oscillators // Zh. Eksp. Teor. Fiz. 1983,
v. 84, p. 1279-1289.
4. V.A. Balakirev, V.A. Buts, A.P. Tolstoluzhskii,
Yu. A. Turkin. Charged-particle dynamics in the
field of two electromagnetic waves // Zh. Eksp. Teor.
Fiz. 1989, v. 95, p. 1231-1245.
5. A.F. Alisov, A.M. Artamoshkin, S.A. Berdina,
V.I. Golota, A.M. Egorov, A.G. Zagorodny,
I.A. Zagrebel’ny, V.I. Zasenko, V.I. Karas’,
I.V. Karas’, I.F. Potapenko, A.N. Starostin. Low-
Pressure Discharge Induced by Microwave Radia-
tion with a Stochastically Jumping Phase // Plasma
Physics Reports. 2010, v. 36, № 8, p. 736-749.
6. V.I. Karas’, A.M. Artamoshkin, A.F. Alisov,
O.V. Bolotov, V.I. Golota, I.V. Karas, A.M. Yegorov,
I.A. Zagrebelny, I.F. Potapenko, A.N. Starostin.
Special Features of Low-Pressure Discharge Initiat-
ed by Microwave Radiation with Stochastic Jumping
Phase // IEEE Transactions on Plasma Science.
2013, v. 41, № 9, p. 1458-1463.
Article received 19.05.2015
СТОХАСТИЧЕСКИЙ НАГРЕВ ЗАРЯЖЕННЫХ ЧАСТИЦ
В ОТСУТСТВИЕ РЕЗОНАНСОВ ПЕРВОГО ПОРЯДКА
А.Н. Антонов, В.А. Буц, Е.А. Корнилов, В.Г. Свиченский, В.В. Кузьмин, А.П. Толстолужский
Исследуется динамика заряженных частиц во внешних электромагнитных полях в отсутствие резонан-
сов, пропорциональных первой степени напряженности электромагнитного поля. Исследуются две схемы. В
первой предполагается наличие нескольких регулярных электромагнитных волн. Предполагается, что часто-
ты и волновые векторы этих волн таковы, что фазовая скорость волн биения близка к тепловым скоростям
частиц. Существенно, что таким образом организованные черенковские резонансы пропорциональны квад-
рату малого параметра (квадрату безразмерной напряженности поля). Во второй схеме предполагается, что
фаза поля волны изменяется скачком по случайному закону. Найдены параметры изучаемых систем, при
которых имеет преимущество одна или другая схема передачи энергии поля частицам.
СТОХАСТИЧНЕ НАГРІВАННЯ ЗАРЯДЖЕНИХ ЧАСТИНОК
У ВІДСУТНОСТІ РЕЗОНАНСІВ ПЕРШОГО ПОРЯДКУ
О.М. Антонов, В.О. Буц, Є.О. Корнілов, В.Г. Свіченський, В.В. Кузьмін, О.П. Толстолужський
Досліджується динаміка заряджених частинок у зовнішніх електромагнітних полях при відсутності резо-
нансів, які пропорційні першому ступеню напруженості електромагнітного поля. Досліджуються дві схеми.
У першій передбачається наявність декількох регулярних електромагнітних хвиль. Передбачається, що час-
тоти і хвильові вектори цих хвиль такі, що фазова швидкість однієї з хвиль биття близька до теплових шви-
дкостей частинок. Істотно, що таким чином організовані черенковські резонанси пропорційні квадрату ма-
лого параметра (квадрату безрозмірної напруженості поля). У другій схемі передбачається, що фаза поля
хвилі змінюється стрибком за випадковим законом. Знайдено параметри досліджуваних систем, при яких
має перевагу одна або інша схема передачі енергії поля частинкам.
introduction
5. experimental INVESTIGATIONS
СТОХАСТИЧЕСКИЙ НАГРЕВ ЗАРЯЖЕННЫХ ЧАСТИЦ В ОТСУТСТВИЕ РЕЗОНАНСОВ ПЕРВОГО ПОРЯДКА
СТОХАСТИЧНЕ НАГРІВАННЯ ЗАРЯДЖЕНИХ ЧАСТИНОК У ВІДСУТНОСТІ РЕЗОНАНСІВ ПЕРШОГО ПОРЯДКУ
|