The influence of thermal motion of particles on the formation of cavities in plasma
The effect of the thermal motion of ions and electrons on the formation of plasma density cavities which appear due to inhomogeneous stochastic electric fields is considered. Using the equation of motion of plasma particles in a constant magnetic field and in an inhomogeneous stochastic electric fie...
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irk-123456789-1959022023-12-08T12:59:50Z The influence of thermal motion of particles on the formation of cavities in plasma Azarenkov, N.A. Chibisov, A.D. Chibisov, D.V. Basic plasma physics The effect of the thermal motion of ions and electrons on the formation of plasma density cavities which appear due to inhomogeneous stochastic electric fields is considered. Using the equation of motion of plasma particles in a constant magnetic field and in an inhomogeneous stochastic electric field with a frequency of the order of lower hybrid oscillations, taking into account the thermal motion of particles, the diffusion coefficients and drift velocities of ions and electrons are obtained. These values are used in the Fokker-Planck equation to determine the stationary distribution of the plasma density due to the effect of an inhomogeneous stochastic electric field. The conditions for the thermal velocities of particles under which the formation of a cavity is possible are obtained. Розглянуто вплив теплового руху іонів та електронів на формування порожнин щільності плазми, що виникають внаслідок впливу неоднорідних електричних стохастичних полів. З рівняння руху частинок плазми у постійному магнітному полі та в неоднорідному стохастичному електричному полі з частотою нижньогібридних коливань з урахуванням теплового руху частинок отримані коефіцієнти дифузії та дрейфові швидкості іонів та електронів. Знайдені значення використовуються у рівнянні Фоккера-Планка для знаходження стаціонарного розподілу щільності плазми внаслідок впливу неоднорідного стохастичного поля. Визначено умови на теплові швидкості частинок, за яких можливе утворення порожнини. 2022 Article The influence of thermal motion of particles on the formation of cavities in plasma / N.A. Azarenkov, A.D. Chibisov, D.V. Chibisov // Problems of Atomic Science and Technology. — 2022. — № 6. — С. 16-20. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.35.Ra, 52.35.Mw DOI: https://doi.org/10.46813/2022-142-016 http://dspace.nbuv.gov.ua/handle/123456789/195902 en Problems of Atomic Science and Technology Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Basic plasma physics Basic plasma physics Azarenkov, N.A. Chibisov, A.D. Chibisov, D.V. The influence of thermal motion of particles on the formation of cavities in plasma Problems of Atomic Science and Technology |
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The effect of the thermal motion of ions and electrons on the formation of plasma density cavities which appear due to inhomogeneous stochastic electric fields is considered. Using the equation of motion of plasma particles in a constant magnetic field and in an inhomogeneous stochastic electric field with a frequency of the order of lower hybrid oscillations, taking into account the thermal motion of particles, the diffusion coefficients and drift velocities of ions and electrons are obtained. These values are used in the Fokker-Planck equation to determine the stationary distribution of the plasma density due to the effect of an inhomogeneous stochastic electric field. The conditions for the thermal velocities of particles under which the formation of a cavity is possible are obtained. |
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Article |
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Azarenkov, N.A. Chibisov, A.D. Chibisov, D.V. |
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Azarenkov, N.A. Chibisov, A.D. Chibisov, D.V. |
| author_sort |
Azarenkov, N.A. |
| title |
The influence of thermal motion of particles on the formation of cavities in plasma |
| title_short |
The influence of thermal motion of particles on the formation of cavities in plasma |
| title_full |
The influence of thermal motion of particles on the formation of cavities in plasma |
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The influence of thermal motion of particles on the formation of cavities in plasma |
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The influence of thermal motion of particles on the formation of cavities in plasma |
| title_sort |
influence of thermal motion of particles on the formation of cavities in plasma |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2022 |
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Basic plasma physics |
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http://dspace.nbuv.gov.ua/handle/123456789/195902 |
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The influence of thermal motion of particles on the formation of cavities in plasma / N.A. Azarenkov, A.D. Chibisov, D.V. Chibisov // Problems of Atomic Science and Technology. — 2022. — № 6. — С. 16-20. — Бібліогр.: 3 назв. — англ. |
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Problems of Atomic Science and Technology |
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2025-07-17T00:10:20Z |
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BASIC PLASMA PHYSICS
ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142).
16 Series: Plasma Physics (28), p. 16-20.
https://doi.org/10.46813/2022-142-016
THE INFLUENCE OF THERMAL MOTION OF PARTICLES ON THE
FORMATION OF CAVITIES IN PLASMA
N.A. Azarenkov1,2, A.D. Chibisov3, D.V. Chibisov2
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine;
2V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
3H.S. Skovoroda Kharkiv National Pedagogical University, Kharkiv, Ukraine
E-mail: dmitriychibisov@karazin.ua
The effect of the thermal motion of ions and electrons on the formation of plasma density cavities which appear
due to inhomogeneous stochastic electric fields is considered. Using the equation of motion of plasma particles in a
constant magnetic field and in an inhomogeneous stochastic electric field with a frequency of the order of lower
hybrid oscillations, taking into account the thermal motion of particles, the diffusion coefficients and drift velocities
of ions and electrons are obtained. These values are used in the Fokker-Planck equation to determine the stationary
distribution of the plasma density due to the effect of an inhomogeneous stochastic electric field. The conditions for
the thermal velocities of particles under which the formation of a cavity is possible are obtained.
PACS: 52.35.Ra, 52.35.Mw
INTRODUCTION
It was shown in [1] that inhomogeneous stochastic
electrostatic fields lead to diffusion and drift motion
of ions from a region with an increased level of
oscillations, which causes here a depletion of the ion
density. This effect is similar to that which occurs in
plasma with an inhomogeneous harmonic electric
field, when the resulting ponderomotive force pushes
electrons out of the region with oscillations [2]. It was
assumed in [1], that there was no magnetic field, so
the ions were unmagnetized. In [3] the effect of
nonuniform stochastic electric fields with a frequency
of the order of lower hybrid oscillations on the
diffusion and drift motion of particles in plasma in a
constant magnetic field was considered. It was
assumed that the characteristic oscillation frequency
significantly exceeded the cyclotron frequency of the
ions; therefore, the ions were considered to be
nonmagnetized. Whereas the cyclotron frequency of
the electrons significantly exceeded the oscillation
frequency and the electrons were magnetized. It was
shown that both the diffusion coefficient and the drift
velocity of electrons significantly exceed these values
for ions, so that the time of formation of the plasma
cavity is determined by the electrons. However, in [1,
3] the thermal motion of plasma particles was not
taken into account, whereas, the thermal motion of
ions and electrons can significantly affect the
transport processes in plasma and lead to restrictive
conditions for the formation of a plasma density
cavity.
In this work, we take into account the thermal
motion of particles and obtain the diffusion
coefficients and drift velocities of ions and electrons,
which include their thermal velocities.
We consider collision less homogeneous plasma
in the constant magnetic field B directed along the z
axis, in which at some point in time a region with a
stochastic electric field appears, which is inhomogeneous
along the x-axis and homogeneous in other directions. It
is assumed that the typical frequency of stochastic
oscillations is of the order of the lower hybrid frequency
lh, which is much lower than the electron cyclotron
frequency ce and much higher than the ion cyclotron
frequency ci.
We study the evolution of the plasma distribution
function due to inhomogeneous electrostatic turbulence
using the one-dimensional Fokker-Planck equation
( )
( ) ( )( )
( )
( )
2
2
,
, ,
2
f x t B x
A x f x t f x t
t x x
= − +
, (1)
where A(x) is the drift velocity and B(x)/2 is the diffusion
coefficient. These values we determine from the particle
motion equation as the averaged over a long time quasi-
linear drift velocity ( ) /A x dx dt= and the velocity of
the squared root-mean-square displacement
B(x)/2 = d⟨x2⟩.
The equation of motion of charged particles in a
magnetic field, taking into account the stochastic electric
field, is
( ) ( ), ,
e edv
F x E r t v B
dt m m
= + , (2)
where ( ),E r t is the electric field strength of electrostatic
turbulence, far from the region with a high level of
turbulence, F(x) 1 is the envelope of turbulent
pulsations having a maximum at x = 0 and F() = 1.
In the first part we determine the diffusion coefficient
of ions in inhomogeneous stochastic electric fields, taking
into account their thermal motion, and also find the rate
of drift motion due to the ponderomotive force in an
inhomogeneous stochastic electric field. In the second
ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142) 17
part, we determine the diffusion coefficient of the
guiding centers of electrons both across and along the
magnetic field and also find their rate of drift
movement. In the third part, we analyze the Fokker-
Planck equation, where we use the obtained values of
the diffusion coefficients, as well as the drift
velocities of ions and electrons and determine the
conditions for the thermal velocities of electrons and
ions under which cavities are formed.
1. DIFFUSION AND DRIFT OF IONS
We find the solution of equation (2) for ions,
neglecting the effect of the magnetic field, since the
frequency of stochastic oscillations significantly
exceeds the cyclotron frequency of ions.
( )
( )
( ) ( )
0
0, = = +
t
i
i
i t
dr t e
v t F x E r t dt v
dt m
, (3)
where t=t0 is the time of occurrence of increased level
of stochastic oscillations in plasma, 0iv is the initial
velocity vector of the ion, which is a random vector
whose components are random variables distributed
according to the normal law. In fact 0i Tiv v= is the
thermal ion velocity. Now we find the value of the
random displacement ( )r t of ion by integrating (3)
over time,
( ) ( ) ( ) ( )
0 0 0
0,
t t t
i
it t t
e
r t v t dt F r E r t dt dt v t
m
= = + ,
(4)
and then obtain the rate of change in mean square
displacement by multiplying (3) by (4) and averaging
over a large time interval
( )
( ) ( )
( )
2
2
2
2
1
2
i
i
d r tdr t e
r t F r
dt dt m
= =
( ) ( )
0 0 0
0 0, ,
t t t
t t t
E r t dt dt E r t dt v t v
+ . (5)
In (5) we have
2 2
0 0 0 Tv t v v t v t = = .
To estimate the integral in equation (5), we represent
( ),E r t in the form of a Fourier integral
( ) ( ) ( ), , expE r t E r i t d
−
= − . (6)
Substituting (6) into (5) and integrating over t and
t we obtain
( )
( )
2
2
2
2
1
2 i
d r t e
F r
dt m
=
( )
( ) ( )
0
2
2
,
exp ,
t
T
t
E r
i t d E r t dt v t
−
− + . (7)
We assume that the width of the spectrum of
stochastic oscillations is of the order of the oscillation
frequency
lh = and replace in (7) 21/ by
21/ :
( )
( )
2
2
2
2 2
1
2 i
d r t e
F r
dt m
=
( ) ( ) ( )
0
2, exp ,
t
T
t
E r i t d E r t dt v t
−
− + . (8)
In (8) we find the inverse Fourier transform
( )
2
2
2 2
1
2
i
i
d r t e
dt m
( ) ( ) ( )
0
2 2, ,
t
T
t
F x E r t E r t dt v t + . (9)
We assume that the electrostatic turbulence satisfies the
conditions
( ), 0,E r t =
( ) ( ) ( ) ( )2 2, , ,E r t E r t E r t E r = = , (10)
where ( )2E r is the square of the amplitude of
stochastic oscillations and we assume that this value does
not depend on time. Taking into account (10) and
integrating over time in (9), we obtain
( )
( ) ( )
2
2
2 2 2
2 2
1
2
i
i T
i
d r t e
D F x E r t v t
dt m
= +
. (11)
Equation (11) determines the diffusion coefficient of ions
in the turbulent field of electrostatic turbulence taking
into account the thermal motion of ions. The first term in
(11) describes diffusion in a stochastic electric field, and
was previously obtained in [1, 3]. The second term
describes diffusion due to thermal motion of ions.
Consider the directed motion of ions caused by the
ponderomotive force due to the radial gradient of the
amplitude of the turbulent field envelope.
The equation of motion of ion along x-axis is
( ) ( )
2
2
,i i x
d x
m e F x E r t
dt
= , (12)
where ( ),xE r t is the projection of the electric field
strength of stochastic oscillations onto the x-axis. Using
the calculations performed in the papers [1, 3] and taking
into account (10) we obtain the equation, describes the
drift motion of an ion due to inhomogeneous electrostatic
turbulent field
( ) ( )
22
2 2
02 2
=
i
i x
i
ed x
m F x E r
dt m
. (13)
The expression on the right side of (13) is the
ponderomotive force arising from the inhomogeneity of
the electrostatic turbulent field along the x-axis.
Integrating (13), we obtain the velocity of the drift
motion of ion along the x-axis
18 ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142)
( ) ( )
0
22
2 2
02 2 2
= =
t
i
x
it
edx d x
dt F x E r t
dt dt m
. (14)
Note, that the thermal motion of ions does not affect
their drift velocity.
2. DIFFUSION AND DRIFT OF
ELECTRONS
The solution of eq. (2) for electrons is found
taking into account the magnetic field. First, we
neglect the effect of the stochastic electric field and
obtain the integrals of motion, which across the
magnetic field are the coordinates of the guiding
center
,
y x
ce ce
v v
X x Y y
= + = − , (15)
which mean the invariability in time of the
coordinates of the guiding center of an electron
rotating in a magnetic field. The integral of motion
along the magnetic field is the electron momentum,
pz = mvz.
The appearance of stochastic electric fields in
plasma leads to slight distortions in these values.
Now we consider the solution of (2) in the first
approximation and obtain the changes in the integrals
of motion caused by stochastic electric fields.
Represent v in the form:
0 1e ev v v= + ,
where 0ev is the initial velocity vector of the electron,
which is a random vector whose components are
random variables distributed according to the normal
law. In fact 0e Tev v= is the thermal electron velocity.
1ev is the fluctuation of the velocity, caused by
stochastic electric fields and which is determined by
the equation
( ) ( )1
1, ,e e
e e
e edv
F x E r t v B
dt m m
= + . (16)
The solutions of (16) for the components of velocity
are
( ) ( ) ( )
( )1 2
,
,
e x e
x y
e cee ce
e F x dE r t e F x
v E r t
dt mm
= + , (17)
( ) ( ) ( )
( )1 2
,
,
ye e
y x
e cee ce
dE r te F x e F x
v E r t
dt mm
= − , (18)
( ) ( )
0
1 ,
t
z z
e t
e
v F x E r t dt
m
= . (19)
Write the X coordinate in the form
0 1X X X= + , (20)
where 1X is the random displacement of the
coordinate of the guiding center due to stochastic
electric fields which is found using (15), (17) and
(18) and obtain the rate of change in mean square
displacement [3]
( )
2 2
1 21
1 2 2
1 1
2
e
e ce
d X edX
X F x
dt dt m
= =
( ) ( )
( )
2
0
,1 1
, , .
2
t
y
y y
ce
d E r t
E r t E r t dt
dt
+
(21)
Neglecting the second term in (21) which is much smaller
than the first one, we obtain
( ) ( ) ( )
2 2
1 2
2 2
0
1 1
, , .
2
t
e
y y
e ce
d X e
F x E r t E r t dt
dt m
= (22)
Taking into account the condition (10) we obtain
( ) ( )
2 2
1 2 2
2 2
1
2 2
e
xe y
e ce
d X e
D F x E r t
dt m
= = . (23)
Equation (23) can also be written as
( ) ( )
2 2
1 2 2
2
1
2 2
y
d X c
F x E r t
dt B
= , (24)
or, label
y
dx
cE
v
B
= , (25)
which is equal to the velocity of the drift motion of a
particle in crossed fields along the x -axis we obtain
( )
( )
2 2
1 21
,
2 2
dx
d X F x
v r t t
dt
= . (26)
Thus, the rate of the root-mean-square displacement of
the coordinate of the guiding center along the x-axis is
determined by the mean value of the square of y-
component of the stochastic electric field, or, otherwise,
by the mean value of the square of the particle drift
stochastic velocity in crossed fields.
Along the z-axis there is no influence of the magnetic
field on the motion of electrons, therefore, similarly to
what was done for ions, for the root-mean-square shift of
the coordinate along the z-axis we obtain
( )
( ) ( )
2
2
2 2 2
02 2
1
2 2
ze z e
e
d z t e
D F x E r t v t
dt m
= +
. (27)
The first term in (27) determines the diffusion of
electrons in a stochastic electric field, and the second one
determines diffusion due to the thermal motion of
electrons. In order for the contribution of the stochastic
term to the diffusion coefficient to exceed the
contribution from thermal motion, it is necessary that the
condition
( ) ( )2 2 2
2 2
02
z
e e
e F x E r
m v
(28)
hold. This means that the change in the momentum of an
electron in a stochastic electric field over a time of the
order of the oscillation period must exceed the thermal
momentum of the electron.
Comparison of (23) and (27) shows, that the diffusion
of electrons along the magnetic field is much greater than
the diffusion across the magnetic field so that at least
2 2/ ~ / 1ze xe ce lhD D . (29)
Moreover, also
2 2~ / 1xe i lh ciD D , (30)
ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142) 19
that is the diffusion coefficient of the guiding centers
of electrons significantly exceeds the diffusion
coefficient of ions.
Now we obtain the velocity of the drift motion of
electrons along the x-axis caused by the
ponderomotive force in an inhomogeneous
electrostatic turbulence. We represent the random
displacement of the coordinate of the guiding center
X1 as the sum of oscillatory X and quasilinear X
components
1X X X= + , (31)
where 1X X= and 0X = . To determine the drift
velocity of the guiding center we use (20) and obtain
[3]
( ) ( ) ( ) ( )
2
0 02 2
0
, , .
t
e
y y
e ce
edX
F X F X E r t E r t dt
dt m
= (32)
Integration (32) using condition (10) yields
( ) ( )
2
2
2
02 2
,
2
e
y
e ce
edX
F X E r t t
dt m
= . (33)
Equation (33) determines the velocity of the drift
motion of the guiding center of electron along the x -
axis. Note, that the thermal motion of electrons does
not affect their drift velocity.
Comparison of (14) and (33) shows, that the drift
velocities of guiding centers of electron is much
greater than the drift velocities of ions
2
2
1
ci
dX dx
dt dt
. (34)
Thus, electrons leave the region of increased
turbulence level much faster than ions.
Note also that the ratio of the diffusion
coefficients of the guiding centers of electrons and
ions is of the same order as the ratio of their drift
velocities, namely
2 2
ci .
3. STATIONARY DENSITY
DISTRIBUTION AND CAVITY
FORMATION CONDITIONS
Inhomogeneous electrostatic turbulence leads to a
change in the plasma density distribution. The
evolution of the distribution function а f(x,t) as a
result of diffusion as well as the drift motion of
particles is governed by the Fokker-Planck equation
(1). Above were obtained the diffusion coefficients
along the x-axis for electrons (23)
( ) ( )
2
2 2
2 22 2
e e
xe y
e ce
B e
D F x E r t
m
= = , (35)
and for ions (11)
( ) ( )
2
2 2 2
2 22
i i
xi x T
i
B e
D F x E r t v t
m
= = +
. (36)
The drift velocities are (33)
( ) ( )
2
2
2
02
,
2
e
e y
e ce
edX
A F X E r t t
dt m
= = (37)
for electrons and (14)
( ) ( )
2
2 2
02 2
i
i x
i
edx
A F x E r t
dt m
= =
(38)
for ions.
We now find the dependence of the plasma density on
the x -axis in a stationary state, n(x) = f(x), assuming that
the evolution of the distribution function has ended.
Equating in (1) the derivative of the distribution function
with respect to time to zero, we obtain the equation
( ) ( )( )
( )
( )
2
2
0
2
B x
A x n x n x
x x
− + =
. (39)
This equation can be simplified
( ) ( )
( )
( ) 0
2
B x
A x n x n x
x
− + =
, (40)
and then
( )
( )
( ) ( )( )
( ) ( )
2 d B x n xA x dr
B x B x n x
= . (41)
Equation (41) is valid for both electrons and ions.
Consider first the solution of the equation for
electrons, since the processes of diffusion and drift of
electrons significantly exceed these processes for ions.
Substituting (35) and (37) into (41) we get
( )
( )
( )( )
( )
2
2
d F r ndF r
F r nF r
= . (42)
Integrating (42), we obtain
( )
( )
C
n x
F x
= . (43)
Here C =n0, since F() =1. Thus the plasma density
distribution long after the appearance of a region with an
increased level of turbulence in homogeneous plasma is
( )
( )
0n
n x
F x
= . (44)
In accordance to (44), the minimum plasma density is
reached in the region with the maximum level of low-
frequency turbulence. Thus a region with a depleted
electron density is formed.
Let us consider a possible limitation of the cavity
formation process. As already mentioned, electron
diffusion occurs not only across but also along the
magnetic field. If it turns out that the longitudinal size L
of the region with an increased level of stochastic
oscillations is not large enough and the electrons leave
this region along the magnetic field rather quickly, then
no electron density cavity is formed. The criterion for the
formation of a cavity can be chosen as follows. The
longitudinal size of the cavity must exceed a certain
value, so that when passing this distance along the z axis,
the electron is displaced across the magnetic field by an
amount exceeding the thermal Larmor radius of the ions
Ti (in this case, the transverse size of the cavity will be
no less than the thermal Larmor radius of the ions). If this
condition is not met and the electrons quickly leave this
region, then no cavity is formed. For the case when the
stochastic field is strong enough so that inequality (28)
holds, the longitudinal size of the plasma must satisfy the
inequality
20 ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142)
ce
Ti
lh
L
. (45)
If inequality (28) is not satisfied, then the longitudinal
dimension must satisfy the inequality
2
Te
Ti
dx
v
L
F v
, (46)
where dxv is determined by (25). If the plasma has
sufficiently large dimensions (infinite in the limiting
case), then the transverse dimensions of the cavity are
determined by (44), and is equal to F(x)/∇F(x).
Let us now assume that the longitudinal
dimensions of the plasma are not sufficient, so that
the inequalities (45) or (46) are not satisfied, and no
electron density cavity is formed. In this case, the
plasma density cavity can be formed due to ions.
Substituting (36) and (38) into (41) we obtain for ions
( )
( )
( ) ( )( )
( ) ( )
22
2 22
d F x n xdF x
F x n x F x
= , (47)
where
( )
( )
2 2 2
02 2
2
2 ,
i
i x
m v
F F x
e E r t
= + . (48)
Integrating (47), we obtain
( )
( )
0=
n
n x
F x
. (49)
For the formation of a cavity in the ion density, it is
necessary to satisfy the condition
( )
2 2 2
0
2
2
1
,
i
i x
m v
e E r t
, (50)
which means that the energy received by the ion from
the stochastic electric field over time, of the order of
the oscillation period, exceeds the thermal energy of
the ion. If the inequality opposite to (50) is satisfied,
then the depth of the cavity turns out to be smaller
and tends to zero as this ratio grows.
CONCLUSIONS
Inhomogeneous stochastic electric fields in the
frequency range of the lower hybrid frequency can
lead to the formation of density cavities in
magnetized plasma. It has been established that the
electron density cavity is formed much faster than the ion
density cavity, since the drift velocity and the electron
diffusion coefficient exceed these values for ions by a
factor of 2 2/ 1lh ci . The formed cavity of electron
density accelerates the formation of the cavity of ion
density due to the arising constant electric field, which
increases the drift velocity of the ions.
However, the thermal motion of plasma particles can
lead to limitations in the formation of cavities. In
particular, it is shown that the thermal motion of electrons
along the magnetic field does not lead to the formation of
electron density cavities if the size of the of region with
an increased level of stochastic oscillations in the
magnetic field does not satisfy the inequalities (45) or
(46), since the electrons leave the structure in the
magnetic field faster than the cavity is formed.
If the electron density cavity is not formed, then the
plasma density cavity can be formed due to ions (49).
However, in this case, the thermal motion of ions along
the direction of the inhomogeneity of the stochastic
electric field also limits the possibility of the formation of
an ion density cavity. It is shown that the cavity is formed
if the energy received by the ion from stochastic electric
fields during the period of lower hybrid oscillations
exceeds the thermal energy of the ion (50). Otherwise, the
cavity depth becomes smaller and tends to zero as this
ratio increases.
REFERENCES
1. D.V. Chibisov. Formation of cavities in the ionospheric
plasma due to localized electrostatic turbulence //
Problems of Atomic Science and Technology. Series
«Plasma Electrons and New Methods of Accelerations»
(116). 2018, № 4, p.131-134.
2. A.V. Gaponov, M.A. Miller. Potential wells for
charged particles in a high-frequency electromagnetic
field // JETP. 1958, v. 7, № 1, p. 168.
3. N.A. Azarenkov, A.D. Chibisov, D.V. Chibisov.
Movement of charged particles in magnetic and
nonuniform stochastic electric fields // Problems of
Atomic Science and Technology. Series «Plasma
Electrons and New Methods of Accelerations» (134).
2021, № 4, p. 112-117.
Article received 11.10.2022
ВПЛИВ ТЕПЛОВОГО РУХУ ЧАСТИНОК НА ФОРМУВАННЯ ПОРОЖНИН У ПЛАЗМІ
М.О. Азаренков, О.Д. Чібісов, Д.В. Чібісов
Розглянуто вплив теплового руху іонів та електронів на формування порожнин щільності плазми, що
виникають внаслідок впливу неоднорідних електричних стохастичних полів. З рівняння руху частинок
плазми у постійному магнітному полі та в неоднорідному стохастичному електричному полі з частотою
нижньогібридних коливань з урахуванням теплового руху частинок отримані коефіцієнти дифузії та
дрейфові швидкості іонів та електронів. Знайдені значення використовуються у рівнянні Фоккера-Планка
для знаходження стаціонарного розподілу щільності плазми внаслідок впливу неоднорідного стохастичного
поля. Визначено умови на теплові швидкості частинок, за яких можливе утворення порожнини.
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