The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields
The problem of determining the distribution function of particles in plasma that is produced in crossed longitudinal magnetic and radial electric fields is solved. It is assumed that the velocity probability distribution functions for the particles of the residual gas before its ionization or initia...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2014
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| Cite this: | The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields / D.V. Chibisov // Вопросы атомной науки и техники. — 2014. — № 6. — С. 55-57. — Бібліогр.: 3 назв. — англ. |
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| citation_txt | The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields / D.V. Chibisov // Вопросы атомной науки и техники. — 2014. — № 6. — С. 55-57. — Бібліогр.: 3 назв. — англ. |
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| description | The problem of determining the distribution function of particles in plasma that is produced in crossed longitudinal magnetic and radial electric fields is solved. It is assumed that the velocity probability distribution functions for the particles of the residual gas before its ionization or initial plasma before it has appeared in crossed fields are Maxwellian with non-zero thermal velocities. Then produced plasma particles are moving in crossed fields without collisions. The obtained distribution function is written in the coordinates of the guiding center. The expressions for the distribution function in the limiting cases of strong and weak electric field are also obtained.
Решается задача об определении функции распределения частиц в плазме, которая создается в скрещенных продольном магнитном и радиальном электрическом полях. Предполагается, что функция распределения по скоростям частиц остаточного газа до его ионизации, или начальной плазмы, прежде чем она появляется в скрещенных полях, является максвелловской с ненулевой тепловой скоростью. Образовавшиеся частицы плазмы затем движутся в скрещенных полях без столкновений. Полученная функция распределения записана в координатах ведущего центра. Получены также выражения для функции распределения в предельных случаях сильного и слабого электрических полей.
Розв’язано задачу про визначення функції розподілу частинок у плазмі, яка створюється в схрещених поздовжньому магнітному і радіальному електричному полях. Вважається, що функція розподілу за швидкостями частинок залишкового газу до його іонізації, або початкової плазми, перш ніж вона з'являлася в схрещених полях, є максвелiвською з ненульовою тепловою швидкістю. Частинки плазми, що утворилися, рухаються далі в схрещених полях без зіткнень. Здобуту функцію розподілу записано в координатах ведучого центра. Здобуто також вирази для функції розподілу в граничних випадках сильного та слабкого електричних полів.
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BASIC PLASMA PHYSICS
ISSN 1562-6016. ВАНТ. 2014. №6(94)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2014, №6. Series: Plasma Physics (20), p.55-57. 55
THE DISTRIBUTION FUNCTION OF PLASMA PARTICLES IN LONGI-
TUDINAL MAGNETIC AND RADIAL ELECTRIC FIELDS
D.V. Chibisov
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
E-mail: chibisovdm@mail.ru
The problem of determining the distribution function of particles in plasma that is produced in crossed longitudi-
nal magnetic and radial electric fields is solved. It is assumed that the velocity probability distribution functions for
the particles of the residual gas before its ionization or initial plasma before it has appeared in crossed fields are
Maxwellian with non-zero thermal velocities. Then produced plasma particles are moving in crossed fields without
collisions. The obtained distribution function is written in the coordinates of the guiding center. The expressions for
the distribution function in the limiting cases of strong and weak electric field are also obtained.
PACS: 52.27.Jt
INTRODUCTION
In the problem of plasma stability in crossed longitu-
dinal magnetic В and radial electric rE fields, it is im-
portant to know the distribution function on the particle
velocity and coordinates. For example, in [1-3], the ion
cyclotron instabilities of plasma in crossed fields were
studied. In [1], the ion cyclotron instability of rotating
plasma was considered in the case of a weak radial elec-
tric field, when the energy of a particle motion in the
electric field was much less than their thermal motion
energy. In [2, 3], the ion cyclotron instability of the
plasma was studied in the opposite case of strong radial
electric field. In each of these limiting cases the particle
distribution function was defined in different way based
on the specific features of the problem. Accordingly, in
each of these cases the excitation of the ion cyclotron
instability is determined by different mechanisms. In
some cases, an intermediate version when the energy of
the particles thermal motion is comparable to the energy
transferred to the particle from the electric field is also
interesting.
In cylindrically symmetric plasma it is convenient to
use independent variables which define the energy of
the transverse particle motion and its generalized
angular momentum instead of the coordinates and
velocities. These new coordinates are the integrals of
motion. Distribution function was chosen in [2, 3] as
follows
zez vaeYvF )(,, . (1)
In (1), Y is the Heaviside step function, is the Dirac
delta function, a is the potential on the external
electrode (anode), /e rcE Br . The radial profile of
the electric field potential was assumed to be parabolic,
2 2( / )r a r a , which ensured the independ-
ence of the angular rotation rate of the particle on the
radius (rigid rotor model). One of the conditions of the
limit of strong electric field is that the initial energy of
ions produced in the plasma as a result of the ionization
should be considerably less than the ion energy obtained
in the radial electric field. Second condition is the ine-
quality cir , where r is the frequency of the
radial ion oscillations (modified cyclotron frequency in
crossed fields), ci is the ion cyclotron frequency. In
this paper, the unperturbed distribution function is de-
termined for an arbitrary ratio of the initial ion energy
and the energy of motion in crossed fields. Furthermore,
the ratio of frequencies r and ci is assumed to be
arbitrary one, which corresponds to an intermediate case
in which the energy of the particles thermal motion is
comparable to the energy obtained by the particle from
the electric field.
1. GENERAL EXPRESSION FOR THE
DISTRIBUTION FUNCTION
We consider the same model of plasma as in the pa-
pers [2, 3]. Cylindrically symmetric collisionless plasma
is placed in crossed longitudinal magnetic and radial
electric fields. In the radial direction, plasma is limited
by metal electrode (anode) with the radius ar . The elec-
trode has a positive potential relative to the axis, so that
the electric field is directed inside the plasma. The po-
tential in plasma is assumed to have a quadratic depend-
ence on the radius
2
0( ) ( )ar r r so that the rota-
tional speed of the particles does not depend on the ra-
dius. This potential distribution arises in negatively
charged plasma with uniform radial distribution of the
electron density. The plasma is considered to be unlim-
ited along the magnetic field. The equilibrium distribu-
tion function of plasma particles of a specie (ion or
electron) should depend on variables , , zv ,
where
2 2
,
2 2
cm v m r
e r m v r
. (2)
To determine the particular form of the dependence
on variables (2) it is necessary to know the conditions
under which the plasma was produced. One of the vari-
ous possible methods of plasma production in crossed
rE and zB fields is a sharp application of an electric
field to plasma produced in advance in a magnetic field.
Yet another method is the ionization of neutral gas in
the system in which the crossed fields already exist.
Such a scenario of plasma production takes place, for
example, in a Penning discharge. We assume both ways
to be identical ones. Further by saying the phrase "the
appearance of a charged particle in crossed fields" we
shall imply either sharp application of a radial electric
field in plasma or ionization of the neutral atoms.
56 ISSN 1562-6016. ВАНТ. 2014. №6(94)
Let assume that the particle of specie appears in
the crossed fields with the initial values of the radial
coordinate 0r and velocity components 0 0 0, ,r zv v v .
The probability that, moving in crossed fields, the parti-
cle resides in the phase volume zd d dv is equal
to:
2 2
0 0
0( ( )) (
2 2
cm v m r
dp e r
0 0 0) ( )z z zm v r v v d d dv , (3)
where
2 2 2
0 0 0rv v v . Such a form of the dependence
of the probability density is a consequence of the con-
servation laws of energy and angular momentum of a
particle. If before the particles appearance in crossed
fields they have a certain distribution on the coordinates
and velocities 0 0 0 0( , , , )r zf r v v v , then the probability
of the particle to belong to the phase volume
0 0 0 0 0z r zd d dv r dr dv dv dv is equal to:
0 0 0 0 0 0 0 0( , , )z r zdP f r v v dpr dr dv dv dv . (4)
To obtain the particle distribution function, it is neces-
sary to integrate the expression (4) over the variables
0 0 0 0, , ,r zr v v v . Let suppose that the distribution func-
tion 0 0 0 0( , , , )r zf r v v v is given by the following ex-
pression:
2 2
0 0 0
0 0 0 0 3 2 3 2 2
0 0 0
( , , , ) exp
(2 ) 2 2
z
r z
T T T
n v v
f r v v v
v v v
0( )aY r r , (5)
where 0n and 0Tv are the density and the thermal ve-
locity of the particles before their appearance in the
crossed fields. If the 0n and 0Tv do not depend on the
radius, then the distribution function of plasma particles
in crossed fields is as follows:
0
0 01 2 22 3 2
0 0
2
, ,
2 2
c
z
T T c
n
f v I
m v m v
c
2
22
0
exp
2
c
T cm v
2
2
2 22
00
22
c z
TT c
v
vm v
1 2
2
c
c
Y
m
2
ar
m
, (6)
where 0I x is the Bessel modified function, is
the angular velocity of the particles drift motion in
crossed fields:
0
2 2
8
1 1
2
c
c a
e
m r
. (7)
For the quadratic dependence of the potential on the
radius, is constant. In the frame of reference rotat-
ing with the angular velocity , a particle of specie
makes a circular motion like Larmor rotation in the
magnetic field. The modified cyclotron frequency
2c c plays the role of the cyclotron fre-
quency in this case. The Larmor radius as well as
the radial coordinate of the center of the Larmor circle
R (variables of the guiding center of the particle) are
related to the variables and by the following
relations [1]
1 2
2 c m
, (8)
1 2
2
c
c
R
m
. (9)
Note that and R are also integrals of motion. Dis-
tribution function in the variables of the guiding center
has the following form
R
v
I
v
n
vRf
T
c
T
z 2
0
03
0
21
0
0
2
,,
2 22 2 2
2 2 2
0 0 0
exp
2 2 2
c z
T T T
R v
v v v
0Y R r . (10)
Equation (10) presents the wanted distribution function
of particles in crossed fields with an arbitrary relation-
ship between the energy of their motion in the electric
field and their energy of thermal motion.
2. EXPRESSIONS FOR THE DISTRIBUTION
FUNCTION IN THE LIMITING CASES
Now we consider the limiting cases of the distribu-
tion function for different limiting ratios of the thermal
velocity 0Tv and the velocity of the drift motion of the
guiding center R .
First, let assume that 00 Tv , while the anode po-
tential 0 is high enough so that the following inequal-
ity is valid for the most of the plasma volume (except of
the paraxial region)
2
0Tc vR . (11)
Then, we can use the asymptotic form for the large val-
ues of the Bessel functions argument, xexI x 20 .
In this case the distribution function (10) has the form
0
0 2
0
, ,
2
z
T c
n
f R v
v R
2
2
2 2
0 0
exp
2 2
c z
T T
R v
v v
0Y R r . (12)
ISSN 1562-6016. ВАНТ. 2014. №6(94) 57
It follows from Eq. (12), that the main contribution to
the equilibrium distribution function comes from parti-
cles, which satisfy the condition:
0Tc vR . (13)
Inequality (13) determines the approximation of a
strong radial electric field at which the thermal velocity
of the particles before their appearance in crossed fields
is much less than the velocity of the particle drift in
crossed fields, c . Note that the approximation
of a strong radial electric field (13) differs from that
approach, taken in [2, 3], where the criterion of a strong
electric field is also the inequality RVT 0 (besides
the condition 0 0Tv ). In contrast to the latter condi-
tion, the criterion (13) admits the same order of magni-
tude for and c . In the limiting case 0 0Tv , the
distribution function (12) degenerates into -function:
2
0 , cRRf . (14)
Similar ion distribution function in the form of -
function was considered in [2, 3], where the problem of
the excitation of the ion cyclotron instability of the
plasma in crossed B and rE fields has been solved.
Let assume now that the inequality opposite to (13)
holds,
2
0Tc vR . (15)
This inequality corresponds to the case of a weak elec-
tric field. In this case, the distribution function (10)
takes the form:
0
0 1 2 3
0
, ,
2
z
T
n
f R v
v
2
0
2
2
0
22
2
0
22
222
exp
T
z
T
c
T v
v
vv
R
0Y R r . (16)
The smallness of the velocity of the drift motion in
comparison to the thermal velocity, 0TvR , (the
approximation of weak radial electric field) is sufficient
condition for the inequality (15). In fact, the inequality
0TvR is equivalent to the condition c ,
that is also a criterion of a weak electric field. We also
note that despite the uniform ionization along the radius,
the distribution function (16) is Gaussian on radial co-
ordinate of the guiding centre.
CONCLUSIONS
The expression for the equilibrium distribution func-
tion for plasma particles in crossed longitudinal mag-
netic and radial electric field is derived using the prob-
abilistic approach. This expression takes into account
non-zero initial velocity of the atoms before the ioniza-
tion and includes the product of the modified Bessel
function and exponential (11) whose arguments are the
coordinates of the guiding centre.
Limiting expressions are obtained from the general
expression for the distribution function for the cases of
strong (12) and weak (16) radial electric field. These
expressions are consistent with the previously obtained
expressions.
REFERENCES
1. D.V. Chibisov, V.S. Mikhailenko, K.N. Stepanov. Ion
cyclotron turbulence theory of rotating plasmas// Plasma
Phys. Control. Fusion. 1992, v. 34, № 1, p. 95-117.
2. V.G. Dem’yanov, Yu.N. Eliseev, Yu.A. Kirochkin, et
al. Equilibrium and non-local ion cyclotron instability of
plasma in crossed longitudinal magnetic and strong ra-
dial electric fields // Fiz. Plazmy. 1988, v. 14, № 10,
p. 840-850.
3. Yu.N. Yeliseyev. Nonlocal theory of the spectra of
modified ion cyclotron oscillations in a charged plasma
produced by gas ionization // Plasma Phys. Rep. 2006,
v. 32, № 11, p. 927-936.
Article received 22.09.2014
ФУНКЦИЯ РАСПРЕДЕЛЕНИЯ ЧАСТИЦ ПЛАЗМЫ В ПРОДОЛЬНОМ МАГНИТНОМ
И РАДИАЛЬНОМ ЭЛЕКТРИЧЕСКОМ ПОЛЯХ
Д.В. Чибисов
Решается задача об определении функции распределения частиц в плазме, которая создается в
скрещенных продольном магнитном и радиальном электрическом полях. Предполагается, что функция рас-
пределения по скоростям частиц остаточного газа до его ионизации, или начальной плазмы, прежде чем она
появляется в скрещенных полях, является максвелловской с ненулевой тепловой скоростью.
Образовавшиеся частицы плазмы затем движутся в скрещенных полях без столкновений. Полученная
функция распределения записана в координатах ведущего центра. Получены также выражения для функции
распределения в предельных случаях сильного и слабого электрических полей.
ФУНКЦІЯ РОЗПОДІЛУ ЧАСТИНОК ПЛАЗМИ В ПОЗДОВЖНЬОМУ МАГНІТНОМУ
І РАДІАЛЬНОМУ ЕЛЕКТРИЧНОМУ ПОЛЯХ
Д.В. Чібісов
Розв’язано задачу про визначення функції розподілу частинок у плазмі, яка створюється в схрещених
поздовжньому магнітному і радіальному електричному полях. Вважається, що функція розподілу за швид-
костями частинок залишкового газу до його іонізації, або початкової плазми, перш ніж вона з'являлася в
схрещених полях, є максвелiвською з ненульовою тепловою швидкістю. Частинки плазми, що утворилися,
рухаються далі в схрещених полях без зіткнень. Здобуту функцію розподілу записано в координатах ведучо-
го центра. Здобуто також вирази для функції розподілу в граничних випадках сильного та слабкого електри-
чних полів.
|
| id | nasplib_isofts_kiev_ua-123456789-81196 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T12:37:19Z |
| publishDate | 2014 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Chibisov, D.V. 2015-05-13T15:21:16Z 2015-05-13T15:21:16Z 2014 The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields / D.V. Chibisov // Вопросы атомной науки и техники. — 2014. — № 6. — С. 55-57. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.27.Jt https://nasplib.isofts.kiev.ua/handle/123456789/81196 The problem of determining the distribution function of particles in plasma that is produced in crossed longitudinal magnetic and radial electric fields is solved. It is assumed that the velocity probability distribution functions for the particles of the residual gas before its ionization or initial plasma before it has appeared in crossed fields are Maxwellian with non-zero thermal velocities. Then produced plasma particles are moving in crossed fields without collisions. The obtained distribution function is written in the coordinates of the guiding center. The expressions for the distribution function in the limiting cases of strong and weak electric field are also obtained. Решается задача об определении функции распределения частиц в плазме, которая создается в скрещенных продольном магнитном и радиальном электрическом полях. Предполагается, что функция распределения по скоростям частиц остаточного газа до его ионизации, или начальной плазмы, прежде чем она появляется в скрещенных полях, является максвелловской с ненулевой тепловой скоростью. Образовавшиеся частицы плазмы затем движутся в скрещенных полях без столкновений. Полученная функция распределения записана в координатах ведущего центра. Получены также выражения для функции распределения в предельных случаях сильного и слабого электрических полей. Розв’язано задачу про визначення функції розподілу частинок у плазмі, яка створюється в схрещених поздовжньому магнітному і радіальному електричному полях. Вважається, що функція розподілу за швидкостями частинок залишкового газу до його іонізації, або початкової плазми, перш ніж вона з'являлася в схрещених полях, є максвелiвською з ненульовою тепловою швидкістю. Частинки плазми, що утворилися, рухаються далі в схрещених полях без зіткнень. Здобуту функцію розподілу записано в координатах ведучого центра. Здобуто також вирази для функції розподілу в граничних випадках сильного та слабкого електричних полів. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Фундаментальная физика плазмы The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields Функция распределения частиц плазмы в продольном магнитном и радиальном электрическом полях Функція розподілу частинок плазми в поздовжньому магнітному і радіальному електричному полях Article published earlier |
| spellingShingle | The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields Chibisov, D.V. Фундаментальная физика плазмы |
| title | The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields |
| title_alt | Функция распределения частиц плазмы в продольном магнитном и радиальном электрическом полях Функція розподілу частинок плазми в поздовжньому магнітному і радіальному електричному полях |
| title_full | The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields |
| title_fullStr | The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields |
| title_full_unstemmed | The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields |
| title_short | The distribution function of plasma particles in longi-tudinal magnetic and radial electric fields |
| title_sort | distribution function of plasma particles in longi-tudinal magnetic and radial electric fields |
| topic | Фундаментальная физика плазмы |
| topic_facet | Фундаментальная физика плазмы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/81196 |
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