Calculations for parameters of a stationary reflex discharge
A variant of the stationary reflex discharge model based on the global (volume-averaged) model is under consideration. The calculation is carried out on the plasma electron density and temperature as a function of the adsorbed power in the reflex discharge, magnetic field value and initial gas press...
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| Cite this: | Calculations for parameters of a stationary reflex discharge / Yu.V. Kovtun, A.N. Ozerov, A.I. Skibenko, E.I. Skibenko, V.B. Yuferov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 201-204. — Бібліогр.: 22 назв. — англ. |
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| author | Kovtun, Yu.V. Ozerov, A.N. Skibenko, A.I. Skibenko, E.I. Yuferov, V.B. |
| author_facet | Kovtun, Yu.V. Ozerov, A.N. Skibenko, A.I. Skibenko, E.I. Yuferov, V.B. |
| citation_txt | Calculations for parameters of a stationary reflex discharge / Yu.V. Kovtun, A.N. Ozerov, A.I. Skibenko, E.I. Skibenko, V.B. Yuferov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 201-204. — Бібліогр.: 22 назв. — англ. |
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| description | A variant of the stationary reflex discharge model based on the global (volume-averaged) model is under consideration. The calculation is carried out on the plasma electron density and temperature as a function of the adsorbed power in the reflex discharge, magnetic field value and initial gas pressure. The density of neutral atoms and cathode material ions coming into the discharge by the cathode sputtering mechanism is determined.
Рассмотрен вариант модели стационарного отражательного разряда на основе global (volume-averaged) модели. Проведен расчет зависимости плотности и температуры электронов плазмы от адсорбированной мощности в отражательном разряде, величины магнитного поля и начального давления газа. Определена плотность нейтральных атомов и ионов материала катода, поступающих в разряд за счет механизма катодного распыления.
Розглянуто варіант моделі стаціонарного відбивного розряду на основі global (volume-averaged) моделі. Проведено розрахунок залежності густини та температури електронів плазми від адсорбованої потужності у відбивному розряді, величини магнітного поля і початкового тиску газу. Визначена густина нейтральних атомів та іонів матеріалу катода, що поступають у розряд за рахунок механізма катодного розпилення.
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ISSN 1562-6016. ВАНТ. 2015. №1(95)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 201-204. 201
CALCULATIONS FOR PARAMETERS OF A STATIONARY REFLEX
DISCHARGE
Yu.V. Kovtun, A.N. Ozerov, A.I. Skibenko, E.I. Skibenko, V.B. Yuferov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: Ykovtun@kipt.kharkov.ua
A variant of the stationary reflex discharge model based on the global (volume-averaged) model is under consid-
eration. The calculation is carried out on the plasma electron density and temperature as a function of the adsorbed
power in the reflex discharge, magnetic field value and initial gas pressure. The density of neutral atoms and cathode
material ions coming into the discharge by the cathode sputtering mechanism is determined.
PACS: 52.80.-s; 52.80.Sm
INTRODUCTION
The reflex discharge, known as a Penning discharge
was first proposed by Penning F.M. [1] for developing a
vacuum manometer. During the many-year history a
large number of various reflex discharge models were
offered and investigated. A series of technical devices
based on the reflex discharge has been developed and
widely used for solving both physical and applied prob-
lems. The main fields of reflex discharge applications
are: vacuum engineering, physics of atomic and electron
collisions, physics of charged particle beams, plasma
physics, applied plasma technologies etc. However, in
spite of long-term and numerous investigations at pre-
sent there is a lack of a sufficiently full physical model
describing the reflex discharge. In particular, it is ex-
plained by the fact that the reflex discharge has several
modes depending on the pressure and magnetic field
values [2]. Moreover, in different modes the discharge
and produced plasma parameters are different. The dis-
charge modes were investigated theoretically and exper-
imentally under low pressure p 1.33·10
-2
Pa e.g. in [2].
Under higher pressures and low plasma ionization level
the reflex discharge is used most often for developing
plasma sources based on this discharge. For this it is
necessary to obtain an optimum relation between the
plasma parameters and the energy contribution. There-
fore it is of interest to consider the reflex discharge
model under pressures p> 1.33·10
-2
Pa.
In the present paper for the stationary case we con-
sider a reflex discharge model based on the global (vol-
ume-averaged) model [3]. Weakly ionized
nonisothermal plasma in a magnetic field and without
magnetic field is studied.
1. CHARGED PARTICLE BALANCE IN THE
PLASMA OF STATIONARY REFLEX
DISCHARGE
The equation of charged particle balance in the
plasma under condition, when the process of diffusion
particle loss dominates over recombination, can be writ-
ten in the following form [3, 4]:
, (1)
where V is the plasma-occupied volume m
3
; Kiz – neu-
tral particle ionization rate constant, m
3
/s; N0 – neutral
particle density, m
-3
; Ne – plasma electron density,
Ne=Ni, m
-3
; vB – Bohm velocity, , m/s;
Te – electron temperature, eV; q – electron charge, C; Mi
– ion mass, kg ; Aeff – effective area of charged particle
loss, m
2
.
The effective area of charged particle loss is equal to
Aeff = ALhL +ARhR, where AL and AR are, respectively,
the base area and the cylinder lateral surface, m
2
; hL and
hR are the axial and radial relations of the plasma densi-
ty at the plasma boundary to the plasma density in the
center.
According to [5, 6] for the case of weakly
nonisothermal plasma the values hL and hR are equal to:
, (2)
, (3)
where hR0 is the value of hR for the case without mag-
netic field, ; λi – ion free path in
gas λi = 1/N0σmi, m; σmi – transport cross-section of ion
scattering on the neutral particle, m
2
. Upon the collision
of ions with the proper gas neutrals σmi = 2 σсх, where
σсх is the charge exchange cross-section, m
2
.
The parameter G is [6]:
, (4)
where λe is the electron free path in gas, λe = 1/N0σme, m;
σme – transport cross-section of electron scattering on
the neutral particle, m
2
; re = vTe/ωce, ri = vs/ωci, ωce and
ωci – electron and ion cyclotron frequencies, respective-
ly; B – magnetic induction, T; me – electron mass, kg;
– electron thermal velocity, m/s; – ion sound
velocity, m/s.
Equation (1) can be reduced to the following form:
. (5)
By solving equation (5) the electron temperature Te
as a function of the neutral particle density can be de-
termined.
2. POWER BALANCE IN THE PLASMA OF
STATIONARY REFLEX DISCHARGE
The total power adsorbed in the discharge equals to:
, (6)
where PVe is the power released in the volume due to the
electron energy loss during collision with neutral parti-
mailto:Ykovtun@kipt.kharkov.ua
204 ISSN 1562-6016. ВАНТ. 2015. №1(95)
cles, W; Ple – plasma power taken by electrons, W; Pli –
plasma power taken by ions, W.
, (7)
where Kex is the excitation rate constant, m
3
/s; Kel – elas-
tic collision rate constant, m
3
/s; εiz, εex, εel – energy be-
ing lost by electrons upon ionization, atomic excitation
and elastic collisions.
The average energy for ion-electron pair formation
or the ionization value is equal to:
. (8)
Taking into account equations (1) and (2), equation
(7) takes the following form:
. (9)
In the reflex discharge the most part of electrons go
away in the radial direction to the anode, and ions are
axially directed to the cathodes. The power taken by
electrons and ions from the plasma can be written in the
general form as:
, (10)
, (11)
where Ee is the average kinetic energy taken by elec-
trons, in the Maxwell distribution assumption its value
is 2Te; Ei – average kinetic energy taken by ions which
equals to the sum of ion energies before coming into the
layer and obtained in the layer, Ei = (Te/2)+Vs, where Vs
is the cathode potential drop approximately equal to the
discharge voltage Vdc.
As a result, equation (6), with taking into account
equations (9-11), takes the form:
. (12)
From equation (12) we can obtain the relationship
connecting the density with the adsorbed power in the
discharge that is equal to:
. (13)
3. CALCULATION OF THE DENSITY AND
ELECTRON TEMPERATURE IN THE
REFLEX DISCHARGE
Let us calculate the density and electron temperature
in the reflex discharge and compare the calculation re-
sults with experimental data given in [7]. The initial
conditions are as follows: plasma column radius
R = 0.05 m; length L = 0.025 m; pressure
Ar p = 4 Pa (N0 = 1.06·10
21
m
-3
); plasma occupied vol-
ume V = 1.963·10
-4
m
3
; AL = 0.016 m
2
, AR = 7.854·10
-4
m
2
,
B = 0 T. The charge exchange cross-section is calculat-
ed on the base of the asymptotic approach developed in
[8] and for the ion thermal energies is σcx =
7.8·10
-19
m
2
, and σmi = 1.56·10
-18
m
2
.
The values of the ionization rate, excitation and elas-
tic collision constants, and the values of εiz, εex are taken
from [9]. The transport cross-section of electron scatter-
ing on the neutral particle is taken from [10].
By solving equation (5), with taking into account the
initial conditions, the electron temperature Te = 2.5 eV
is obtained. The plasma density was calculated by for-
mula (13) with taking into account the calculated elec-
tron temperature. The calculated curves of the plasma
density as a function of the adsorbed power in the dis-
charge in comparison with the experimental data of [7]
are represented in Fig. 1.
By solving equation (5) with taking into account the
initial conditions, the plasma electron temperature in a
magnetic field and without magnetic field as a function
of the pressure was obtained (shown in Fig. 2).
Fig. 1. Plasma density as a function of adsorbed power
in the discharge
Fig. 2. Electron temperature as a function of pressure
The calculated curves of the plasma density as a
function of the magnetic induction at the absorbed pow-
er in the discharge of 10 W (p = 4 Pa, Vs = 0) is present-
ed in Fig. 3.
Fig. 3. Plasma density as a function of magnetic induc-
tion
202
ISSN 1562-6016. ВАНТ. 2015. №1(95) 203
4. CONSIDERATION OF THE NEUTRAL
ATOMS PRODUCED BY THE CATHODE
SPUTTERING WITH THEIR SUBSEQUENT
IONIZATION IN THE REFLEX DISCHARGE
The flux of neutral atoms arriving into the discharge
chamber volume as a result of cathode sputtering can be
determined as:
, (14)
where Υ is the cathode material sputtering coefficient;
Ng
+
is the gas ion density, m
-3
.
The balance of neutral atoms coming into the reflex
discharge due to the cathode sputtering, with taking into
account equation (14), is written in the following form:
, (15)
where Nm is the density of neutral atoms from the cath-
ode material, m
-3
; τD – particle diffusion characteristic
time, , where Λ is the characteristic diffusion
length, m; Dm – diffusion coefficient of the atom scat-
tered in gas, m
2
/s; τi – neutral atomic ionization time, s.
By solving equation (15) the density of neutral metal
atoms can be determined.
Taking into account that the ion of sputtered cathode
material is formed by the electron impact ionization and
gas ion charge exchange we can write the equation of
metal ion balance in the following form:
, (16)
where Nm
+
is the metal ion density, m
-3
; vB
m
– Bohm
velocity of metal ion, m/s; KCT – constant of gas ion
charge exchange on the neutral metal atom, m
3
/s.
To calculate the density of neutral atoms and cath-
ode material ions let us taken the following initial con-
ditions: energy of argon ions colliding with the cathode
surface EAr = 360 eV; gas ion density Ng
+
= 1.9·10
16
m
-3
(calculated value for Pabs = 10 W, p = 4 Pa); electron
temperature Te = 2.5 eV (calculated value for p = 4 Pa);
target material is stainless steel).
To calculate the sputtering ratio, the diffusion time
of a neutral atom and the time of atomic ionization we
use the model proposed in [11]. The values of constants
for the iron atomic ionization rate by the electron impact
and the velocity of charge exchange of gas ion on iron
atom, taken from [12, 13], are Kiz = 2·10
-15
m
3
/s [12],
KCT = 13.76·10
-15
m
3
/s [13]. The calculated sputtering
ratio in the case of normal ion incidence onto the target
[14] for Ar
+
→Fe is 0.698 for Fe
+
→Fe = 0.677.
The calculated value τD of the particle diffusion time
is 7.746·10
-6
s, the value of the ionization time is τi =
3.417·10
-3
s.
As τD<τi, a minor part of sputtered atoms will be
ionized, and a major part will go onto the discharge
chamber surface due to the diffusion.
By solving equation (15) we find the density value
for iron neutral atoms Nm = 3.5·10
15
m
-3
, which is less
by a factor of 10
5
than that of neutral argon particles.
When the iron ion density is determined from (16) we
obtain Nm
+
= 2.8·10
13
m
-3
, that is by ~3 order of magni-
tude less than the density of Ar ions. The metal atomic
ionization level is 8·10
-3
, for gas it is 1.8·10
-5
. It should
be noted that when the plasma density is << 10
17
m
-3
the
Penning ionization can made a significant contribution
into the metal atomic ionization level [15].
5. DISCUSSION OF CALCULATION
RESULTS
The comparison of calculation results with experi-
mental data [7] (see Fig.1) shows that they differ by a
factor of ~2. May be this difference is related with some
unaccounted factors in the model under consideration.
In the above calculations it has been assumed that the
electron energy distribution function (EEDF) is
Maxwellian. In the plasma discharge several group of
electrons with different energy can exist. For example,
in [7] the EEDF bi-Maxwellian with a group of cold and
hot electrons was observed. In [16] the calculations
have been performed for a series of model EEDF’s
which were changing from the Maxwell function to the
Druyvesteyn function and the EEDF influence on the
calculated parameters Te, Ne, Eiz was shown.
In the reflex discharge a group of fast primary elec-
trons is produced as a result of the second emission
from the cathode. These electrons are accelerated at the
cathode voltage drop Vs being approximately equal to
the discharge voltage Vdc.
Primary electrons, moving in a magnetic field, fall
into a potential well between the cathodes and can oscil-
late between them. The primary electrons lose their en-
ergy during elastic and inelastic collisions with gas at-
oms. A part of the electron energy remains in the sec-
ondary electron knocked from the atom during ioniza-
tion. The secondary electron energy is within the range
from 0 to E – εiz, where E is the primary electron energy
[17]. A part of knock-on (secondary) electrons, of an
energy exceeding εiz, also participates in the ionization
process. The average number of ions produced in the
process is , where Wiz is the average ener-
gy spent for ion-electron pair formation or the ionization
value. Though Wiz and Eiz have a similar physical sense
(see Eq.8), in the general case their values are not equal.
The comparison carried out in [18] for the water mole-
cule has shown that with E > εiz the values of Wiz > Eiz,
at E = εiz, are, consequently, . It is related with
different approaches to the determination of these val-
ues. According to [19] under pressure p> 1.33·10
-2
Pa
the flux of emitted electrons is proportional to the flow
of ions falling onto the cathode. Hence the primary elec-
tron flux is equal to , where γ is the
ion-electron emission coefficient. So, the number of fast
primary electrons being in the plasma volume can be
determined as , where τD
is the characteristic energy confinement time of fast
electron diffusion to the anode and τE is the characteris-
tic energy confinement time of the fast electron.
By calculating the reflex discharge parameters it is
necessary to consider in the energy balance, besides the
EEDF effect, the values of average kinetic energy taken
with ions and, consequently, the cathode drop of poten-
tial Vs (see Fig. 1).
204 ISSN 1562-6016. ВАНТ. 2015. №1(95)
From the condition of self-maintained discharge
stationarity, taken in [20], it follows that
,
consequently, , and hence
. Using in calculations the experimentally
measured values of Vdc [7] and taking , as well
as, , we obtain the discrepancy between the calcu-
lation data and experimental results by a factor of ~ 2.
Analysis of the plasma density dependence on the
magnetic induction (see Fig. 3) shows that the plasma
density increases with magnetic field increasing and,
consequently, the particle loss decreases across the
magnetic filed. In the case under consideration a classic
diffusion has been assumed and corresponding values hL
and hR were taken [5, 6]. It should be noted that in the
reflex discharge, after reaching some critical value of
the magnetic field Bcr [21], an anomalous diffusion with
the transverse diffusion coefficient, close to the Bohm
diffusion coefficient, is observed [22].
CONCLUSIONS
Within the framework of the model under considera-
tion the plasma electron density and temperature as a
function of the adsorbed power in the reflex discharge,
magnetic field value and initial gas pressure are calcu-
lated. The processes of cathode material sputtering with
subsequent ionization of sputtered atoms in the plasma
are investigated. The density of neutral particles and
sputtered cathode material atomic ions is determined.
The comparison between the calculation data and exper-
imental results has been performed. The calculation data
differ from the experimental results by a factor of ~2.
Such accordance between the results is satisfactory for a
sufficiently simple reflex discharge model under
4onsideration. Further development of the model can
give more accurate coincidence between the experi-
mental and calculation results.
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Article received 20.12.2014
РАСЧЕТ ПАРАМЕТРОВ СТАЦИОНАРНОГО ОТРАЖАТЕЛЬНОГО РАЗРЯДА
Ю.В. Ковтун, А.Н. Озеров, А.И. Скибенко, Е.И. Скибенко, В.Б. Юферов
Рассмотрен вариант модели стационарного отражательного разряда на основе global (volume-averaged)
модели. Проведен расчет зависимости плотности и температуры электронов плазмы от адсорбированной
мощности в отражательном разряде, величины магнитного поля и начального давления газа. Определена
плотность нейтральных атомов и ионов материала катода, поступающих в разряд за счет механизма катод-
ного распыления.
РОЗРАХУНОК ПАРАМЕТРІВ СТАЦІОНАРНОГО ВІДБИВНОГО РОЗРЯДУ
Ю.В. Ковтун, О.М. Озеров, А.І. Скибенко, Е.І. Скібенко, В.Б. Юферов
Розглянуто варіант моделі стаціонарного відбивного розряду на основі global (volume-averaged) моделі.
Проведено розрахунок залежності густини та температури електронів плазми від адсорбованої потужності у
відбивному розряді, величини магнітного поля і початкового тиску газу. Визначена густина нейтральних
атомів та іонів матеріалу катода, що поступають у розряд за рахунок механізма катодного розпилення.
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| id | nasplib_isofts_kiev_ua-123456789-82243 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-26T12:48:42Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kovtun, Yu.V. Ozerov, A.N. Skibenko, A.I. Skibenko, E.I. Yuferov, V.B. 2015-05-27T08:37:02Z 2015-05-27T08:37:02Z 2015 Calculations for parameters of a stationary reflex discharge / Yu.V. Kovtun, A.N. Ozerov, A.I. Skibenko, E.I. Skibenko, V.B. Yuferov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 201-204. — Бібліогр.: 22 назв. — англ. 1562-6016 PACS: 52.80.-s; 52.80.Sm https://nasplib.isofts.kiev.ua/handle/123456789/82243 A variant of the stationary reflex discharge model based on the global (volume-averaged) model is under consideration. The calculation is carried out on the plasma electron density and temperature as a function of the adsorbed power in the reflex discharge, magnetic field value and initial gas pressure. The density of neutral atoms and cathode material ions coming into the discharge by the cathode sputtering mechanism is determined. Рассмотрен вариант модели стационарного отражательного разряда на основе global (volume-averaged) модели. Проведен расчет зависимости плотности и температуры электронов плазмы от адсорбированной мощности в отражательном разряде, величины магнитного поля и начального давления газа. Определена плотность нейтральных атомов и ионов материала катода, поступающих в разряд за счет механизма катодного распыления. Розглянуто варіант моделі стаціонарного відбивного розряду на основі global (volume-averaged) моделі. Проведено розрахунок залежності густини та температури електронів плазми від адсорбованої потужності у відбивному розряді, величини магнітного поля і початкового тиску газу. Визначена густина нейтральних атомів та іонів матеріалу катода, що поступають у розряд за рахунок механізма катодного розпилення. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Низкотемпературная плазма и плазменные технологии Calculations for parameters of a stationary reflex discharge Расчет параметров стационарного отражательного разряда Розрахунок параметрів стаціонарного відбивного розряду Article published earlier |
| spellingShingle | Calculations for parameters of a stationary reflex discharge Kovtun, Yu.V. Ozerov, A.N. Skibenko, A.I. Skibenko, E.I. Yuferov, V.B. Низкотемпературная плазма и плазменные технологии |
| title | Calculations for parameters of a stationary reflex discharge |
| title_alt | Расчет параметров стационарного отражательного разряда Розрахунок параметрів стаціонарного відбивного розряду |
| title_full | Calculations for parameters of a stationary reflex discharge |
| title_fullStr | Calculations for parameters of a stationary reflex discharge |
| title_full_unstemmed | Calculations for parameters of a stationary reflex discharge |
| title_short | Calculations for parameters of a stationary reflex discharge |
| title_sort | calculations for parameters of a stationary reflex discharge |
| topic | Низкотемпературная плазма и плазменные технологии |
| topic_facet | Низкотемпературная плазма и плазменные технологии |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82243 |
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