Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal
This report is devoted to the investigation of the non-linear mechanism of plasma electrons heating on dispersion properties of potential surface waves (SWs) that propagate along interface of metal-magnetized plasma of a finite pressure. The external steady magnetic field is perpendicular to the med...
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irk-123456789-1104872017-01-05T03:03:29Z Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal Akimov, Yu.A. Azarenkov, N.A. Olefir, V.P. Basic plasma physics This report is devoted to the investigation of the non-linear mechanism of plasma electrons heating on dispersion properties of potential surface waves (SWs) that propagate along interface of metal-magnetized plasma of a finite pressure. The external steady magnetic field is perpendicular to the medium interface. The different mechanisms of electron energy losses in approach of a weak heating are considered. The spatial distribution of plasma electron temperature on the basis of energy balance equation in framework of the non-local electrons heating is obtained. The nonlinear shift of wavenumber and spatial attenuation coefficient at different plasma parameters are researched. The obtained results are valid both for semiconductor and gas plasma. Дана робота присвячена вивченню впливу нелінійного механізму нагріву електронів плазми на дисперсійні властивості потенціальних поверхневих хвиль (ПХ), що поширюються уздовж межі метал - магнітоактивна плазма кінцевого тиску. Зовнішнє стале магнітне поле спрямоване перпендикулярно межі розподілу середовищ. В наближенні слабкого нагріву розглянуто різні механізми втрати енергії електронів. Отримано просторовий розподіл температури плазми в рамках нелокального нагріву електронів на основі рівняння балансу енергії. Досліджено нелінійний зсув хвильового числа та нелінійний декремент просторового загасання ПХ в залежності від параметрів плазми. Отримані результати справедливі як для напівпровідникової, так і для газової плазми. Данная работа посвящена изучению влияния нелинейного механизма нагрева электронов плазмы на дисперсионные свойства потенциальных поверхностных волн (ПВ), распространяющихся вдоль границы металл - магнитоактивная плазма конечного давления. Внешнее постоянное магнитное поле направлено перпендикулярно границе раздела сред. В приближении слабого нагрева рассмотрены различные механизмы потери энергии электронов. Получено пространственное распределение температуры плазмы в рамках нелокального нагрева электронов на основе уравнения баланса энергии. Исследованы нелинейный сдвиг волнового числа и нелинейный декремент пространственного затухания ПВ в зависимости от параметров плазмы. Полученные результаты справедливы как для полупроводниковой, так и для газовой плазмы. 2003 Article Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal / Yu.A. Akimov, N.A. Azarenkov, V.P. Olefir // Вопросы атомной науки и техники. — 2003. — № 1. — С. 70-73. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.35.-g http://dspace.nbuv.gov.ua/handle/123456789/110487 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Basic plasma physics Basic plasma physics Akimov, Yu.A. Azarenkov, N.A. Olefir, V.P. Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal Вопросы атомной науки и техники |
| description |
This report is devoted to the investigation of the non-linear mechanism of plasma electrons heating on dispersion properties of potential surface waves (SWs) that propagate along interface of metal-magnetized plasma of a finite pressure. The external steady magnetic field is perpendicular to the medium interface. The different mechanisms of electron energy losses in approach of a weak heating are considered. The spatial distribution of plasma electron temperature on the basis of energy balance equation in framework of the non-local electrons heating is obtained. The nonlinear shift of wavenumber and spatial attenuation coefficient at different plasma parameters are researched. The obtained results are valid both for semiconductor and gas plasma. |
| format |
Article |
| author |
Akimov, Yu.A. Azarenkov, N.A. Olefir, V.P. |
| author_facet |
Akimov, Yu.A. Azarenkov, N.A. Olefir, V.P. |
| author_sort |
Akimov, Yu.A. |
| title |
Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal |
| title_short |
Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal |
| title_full |
Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal |
| title_fullStr |
Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal |
| title_full_unstemmed |
Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal |
| title_sort |
heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2003 |
| topic_facet |
Basic plasma physics |
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http://dspace.nbuv.gov.ua/handle/123456789/110487 |
| citation_txt |
Heat nonlinearity of surface waves at interface between finite gas pressure magnetized plasma and metal / Yu.A. Akimov, N.A. Azarenkov, V.P. Olefir // Вопросы атомной науки и техники. — 2003. — № 1. — С. 70-73. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
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| first_indexed |
2025-07-08T00:39:48Z |
| last_indexed |
2025-07-08T00:39:48Z |
| _version_ |
1837037193290317824 |
| fulltext |
HEAT NONLINEARITY OF SURFACE WAVES AT INTERFACE BETWEEN
FINITE GAS PRESSURE MAGNETIZED PLASMA AND METAL
Yu.A. Akimov, N.A. Azarenkov, V.P. Olefir
Department of Physics and Technology, Kharkov National University, Kharkov, Ukraine,
E-mail: olefir@pht.univer.kharkov.ua; Fax: (0572)353977; Tel: (0572)350509
This report is devoted to the investigation of the non-linear mechanism of plasma electrons heating on dispersion prop-
erties of potential surface waves (SWs) that propagate along interface of metal-magnetized plasma of a finite pressure.
The external steady magnetic field is perpendicular to the medium interface. The different mechanisms of electron ener-
gy losses in approach of a weak heating are considered. The spatial distribution of plasma electron temperature on the
basis of energy balance equation in framework of the non-local electrons heating is obtained. The nonlinear shift of
wavenumber and spatial attenuation coefficient at different plasma parameters are researched. The obtained results are
valid both for semiconductor and gas plasma.
PACS: 52.35.-g
1. INTRODUCTION
The properties of SW in plasma-metal structures are
subject of intensive theoretical and experimental research-
es during last 20 years. The interest to these structures is
stipulated by their numerous applications in plasma and
semiconductor electronics, gas discharge etc.[1]. The lin-
ear theory of SW in such structures is developed rather
completely. However the SW behavior can become essen-
tially nonlinear even at rather small wave field amplitudes
[2]. In dependence from plasma parameters those or other
nonlinear self-interaction mechanisms can be dominating.
Among of research directions of various nonlinear effects
that determine the SW properties in plasma waveguide
structures, one can be noted the following ones. So, for
example, the resonance excitation of the second harmonic
studied in [2], nonlinear damping of SWs [3], self-interac-
tion of SW due to the nonlinearity of a quasihydrodynam-
ics equations [2], the ionization nonlinearity and heating
one [4].
The aim of this report is the study of heating nonlin-
earity of SWs that propagate at interface ‘plasma – metal’
at a presence of external steady magnetic field.
2. TASK STATEMENT
Let us consider the nonlinear process of self-interac-
tion of potential SW due to the plasma electron heating in
a field of finite amplitude wave. Let us assume that the
wave propagates along interface of metal and plasma of a
finite pressure. The warm magnetized plasma occupies a
half-space 0>x and bounded by perfect conducting met-
al in the plane 0=x . The plasma – metal interface is sup-
posed sharp that is valid, when the transitional layer size
is significantly less than the penetration depth of SW field
into plasma. The external steady magnetic field 0H
is di-
rected perpendicularly to the mediums interface (axes x ).
Such magnetic field is typical for HF and UHF dis-
charges, magnetrons, Penning sources, magneto-discharge
pumps, Hall sensors, fusion devices (divertor, limiter) etc.
Let us consider the properties of SW that propagates
in weakly collision plasma with effective electron - scat-
tering centers collision frequency icol ν+ν+ν=ν * (
icol ννν ,, * are elastic collisions, excitation and ioniza-
tion frequency respectively). We assume that ν is much
less than the wave frequency ω . The scattering centers in
the case of gas plasma are the ions and atoms of working
gas, impurity. In the case of semiconductor plasma they
are optical and acoustic phonons also.
The mechanism of SW self-interaction consists in that
the plasma electrons receive from wave an additional en-
ergy and then return its to scattering centers as a result of
collisions. It leads to the spatial distribution of electron
temperature that determines both electron collision fre-
quency and plasma pressure is changed. It results in the
electrodynamic plasma properties and SW dispersion ones
are varied also.
It is necessary to note that the heating mechanism of
self-interaction is similar to ionization nonlinearity [2, 4].
The growth of SW amplitude results in modifications of
spatial distribution of electron temperature and coeffi-
cients of elementary processes in plasma. It leads to modi-
fication of plasma density distribution and the SW disper-
sion properties consequently. In the case of weak nonlin-
earity the wave amplitude is rather small and the perturba-
tions of plasma parameters (temperature and pressure of
electrons, collision frequency etc.), caused by wave, are
much less than nonperturbed ones. The influence of ion-
ization and heating nonlinearities on wave dispersion can
be taken into account by additionally. It allows to study
these mechanisms independently from each other.
3. LINEAR THEORY RESULTS
According to the linear theory [5], the considered SWs
exist in a frequency region 22
ceω>ω ( ceω is electron cy-
clotron frequency) and necessary condition of their exis-
tence is the finite value of electron thermal velocity
eTe mTV /2= (T is plasma electron temperature). It is
necessary to note that the account of plasma electron ther-
mal motion even in a linear approach the expressions of
wave potential and wavenumber are cumbersome. There-
fore further research of self-interaction of SW will be car-
ried out for rather dense plasma, when condition
0
222 / εω< <ω<ω pece is valid. Here peω is electron
Langmuir frequency and 0ε is dielectric permittivity of
semiconductor lattice (in the case of gas plasma 10 =ε ).
At the above mentioned conditions the potential of
SW can be written in form
70 Problems of Atomic Science and Technology. 2003. № 1. Series: Plasma Physics (9). P. 70-73
)(),,( 211
xx eeAtyx λ−λ− −=Ψ , (1)
where 1A is a wave amplitude. The parameters 2,1λ de-
termine the spatial distribution of SW field in plasma:
ω
ν+
ω
ωε−
ε
ω
=λ+λ=λ i
V
i
peTe
pe 1
2
11
2
2
0
0
''
1
'
11 , (2a)
ω
ν+
ω
ωεω=λ+λ=λ i
V
i
peTe
10
''
2
'
22 . (2b)
The complex value of wave number is equal
ω−ω
ω
ω
ν+
εω
ω−ωω=+=
22
2
0
2
22
''
2
'
22 1
/ cepe
ce
Te
i
V
ikkk (3)
4. TEMPERATURE SPATIAL DISTRIBUTION
Let us consider weak heating nonlinearity, when the
modification of electron temperature Tδ in SW field is
much less than its equilibrium value oT : TTT o δ+= ,
oTT < <δ . Let us assume also that the modification of
collision frequency icol δ ν+δ ν+δ ν=δ ν * is small
enough in comparison with its nonperturbed value ν at
absence of SW.
We suppose that the wave frequency ω is much more
than characteristic frequency of electron energy transmis-
sion ν~ into plasma. In this case the process of electron
energy transmission to scattering centers can be consid-
ered as quasistationary. The perturbation of electron tem-
perature will depend on coordinates and square of wave
amplitude module, averaged on wave period:
),,( 2
1AyxTT δ=δ . It can be obtained from stationary
equation of energy balance:
)()(Re3/1 * TPQdivEj −=
, (4)
where Q
is heat flux vector, j
is a high-frequency elec-
tron current density, *E
is a complex conjugate wave
electric field.
The term ( ) ( ) )(~
ooo TTTnTP −ν−= in (4) determines
the energy that electrons transmit in a unit of volume to
scattering centers with characteristic frequency
( )
oo T
i
i
T
ocolo T
U
T
UTT
∂
ν∂
+
∂
ν∂
+νγ=ν *
*)(~
, (5)
where on is nonperturbed plasma density, the parameter
2)/(2 MmMm ee +=γ is a part of electron energy trans-
mitted to scattering centers (with mass M ) by electrons
at elastic collisions, and *U , iU are excitation energy of
the first atom level and ionization one. It is necessary to
note that in general the characteristic frequency ν~ is de-
termined by frequencies of elastic collisions, excitation
and ionization of atoms.
The components of heat flux Q
in equation (4) are
given by expression jiji TQ ξ∂∂χ−= / , where ijχ is a
tensor of electron thermal conductivity of plasma, vector
),( yx=ξ
. The left part of the balance energy equation
(4) describes the dissipative heating of plasma electrons
in SW field. The terms in right part (4) describe the elec-
tron energy losses in a unit volume due to finite thermal
conductivity and energy transmission to the scattering
centers.
The energy balance equation can be simplified by as-
suming that the heat transport occurs mainly along mag-
netic field: yyxyyxxx χχχ> >χ=χ ,, . This condition is
valid at collision frequencies are much less than electron
cyclotron frequency ( ceω< <ν ). Taking into account
these assumptions, the equation (4) can be written in fol-
lowing form:
locoooT T
T
T
T
T
T
x
δ=δ+δ
∂
∂
λ
−
2
2
2
1
, (6)
where )5/(~3/11
oeT Tm νν=λ − is a characteristic length
of electron thermal conductivity and
)~3/()Re()/( *
oeloco TEVeTT ν−=δ
(7)
is a relative modification of electron temperature in a lo-
cal heating approach. It is necessary to note that the local
heating approach is used in many papers. But, at the made
above assumption about smallness of frequencies ν and
ν~ the condition of local heating can be reduced to in
equation 1)~/(2 < <ννω pe that is not valid in the case con-
sidered. Moreover, the heating of electrons in the consid-
ered task has essentially non-local character [4]. There-
fore expression (7) in the considered task characterizes
only the spatial distribution of a wave power absorbed by
plasma electrons as a result of collisions with scattering
centers, and doesn't describe the spatial distribution of
temperature.
To determine the spatial distribution of plasma tem-
perature in conditions of non-local heating it is necessary
to use the equation (4), solving it together with (7):
)(3/2/
'
2
''
2 222 xxyk
o eeePTT T λ−λ−− −µ≈δ , (8)
where parameter )/( 2
1 TeeVmAe=µ represents a ratio of
electron energy in wave field to its thermal energy,
)](/[6.0 22222
ceP ω−ωωων= .
The relative variation of electron temperature achieves
the maximum value
yk
o ePTT
''
222
max 3/2)/( −µ≈δ (9a)
at some distance from the plasma – metal interface:
)/2(ln)2( '
2
1'
2max Tx λλλ= − . (9b)
Such behavior is determined by spatial distribution of
power that electrons receive from SW field.
The condition of weak heating oTT < <δ ,
)( oTν< <δ ν corresponds to the following limitation on
the wave amplitude:
1/3/2 *,
2 < <µ oi TUP . (10а)
At the same time the results of linear theory are valid at
1)]/([ 2/1222 < <ω−ωωµ ce . (10b)
71
The numerical calculation (see Figure) has shown that
these conditions are fulfilled at the wave amplitudes µ ≤0,1.
0 50 100 150 200
0,00000
0,00005
0,00010
0,00015
7
6
x/r
de
δT/T
0
5
4
3
2
1
No
curve pece ωωε /0 peωωε /0 ων / νν ~/ µ
1 0,05 0,2 0,1 103 0,1
2 0,05 0,4 0,05 103 0,1
3 0,1 0,2 0,1 103 0,1
4 0,05 0,2 0,2 103 0,1
5 0,1 0,4 0,1 103 0,1
6 0,05 0,2 0,1 103 0,15
7 0,05 0,2 0,2 2*103 0,1
Spatial electron perturbation temperature distribution
The growth of wave amplitude ( µ ) and collision fre-
quency ( ων / ) results in increase of Joule SW losses and
essentially influences on plasma heating. It is necessary to
note that the effective transmission of SW energy into
plasma takes place when the wave frequency come close
to electron cyclotron one. The growth of νν ~/ leads to the
increase of characteristic length of thermal conductivity (
νν∝λ − ~/1
T ). It leads to more smooth decrease of tem-
perature in plasma.
5. SW SELF-INTERACTION
As noted above the perturbation of electron tempera-
ture results in collision frequency modification δ ν . Tak-
ing into consideration the value δ ν and the variation of
plasma electron pressure Tnp oδ=δ in the equation of
electron motion and solving its together with the continu-
ity and Poisson's equations one can obtain the expression
of wavenumber in following form:
)1(22 pLNL SSkk δδ ν ++= . (11)
Here Lk2 is the wavenumber value (3) obtained from the
linear theory and δ νS , pSδ describe the influence of
plasma electron heating.
The analysis has shown that the nonlinear addition to
the complex wavenumber is determined mainly by δ ν
value. The growth of magnetic field value leads to in-
crease of the nonlinear addition to the real part of
wavenumber. The magnetic field influence on the nonlin-
ear addition to the imaginary part of wavenumber has a
more complicated character. So, at weak magnetic fields
nonlinear addition increases with magnetic field growth.
But at rather strong fields, when the wave frequency
comes close to electron cyclotron one, it decreases and
can changes its sign. And when it is closed to zero, the
value of pδ becomes important for the SW attenuation.
The influence of δ ν on SW dispersion is determined by
dependence )(Tν . So, if 0>∂ν∂
oTT then the nonlinear
shift of real part of wavenumber is negative. At weak
magnetic fields the nonlinear decrement decreases and at
rather strong fields it decreases in comparison with its lin-
ear value. In opposite case when 0<∂ν∂
oTT there is a
contrary dependence.
The analysis of parameter νν ~/ influence has shown
that the nonlinear correction to spatial damping factor due
to electrons heating are most essential under rather high
gas pressure.
6. CONCLUSIONS
It is obtained and investigated the linear dispersion equa-
tion of considered SWs. In an approach of non-local plasma
heating the spatial distribution of electron temperature is ob-
tained. The nonlinear dispersion equation of SW considered is
investigated. The perturbation of electron temperature results
in collision frequency modification δ ν and in the variation of
plasma electron pressure Tnp oδ=δ . It is shown that the non-
linear addition to the complex wavenumber is determined
mainly by δ ν value. The growth of magnetic field value leads
to increase of the nonlinear addition to the real part of
wavenumber. The dependence of nonlinear addition to the
imaginary part of the wavenumber on magnetic field value is
more complicated. The results of the carried out researches are
valid both for semiconductor and gas plasma.
This work is partially supported by the Science and
Technology Center in Ukraine (STCU, Project # 1112).
REFERENCES
[1] M. Moisan, J. Hurbert, J. Margot, Z. Zakrzewski.//
Amsterdam: Kluwer Academic Publisher, 1999, pp.
1-42.
[2] N.A. Azarenkov, K.N. Ostrikov.// Physics Reports
308 ,1999, p.333.
[3] A.N. Kondratenko Plasma Waveguides./ Moscow:
Atomizdat. 1976(in Russian).
[4] Yu.M. Aliev, H. Schluter, A. Shivarova.// Plasma
Sources Sci. Technol. 5, 1996, p.514.
[5] N.A. Azarenkov, N.A. Kondratenko, Yu.O. Tysheck-
ij// Sov. J. JTF 69, 1999, p.30.
НАГРІВНА НЕЛІНІЙНІСТЬ ПОВЕРХНЕВИХ ХВИЛЬ НА МЕЖІ МАГНІТОАКТИВНОЇ ПЛАЗМИ
КІНЦЕВОГО ТИСКУ З МЕТАЛОМ
72
Ю.О. Акімов, М.О. Азаренков, В.П. Олефір
Дана робота присвячена вивченню впливу нелінійного механізму нагріву електронів плазми на дисперсійні
властивості потенціальних поверхневих хвиль (ПХ), що поширюються уздовж межі метал - магнітоактивна
плазма кінцевого тиску. Зовнішнє стале магнітне поле спрямоване перпендикулярно межі розподілу середовищ.
В наближенні слабкого нагріву розглянуто різні механізми втрати енергії електронів. Отримано просторовий
розподіл температури плазми в рамках нелокального нагріву електронів на основі рівняння балансу енергії.
Досліджено нелінійний зсув хвильового числа та нелінійний декремент просторового загасання ПХ в залеж-
ності від параметрів плазми. Отримані результати справедливі як для напівпровідникової, так і для газової
плазми.
НАГРЕВНАЯ НЕЛИНЕЙНОСТЬ ПОВЕРХНОСТНЫХ ВОЛН НА ГРАНИЦЕ МАГНИТОАКТИВНОЙ
ПЛАЗМЫ КОНЕЧНОГО ДАВЛЕНИЯ С МЕТАЛЛОМ
Ю.А. Акимов, Н.А. Азаренков, В.П. Олефир
Данная работа посвящена изучению влияния нелинейного механизма нагрева электронов плазмы на диспер-
сионные свойства потенциальных поверхностных волн (ПВ), распространяющихся вдоль границы металл - маг-
нитоактивная плазма конечного давления. Внешнее постоянное магнитное поле направлено перпендикулярно
границе раздела сред. В приближении слабого нагрева рассмотрены различные механизмы потери энергии
электронов. Получено пространственное распределение температуры плазмы в рамках нелокального нагрева
электронов на основе уравнения баланса энергии. Исследованы нелинейный сдвиг волнового числа и нелиней-
ный декремент пространственного затухания ПВ в зависимости от параметров плазмы. Полученные результаты
справедливы как для полупроводниковой, так и для газовой плазмы.
73
Yu.A. Akimov, N.A. Azarenkov, V.P. Olefir
Department of Physics and Technology, Kharkov National University, Kharkov, Ukraine,
E-mail: olefir@pht.univer.kharkov.ua; Fax: (0572)353977; Tel: (0572)350509
2. Task statement
3. Linear theory results
4. Temperature spatial distribution
Spatial electron perturbation temperature distribution
References
Ю.О. Акімов, М.О. Азаренков, В.П. Олефір
Ю.А. Акимов, Н.А. Азаренков, В.П. Олефир
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