Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions
The low-frequency spectrum of axially-symmetric modes of oscillations of the nonneutral plasma completely filling a waveguide is evaluated. Plasma consists of cold electrons and a small additive of ions produced by ioniza-tion of neutrals of background gas by electron impact. Ions are described by t...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
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| Cite this: | Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions / Yu.N. Yeliseyev // Вопросы атомной науки и техники. — 2015. — № 1. — С. 93-96. — Бібліогр.: 7 назв. — англ. |
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| citation_txt | Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions / Yu.N. Yeliseyev // Вопросы атомной науки и техники. — 2015. — № 1. — С. 93-96. — Бібліогр.: 7 назв. — англ. |
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| description | The low-frequency spectrum of axially-symmetric modes of oscillations of the nonneutral plasma completely filling a waveguide is evaluated. Plasma consists of cold electrons and a small additive of ions produced by ioniza-tion of neutrals of background gas by electron impact. Ions are described by the equilibrium distribution function adequately taking into account the peculiarity of their formation. The spectrum of oscillations consists of the family of lower hybrid modes and families of "modified" of ion cyclotron (MIC) modes. MIC modes intersecting lower hybrid modes are unstable within a wide range of changing of fields and plasma density due to anisotropy of the ion distribution function.
Численно определены низкочастотные спектры аксиально-симметричных мод колебаний заряженной плазмы, полностью заполняющей волновод. Плазма состоит из холодных электронов и малой добавки ионов, которые образовались ионизацией нейтралей фонового газа электронным ударом. Ионы описываются равновесной функцией распределения, адекватно учитывающей эту особенность их образования. Спектр состоит из семейства нижнегибридных мод и семейств модифицированных» ионных циклотронных (МИЦ) мод. МИЦ-моды, которые пересекаются с нижнегибридными модами, неустойчивы в широком диапазоне изменения полей и плотности плазмы из-за анизотропии функции распределения ионов.
Чисельно визначені низькочастотні спектри аксіально-симетричних мод коливань зарядженої плазми, що повністю заповнює хвилевід. Плазма складається з холодних електронів і малої добавки іонів, які утворилися іонізацією нейтралів фонового газу електронним ударом. Іони описуються рівноважною функцією розподілу, що адекватно враховує цю особливість їх утворення. Спектр складається з сімейства нижньогібридних мод і сімейств «модифікованих» іонних циклотронних (MIЦ) мод. МІЦ-моди, які перетинаються з нижньогібридними модами, нестійкі в широкому діапазоні зміни полів і щільності плазми із-за анізотропії функції розподілу іонів.
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ISSN 1562-6016. ВАНТ. 2015. №1(95)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 93-96. 93
STABILITY OF AXIALLY–SYMMETRIC LOWER HYBRID MODES
OF SINGLE COMPONENT ELECTRON PLASMA
WITH ADDITIVE OF BACKGROUND GAS IONS
Yu.N. Yeliseyev
Institute of Plasma Physics of the NSC KIPT, Kharkov, Ukraine
E-mail: eliseev2004@rambler.ru
The low-frequency spectrum of axially-symmetric modes of oscillations of the nonneutral plasma completely
filling a waveguide is evaluated. Plasma consists of cold electrons and a small additive of ions produced by ioniza-
tion of neutrals of background gas by electron impact. Ions are described by the equilibrium distribution function
adequately taking into account the peculiarity of their formation. The spectrum of oscillations consists of the family
of lower hybrid modes and families of "modified" of ion cyclotron (MIC) modes. MIC modes intersecting lower
hybrid modes are unstable within a wide range of changing of fields and plasma density due to anisotropy of the ion
distribution function.
PACS: 52.20.Dq; 52.25.Dg; 52.27.Jt; 52.35.-g; 52.35.Fp; 52.35.Qz; 41.20.Cv
1. The misconception has established after paper [1]
in the theory of non-neutral plasma that a diocotron
mode is the only low-frequency electron mode capable
to interact with ions. The low-frequency volume (plas-
ma) modes have escaped from the investigation and
experimental results were interpreted exclusively in a
spirit of paper [1]. However the direct graphings carried
in [2, 3], have shown that the frequencies of volume
electron modes of the waveguide completely filled with
nonneutral plasma, equal to upper hybrid (UH) and
lower hybrid (LH) frequencies with the Doppler shift
[4], also get into the low–frequency range (Fig. 1). It
occurs when the Doppler shift caused by rotation of
electrons in crossed fields, is compensated by hybrid
frequency. When 1m , only lower hybrid modes can
be low-frequency. When 2m – both lower hybrid,
and upper hybrid modes can be low-frequency. Presence
even a small additive of ions of background gas, which
always are present in plasma, leads to interaction of
electron and ion modes and, probably, instabilities of
plasma. Frequency spectra of such plasma for modes
having azimuth numbers 1, 2m have been evaluated
in [2, 3] .
The axially-symmetric lower hybrid modes ( 0m )
do not contain the Doppler shift, but also pass through
the low-frequency region at small values of the prob-
lem parameter
2 22 /pe ceq and at values of q of the
order of Brillouin value 1q (regions I and II in
Fig. 1). In the present paper the frequency spectrum of
axially-symmetric modes of oscillations ( 0m ) of
nonneutral plasma are evaluated using the dispersion
equation derived in [2, 3]. Spectrum is found within the
entire range of allowable values of strengths of electric
and magnetic fields, for magnetized and unmagnetized
ions.
2. We consider plasma completely filling a metal
waveguide of radius a and consisting of "cold" magnet-
ized electrons, homogeneously distributed on radius,
and a small additive of ions produced by ionization of
atoms, molecules of background gas by electron impact.
Equilibrium distribution function of such ions is equal
[2, 3]:
( , , )zF M v
,i
rot z
i ci
mN
Y e a M v
T
(1)
where N const is a density of ion production, Y is
the Heaviside function, is the Dirac delta function,
, , ,i zm M v are the mass, transverse energy, general-
ized momentum, and longitudinal velocity of ions,
ci
is
the ion cyclotron frequency, 2 /i iT const is the
period of radial ion oscillations in the crossed fields,
1/2
2 4 /i ci r ieE m r const is the MIC frequency,
the coefficient ( ) 0rot rc E Br const , ( )r is
the electric potential, that is supposed to be a quadratic
function of the radius 2 2( ) ( )( / )r a r a , ( ) 0a .
The radial electric field is determined by the space
charge of electrons and ions, 22 ( ) /rE a a r
2/ 2 (1 ) 0e pem e f r .
The dispersion equation of oscillations of the consid-
ered plasma, that is contained in papers [2, 3], has been
solved numerically for azimuth number 0m . The
other parameters have the same values, as in [2, 3]: the
ion mass equal to that of a nitrogen atom ( 14m
i
amu,
5/ 4 10e im m ), the oscillations are strongly elongated
along cylinder axis, 0.1zk a (
zk is the longitudinal
wave number), the ion density is small,
/ 0.01ef N n (
en is a density of electrons). The
results are presented in Figs. 2, 3 as dependencies of
normalized frequencies of oscillations / i
versus
parameter q .
3. The spectrum of oscillations consists of the family
of lower hybrid modes (Fig. 2) and families of MIC
modes. Due to the low ion density MIC modes are lo-
cated in close vicinities of harmonics of MIC frequen-
cies and in a scale of Fig. 2 they are undistinguishable
from these harmonics. Only the mode LH1 passing near
94 ISSN 1562-6016. ВАНТ. 2015. №1(95)
to the second harmonic perturbs the mode MIC1 consid-
erably (see Fig. 2), and both modes remain stable. The
behaviour of modes is traced in a more large-scale
Fig. 3. Because of limited article volume we give pat-
terns of behaviour of modes only near to the first har-
monic of MIC frequency ( / 1i
).
0,0 0,5 1,0
-1
0
1
2
e
m=0
m=0
m=1
m=2
m=2
m=1
m=2
m=1
m=0
(SUH)
(LH)
(FUH)
q
/
ce
, m
e
/
ce
SF
e
I II
Fig. 1. Behaviour of fast (F) and slow (S) lower hybrid
(LH) modes and upper hybrid (SUH, FUH) modes of
single component electron plasma ( 0f ) with azi-
muth numbers 0, 1, 2m . In regions I and II electron
modes are low-frequency
As it is seen from Figs. 2, 3,a; 3,b the first harmonic
intersects with three radial lower hybrid modes LH0,
LH1, LH2 in regions I and II. In the region I the inter-
sections with modes LH0 and LH1 occur in a weak radi-
al electric fields ( 5/ 4 10e iq m m ) when ions of
background gas are magnetized. The intersection with
the mode LH2 occurs in the field of the order of critical
value ( ~ /e iq m m ) when ions are unmagnetized. In the
region II intersections occur in extremely strong radial
electric fields, of the order Brillouin value and even
close to it.
Near to each harmonic of MIC frequency there are
two MIC modes with the identical number of the radial
mode. One mode is located above, another mode – be-
low the harmonic (see Figs. 3,a; 3,b). MIC modes which
are crossed with lower hybrid modes with the same ra-
dial number are unstable with slow growth rates within
a wide range of changing of parameter q , located with-
in a crossing interval.
This peculiarity of spectrum of MIC modes, their
instability in a wide range of changing of parameter q
are caused by anisotropy of the distribution function of
ions of background gas (1). It shows itself only under
the kinetic description of ions, both magnetized, and in
a greater degree unmagnetized ions. For this cause the
existence of instability of MIC modes in a wide range of
fields changing could not be found out in [1] where the
hydrodynamic description has been used for ions.
Radial mode MIC2 has the fastest growth rate (both
normalized, and absolute) in the region II near to the
crossing with the radial electron mode (LH2). It equals
2Im 0.004 0.64 0.9 10i ci pi .
Near to the second harmonic of MIC frequency the
mode MIC1 in the region II has the fastest growth rate
almost on two orders smaller, and in the region I – still
on the order slower.
1E-7 1E-6 1E-5 1E-4 1E-3 0,01 0,1 1
0
1
2
3
4
5
LH
3LH
2
LH
1
q
/
i
LH
0
Fig. 2. Behaviour of the family of lower hybrid axially-
symmetric modes. MIС modes on the scale of drawing
coincide with harmonics MIC of frequency
( / 1, 2, 3, 4i n ). Square markers designate
neighbourhoods of crossings of electron modes with
MIC modes. Coefficients in labels of hybrid modes des-
ignate numbers of radial modes
Near to zero harmonic of MIC frequency
( / 0i
) the ion modes are missing. Presence of ions
leads to only small corrections to frequencies of electron
modes.
4. Modes having 0m were observed in experi-
ments with the electron beams propagating through re-
sidual gas (see, for example, [5]). Results of experi-
ments were interpreted within the frame of the local
theory. However ions of secondary plasma were
unmagnetized in experiments and the local theory was
inapplicable for their description. Bad accordance be-
tween a theory and experiment has been marked that has
forced the author [5] to compare not spectra, but thresh-
olds of excitation of unstable oscillations for which
there has been better accordance.
During excitation of oscillations having asimuth
numbers 0, 1m in the experiments described in [5],
there was a multiple increase of transversal energy of
secondary ions. It means that ions were in resonance
(
in ) with excited unstable MIC modes. When
0m - the frequency of mode is close to the first
harmonic of MIC frequency ( 1, in ), and when
1m the frequency is close to a zero harmonic
( 0, 0 in m ), so the frequency is
equal / 2ci im . Thus the frequency
of mode with 0m appears greater than the frequency
of mode with 1m : ( 0) 2 ( 1)m m . The same
correspondence takes place in an experiment (Fig. 3,b in
[5]). Frequencies of modes are proportional to the MIC
frequency
i
. The same dependencies of frequencies
ISSN 1562-6016. ВАНТ. 2015. №1(95) 95
on the parameters of plasma and fields were observed in
an experiment (Fig. 57 in [5]).
In general, experiments that were carried before
80th were interpreted as a rule on the basis of model of
boundless plasma in cartesian geometry, i.e. plasma is
not rotating, but is moving rectilineally. In such a model
the relation takes place
i ci
, while the relation
i ci
is more often fulfilled in experiments. Such a
model is unable to predict MIC modes excited in an
experiment, and to interpret correctly the results of ex-
periments. So, the interpretation of many such experi-
ments must be revised in spirit of theory of nonneutral
plasmas [4].
Fig. 3,a. Behaviour of frequencies and growth rates (in the bottom of Fig.) of modes near to the first harmonic
of MIC frequency
Fig. 3,b. The same, as in Fig. 3,а, but on the large
scale in the region II ( ~ 1q )
In [2, 3] and in the present paper stability of
nonneutral plasma is considered within the framework
of cylinder model, the adequate kinetic description of
ions, produced by ionisation of neutrals of background
gas by an electron impact, is given. It is valid for mag-
netized and unmagnetized ions, for long and short
waves in longitudinal and transversal directions. It is
applicable for interpretation of spectra of oscillations in
these ([5]) and other similar experiments.
We will notice, secondary ions that appear as a result
of charge exchange of ions of primary beam on neutrals
of background gas [6], also form the distribution func-
tion (1), if the radial electric field is directed to the axis
of ion beam (excess of electrons). Obtained in [2, 3] and
in present paper results give description of such ions
without any changes and determine their contribution to
dispersion equation.
5. Not only ions can be unmagnetized, but also elec-
trons. Such a situation takes place when a magnetic field
is absent and the ion beam propagates through back-
ground gas, ionizes it, forming secondary plasma [7]. If
the ion charge dominates in the beam channel, and
mostly it so, secondary electrons move in a transversal
plane on trajectories strongly elongated on radius, as
well as considered ions of background gas. Therefore
the used description of ions can be extended to these
electrons.
Spectra of axially-symmetric oscillations of such
plasma have been studied in [7] experimentally and
theoretically. The kinetic equation for electrons has
been solved by the same method, as in [1]. The solution
has been obtained under strong restrictions on ,
zk ,
electron temperature. When
zk is small, a reduction of
an oscillation frequency has been found out in compari-
son with the electron Langmuir frequency (the formula
(16) and Fig. 5 in [7]). According to authors opinion this
phenomenon is caused by radial oscillations of elec-
96 ISSN 1562-6016. ВАНТ. 2015. №1(95)
trons. Between theoretical and observational results [7]
there is an appreciable discrepancy.
Obtained in [2, 3] expressions allow to determine the
coefficient U of reduction of oscillations frequency
due to radial oscillations, introduced in [7], for the mod-
el of homogeneous beam, completely filling a wave-
guide, without any restrictions used in [7]. It is deter-
mined by the summand in a diagonal matrix element
11A (see [2, 3]), in which it is necessary to put
0m p , that corresponds to the axially-symmetric
mode and its low frequency ( / 1e
). A coefficient
of reduction of oscillation frequency is equal
1
2 22
0,1 ,z z zU I k a k a (2)
where
1
2
1 4
0 0 0,1
0
1 2 1,9zI N xdx J x ,
0,1 2,4 ,
2 21
0 1 0,1( ) 2N J .
0 2 4 6 8 10
0
1
2
k
z
a
U
1,9
1/2 2
1
Fig. 4. The behaviour of the coefficients U of reduc-
tion of oscillation frequency due to radial oscillations:
curve 1 is the theoretical result of paper [7], cirve 2 is
based on formula (2) of present paper
Its behavior versus a parameter
zk a is presented in
Fig. 4 (curve 2). Here it is also presented the behavior of
coefficient U , obtained theoretically in [7] (curve 1).
Comparing to the experimentally measured curve de-
pendence (see Fig. 4, 5 in [7]) testifies, that curve 2 cor-
rectly approximates behavior U both in the area of
small
zk a and in the area of 1zk a . A conversion from
one dependence to another takes place at
0,1~ 2 ~zk a ,
as well as in an experiment [7]. However at
0,1zk a
coefficient 1U , so that reduction of frequency due to
radial oscillations of electrons does not take place in the
considered model of homogeneous plasma completely
filling a waveguide. During radial oscillations every
particle submerges into the region, where the field of
wave
zE is stronger. This increases the induced space
charge of particles comparing to a case, when particles
are magnetized and rotate, being on the same radius.
REFERENCES
1. R.H. Levy, J.D. Daugherty, O.Buneman // Phys. Fl.
1969, v. 12, p. 2616.
2. Yu.N. Yeliseyev // Plasma Phys. Rep. 2010, v. 36,
p. 607.
3. Yu.N. Yeliseyev // Plasma Phys. Rep. 2014, v. 40,
p. 442.
4. R.C. Davidson. Theory of Nonneutral Plasmas. Ben-
jamin, New York, 1974.
5. M.V. Nezlin. Dynamika puchkov v plasme. М., 1982
(in Russian).
6. M.D. Gabovich // Uspekhi Fizicheskikh Nauk. 1977,
v. 121, issue 2, p. 259 (in Russian).
7. L.P. Katsubo, V.V. Lisitchenko, I.A. Soloshenko,
Ya.M. Shuba // JETP. 1979, v. 77, p. 1372.
Article received 03.12.2014
УСТОЙЧИВОСТЬ АКСИАЛЬНО-СИММЕТРИЧНЫХ НИЖНЕГИБРИДНЫХ МОД
ОДНОКОМПОНЕНТНОЙ ЭЛЕКТРОННОЙ ПЛАЗМЫ С ДОБАВКОЙ ИОНОВ ФОНОВОГО ГАЗА
Ю.Н. Елисеев
Численно определены низкочастотные спектры аксиально-симметричных мод колебаний заряженной
плазмы, полностью заполняющей волновод. Плазма состоит из холодных электронов и малой добавки ио-
нов, которые образовались ионизацией нейтралей фонового газа электронным ударом. Ионы описываются
равновесной функцией распределения, адекватно учитывающей эту особенность их образования. Спектр
состоит из семейства нижнегибридных мод и семейств «модифицированных» ионных циклотронных (МИЦ)
мод. МИЦ-моды, которые пересекаются с нижнегибридными модами, неустойчивы в широком диапазоне
изменения полей и плотности плазмы из-за анизотропии функции распределения ионов.
СТІЙКІСТЬ АКСІАЛЬНО-СИМЕТРИЧНИХ НИЖНЬОГІБРИДНЫХ МОД ОДНОКОМПОНЕНТНОЇ
ЕЛЕКТРОННОЇ ПЛАЗМИ З ДОБАВКОЮ ІОНІВ ФОНОВОГО ГАЗУ
Ю.М. Єлісеєв
Чисельно визначені низькочастотні спектри аксіально-симетричних мод коливань зарядженої плазми, що
повністю заповнює хвилевід. Плазма складається з холодних електронів і малої добавки іонів, які утворили-
ся іонізацією нейтралів фонового газу електронним ударом. Іони описуються рівноважною функцією розпо-
ділу, що адекватно враховує цю особливість їх утворення. Спектр складається з сімейства нижньогібридних
мод і сімейств «модифікованих» іонних циклотронних (MIЦ) мод. МІЦ-моди, які перетинаються з нижньо-
гібридними модами, нестійкі в широкому діапазоні зміни полів і щільності плазми із-за анізотропії функції
розподілу іонів.
|
| id | nasplib_isofts_kiev_ua-123456789-82097 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:07:26Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Yeliseyev, Yu.N. 2015-05-25T05:50:54Z 2015-05-25T05:50:54Z 2015 Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions / Yu.N. Yeliseyev // Вопросы атомной науки и техники. — 2015. — № 1. — С. 93-96. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 52.20.Dq; 52.25.Dg; 52.27.Jt; 52.35.-g; 52.35.Fp; 52.35.Qz; 41.20.Cv https://nasplib.isofts.kiev.ua/handle/123456789/82097 The low-frequency spectrum of axially-symmetric modes of oscillations of the nonneutral plasma completely filling a waveguide is evaluated. Plasma consists of cold electrons and a small additive of ions produced by ioniza-tion of neutrals of background gas by electron impact. Ions are described by the equilibrium distribution function adequately taking into account the peculiarity of their formation. The spectrum of oscillations consists of the family of lower hybrid modes and families of "modified" of ion cyclotron (MIC) modes. MIC modes intersecting lower hybrid modes are unstable within a wide range of changing of fields and plasma density due to anisotropy of the ion distribution function. Численно определены низкочастотные спектры аксиально-симметричных мод колебаний заряженной плазмы, полностью заполняющей волновод. Плазма состоит из холодных электронов и малой добавки ионов, которые образовались ионизацией нейтралей фонового газа электронным ударом. Ионы описываются равновесной функцией распределения, адекватно учитывающей эту особенность их образования. Спектр состоит из семейства нижнегибридных мод и семейств модифицированных» ионных циклотронных (МИЦ) мод. МИЦ-моды, которые пересекаются с нижнегибридными модами, неустойчивы в широком диапазоне изменения полей и плотности плазмы из-за анизотропии функции распределения ионов. Чисельно визначені низькочастотні спектри аксіально-симетричних мод коливань зарядженої плазми, що повністю заповнює хвилевід. Плазма складається з холодних електронів і малої добавки іонів, які утворилися іонізацією нейтралів фонового газу електронним ударом. Іони описуються рівноважною функцією розподілу, що адекватно враховує цю особливість їх утворення. Спектр складається з сімейства нижньогібридних мод і сімейств «модифікованих» іонних циклотронних (MIЦ) мод. МІЦ-моди, які перетинаються з нижньогібридними модами, нестійкі в широкому діапазоні зміни полів і щільності плазми із-за анізотропії функції розподілу іонів. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Фундаментальная физика плазмы Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions Устойчивость аксиально-симметричных нижнегибридных мод однокомпонентной электронной плазмы с добавкой ионов фонового газа Стійкість аксіально-симетричних нижньогібридных мод однокомпонентної електронної плазми з добавкою іонів фонового газу Article published earlier |
| spellingShingle | Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions Yeliseyev, Yu.N. Фундаментальная физика плазмы |
| title | Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions |
| title_alt | Устойчивость аксиально-симметричных нижнегибридных мод однокомпонентной электронной плазмы с добавкой ионов фонового газа Стійкість аксіально-симетричних нижньогібридных мод однокомпонентної електронної плазми з добавкою іонів фонового газу |
| title_full | Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions |
| title_fullStr | Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions |
| title_full_unstemmed | Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions |
| title_short | Stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions |
| title_sort | stability of axially–symmetric lower hybrid modes of single component electron plasma with additive of background gas ions |
| topic | Фундаментальная физика плазмы |
| topic_facet | Фундаментальная физика плазмы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82097 |
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