Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2)
The spectrum of nonneutral plasma oscillations with azimuth number m = 2 is computed. Plasma, completely filling a wave guide, is composed of "cold" magnetized electrons and a small density fraction of ions, produced by ionization of background gas atoms by electron impact. The distributio...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2010
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| Cite this: | Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) / Yu.N. Yeliseyev // Вопросы атомной науки и техники. — 2010. — № 6. — С. 79-81. — Бібліогр.: 6 назв. — англ. |
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Yeliseyev, Yu.N. 2011-02-26T21:13:06Z 2011-02-26T21:13:06Z 2010 Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) / Yu.N. Yeliseyev // Вопросы атомной науки и техники. — 2010. — № 6. — С. 79-81. — Бібліогр.: 6 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/17464 The spectrum of nonneutral plasma oscillations with azimuth number m = 2 is computed. Plasma, completely filling a wave guide, is composed of "cold" magnetized electrons and a small density fraction of ions, produced by ionization of background gas atoms by electron impact. The distribution function of ions is anisotropic one. The spectrum obtained consists of the families of "modified" ion cyclotron (MIC) modes and also of the families of electron upper hybrid (UH) and low hybrid (LH) modes (Trivelpiece–Gould modes) Doppler shifted by the electron rotation. Due to the Doppler shift the frequencies of electron modes fall into a region of ion frequencies and instabilities arise. The growth rates of UH modes are much faster than the growth rates of LH and MIC modes. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Фундаментальная физика плазмы Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) Спектр колебаний электронной плазмы с добавкой ионов фонового газа (азимутальное число m = 2) Article published earlier |
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Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) |
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Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) Yeliseyev, Yu.N. Фундаментальная физика плазмы |
| title_short |
Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) |
| title_full |
Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) |
| title_fullStr |
Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) |
| title_full_unstemmed |
Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) |
| title_sort |
oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) |
| author |
Yeliseyev, Yu.N. |
| author_facet |
Yeliseyev, Yu.N. |
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Фундаментальная физика плазмы |
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Фундаментальная физика плазмы |
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2010 |
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English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| title_alt |
Спектр колебаний электронной плазмы с добавкой ионов фонового газа (азимутальное число m = 2) |
| description |
The spectrum of nonneutral plasma oscillations with azimuth number m = 2 is computed. Plasma, completely filling a wave guide, is composed of "cold" magnetized electrons and a small density fraction of ions, produced by ionization of background gas atoms by electron impact. The distribution function of ions is anisotropic one. The spectrum obtained consists of the families of "modified" ion cyclotron (MIC) modes and also of the families of electron upper hybrid (UH) and low hybrid (LH) modes (Trivelpiece–Gould modes) Doppler shifted by the electron rotation. Due to the Doppler shift the frequencies of electron modes fall into a region of ion frequencies and instabilities arise. The growth rates of UH modes are much faster than the growth rates of LH and MIC modes.
|
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1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/17464 |
| citation_txt |
Oscillation spectrum of electron plasma, containing additive of background gas ions (azimuth number m=2) / Yu.N. Yeliseyev // Вопросы атомной науки и техники. — 2010. — № 6. — С. 79-81. — Бібліогр.: 6 назв. — англ. |
| work_keys_str_mv |
AT yeliseyevyun oscillationspectrumofelectronplasmacontainingadditiveofbackgroundgasionsazimuthnumberm2 AT yeliseyevyun spektrkolebaniiélektronnoiplazmysdobavkoiionovfonovogogazaazimutalʹnoečislom2 |
| first_indexed |
2025-11-24T16:10:08Z |
| last_indexed |
2025-11-24T16:10:08Z |
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1850851076652138496 |
| fulltext |
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2010. 6. 79
Series: Plasma Physics (16), p. 79-81.
OSCILLATION SPECTRUM OF ELECTRON PLASMA, CONTAINING
ADDITIVE OF BACKGROUND GAS IONS (AZIMUTH NUMBER m=2)
Yu.N. Yeliseyev
Institute of Plasma Physics NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: eliseev2004@rambler.ru
The spectrum of nonneutral plasma oscillations with azimuth number m = 2 is computed. Plasma, completely fill-
ing a wave guide, is composed of "cold" magnetized electrons and a small density fraction of ions, produced by ioniza-
tion of background gas atoms by electron impact. The distribution function of ions is anisotropic one. The spectrum
obtained consists of the families of "modified" ion cyclotron (MIC) modes and also of the families of electron upper
hybrid (UH) and low hybrid (LH) modes (Trivelpiece–Gould modes) Doppler shifted by the electron rotation. Due to
the Doppler shift the frequencies of electron modes fall into a region of ion frequencies and instabilities arise. The
growth rates of UH modes are much faster than the growth rates of LH and MIC modes.
PACS: 52.20.Dq; 52.25.Dg; 52.27.Jt; 52.35.-g; 52.35.Fp; 52.35.Qz; 41.20.Cv
The nonneutral plasma with magnetized electrons and
unmagnetized ions, produced by ionisation of atoms
(molecules) of background gas by electron impact, is cre-
ated and used in many experiments and technological
devices (plasma lenses, ion sources made on the basis of
Penning sell, beam experiments etc). Oscillations excited
in such plasma, affect the operation regimes of such de-
vices and even hinder them. An interpretation of oscilla-
tion spectra of such a plasma, observed over a wide range
of electric and magnetic fields, electron and ion densities,
is still lacking.
For determining the spectrum of plasma oscillations in
such devices it is necessary to solve the non local stability
problem in cylindrical geometry, using the kinetic de-
scription of equilibrium and stability of unmagnetized
ions of background gas concerning helical perturbations
[ ]( ) exp ( )m zr i m k z tΦ = Φ ϕ + − ω% % . Attempts to solve this
problem, go back to the known article [1] (see also [2]).
However, only in [3] the adequate kinetic description of
an equilibrium state of unmagnetized ions of the back-
ground gas, taking into account the peculiarity of ion
production, has been yielded, in [4] their perturbed state
has been worked out, in [5, 6] the non local dispersive
equation for oscillation frequencies of the electron
plasma, containing a small density fraction of such ions,
has been obtained. It is valid over the entire range of al-
lowable electric and magnetic field strengths, for magnet-
ized and unmagnetized ions. In [5, 6] the electrons are
supposed to be "hot" and "cold" consequently.
As it has been shown in [3, 6], the frequencies of vol-
ume electron modes, which are high-frequency in neutral
plasma, in nonneutral plasma can fall into the region of
low (ion) frequencies due to the Doppler shift caused by
the electron rotation in crossed fields. This peculiarity of
behaviour of electron modes is important for the stability
of nonneutral plasma. If there is even a small additive of
ions in plasma it leads to interaction of electron and ion
modes and plasma instability This peculiarity has not
been revealed in [1, 2].
The frequencies of volume oscillations of the
waveguide completely filled with the homogeneous non-
neutral "cold" electron plasma are equal (neglecting the
ions influence) the upper hybride (UH) and low hybride
(LH) frequencies with Doppler shift [2]:
{ }
1
1 2
2 2 2 2 2 2 2 2 2
1
2
1 ( ) 4 cos .
2
e
e pe e pe pe e
mω = ω ±
± Ω + ω ± Ω + ω − ω Ω θ
(1)
In (1) peω is Lengmuir frequency of electrons,
( )1/ 22 +4 /e ce r eeE m rΩ = ω - their "modified" cyclotron
frequency in the crossed fields, 0e > , radial electric field
0rE < , ceω is the electron cyclotron frequency,
2 2 2 2 2 2 2
,cos / ( ), /z z m lk k k k a⊥ ⊥θ = + = κ , zk is a
longitudinal wave vector, ,m lκ – the lth root of Bessel
function mJ , a is the plasma radius, 0eω > is the angular
velocity of electron rotation in the crossed fields. For
unambiguity it is considered 0m > , and a frequency sign
ω is arbitrary. The signs « ± » in braces define the
frequencies of UH and LH modes, and signs « ± » before
braces – the frequencies of the fast (F) and slow (S)
modes.
Thus, in the nonneutral plasma four families of elec-
tron modes – FUH, FLH, SUH, SLH - are available
(Fig. 1). The frequencies of fast modes (FUH and FLH)
are always positive. The frequencies of slow modes (SUH
and SLH) due to Doppler shift can change a sign, crossing
zero of frequency and region of ion frequencies. The SLH
modes cross the region of ion frequencies at any azimuth
wave numbers m (region I in Fig. 1 and Fig. 2), the SUH
mode – at azimuth numbers 2m ≥ (region II in Fig. 1).
For azimuth wave number 1m = the spectrum of
modes of plasma, containing an additive of background
gas ions, has been computed in [6]. In the present work
the computation results of oscillation spectrum for
azimuthal wave number 2m = are presented.
The normalized frequencies '/ iω Ω ( ' = - m +ω ,
( ) / 2ci i+ω = −ω + Ω , ( )1/22 4 /i ci r ieE m rΩ = ω − –
"modified" ion cyclotron (MIC) frequency) versus parameter
2 22 /pe ceq = ω ω were computed at the same numerical values
of parametres, as in [6]: the ion mass is chosen equal to the
mass of atomic nitrogen ion ( 14mi = u), 0.1zk a = ,
coefficient of charge neutralisation [2] / 0.01ef N n= =
( N , en – the ion and electron densities).
mailto:eliseev2004@rambler.ru
80
0 1
-1
0
1
2
2ωe
m=0
m=0
m=1
m=2
m=2
m=1
m=2
m=1
m=0
(SUH)
(LH)
(FUH)
q
ω/ωce, mωe/ωce
SF
ωe
I II
1E-6 1E-5 1E-4 1E-3 0,01
-2
-1
0
1
2
3
SLH1
SLH0
FLH1
FLH0
FLH
SLH
2ωe'
q
ω'/Ωi
Fig. 1. Behaviour of modes (1) with azimuth numbers
0, 1, 2m = in pure electron plasma ( 0f = ). The
modes intersect the frequency zero in regions I and II.
The shaped lines denote Doppler shifts emω ,
2 2cos 10−θ =
Fig. 2. Spectrum of electron modes with 2m = in re-
gion I. MIC modes on the scale of figure coincide with
the harmonics of MIC frequency. Square markers de-
note the regions of intersection of electron modes with
MIC modes, ' -e e +ω = ω ω
0,70 0,75 0,80 0,85 0,90 0,95 1,00
-1,02
-1,01
-1,00
-0,99
-0,98
SUH0
MIC0
MIC0
MIC0
MIC0
SUH0
qq*
ω'/Ωi
A
B
0,936 0,938 0,940 0,942
-1,4
-1,2
-1,0
-0,8
-0,6
SU
H 1
SU
H 1
MIC1
MIC1
SUH 1
SUH 0
SU
H 0
MIC0
MIC0 MIC0
MIC0
SU
H 0
q
ω'/Ωi, Re(ω'/Ωi)
C
0,70 0,75 0,80 0,85 0,90 0,95 1,00
1E-5
1E-4
1E-3
0,01
0,1
MIC1
MIC3
MIC2
q*
A q
Im(ω'/Ω i)
B
SUH2
MIC0
MIC0
0,936 0,938 0,940 0,942
1E-3
0,01
0,1
MIC1
q
Im(ω '/Ω i) SUH0
SUH1SUH2
SUH3
MIC2
MIC3
C
Fig. 3. The spectrum of oscillations with azimuthal number 2m = in a vicinity of MIC frequency harmonic
/ 1i n′ω Ω = = − in region *q q> : a) the modes with the real frequencies are shown, b) the real frequencies and real
parts of complex frequencies of SUH modes in the region II are presented on a large scale in q-axis, c) the imagi-
nary parts of complex frequencies (growth rates) of SUH modes are shown, d) the same growth rates, as in (c), are
presented on a large scale, corresponding to the scale of (b). Real and imaginary parts of complex frequencies are
denoted by open markers. Indexes 0, 1, 2, 3 denote the numbers of radial modes. The shaped lines *q q= in (a) and
(c) specify the position of the peculiarity of MIC modes behaviour, where the relation 2 2 2
e em ω ≈ Ω is satisfied
II
ba
II
I
d
c
81
Results of computations on dispersion equation are pre-
sented in Fig. 2, 3. The LH mode behavior in the region
of intersection with MIC frequencies (region I in Fig. 1,
Fig. 2) is similar to the case 1m = [6]. The spectrum
consists of the families of FLH and SLH modes and fami-
lies of MIC modes. Owing to the smallness of ion density
the MIC modes are located in a narrow vicinity of har-
monics of MIC frequency and coincide with them on a
scale of Fig. 2. Because of the strictly circumscribed pa-
per size we do not present here the figures, demonstrating
behavior of modes within the region I on a large scale.
We enumerate here only main results. FLH modes cross
only positive harmonics of MIC frequency (Fig. 2) and
remain stable. SLH modes cross the harmonics of MIC
frequency with numbers 1n ≥ − and become unstable in a
vicinity of intersection with the non-negative harmonics
of MIC frequency. The fastest growth rate, equal
( )Im ' 0.04iω Ω ≈ , has the lowest (zero) radial mode
SLH0 near to intersection with zero harmonic of MIC
frequency. It is almost twice less than the corresponding
growth rate in the case 1m = .
The MIC modes are unstable over a wide range of
field changing in region I. Their fastest growth rates reach
the value ( )Im ' 0.002iω Ω ≈ . The behavior of MIC
modes is similar to their behavior in the case 1m = . The
oscillations of small amplitude are observed on frequency
profiles. They are similar to oscillations on dispersive
curves of plasma in metals. This peculiarity is caused by
the resemblance of the distribution function of back-
ground gas ions with the degenerate Fermi-Dirac distribu-
tion. There is also another peculiarity of MIC mode be-
havior, located in a case 2m = near the value
* 0.75q q= ≈ (Fig. 3,a, c). At this value of q the relation
2 2 2
e em ω ≈ Ω is satisfied and a transverse component of a
tensor of dielectric permeability of electrons 1ε has a pole.
The interaction of SUH and MIC modes in the region
of stronger electric fields ( *q q> ), including region II, is
presented in Fig. 3. Because of the circumscribed paper
size we present behavior of modes in a vicinity of the
harmonic of MIC frequency 1n = − only. Near the other
harmonics they behave in the same manner. Different
radial SUH modes are arranged very closely to each other
(Fig. 3,a, b).They cross the region of ion frequencies
nearly erect. The SUH modes are unstable near the inter-
section with all harmonics of MIC frequency – both posi-
tive, and negative, and zero. The growth rates of SUH
modes (Fig. 3,c, d) are much faster than the growth rates
of SLH and MIC modes. The fastest value
( ( )Im ' 0.27iω Ω ≈ ) has the lowest radial mode SUH0 in
a vicinity of intersection with zero harmonic of MIC fre-
quency ( 0n = ).
The causes of instabilities of both electron and MIC
modes are a relative azimuth motion of electron and ion
components of nonneutral plasma (a cross current) and
anisotropy of the distribution function of ions, produced
by ionization of background gas in crossed fields.
The modes with 2m > behave in the same manner as
the modes with 2m = . The modes with azimuth numbers
m = 1, 2 exhaust the all variety of behavior types of non-
neutral plasma modes. Thus, the spectra, obtained in [6]
and in present article, yield the complete solution of
nonlocal stability problem of nonneutral plasma with an
additive of background gas ions, discussed in [1], in that
specific case of plasma, completely filling a waveguide.
REFERENCES
1. R.H. Levy, J.D. Daugherty, O. Buneman // Phys. Fl. 1969,
v. 12, p. 2616.
2. R.C. Davidson. Theory of Nonneutral Plasmas, New
York: “Benjamin”, 1974.
3. V.G. Dem’yanov, Yu.N. Eliseev, Yu.A. Kirochkin et al. //
Fiz. Plazmy. 1988, v. 14, p. 840.
4. D.B. Chibisov, V.S. Mikhailenko, K.N. Stepanov //
Plasma Phys. Controlled Fusion. 1992, v. 34, p. 95.
5. Yu.N. Yeliseyev // Plasma Phys. Rep. 2006, v. 32, p. 927.
6. Yu.N. Yeliseyev // Plasma Phys. Rep. 2010, v. 36, p. 607.
Article received 29.09.10
m = 2 )
.
m = 2 ,
, « » ,
, .
« » (MIC)
(UH) (LH) ( ) ,
, ,
. UH- LH- MIC- .
m = 2)
.
m = 2 o ,
« » , -
, . « -
» (MIC) (UH) (LH)
( ) , , -
, . UH- -
LH- MIC- .
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