Long-wavelength ion-sound parametric instability of plasma in magnetic field
We have studied numerically the dependence of the frequency and growth rate of the ion sound wave on the wave vector in plasma in the electric field of a helicon for long-wavelength oscillations when the wavelength of ion sound wave is equal to the order of magnitude to or exceeds the electron Larmo...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Цитувати: | Long-wavelength ion-sound parametric instability of plasma in magnetic field / V.S. Mikhailenko, V.V. Ol'shansky, K.N. Stepanov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 43-45. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-816062015-05-19T03:01:57Z Long-wavelength ion-sound parametric instability of plasma in magnetic field Mikhailenko, V.S. Ol'shansky, V.V. Stepanov, K.N. Газовый рaзряд, ППР и их применения We have studied numerically the dependence of the frequency and growth rate of the ion sound wave on the wave vector in plasma in the electric field of a helicon for long-wavelength oscillations when the wavelength of ion sound wave is equal to the order of magnitude to or exceeds the electron Larmor radius. Such instability may be separated if the pumping wave frequency (helicon) is equal to the order of magnitude to or less than the lower hybrid frequency. The growth rate approaches some tenth parts of the lower hybrid frequency if the electron drift velocity in the field of the helicon exceeds the ion sound velocity by some times. As for short wavelength oscillations, one should expect the anomalous damping of the helicon, turbulent heating of electrons and sustainment of gas discharge at the nonlinear stage of such instability. 2000 Article Long-wavelength ion-sound parametric instability of plasma in magnetic field / V.S. Mikhailenko, V.V. Ol'shansky, K.N. Stepanov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 43-45. — Бібліогр.: 6 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/81606 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Газовый рaзряд, ППР и их применения Газовый рaзряд, ППР и их применения |
| spellingShingle |
Газовый рaзряд, ППР и их применения Газовый рaзряд, ППР и их применения Mikhailenko, V.S. Ol'shansky, V.V. Stepanov, K.N. Long-wavelength ion-sound parametric instability of plasma in magnetic field Вопросы атомной науки и техники |
| description |
We have studied numerically the dependence of the frequency and growth rate of the ion sound wave on the wave vector in plasma in the electric field of a helicon for long-wavelength oscillations when the wavelength of ion sound wave is equal to the order of magnitude to or exceeds the electron Larmor radius. Such instability may be separated if the pumping wave frequency (helicon) is equal to the order of magnitude to or less than the lower hybrid frequency. The growth rate approaches some tenth parts of the lower hybrid frequency if the electron drift velocity in the field of the helicon exceeds the ion sound velocity by some times. As for short wavelength oscillations, one should expect the anomalous damping of the helicon, turbulent heating of electrons and sustainment of gas discharge at the nonlinear stage of such instability. |
| format |
Article |
| author |
Mikhailenko, V.S. Ol'shansky, V.V. Stepanov, K.N. |
| author_facet |
Mikhailenko, V.S. Ol'shansky, V.V. Stepanov, K.N. |
| author_sort |
Mikhailenko, V.S. |
| title |
Long-wavelength ion-sound parametric instability of plasma in magnetic field |
| title_short |
Long-wavelength ion-sound parametric instability of plasma in magnetic field |
| title_full |
Long-wavelength ion-sound parametric instability of plasma in magnetic field |
| title_fullStr |
Long-wavelength ion-sound parametric instability of plasma in magnetic field |
| title_full_unstemmed |
Long-wavelength ion-sound parametric instability of plasma in magnetic field |
| title_sort |
long-wavelength ion-sound parametric instability of plasma in magnetic field |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2000 |
| topic_facet |
Газовый рaзряд, ППР и их применения |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/81606 |
| citation_txt |
Long-wavelength ion-sound parametric instability of plasma in magnetic field / V.S. Mikhailenko, V.V. Ol'shansky, K.N. Stepanov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 43-45. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT mikhailenkovs longwavelengthionsoundparametricinstabilityofplasmainmagneticfield AT olshanskyvv longwavelengthionsoundparametricinstabilityofplasmainmagneticfield AT stepanovkn longwavelengthionsoundparametricinstabilityofplasmainmagneticfield |
| first_indexed |
2025-07-06T06:47:19Z |
| last_indexed |
2025-07-06T06:47:19Z |
| _version_ |
1836879124135673856 |
| fulltext |
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ 2000. №1.
Серия: Плазменная электроника и новые методы ускорения (2), с. 43-46.
43
UDK 533.9
LONG-WAVELENGTH ION-SOUND PARAMETRIC INSTABILITY
OF PLASMA IN MAGNETIC FIELD
Mikhailenko V.S.*, Ol'shansky V.V. **, Stepanov K, N. **
* V.N.Karazin Kharkov National University, Kharkov, Ukraine
** Institute of Plasma Physics, NSC «Kharkov Institute of Physics and Technology»,
Kharkov, Ukraine, e-mail olshansky@ipp.kharkov.ua
We have studied numerically the dependence of the frequency and growth rate of the ion sound wave on the
wave vector in plasma in the electric field of a helicon for long-wavelength oscillations when the wavelength of
ion sound wave is equal to the order of magnitude to or exceeds the electron Larmor radius. Such instability may
be separated if the pumping wave frequency (helicon) is equal to the order of magnitude to or less than the lower
hybrid frequency. The growth rate approaches some tenth parts of the lower hybrid frequency if the electron drift
velocity in the field of the helicon exceeds the ion sound velocity by some times. As for short wavelength
oscillations, one should expect the anomalous damping of the helicon, turbulent heating of electrons and
sustainment of gas discharge at the nonlinear stage of such instability.
1. Introduction
The papers [1,2] have shown that high efficiency of
helicon plasma sources, in which heating is performed by
a helicon, may be due to the short wavelength kinetic ion-
sound parametric instability. For this instability the
wavelength of unstable ion-sound oscillations 1 k is
considerably less than the electron Larmor radius,
k eρ >>1. As these oscillations are excited at the pumping
frequency ω ω0 > m ( m is an integer) and
a kuE = >ω0 1 , then the condition k eρ >>1 holds for
ω ω ω0 >> ce ci , m ~ 1 (here k is the wavenumber,
v T ms e i= is the velocity and ω ~ kvs is the frequency
of the ion-sound wave, ( )u c E B= ⊥ 0 0 is the velocity of
electron oscillations in the field of the helicon with the
frequency ω0 , wave vector k0 and the strength of the
transverse electric field of the wave E⊥ 0 ). It is assumed
that u vs>~ . Thus the separation of unstable short
wavelength oscillations is possible only for «weak»
magnetic fields. Such condition holds in a number of
experiments by F.Chen et al [3], for which the
correspondence between theoretical data [1,2] and
measured ones is obtained. However there are a number
of experiments with helicon sources in which this
condition does not hold and the magnetic field is «strong»,
ω ω ω0 >~ ce ci . In this case it is necessary to consider
the long wavelength, k eρ <~ 1 , ion-sound instability. The
present paper is devoted to the theoretical treatment of
this problem.
1. Dispersion equation
If the plasma is immersed in the electric field of the
pumping wave (helicon) with the frequency ω0 ,
considerably below the electron cyclotron frequency
ωce , but considerably more than the ion cyclotron
frequency ωci , then with u vs>~ in the plasma the ion-
sound oscillations may be excited for which one can
neglect the plasma and pumping field nonuniformity in the
direction perpendicular to the magnetic field. As the
oscillations may be excited that propagate almost across
the magnetic field, then the longitudinal wavenumbers of
the unstable oscillation k || and the helicon k||0 may be the
same to the order of magnitude, generally speaking.
Therefore we will take into account the nonuniformity of
the helicon electric field along the magnetic field.
We describe the linear stage of the ion-sound
parametric instability with the following infinite set of
difference equations for the Fourier components of the
electric potential of the ion-sound wave determined in the
reference frame oscillating together with ions in the
pumping wave field (see. [4,5]):
( )( ) ( ) ( ) ( )
( ) ( )
1
00 0 0 0
+ + ⋅
⋅ − − + + =
=−∞
∞
=−∞
∞
+∑∑δε ω ϕ ω
δε ω ω ϕ ω ω
δ
i p E
pn
p n E
in
e
J a J a e
p p n n
, ,
, ,
k k
k k k k
(1)
where δεα is the contribution to the longitudinal dielectric
permittivity made by α -species particles. For the
oscillations considered the growth rate exceeds the ion
cyclotron frequency substantially and the wavelength is
much less than the ion Larmor radius. In this case we can
neglect the action of the magnetic field on ion motion, so
that
( ) ( )[ ]δε ω
ω
πi
pi
Ti
i ik v
i z W z,k = +
2
2 2 1 . (2)
For electrons, neglecting the small terms proportional to
ω ω2 2
ce and ( )k vTe ce||
2 2 2ω , we obtain
44
( ) ( ) ( )[ ]δε ω
ω
π ρe
pe
Te
e e ek v
i A k z W z,k = +
2
2 2 0
2 21 . (3)
Here the following notation is introduced
ω πα α αp n e m= 4 0
2 is the Langmuir frequency of α -
species particles, v T mTα α α= is their thermal
velocity, ( )z k vi Ti=ω 2 ,
( )z k ve Te=ω 2 || , ( )W z i e dtt
z
= +
∫1 2 2
0π
.
Making in eq.(1) the substitution ω ω ω→ −m 0 ,
k k mk|| || ||→ − 0 , we obtain the set of difference equations,
whose solubility criterion is the vanishing of the
determinant
D amn m n= ==−∞
∞det , 0 , (4)
where
( )( )[ ]
( ) ( )
( ) ( )
a
i n m
m
J a
J a p p
mn mn
i o
p m e
p
p n E e o
= +
− − +
+ −
⋅
⋅ + +
+
=−∞
∞
+
∑δ
δ π
δε ω ω
δε ω ω
exp
,
, ||
1
0
k
k k
.
For short wavelength oscillations with k eρ >>1 there
immediately follows from eq. (4) the vanishing of any
diagonal element, i.e.,
( ) ( )
( )
D
m
J a
p p
i
p m E
p
e
~
,
, ||
1 1
1
0
0
2
0 0
+
+ −
⋅
⋅ + + =
+
=−∞
∞
∑δε ω ω
δε ω ω
k
k k
(5)
This equation was studied in paper [4] at k||0 0= and in
paper [2] at k||0 0≠ . Our problem consists in studying
equation (4) for k eρ <~ 1 and pumping frequencies
ω ω ω0 >~ ce ci .
2. Numerical solution of the dispersion equation
for long wavelength oscillations
On solving eq. (4) numerically we assumed
ω ω ω0 0 6ce ci = , , ω ωpe ce
2 2 25= (helicon branch of
oscillations is usually treated under the condition
ω ωpe ce
2 2>> ), T Te i = 20 , u vs = 5 , the operating gas is
hydrogen. Conventionally one uses argon in helicon
sources as an operating gas. It is easy to prove that under
the condition krDe << 1 the result of calculation does not
depend on the mass of the operating gas ions if the
quantities ω ω ω0 ce ci and u vs are regarded as fixed.
Figure 1 shows the dependence of the oscillation
frequency on the wave vector. For comparison the same
figure shows the dependence of the ion-sound frequency
ω = kvs . It is seen that the frequency depends actually
linearly on the wave vector. The difference of the
frequency of unstable oscillations on the frequency of ion-
sound oscillations of the unmagnetized plasma is
associated with the effects of finite electron Larmor radius
and the presence of the pumping wave.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
ω ω ωce ci
k eρ
ion sound
frequency
Fig.1. Frequency of parametrically unstable ion-sound
oscillations in the helicon field against the wavenumber
(ω ωpe ce
2 2 25= , u vs = 5 ,
T Te i = 20 ,. ω ω ω0 0 6ce ci = , , k ||0 0= )
0.0 0.5 1.0
0.0
0.2
0.4
γ ω ωce ci
k eρ
Fig. 2. Growth rate of unstable ion-sound oscillations in
the helicon field against the wavenumber for different
cosθ values (the parameters are the same as in Fig.1).
The curves correspond to various cosθ values from
0,01to 0,05 wth the step 0,005
45
Figure 2 depicts the growth rate against different
values of the angle θ between the magnetic field and the
wave vector. We see that on increasing cosθ the
maximum value of the growth rate increases up to the
value cos .θ = 0 045 , at which the maximum value of the
growth rate, γ ω ωmax ,≈ 0 3 ce ci ,appears to be the
biggest. At larger cosθ values the maximum value of the
growth rate decreases with the quantity cosθ increasing.
This biggest value is achieved at k eρ ≈ 0 75, . In this case
the frequency is ω ω ω≈ 0 9, ce ci and the excitation of
oscillations is due to the beats with p =2 ( pω ω0 > ). The
maximum value of the growth rate in Fig. 2 is related with
the value zep ~ 1 . In this case the interaction of resonant
electrons possessing the velocity of the order of the
thermal velocity with the p -th beats appears to be
important.
Conclusion
The study performed shows that in the case of
«strong» magnetic field the ion-sound parametric
instability in the helicon field also sets in. And it is even
stronger than in the case of «weak» magnetic field for
which γ ω ωmax ~ ,0 05 ce ci . Therefore one can also
expect in the «strong» magnetic field case the appearance
of a strong ion-sound turbulence leading to electron
heating and discharge sustainment in the alternating field
of the helicon.
Note that recently there is actively discussed (see
review [6]) the mechanism of strong absorption of the
pumping wave in the helicon source due to the excitation
of electrostatic oscillations called Gould-Trivelpiece
waves in the surface layer of plasma and the coupling of
these modes with helicons due to plasma nonuniformity.
In the presence of such mechanisms the parametric
excitation of ion-sound oscillations is also possible for the
strong as well for the weak magnetic field. In this case the
absorption of the Gould-Trivelpiece mode is determined
by the effective collision frequency depending on the level
of the ion-sound turbulence.
References
1. O.I.Akhiezer, V.S.Mikhailenko, K.M. Stepanov Ion-
Sound Plasma Turbulence with Transverse Current in
Magnetic Field //Ukrainian Journal of Physics, 1997 vol.
42, No. 8, p. 990.
2. A.I. Akhiezer, V.S. Mikhailenko, K.N. Stepanov Ion-
sound parametric turbulence and anomalous electron
heating with application to helicon plasma sources
//Physics Letters A, 1998, vol. 245, No. 1-2, p. 117.
3. F.F. Chen Physics of helicon discharges //Physics of
Plasmas, 1996, vol. 3, p. 1783.
4. A.B. Kitsenko, V.I. Panchenko, K.N. Stepanov, F.F.
Tarasenko Parametric instabilities and turbulent heating of
a plasma in the field of a fast magneto-acoustic wave
//Nuclear Fusion, 1973, vol. 23, p. 527.
5. A.F. Korzh, V.S. Mikhailenko, K.N. Stepanov
Parametric Ion Cyclotron Instability of Plasma in the
Electric and Magnetic Fields of the Nonuniform MHD
wave //Sov. Plasma Physics, 1989, vol. 15, p. 413.
6. K.P. Shamrai Collective mechanisms of absorbing RF
power in helicon plasma sources //Fizika plazmy, 1999,
vol. 25, p. 934.
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