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Submitted on 1 Jan 1979
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ON THE LINEAR STABILITY OF A
COLLISIONLESS SINGLE-ENDED Q-MACHINE
M. Fang, S. Kuhn
To cite this version:
M. Fang, S. Kuhn. ON THE LINEAR STABILITY OF A COLLISIONLESS SINGLE- ENDED Q-MACHINE. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-579-C7-580.
�10.1051/jphyscol:19797280�. �jpa-00219268�
JOURNAL DE PHYSIQUE CoZloque C7, suppZdment au n07, Tome 40, JuiZZet 1979, vage c7- 579
ON THE LINEAR STABILITY OF A COLLlSIONLESS SINGLE-ENDED Q-MACHINE
M.T.C. Fang and S. ~ u h n " .
Department of EZectricaZ Eng-ineering and EZectronics, University o f LiverpooZ, LiverpooZ L69 3BX, Xl?ngZand U. K.
I n s t i t u t e for TheoreticaZ Physics, University of Innsbruck, A-6020 Innsbrmk, Austria.
1. Introduction. The electron-current driven "ion-acoustic" instability in colli-
sionless single-ended 8-machines has for many years been the subject of extensive experimental investigations /I-3/. It is excited by applying a sufficiently large positive d.c. bias to the cold plate (or to some grid) and usually exhibits the charac- ter of a strongly nonlinear standing longi- tudinal wave localized between the hot plate and the exciting electrode. Similar oscillations have been observed in certain plasma diodes /4/.
Tne present work 1s intended as a con-- tribution towards clarifying the o n s e t of these oscillations. We investigate the lin- ear stability of low-frequency, long-wave- length longitudinal waves in a collision- less single-ended Q-machine by solving Lan- dau's dispersion relation in the appropri- ate limit. The time-independent state is characterized by the strongly non->laxwell- ian velocity distribution functions associ- ated with the monotonic potential distribu- tions of a collisionless olane one-emitter
v:~=- (ncp-np) 'I2 for nonotonic in- creasing ones. Here, q p and
n
respective-CP
ly denote the plasma potential and the cold- plate bias normalized to KT/e, e being the magnitude of the electric elenentary charge.
The normalized plasma potential (as well as other quantities of interest) can be calcu- lated from the neutral-flux density irradi- ating the hot plate, from the hot-plate and ionic properties, and from the applied bias / 6 , 7 / -
We are looking for small-amplitude wave perturbations proportional to exp[i (kx-wt)]
,
where the (real) wavenumber k is given by r/L, and the (complex) frequency w is to be computed from Landau's dispersion relation
/8/. Using (1) and restricting ourselves to
low-frequency, long-wavelength modes (i.e., neglecting w/ka, and k(~1/4rn-.e~)"~, where
P
np is the plasma density), we obtain
1
-
Se/vCe = 2 ' (vCiI < ) /erfcuci. (2) Here,-
ve= (a'fzerfcvce) "exp(-vc$) is thenormalized average velocity of the elec- trons, <=w/kai, Z1=aZ/ar, and Z is the in- complete plasma dispersion function as dis- diode. Apart from this choice, the bounded
cussed by Franklin / 9 / .
nature of the system is only accounted for
The subsequent stability analysis is by postulating that the wavelength equals
based on the steady-state results of Fig.1, twice the system length L / 2 / .
which shows a section of the ( q c p t ~ ) para- 2. Theoretical background. The steady-
state velocity distribution functions (nor- malized to unity) are given by /5,6/
where s denotes the particle species (e for the electrons, i for the ions) , As= (vu2 as.
.erfcvCs)
-',
aS=(2KT/m,) ' I 2 , K is ~ o l t z - mann's constant, T is the hot-plate temper- ature, ms is the particle mass, erfc = 1-
erf is the complementary error function, vs
= v/a,, vcs= vcs/aS is the normalized cut- off velocity, and U is 'the unit step func- tion. The cutoff velocities are given by
v&=- (qp-n )
* vii=-
( q B ) ' f i for monotonicP
decreasing potential distributions, and by
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797280
meter plane for a collisionless plane one- emitter diode /5-7/. The "neutralization pa- rameter" a is the ratio of emitted particle densities at the hot-plate surface (a =
nro/nz0)
.
The figure shows type-separating curves (vA,Vo,aop),
curves of constant nor- malized plasma density vp (in units of n:,), of constant normalized plasma potentialn
P' of zero net electric current (af), and of constant normalized neutral flux (aX).
3. Discussion of results. To be specific, we solve the dispersion relation (2) for pa- rameters along the a -curves of Fig.1, which
X
correspond to experimental situations where the cold-plate bias is varied, everything else being kept constant /6/. The dashed portions of these curves are outside the pa- rameter domains for monotonic potential dis- tributions and will not be considered here.
In the negative-bias region, Eq. (2) al- ways yields one undamped "slow" mode (with a phase velocity lower than the ion cutoff velocity) and an infinite number of Lancau- damped "fast" modes (with Re< > vci), but no growing modes. The results for the slow node and one of the fast modes are shown in
I
-__---
ion average-
velociQ--- - - -
-_____----_-_-_---.__
2. lon cutoff velocity
Fig. 2
In the positive-bias region considered we' always find one unstable mode, cf. Fig.3.
Note that for sufficiently high n we al- c P
ways have Imc > > Re<. The situation is es- sentially that of a Buneman two-stream in- stability /lo/, where the large growth rate is known to be due to the resonant transfer of energy from the negative-energy, Doppler-
10-
0 10 20 30
9rp Fig. 3
shifted electron plasma oscillations to the positive-energy ion mode. The real part of the phase velocity is far too small to ac- count for the observed oscillation frequen- cies /I-3/, which clearly demonstrates that the results of our linear analysis cannot he e x t r a p t l.ated to descr j he the nonlinear stage. However, our results are in qualita- tive agreement with experimental observa- tions in that the growth rates are of the order of the ion-sound velocity, (~T/m~)'l~
/2,11/.
Acknowledgement. This work was supported by the Fonds zur Forderung der wissenschaft lichen Forschung (Austria) under grant nos.
2781/S and S-18/02.
References.
/I/ N.S.Buchel'nikova, R.A.Salimov, Sov.
Phys. JETP
24
(1969) 595/2/ P-Michelsen et al., Plasma Phys. (1979) /3/ R.Schrittwieser et al., this conference /4/ V.I.Kuznetsov, A.Ya.knder, Sov. Phys.
Tech. Phys.
22
(1977) 1295/5/ W.Ott, Z. Naturforschuny
3
(1967) 105 /6/ S.Kuhn et al., Rept. UigICP-FSP78/1,Innsbruck Univ. (July 1978)
/7/ S.Kuhn, submitted for publication /8/ L.D.Landau, J. Phys. USSR
10
(1946) 25 /9/ R.N.Franklin, Proc. 10th ICPIG, Oxford(1971) 269
/lo/ T.D.Mantei et al., Plasma Phys.
18
(1976) 705
/11/ E.Mravlag, to be published
