J. Plasma Fusion Res. SERIES, Vol.4 (2001) 317-321

On the Excitation of Low-Frequency Instabilities
in Magnetized Plasmas
SANDULOVICIU Mircea*, SCHRITTWIESER Romanl, IONITA Codrinal.
LOZNEANU Erzilia, DIMITRIU Dan and IGNATESCU Valerian
Department of Plasma Physics, "Al. I. Cuza" University, Iasi, Romania
llnstiutefor lon Physics University of Innsbruck, Austria(Received: 5 December 2000 / Accepted: 3 October 2001)

Abstract
We propose a new phenomenological model, based on self-organization phenomena, which is able
to explain the role of complex space charge configurations in the excitation of low frequency instabilities
in a Q machine. Our model can also explain the recently observed interaction between them. We assume
that for instabilities nonlinear space charge accumulations are relevant. These are created due to thedifferent locations where the neutrals excitation and ionization cross section reach their respective
maxima in a region where a gradient of the electron kinetic energy is present.

Keywords:
self-organization, double layer, Q-machine, ion instabilities, interaction between PRI and EICI,
hvsteresis. sidebands

1. lntroduction

field of each DL) originating from the external dc power
supply during the generation of each DL. The matterand energy are released into the electrical circuit wheneach DL disrupts [1,2]. The generation process of astationary DL actually represents the creation of "longrange spatial order", and its successive generation anddecay, triggered by internal processes, reveals theappearance of "spatio-temporal order". Therefore bothof these phenomena prove the presence of selforganization [3]. The hysteresis phenomenon, also a

Plasmas are systems with strong nonlinearities,
which are manifested, among others, by abrupt changesof the electrical transport properties, when an extemallyapplied constraint is gradually varied []. In these cases
the constraint is an external electric field. The abruptchanges are shown by critical points in the static

current-voltage characteristic (1-V trace) where 1
suddenly changes for certain critical values of V. The
following reasons for the nonlinear properties ofplasmas were identified |,21: (i) the spontaneous
generation of a stationary electric double layer (DL) in

specific property of a self-organized complexity,
demonstrates the bistability of the plasma state, which isfront of a current-collecting electrode; (ii) the transitionof the DL into a propagating state so that new DLs are
successively generating and decaying. All nonlinear
variations of 1 show hysteresis when V is gradually
increased and decreased. The DLs stimulate oscillations,which are related to the accumulation of matter(electrons and ions) and energy (stored in the electrica premise for its ability to stimulate oscillations. The
oscillations appear when, besides the DL, which is
localized in a certain region of the plasma, another partof the plasma works as a resonant system able toperform natural oscillations [1,2]. The shape of the DL
depends on the plasma device, in which it is produced.
@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research*Corresponding author's e-mail : msandu@ uaic.ro317

Sanduloviciu M. et al., On the Excitation of Low-Frequency Instabilities in Magnetized Plasmas

considered as the cause for the current limitation [6].
The phenomenology of such a virtual cathode in theframework of collisionless plasma models is far from

When the positively biased electrode is on all sides
surrounded by plasma, the DL appears at its border as anearly spherical luminous space charge configuration
called fireball [4]. When an external magnetic field
confines the plasma in the form of a column, the DL
appears at the border of a U-shaped luminous spacecharge configuration dubbed firerod [5]. The latter spacecharge configuration was observed in a Q-machine filledwith different noble gases at a pressure of the order ofbeing well understood.
As recently shown [7,8], there is another possibility
to explain the current limitation in a Q-machine whencollective effects related to the spatial separation ofneutral excitation and ionization cross sections byelectrons impact are taken into account. Suchphenomena can appear in a Q-machine in front of the

lO-a mbar.

cold plate (CP), where thermal plasma electrons are

2. Double Layers Acting as Stimulators ofaccelerated. Experimental arguments supporting thispresumption are the lcr-Vg characteristics shown in
Fig. l(a) and (b), which are taken when the potential
relaxation instability (PRI) is excited [7].
The nonlinearity of the lcrVcp characteristic in Fig

Oscillations

As recently shown, the most importantphenomenon in connection with the dynamics of aplasma diode, is the bifurcation of states, a process theunderstanding of which is crucial for all systems
exhibiting a space-chargelimited flow. A typical device,in which current limitation was observed, is the Q-

1(a) yields important information concerning the
succession of processes which lead to the excitation of

PRI. This instability appears when V6p surpasses thecritical value V3. Essential for a complete understandingof PRI is also the identification of the causes that
produce the current jumps before the PRI appearing.
since these jumps are related to phenomena that create
the necessary conditions for the PRI. These phenomena
successively appear when V6p is gradually increased in
the voltage range V 1 3 Vco ! V3. First, we note that thecollection of plasma electrons by the CP, indicated by
the general rise of the current from the ion saturation
regime, proves that the potential of the CP is positive
with respect to the unperturbed plasma potential [7]. So,

machine, the plasma of which is presumed collisionless.

Usually the generation of a so-called "virtual cathode" islcpGrA)

150

I

I

tm

(a)

{'

t

I

50

I

-5

v6vtl othermal plasma electrons are accelerated towards CP.
Fig. 1(a) shows that, before the current limitation at
Vcp = Vs, a sudden decrease of 1 occurs for V", = I/t.
This has its origin in the formation of a net negative

f"(V)

V,

5

0

Vt Vs

vs v2

space charge in front of CP by accumulation of thoseelectrons that have lost their kinetic energy after theexcitation of neutrals. Thereby a localized space charge
barrier for the electron current.I is created causing its
drop. Another important nonlinear phenomenon which
precedes the appearance of the PRI is the current jumpfor V6p - Vz. Such a sudden increase becomes possible

vro

5

l0

e.g. when a new source of charged particles appears in
the plasma column. The sole effect able to produce new
charged particles is ionization of neutrals. Ionization canappear when a DL with a potential drop equal or greaterthan the ionization potential of the potassium atoms ispresent in the plasma column. Based on experimental
results from a collisional plasma diode [,9], we
presume that such a DL is created by self-organizationin front of the CP. Under such conditions, we explain

15

Fis. 1 a, b: Current voltage characteristic of the coldplate of a O-machine for two different plasma
densities. For (a) the density is npr 107 cm-3, for=(b) n", 10e cm-3. The diameter of the plasma=
column is 3.6 cm and the diameter of the coldplate 7 cm.
318

Sanduloviciu M. et al., On the Excitation of Low-Frequency Instabilities in Magnetized Plasmas

the current limitation for Vsp = Vs by relating it to theexperimentally proved by the Isp-Vsp trace shown in
Fig. 2. This trace was obtained in the Q-machine of the
University of Innsbruck under the conditions that thediameter of the magnetized plasma column exceeded thediameter of CP. In that case in front of the CP a DL istransition of the DL from a stable state in the voltage
runge V2 3 Vcp 3 V3 into an unstable state that appears
for Vgp 2 V3. This transition involves the successive
formation, detachment and disruption of DLs formed in
front of CP, i.e. the PRI appearing, the temporal
evolution of which matches the above sequence of
events [9,10]. The barriers for 16.p that on the time

formed, which divides the column in two different
regions. One of these regions is located between the CP
and the DL, which performs its self-sustained dynamics(its position depending on V6.p). The other part is placed

average produce the current limitation for V6p) Vs, arethe negative dips at the low potential side of each DL.
We point out that until recently in the phenomenologicalmodel of the PRI only lcrVcp characteristics have been
considered with a single current limitation as that one

between the region where the DL disrupts and the HP.

The Isp-Vgp characteristic in Fig 2 shows that theappearance of the EICI is accompanied by a sudden
decrease of the time-averaged current for Vsp > % (Fig.
2 - left-hand side of the vertical dashed line) [7]. This

shown in Fig. 1(b). Such 16.p-V6,p traces appear for

relatively high plasma densities where the PRI is

can also be seen in the observed frequency spectrum[Fig. 3(a)], which shows a peak at 57.6 kHz; thiscorresponds well to the gyrofrequency of a K+ ion at theapplied magnetic field (B = 0.13T). Such a current drop
can be explained by the transition of the DL, generated
in front ofthe CP, into an unstable state. In this state thebarriers for I are successively self-assembled andobviously excited for lower values of Vgp, and where the
Icp-Vcp trace shows a negative differential resistance(NDR) around % [9]. The lack of current jumps in Fig.
1(b), like those observed in Fig. 1(a) for V1 and V2, is
not an argument that the above described effects (whichgive rise to the jumps) would not take place also in thecase of higher densities, but because of the earlier onset

disrupted at the low potential side of the DL. The
plasma potential in front of CP jumps as soon as a DL isof the PRI they are not observable. This is easy tounderstand since the accumulation of opposite spacecreated. The sudden increase of the plasma potential
gives rise to the simultaneous ejection of ions from thecharges in the DL strongly depends on the excitationand ionization rates, which depend on I6p, but also on
the neutral density in front of CP. For higher plasma

plasma channel between the DL and the CP, almost
perpendicular to the magnetic field lines [14,15]. After
this ejection the ions perform a cycloidal motion so thatdensities additional neutrals can be created byrecombination of ions and electrons, the concentrationof both being high in front of CP.
Based on the above-described phenomenologicalmodel we tentatively explain the recently re-investigatedinteraction between the PRI and the electrostatic ioncyclotron instability (EICI) in a Q-machine [11-13]. Webase our explanation on the presumption that bothinstabilities involve space charge DLs, which, during the
temporal evolution ofthe instabilities, are first created at
the positively biased CP, then start propagating awayfrom CP. The decay of the DL at a short distance fromCP determines a sudden increase of the plasma column

potential shifting in front of HP the region where
electrons are accumulated after neutral excitation.
Consequently, a barrier for 1 appears in front of HP. Its-5development and propagation towards CP, as abipotential structure, determine the main part of the PRI
period [9].

-2.5 VoVz Zs

5

Fig. 2 Typical l"r-V"rcharacteristic. CP diameter (1 cm) is
smaller than that of the plasma column (3.6 cm),when both instabilities, the ElCl and the PRl, areexcited. The two vertical dashed lines show thethreshold values of Vrr, above which the ElCl and3. Interactions between Low Frequency
Instabilities in a O-machine

the PRl, respectively, start. Above the PRI

The interaction between the PRI and the EICI isthreshold. both instabilities exist.
319

Sanduloviciu M. et al., On the Excitation of Low-Frequency Instabilities in Magnetized Plasmas

Above the PRI threshold there is a region where bothinstabilities appear with comparable amplitude, stronglyinteracting with each other. This is seen in the spectrum
tFig. 3(b)l not only by the appearance of a peak at the
expected PRI frequency under these conditions (12.8
kHz) but also of sidebands on both sides of the EICI
frequency peak at a distance of +12.8 kHz [13]. In thisway, the EICI is strongly modulated by the PRI [12].
The experimental results presented in Fig. 3(b)prove the presence of a system, which is able to perform
two independent natural oscillations. One oscillation
involves the entire plasma column between the DL in
front of CP and the HP, where the PRI is excited, andthe other one involves only the region in front of CP,
where the EICI appears. In our picture, both instabilities
are sustained by the dynamics of the DL. During the
Dl-self-assembling process energy from the extemal dc

ci
LL
lJ-

50

16

150

26

250

Frequency (kHz)

50kHz

.t.

ll
I

ci

F
L

power supply is accumulated. This energy is released in
the circuit during DL decay. This dynamics is triggeredby the natural oscillations above mentioned.
As we have described above, the main feature of
the EICI is the periodical expulsion of a group of ions,originating mostly from the region in front of the CP,

L

Frequency (kHz)from the current channel into the surrounding

Fig. 3 Two typical frequency spectra obtained for asmall CP, taken for the same conditions as the /cpVrrlrace of Fig. 2, once above the ElCl threshold(a), once above the PRI threshold (b) where bothinstabilities are present, i.e. for different currents

unperturbed plasma during every maximum of the EICLThis ion expulsion occurs because of a periodicallycreated radial electric field at the edge of the current
channel, which is due to a periodically appearing

collected by CP.

positive space charge inside it. The space charge appears

in front of the positively biased CP simultaneously withthey return into the channel after one gyroperiod. In thisway the synchronous cycloidal motion of the ions gives
rise to the EICI oscillations, while after each cycle the
DL self-assembles and decays. The return of the ions
into the channel produces a fast re-increase of thepotential drop. The accumulated matter and energy inthe formation of a frontal double layer. Thus, the spacecharge in front of the CP is confined both axially andradially by double layers. For higher collected currents
the number of electron impact ionizations in front of the
CP increases. Consequently, the electron density in thecurrent channel between the double layer and the CP ishigher and the radial electric field smaller, so that theions are ejected from the current channel with lower
energy. This leads to a decrease of the EICI frequency,
as emphasized by our experimental data (obtained foreach DL is periodically transferred to the EICI
oscillations. Simultaneously with the DL decay during
each EICI cycle, the channel opens for the current sothat a new DL can be built up in front of the CP and thefixed magnetic field strength) in Fig. 4. We see that the
EICI frequency is inversely proportional to the current

entire cycle repeats itself [9].
On the other hand, the PRI does not appear before
16p jumps up again for V6p > V2, followed by a region of
NDR (Fig. 2 - right-hand side of the vertical dashedline). This part of the characteristic is very similar tothat one shown in Fig. l(b) above V3, so that we are led
to the conclusion that the PRI is excited [10]. Also thismechanism of the PRI has recently been linked to theperiodical generation of DLs in front of the CP by themechanism already described above (cf. section ID t7l.

16p. Under our conditions the PRI, taking place between

the DL and HP, produces a strong modulation of thecurrent collected by the CP, and since the EICI
frequency depends on the current, the appearance of the
sidebands around the EICI frequency is also plausible.
As shown in Fig. 3(b), sidebands were also observed
around the second harmonic of EICI at 124.6 kHz. The
presence of the large hysteresis loop, shown in Fig. 2,
320

Sanduloviciu M. et al., On the Excitation of Low-Frequency Instabilities in Magnetized Plasmas

necessary. This density modulation of the positive ionsproduces a corresponding modulation of the current
collected by the CP. As we have seen, the EICI
frequency is inversely proportional to the current

collected by the CP (Fig. 4). The linear fit of theexperimental data in Fig. 4 allows for the calculation of
the amplitude of the current modulation needed toobtain a shift of the EICI frequency with the PRI

x

t

el

o

frequency (sidebands). This result is in good agreementwith the observed amplitude modulation of the current,-trthus demonstrating that the measured frequency- usually related to the cycloidal motion of individual

52 il 56 qs o0 62 04 s6 0s 7,o 12ions - is actually due to the gyromotion of ,,bunches of

ArErt(mAions", which can be considered as complex space charge
configurations, the properties of which also depend on
the overall ion density.

Fig. 4 Dependence of the ElCl frequency on the current/"" collected by the CP.

References

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We believe that unstable complex self-organizedspace charge configurations in the form of DLs,
generated in front of the CP of a Q-machine, can act assources of matter and energy, which in turn are able toexcite different kinds of low frequency instabilities. If
the current variations, related to the DL dynamics,
triggered by one of the instabilities, cover the current
range of another instability, two in principle independent
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EICI frequency, because the EICI frequency depends
not only of the magnetic field induction B, but also on
the current collected by CP (Fig. a). This current

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modulation is produced by the PRI oscillations that takeplace in the plasma column between the DL and HP. To
sustain them, a periodical extraction of positive ionsfrom the current channel between the DL and CP is321