J A N U A R Y 2 0 1 1 VOLUME 18 NUMBER 1

Transit time instabilities in an inverted fireball. II. Mode jumping and nonlinearitiesby R. L. Stenzel, J. Gruenwald, B. Fonda, C. Ionita, and R. Schrittwieser

Transit time instabilities in an inverted fireball. I. Basic propertiesR. L. Stenzel,1 J. Gruenwald,2 B. Fonda,2 C. Ionita,2 and R. Schrittwieser2

1Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA2Institute for Ion Physics and Applied Physics, Leopold-Franzens University of Innsbruck, Innsbruck,Technikerstr. 25, A-6020 Innsbruck, Austria

�Received 15 October 2010; accepted 13 December 2010; published online 11 January 2011�

A new fireball configuration has been developed which produces vircator-like instabilities. Electronsare injected through a transparent anode into a spherical plasma volume. Strong high-frequencyoscillations with period corresponding to the electron transit time through the sphere are observed.The frequency is below the electron plasma frequency, hence does not involve plasma eigenmodes.The sphere does not support electromagnetic eigenmodes at the instability frequency. However, therf oscillations on the gridded anode create electron bunches which reinforce the grid oscillation afterone transit time or rf period, which leads to an absolute instability. Various properties of theinstability are demonstrated and differences to the sheath-plasma instability are pointed out, one ofwhich is a relatively high conversion efficiency from dc to rf power. Nonlinear effects are describedin a companion paper �R. L. Stenzel et al., Phys. Plasmas 18, 012105 �2011��.© 2011 American Institute of Physics. �doi:10.1063/1.3533437�

I. INTRODUCTION

Electron transit-time or inertial instabilities have beenstudied for a long time. Such instabilities can arise when theelectron transit time across a resonant structure correspondsto the rf period. Early research was done in vacuum tubedevices such as a diode incorporated into a microwave reso-nator producing monotron oscillations.1–3 Related devices in-clude klystrons and traveling wave tubes. Vircator reflex os-cillations are also transit-time instabilities based on the sameprinciple as reflex klystrons.4 However, the reflecting elec-trode is replaced by a space charge layer, a “virtual cathode,”produced by a high current relativistic electron beam. Inplasma physics, transit-time instabilities can occur inelectron-rich sheaths of electrodes, producing a sheath-plasma instability near the electron plasma frequency.5 Eventhe basic beam-plasma instability can also be understood as atransit-time instability, where the electron transit timethrough one wavelength corresponds to the instability periodwhich can create a positive feedback between particles andwave.

Fireballs are ionization phenomena near positively bi-ased conductors in plasmas. They have been first observed inconnection with plasma sheath investigations.6 Further atten-tion arose in connection with ionization double layers.7

Many subsequent studies have been concerned with thephysics of spherical fireballs,8–13 in particular, the nonlinearproperties of fireballs.14,15 Most of the previous experimentscreated fireballs with solid conductors of plane, cylindrical,or spherical shape. The fireballs form outside the conductor.The fireball size and position depend on several parameters,which control the production and loss of particles in the fire-ball. The double layer at the fireball boundary has usually apotential drop close to the ionization potential.

In order to control the size and position of a fireball, wehave modified the electrodes. First, two parallel plane elec-trodes were used which trapped the fireball between them.

Then we used a cylindrical electrode which formed a fireballinside the cylinder. Finally, we built a spherical electrode andproduced a fireball inside the sphere which, of course, wasmade of a highly transparent grid. This trapped or “inverted”fireball will be described in this paper. Particular attentionwill be focused on a very strong high-frequency instabilityobserved inside the inverted fireball. It has many similaritiesto the vircator reflex instability4,16–24 but has also significantdifferences. Similarities include the instability mechanism ofa transit-time effect, i.e., the frequency is determined by thetransit time of the electrons through the fireball. Differencesinclude the absence of a negative space charge layer and thatthe sphere does not support electromagnetic or plasma wavesat the instability frequency. The spherical geometry producescounterstreaming electrons which is analogous to electronreflection from a negative space charge layer or a negativelybiased electrode. The instability is extremely strong in com-parison to the sheath-plasma instability, which is also ob-served. Many properties of the instability will be presentedin this and a companion paper.25

The paper is organized as follows. After describing theexperimental setup in Sec. II, the observations of the insta-bility properties are presented in subsections in Sec.III A–III E and summarized in Sec. IV.

II. EXPERIMENTAL ARRANGEMENT

The experiments are performed in two similar plasmadevices, one at UCLA and one at Innsbruck, shown sche-matically in Fig. 1�a�. These produce unmagnetized dc dis-charge plasmas in a parameter regime of electron densityne�108

¯109 cm−3, electron temperature kTe�4 eV, ar-gon gas pressure p�10−5

¯10−3 mbar, discharge voltageVdis=30¯200 V and current Idis�0.1¯1 A, and dimen-sions of 45 cm diam, 100 cm length. Two types of filamen-tary cathodes have been used: standard incandescent tung-sten wires �0.2 mm diam, 5 cm long� and barium-oxide

PHYSICS OF PLASMAS 18, 012104 �2011�

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coated nickel wires �5 cm long�. The former have atemperature-limited emission, i.e., the cathode current isnearly independent of cathode voltage, while the latter havea space-charge-limited emission, where Idis progressively in-creases with Vdis or plasma potential. The fireball electrode isbiased either with a constant dc voltage or pulsed with afast-rise �100 ns� transistor switch.

Plasma diagnostics consist of movable Langmuir probes,one of which is fed by a coaxial cable and serves to detect rfpotential variations with respect to ground. The data acqui-sition was performed with a digital oscilloscope �4 channel,2 GHz, �106 samples�. The light from fireballs is recordedwith a photodiode and a digital camera.

Fireballs are usually formed with positively biased elec-trodes such as solid disks, cylinders, or spheres.14 The fire-balls form outside the electrode. In the present experiments,a new mode of fireballs has been developed, termed “in-verted fireball.” An inverted fireball forms inside a hollowspherical anode, made of a highly transparent grid. Twospheres of approximately 5 and 10 cm diam have been con-structed by spot welding 0.2 mm diam Ta wire with approxi-mately 6 mm spacing. Figure 1�b� shows schematically someequipotential lines when the grid is biased positively. Eachwire is surrounded by an electron-rich sheath. When the wirespacing and Debye lengths are comparable, the sheaths over-lap and form approximately spherical equipotential surfaces.The plasma potential inside the sphere is positive relative tothe ambient plasma potential. The potential drop across thegaps of the grid forms a double layer. Electrons are acceler-ated into the fireball while ions are ejected from the fireball.Similarities and differences to open fireballs will be dis-cussed below.

III. EXPERIMENTAL RESULTS

A. Inverted fireball properties

Fireballs form in a limited parameter regime where elec-tron acceleration and ionization can occur. A positively bi-ased electrode is required to accelerate electrons above theionization potential �15.7 eV in Ar�. Ionization in anelectron-rich sheath leads to sheath expansion and the forma-tion of a double layer at the boundary of the fireball. If theoutflow of ions is balanced by ionization a steady-statedouble layer is formed. The double layer potential drop isclose to the ionization potential.

Φ =const

Fireball

Plasma

Grid

(a)

(b)

Vsphere

dis

CathodeRf probe

Vfil

ekT = 3 eV

B = 0

Vcm

8 -3n = 10 ... 10

9

Ar 10-4

...10-5

mbar

Spherical grid

Fireball

FIG. 1. �a� Schematic of the experimental setup and basic parameters. �b�Schematic equipotential lines around a transparent spherical grid for creat-ing an inverted fireball.

FIG. 2. �Color online� Pictures of inverted fireballs in argon. �a� 10 cm fireball at low pressures �10−5 mbar regime� showing light from 100 eV electronsinside the spherical grid �light is blue in color�. �b� 5 cm fireball at intermediate pressures �10−4 mbar regime� showing light from 15 eV electrons inside thespherical grid �light is white in color�. The fireball boundary may expand beyond the grid. �c� Lack of a fireball at high densities where the sheath thicknessis small compared to the wire spacing. 10 cm diam grid glows due to electron collection from the dark ambient plasma.

012104-2 Stenzel et al. Phys. Plasmas 18, 012104 �2011�

In an inverted fireball, the grid usually forms the bound-ary of the fireball. Figure 2�a� shows a picture of an invertedfireball at low neutral pressures. The grid is biased positively,collects electrons, which raises the plasma potential insidethe sphere. The plasma potential establishes equal electronand ion losses so as to maintain charge neutrality. Drawingelectrons from the grid must be balanced by an enhanced ionloss, which is accomplished by a potential rise in the fireballleading to ion ejection. When ionization balances the loss ofions, a steady-state fireball is maintained. This requirementalso determines the plasma potential inside the fireball. Atlow neutral densities, sufficient ionization requires electronenergies well above the ionization energy since the ioniza-tion cross section increases with energy. At higher neutralpressures, the potential drop decreases to just above the ion-ization. When seen in color, the Argon fireball changes fromblue to white with decreasing electron energy. The fireballmay also expand beyond the grid boundary. Finally, at higherpressures and/or plasma densities, the Debye length becomesso small that the grid does not establish a spherical equipo-tential surface but only a thin sheath around each wire. Theaccelerated electrons inside the sheath are collected and theirenergy is dissipated in the wire causing it to glow, as shownin Fig. 2�c�.

Langmuir probe measurements in an inverted fireball areshown in Fig. 3. First, the plasma properties in the center ofthe fireball �r=0� have been investigated for different gridvoltages. Figure 3�a� shows that with increasing Vgrid, theplasma potential shifts positive but generally remains belowVgrid. The traces reveal a tail of energetic electrons as well asa population of low energy electrons just below the plasmapotential. Figure 3�b� shows probe traces inside and outsidethe fireball at the same grid voltage. The current is displayedon a logarithmic scale and both ion currents and energeticelectron current contributions have been subtracted. Straight-line fits are used to determine the plasma potentials. Thedifference of plasma potential inside and outside the fireballis ���57 V, which is much higher than that of typicalfireballs �15 V�.

The fact that the plasma potential is positive inside thefireball shows that no virtual cathode is formed. However,the electron distribution is similar to that in a vircator orreflex klystron. Electrons are injected radially into the fire-ball. After passing through the center, they diverge radiallyoutward forming counterstreaming electron populations. Dueto the high transparency of the anode, most electrons passthrough the grid, lose energy, and return to the ambientplasma. In steady-state the spherical fireball cannot have aradial current. Since the grid is positive all ions are ejectedradially. A corresponding number of electrons must be lost tothe grid. From the measured fireball plasma parameters, theion current through the surface of a 10 cm diam sphere isIion�82 mA, comparable to the current collected by thegrid, Igrid�120 mA at Vgrid�100 V. The grid collectselectrons from both the ambient plasma and the fireball. Ifall the electrons entering the spherical surface area 4�r2

=314 cm2 were collected, the current would be Iel�1.7 A,which confirms that most of the electrons just traverse thefireball. Consequently, the double layer between the grid

wires is essentially current free. It does not obey the Bohmcriterion Ie / Ii= �mi /me�1/2 but the stability of the double layeris provided by the rigid grid.

It should be pointed out that the above considerationonly applies to the time-average conditions. In the presenceof the strong transit-time instability discussed below, poten-tials and currents can oscillate with amplitudes comparableto the dc values which changes many considerations.

B. Instability properties

When the spherical grid is biased positively, the gridcurrent develops high-frequency fluctuations above a certainthreshold. By inserting a rf transformer into the current path,the fluctuations have been observed in the time and fre-quency domain. Two characteristic instabilities can be iden-tified: the sheath-plasma instability and the fireball transit-time instability. Figure 4 shows the distinction between thesetwo instabilities: the fireball instability has a very large am-plitude compared to the sheath-plasma instability �see inset�.

Iprobe

(mA)

(V)Vprobe

Vgrid

0

1

2

3

100-100

20 80

rprobe

= 0 120V0 9060

(a)

0-40 80 100

ln {I (mA)}probe

Voltage V (V)probe

20

-6

-8

-10

-11

-9

ΔΦ plasma

ekT = 4 eVekT = 14 eV

outside

fireball

inside

fireball

gridV = 120 V

57 V

(b)

FIG. 3. �Color online� Time-averaged Langmuir probe traces in an invertedfireball for �a� different grid voltages with the probe at the center and �b� forthe same grid voltage and the probe inside and outside the fireball. The latterprobe traces display only the current of the electrons �ions and energeticelectron currents subtracted�. Straight-line approximations to ln I vs V yieldplasma potentials, electron temperature, and density inside the fireball �92 V,4 eV, 5.2�109 cm−3� and outside �35 V, 5.3�108 cm−3, 14 eV�. The po-tential drop �� across the double layer well exceeds the ionization potential�15.7 V�. The energetic electrons �57 eV� inside the fireball have a density1.1�107 cm−3. Discharge parameters: Vdis=44 V, Idis=0.4 A, p=5.9�10−5 mbar Ar, BaO cathode, 10 cm diam sphere.

012104-3 Transit time instabilities in an inverted fireball. I. Phys. Plasmas 18, 012104 �2011�

The frequency of the sheath-plasma instability is propor-tional to the electron plasma frequency while the fireball in-stability is nearly density independent. Since the sheath-plasma instability has been studied before, we now focus onthe transit-time instability of the inverted fireball.

The frequency is determined by the transit time througha resonant structure, i.e., it depends on electron velocity andsize of the fireball, f �vel /dgrid. The former can be varied bythe grid velocity. But since the grid current is comparable tothe discharge current, varying the grid voltage also affectsthe discharge properties. For example, increasing the gridvoltage raises the plasma potential not only inside the fireballbut also in the ambient discharge plasma such that vel doesnot simply scale as Vgrid

1/2 . Figure 5 displays the dependence ofinstability frequency and amplitude versus grid voltage. Thefrequency increases with Vgrid but the accelerating doublelayer potential rises less than Vgrid. The increase in the am-bient plasma potential also increases the discharge current ofthe space-charge-limited cathode, thereby raising the plasmadensity and grid current. While the density increase has littleinfluence on the frequency, it does affect the amplitude. As

the density increases, the Debye lengths decrease and thegaps between the grid wires become too wide to form thespherical equipotential surface. The fireball vanishes �seeFig. 2�c�� and so does the instability. The instability also hasa threshold �Vgrid,min�60 V� below which no oscillationsare excited. The threshold is above the potential drop of or-dinary fireballs and no transit-time oscillations have beenfound yet.

Two spheres with different diameters have been built.Figure 6 shows that the smaller sphere oscillates at a higherfrequency than the larger sphere. The frequency ratio isnot given by the size ratio because a higher voltage wasapplied to the 5 cm diam sphere. Using the scaling fromFig. 5, the small sphere would resonate for Vgrid=80 V atf �42 MHz or twice the frequency of the larger sphere, aspredicted by the scaling f �vel /dgrid.

C. Spatial and temporal dependencies

A movable rf probe has been used to measure the radialdependence of the rf oscillations inside the fireball. Figure 7shows that the potential oscillations peak in the center of thesphere. This is in contrast to the sheath-plasma oscillationswhich peak near the grid. Phase and amplitude show no in-dications for radial waves with short wavelengths, which arenot expected since the frequency is well below the electronplasma frequency. The only eigenmode could be the beammode with dispersion �=kvbeam, in which case the rf profilecould be interpreted as a fundamental radial standing wave inspherical geometry. In the companion paper,25 it will beshown that higher order modes are also excited, which ex-hibit a minimum at the center and maxima off center.

The inset in Fig. 7 shows the method to study the timeevolution of the instability. A fast-rise �trise�100 ns� voltagepulse is applied to the grid and the time dependence of theoscillations in the grid current and rf probe signal is re-corded. These reveal the changes in the plasma parametersdue to the fireball, the formation and decay of the potentialstructure, and of the transit-time instability. It will be shown

3 130

140

f

(MHz)

40

80

4 6 12(I ) ~ fdis1/2

pe10

Sheath-plasmainstability

0 187f (MHz)

Transit time

Sheath

Transit timeinstability

FIG. 4. �Color online� Frequency dependence of sheath-plasma oscillationsand transit-time oscillations on electron plasma frequency.

Grid voltage V (V)grid25 20050 225

f

(MHz)

0

20

40

60

Frequency f

I (A)grid

0.5

0.2

0.4

0

(arb. units)

Vrf

FIG. 5. �Color online� Dependence of instability on Vgrid applied in steady-state. The density increases with Vgrid as indicated by the growing gridcurrent. As the Debye length decreases, the double layer collapses and thefireball and instability disappear. Parameters: 5 cm diam sphere, BaO cath-ode, Vdis=30 V, Idis=100 mA, p=9�10−5 mbar Ar.

0 80Frequency f (MHz)

10 cm sphere 5 cm sphere

Vgrid = 140 V

f = 21.4 MHz

Vgrid = 80 V

f = 51.6 MHz

FIG. 6. �Color online� Frequency dependence of transit-time oscillationson size of the fireball. Discharge parameters: Vdis=15 V, Idis=0.12 A,p=9�10−5 mbar Ar.

012104-4 Stenzel et al. Phys. Plasmas 18, 012104 �2011�

that the instability may be present in the transient phase butnot in steady-state. Vice versa, the instability is absent justafter the turn-on of the grid voltage since the double layerforms on an ion time scale as explained below. Finally, itshould be pointed out that the ac grid current is not a con-duction current but primarily a displacement current. This isevident from the large current spike at the voltage turn-onand a similar one of opposite sign at the turn-off. These arecapacitive currents I=CdV /dt due to the sheath and straycapacitances. In the presence of the transit-time instability,the grid voltage oscillates rapidly which also produceslarge rf displacement current. Thus, the observed waveformfor Igrid does not imply a 100% modulation of the electroncurrent.

Figure 8�a� shows the turn-on phase on an expandedtime scale. Grid voltage and current have been smoothed tosuppress the rf oscillations which are displayed in the signalfrom the rf probe in the center of the sphere. Also shown isthe smoothed probe current to show the gradual change fromelectron to ion collection. The probe is terminated into 50 without applying a bias to ground.

The instability onset is delayed for several microsecondswith respect to the start of grid voltage and current. Further-more, the plasma potential inside the sphere rises slowlycompared to the voltage rise time. These slow changes occuron the time scale of ion motions over few millimeters, whichis the dimension of grid spacing. In order to form anelectron-rich sheath or double layer, the ions have to be dis-placed by several Debye lengths. This is the time scale toform a spherical equipotential surface, which allows theplasma potential to rise and the transit-time instability todevelop.

The rf waveform has been sampled such that all frequen-cies are below the Nyquist limit �fNyq=1 / �2tsample��. For suc-cessive 1 s intervals, a fast Fourier transform �FFT� hasbeen performed. The inset of Fig. 8�b� displays the temporalchange of the instability frequency. The gradual frequency

rise shows that the electron velocity, hence double layer po-tential, increases gradually. In steady-state the spectrumshows a single line and a small second harmonic line.

The explanation that the formation of the double layerrequires ion expulsion from the grid spacing is supported bya double pulse experiment. As shown in Figs. 9�a�–9�c�, twoidentical grid voltage pulses are applied successively withdifferent delay times. For the first pulse, the instability delaytime remains unchanged, ton,1�2.1 s �Fig. 9�d��. But forthe second pulse, the instability onset depends on thepulse delay. After the end of the first pulse, ions begin toreturn into the space between the grid wires. The time scalefor this process is �ion��r /cs�2 s, where cs�5 mm /3�105 cm /s is the sound speed and �r / �6 mm is the wirespacing. The ion dynamics may also change the size of thesecond fireball since the spectrum of the second instabilitydiffers from that of the first as discussed below.

Useful information about the decay of the fireball can beobtained from the instability properties at the switch-off ofthe grid voltage. Figure 10�a� shows the waveforms of thegrid voltage and the oscillation at the rf probe. The gridvoltage decays on a time scale shorter than the ion transit

Radius r (cm)

Vrf

(arbunits)

0

1

0 2 4

Vrf

Vgrid

Igrid

t (μs)0 320

FIG. 7. �Color online� Radial profile of the rf amplitude in aninverted fireball. The inset shows steady-state oscillations on grid andprobe after applying a positive pulse to the 10 cm diam grid. Parameters:Vgrid=79 V, Igrid=270 mA, BaO cathode, Vdis=38 V, Idis=100 mA,p=7.4�10−5 mbar Ar.

0

Time t (μs)0 18

200

Vgrid (V)

5 15

140

Frequency f (MHz)0 12510050

Vrf probe (ω)

(a)

(b)

Igrid

Vrf, probe

0.4A

V

Iprobe

mV

μA50

12345678910111213141516171819202122 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11

t (μs)7 18

f

MHz

20

40

f = 33.7 MHzinst

FIG. 8. �Color online� Onset of transit-time oscillations for a pulsed gridvoltage. �a� Waveforms of grid voltage, current and rf probe voltage, whichshows a delayed instability onset due to the gradual rise of the fireballpotential as indicated by the dc probe current. �b� Spectrum of the probevoltage showing in an inset the rise in frequency due to the growingpotential across the double layer. Parameters: 10 cm sphere, Vdis=30 V,Idis=80 mA, p=7.5�10−5 mbar Ar.

012104-5 Transit time instabilities in an inverted fireball. I. Phys. Plasmas 18, 012104 �2011�

time through the double layer. The instability ceases whenthe grid voltage falls below the threshold voltage Vgrid

�60 V. But in spite of the large drop in grid voltage, theoscillation frequency remains constant. Here the frequency isdetermined from the slope of the phase versus time �Fig.10�b�� since the time span is too short for a time-resolvedFFT. The constant frequency implies a constant electron tran-sit time, i.e., no change in the electron velocity and potentialstructure. The double layer potential structure is maintaineduntil ions flow back into the gap between the grid wires.Prior to that, the fireball potential remains positive since anoutflow of energetic electrons in excess of the ion outflowwould charge the fireball positively.

Figure 11 shows a different behavior when the grid volt-age is switched off slowly compared to the potential relax-

ation time. Grid current and voltage are nearly proportional,the probe “dc” current changes from ion to electron collec-tion as the plasma potential decays in the fireball. The rfamplitude has a maximum at an intermediate double layerpotential, as also noticed in Figs. 5 and 8. The instabilityfrequency decreases with Vgrid and for low voltages scales asf �Vgrid

1/2 , which is expected if the double layer potential fol-lows Vgrid.

D. Beat modes

The inverted fireball has also been observed to oscillatesimultaneously at two different frequencies. For example, inthe double pulse experiments the instability of the secondpulse can exhibit a beat phenomenon which is not seen forthe first pulse �see Fig. 9�c��. It suggests that the first pulsemodifies the density profile such that the second fireball isnot spherically symmetric, which can also be seen visually�Fig. 2�b��. If the transit time differs along two orthogonalaxes, the fireball will oscillate at two different frequencies,which produces a beat phenomenon in the time domain �Fig.12�a�� or two spectral lines in the frequency domain �Fig.2�b��. The upper frequency coincides with that produced inthe first pulse. The lower frequency is due to a perturbation,for example, ion expulsion, which temporarily expands thesphere and increases the transit time. The beat frequency��f �7 MHz� is well above the ion plasma frequency, hencenot due to ion oscillations.

Beat amplitude and frequency depend on pulse separa-tion, as shown in the inset. The beat phenomenon arises andchanges on an ionic time scale �t 2� /�pi�1 s�. Al-

Time t (μs)0 164 12

Vgrid, rf

Vgrid, dc(a)

(arb units)

200V

on,1t

tdelay

on,2t

(b)

(c)

(d)

Pulse delay time t (μs)0 100

t

(μs)on

2

1

First pulse

Second pulse

2 8

FIG. 9. �Color online� ��a�–�c�� Double grid pulse experiment showing thatthe instability onset for a second pulse varies with pulse spacing. �d�Delay of instability onset for the second grid pulse vs pulse spacing.Parameters: 5 cm diam sphere, W filaments, Vgrid=212 V, Vdis=60 V,Idis=142 mA, 8�10−5 mbar Ar.

V

V

Time t (μs)

grid, rf

grid

0 1.8

φ/2π

10

0

30

0.4 1.61.2

0

100

200

(V)

(arb units)

(a)

(b)

FIG. 10. �Color online� Fast switch-off of the grid voltage. �a� Waveform ofdc and rf grid voltage. �b� Phase of the rf oscillation, which shows nofrequency shift since the double layer potential changes slower thanVgrid. Parameters: 5 cm diam sphere, W filaments, Vgrid=212 V,Vdis=60 V, Idis=120 mA, p=8�10−5 mbar Ar.

012104-6 Stenzel et al. Phys. Plasmas 18, 012104 �2011�

though the electron transit time does not directly depend ondensity, the latter may affect the fireball geometry andthereby create the frequency shifts.

E. Frequency locking and dc-rf conversion efficiency

Although the spherical electrode is not an electromag-netic cavity, it forms a resonant structure for the only eigen-mode of the system, the longitudinal space charge wave onan electron beam. The beam is created at the double layer bya radial electric field. Outside the fireball, there is no beam,no eigenmode, and the field is evanescent. When the gridvoltage oscillates, it modulates the electron velocity,which creates beam bunching and a radial longitudinalelectric field. When the electron transit time through thesphere corresponds to a rf period, the electron space chargefield enhances the grid potential which creates an absoluteinstability.

The rf grid voltage plays a crucial role as the boundarycondition of the instability. Since rf voltage and current notonly depend on the plasma parameters but also on the im-pedance between electrode and ground, the instability can beinfluenced by the external circuit. It has been observed that a

parallel resonant circuit between the grid and ground �withsuitable dc bypass� can be used to determine frequency andboundary conditions.

Figure 13�a� shows schematically the experimental ar-rangement and the resultant instability spectrum. The insta-bility occurs at the resonance frequency of the external cir-cuit, which may differ from the natural transit-timeinstability. The spectrum exhibits a �-function line. Thus, theinstability has been locked and can be tuned over a limitedrange away from the natural instability frequency.

Tuning the resonant circuit affects the rf voltage andcurrent of the grid, as shown in Fig. 13�b�. When the circuitresonance is tuned to the natural transit-time frequency��61 MHz�, the grid voltage assumes a maximum andthe current has a minimum, hence the impedanceZ=Vgrid,rf /Vgrid,rf peaks. Note that this is the source imped-ance, not the resonant circuit impedance, which alwayspeaks at the frequency it is tuned to. The grid current exhibitsa maximum at a lower frequency ��55 MHz� where the rfvoltage is small.

At resonance, the rf grid voltage has a peak-to-peak volt-age comparable to the dc voltage. The net rf current is smallcompared to the dc current. The displacement current is neg-ligible when the plasma potential and grid potential are es-sentially identical. As the circuit frequency is lowered, thesource characteristics change from a voltage source to a cur-rent source, i.e., the rf grid voltage vanishes while the gridcurrent peaks. The peak-to-peak current exceeds the dc cur-rent, implying that the current temporarily reverses sign. Ion

5.5-65-5.54.5-54-4.53.5-43-3.52.5-32-2.51.5-21-1.50.5-10-0.5

Vgrid( )1/23 157 11

f

(MHz)

f

(MHz)

0

V( )1/216

32

24

Time t (μs)

0 80

5 6520 60

40

15

VgridIgrid

Iprobe

Vrf

(a)

(b)

(c)

0

200V0.4 A

20

FIG. 11. �Color online� Slow switch-off of the grid voltage �a� allows thedouble layer to change adiabatically, which decreases the frequency �b� andscales as Vgrid

1/2 at low Vgrid �c�. Parameters: 10 cm diam sphere, BaO cathode,Vdis=30 V, Idis=60 mA, p=7.5�10−5 mbar Ar.

Time t (μs)

Frequency f (MHz)0 125

0 2.6

(a)

(b)

Vgrid (f )

200 V

0

Vgrid, dc (t)

Vgrid, rf (t)

60 V

10025

1 2

1

(c)

f (MHz)60 80

13μs

Δt=6μs

9

FIG. 12. �Color online� Beat phenomenon in transit-time oscillations. �a� dcand rf grid voltages vs time. Beat remains unchanged as voltage is switchedoff rapidly. �b� Spectrum of Vgrid,rf showing two lines separated by the beatfrequency ��f �7 MHz�. �c� Dependence of beat spectrum on double pulseseparation. Parameters: 5 cm diam sphere, W filaments, Vgrid,dc=200 V,Vdis=60 V, Idis=140 mA, p=8�10−5 mbar Ar.

012104-7 Transit time instabilities in an inverted fireball. I. Phys. Plasmas 18, 012104 �2011�

collection cannot account for the current, hence it is a dis-placement current created by an oscillating plasma potentialrelative to the grounded grid. The rf current between thesphere and ground must close outside the sphere through theambient plasma. Local rf magnetic fields must be producedbut do not excite electromagnetic waves since they areevanescent.

A frequent question in oscillator applications is the effi-ciency of converting dc power to rf power. For this purpose,a resistor has been placed in parallel with the resonant cir-cuit. For matching purpose, its value is chosen to be the ratioof open loop voltage to short circuit current �50 V /40 mA�1.25 k�. The measured rf power was found to be�Prf�=Vrf,rms

2 /R=282 /1250=0.63 W for a dc powerPdc=Vgrid,dcIgrid,dc=140 V�86 mA=12 W, which yields aconversion efficiency �= Prf / Pdc�5.3%.

IV. SUMMARY AND CONCLUSIONS

A new fireball configuration has been developed. It isformed by a highly transparent spherical anode, immersed in

a dc discharge plasma, and biased positively. Electrons areinjected into the sphere and ionize the gas which compen-sates for the loss of ions ejected from the spherical plasma.At low pressures, the potential drop between the sphericalplasma and the ambient plasma is well above the ionizationpotential which differs from previous fireballs.

The injection of counterstreaming electrons gives rise toa high-frequency instability with a similar mechanism as inreflex klystrons or vircators. The instability has a periodgiven by the electron transit time through the sphere. Theelectrons bunch due to the modulation by a rf electric field inthe double layer at the fireball boundary. The electronbunches reinforce the electric field and an absolute instabilitydevelops. The frequency of the instability can be stabilizedand tuned by inserting a resonant circuit between the gridand ground. Growth and decay of the fireball potentials havebeen inferred from the properties of the transit-time instabil-ity. Further properties such as frequency jumps, multipleeigenmodes, and nonlinear harmonic generation will be de-scribed in a companion paper.

ACKNOWLEDGMENTS

One of the authors �R.L.S.� gratefully acknowledges thesupport and hospitality while staying as guest professor atthe University of Innsbruck in June/July 2010. This workwas supported in part by the Austrian Science Funds underGrant No. P19901 and in part by NSF/DOE grant �ContractNo. 20101721�.

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Frequency f (MHz)0 10020 80

Vrf (ω)

(a)

Frequency f (MHz)

(kΩ)

0

10

15

5

45 50 65 70

Vgrid, rms (V)

0

Z

Igrid, rms80

40

(mA)

50

25

(b)

Irf

dcV

Vrf

FireballGriddedHollowAnode

FIG. 13. �Color online� Frequency control of the transit-time instability withan external L-C resonance circuit. �a� Schematic diagram of the experimen-tal setup and a characteristic spectral line of the oscillation which is lockedto the constant external resonator. �b� Grid voltage, current and impedanceZ=Vgrid,rf / Igrid,rf vs frequency. The resonance occurs at the natural instabilitywithout external circuit. Parameters: 5 cm diam sphere, Vgrid,dc=140 V,Igrid,dc=86 mA, Vdis=30 V, Idis=90 mA, p=6�10−5 mbar Ar.

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012104-9 Transit time instabilities in an inverted fireball. I. Phys. Plasmas 18, 012104 �2011�