Necessary conditions for the homogeneous formation of pulsed avalanche discharges at high gas pressures

Necessary conditions for the homogeneous formation of pulsed avalanche dischargesat high gas pressuresJeffrey I. Levatter and ShaoChi Lin

Citation: Journal of Applied Physics 51, 210 (1980); doi: 10.1063/1.327412 View online: http://dx.doi.org/10.1063/1.327412 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/51/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Positive streamer formation in cathode region of pulsed high-pressure discharges for transversely excitedatmosphere laser applications J. Appl. Phys. 110, 053303 (2011); 10.1063/1.3630015 Growth of arc in high-pressure, pulsed glow discharge by gas density depletion J. Appl. Phys. 88, 4531 (2000); 10.1063/1.1314328 Study of xray preionized avalanche discharge XeCl laser at high gas pressures Appl. Phys. Lett. 38, 328 (1981); 10.1063/1.92358 The glowtoarc transition in a pulsed highpressure gas discharge J. Appl. Phys. 52, 681 (1981); 10.1063/1.328747 Effect of electrode surface conditions on the selfbreakdown strength and jitter of a highpressure pulsed gasswitch J. Appl. Phys. 47, 1925 (1976); 10.1063/1.322914

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N.ecessary con~itions for the homogeneous formation of pulsed avalanche discharges at high gas pressuresa)

Jeffrey I. Levatter and Shao-Chi Lin Institutefor Pure and Applied Physical Sciences, University of California, San Diego, La Jolla, California 92093

(Received 6 August 1979; accepted for publication 12 September 1979)

The preionization level and other initial conditions necessary for the formation of spatially homogeneous pulsed avalan~he discharges at high gas pressures are examined. Assuming properly shaped ele~trodes w~t~ ~o ~tro?g edge effects, the minimum preionization level required for homogeneous dIscharge 111ltiatlOn IS found to depend on the voltage rise time across the el~ctrodes ~s well as on the total pressure and various electrochemical properties of the gas mIxture w~lch govern the net rate of change of the first Townsend coefficient with respect to the local electnc field strength. Our predictive results are found to be consistent with experimental observations.

PACS numbers: 52.80. - s, 84.70. + p, 42.55.Hq, 51.50. + v

I. INTRODUCTION

A self-sustained volume discharge initiated by a homogeneous electron avalanche is a convenient method of gas laser excitation. The method involves the sudden application of a strong electric field considerably in excess of the breakdown threshold across the laser gas. This causes the initially low level of ionization within the discharge gap to grow exponentially via the electron avalanche process until the plasma impedance becomes limited by the output impedance of the driving electrical circuit. The driving circuit impedance, in turn, must be so chosen that electrical excitation of the laser gas mixture can progress at the desired rate during the main part of the current pulse. This method of excitation has been successfully employed in many transversely excited atmospheric-pressure (TEA) lasers such as CO2 , N 2' and HF,t-J and more recently, in the rare-gas halide and metal halide excimers such as KrF, XeF, XeCI, ArF, HgCI, HgBr, HgI, etc. 4-10 These lasers have proven to be a powerful source of infrared, visible, and ultraviolet coherent radiation due to the high-density, and hence high-energy handling capacity of the high-pressure gas.

The high-pressure gas lasers based on the pulsed discharge method of excitation just described are usually referred to as fast discharge lasers. This nomenclature is used since it has been found from experience that the high-voltage electrical pulse creating the electron multiplication in the laser gas must have a very fast rise time and also a relatively short duration (of the order of 10 --8 sec) in order to avoid the formation of filamentary arcs within the discharge volume. Efficient laser pumping in the presence of severe arcing is, of course, impossible due to the inadequate excitation rates in regions of low current density and the too rapid thermal equilibration rate (which tends to destroy population inversion) in regions of very high current density. In addition, the resultant nonuniform distributions of refractive index within the inhomogeneously excited medium make formation of

'''Work supported by the Defense Advanced Research Projects Agency under Contract NOOOI4-76-C-0116 monitored by the Office of Naval Research.

high optical quality laser beams difficult. Another problem associated with the onset of severe arcing in discharge pumped lasers is the very low plasma resistivity within the concentrated arcs. The sudden decrease in plasma resistivity with rapidly increasing local current densities may cause the total load resistance across the discharge gap to collapse momentarily to values far below that of the driving circuit output impedance. When this occurs the discharge will either self terminate or go into an unstable oscillatory arc discharge mode, depending on the characteristics of the driving circuit and on the extent of the impedance mismatch. Such premature termination or interruption of the discharge, together with ineffective excitation due to nonuniform current distribution in the presence of severe arcing, tends to limit the total energy output of the laser. This arcing problem has been found to be especially troublesome in the generation of rare-gas halide and metal halide lasers,4-1O where efficient formation of the excimers generally requires a high uniform rate of electron impact excitation. The very presence of the electrophilic halogen molecules and excited metastable raregas atoms tends to make the electron multiplication and disappearance rates very sensitive functions of the local electric field strength and current density. 1 1-13 Accordingly, with the exception of some very recent work performed at our laboratory,9 the effective discharge volume, pulse duration, and maximum laser energy output have all been very low (typically of the order of a.lHter, 10 nsec, and 0.1 J/pulse, respectively).4-8

The above cited difficulties of high-pressure discharges undoubtedly are related to the absence of strong electronic and ionic diffusion effects responsible for homogenization and stabilization oflow-pressure glow discharges. To bypass these difficulties, high-energy electron beams have been successfuly utilized both as a direct pumping source l4

-17 and as

a stabilizing ionization source l8-

21 for homogeneous excitation of high-pressure gas lasers over extended volumes. In many practical applications (for example, applications requiring high-pulse repetition rates), however, the avalanche/self-sustained discharge method has the advantage of relative simplicity when compared with either the elec-

210 J. Appl. Phys. 51(1), January 1980 0021-8979/80/010210-13$01.10 © 1980 American Institute of Physics 210


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tron-beam-pumped or the electron-beam-sustained discharge method. The simplicity of the avalanche/self-sustained discharge method is mainly a result of the lower voltage requirement and the absence of fragile foil windows needed for transmission of the high -energy electron beams at high current densities. The advantages of this method become especially significant if it can be extended to large excitation volumes and long excitation times.

Extension of homogeneous avalanche/self-sustained discharges to large volumes and long times clearly involves two consecutive steps. The first and more crucial step is initiating and sustaining the electron avalanche process until the desired transient state of the plasma is uniformly reached throughout the discharge volume. The second step is then simply a matter of maintaining the plasma spatial homogeneity and temporal stability over the desired length of time. Due to the nonsteady nature of the pulsed discharge, the desired transient state of the plasma is generally not in thermodynamic equilibrium, but can be characterized by a certain range of electron mole fraction or number density, mean energy and velocity distribution, excited species concentrations, excitation rates, etc. Once spatial homogeneity is achieved, maintenance of temporal stability during the postavalanche period is required only in the sense that the discharge must not be allowed to terminate prematurely due to impedance mismatch or any other extraneous factors, and the already homogeneous plasma must not be allowed to develop any spatial instability mode which could break up or collapse the homogeneous current distribution into concentrated arc channels within the time scale of interest. The latter phenomenon, often called the glow-to-arc transition,22.23 is generally associated with some relatively slow processes such as growth of thermal instability modes within the plasma,24 development of hot spots on the electrode surfaces,25 etc., and is therefore likely to be important only when the pulse duration is exceedingly long. Filamentary or arclike discharges, of course, may also result directly from the avalanche initiation process if the voltage rise time is too slow in relation to the arc channel formation time, or if insufficient care has been taken to insure homogeneous avalanche formation. Obviously, such inhomogeneities in the discharge should be considered as being caused by an improper formation process and should not be confused with the glowto-arc transition phenomenon.

In view of the very limited diffusive spreading ability of the secondary electrons and ions associated with each primary electron avalanche during on over-voltaged breakdown at high gas pressures, homogeneous discharge formation would require an initial distribution of free electrons within the discharge volume at the onset of the breakdown process. Such an initial distribution of free electrons, or preionization, can be provided by any suitable volume ionization source such as ultraviolet light, x rays, r rays, a particles, electron beams, corona discharge, etc. However, due to the limited mass-penetration power, intensity, or generation efficiency for some of these sources, it is generally desirable to keep the level of preionization required for homogeneous discharge formation as low as possible. In a short article published in 1974, A.I. Palmer26 proposed a simple physical

211 J. Appl. Phys., Vol. 51, No.1, January 1980

model for estimating the preionization requiremetns for initiating a volume-stabilized glow discharge. The basic requirement of the model was that the preionization density be large enough to cause appreciable spatial overlap of the primary electron avalanches and consequent smoothing of space-charge field gradients at the stage when streamer formation would otherwise occur. Using such a model, a minimum required preionization electron density of about 104

cm - 3 was predicted for a typical CO2 TEA laser discharge, and was found to be generally consistent with experimental observations.

While the physical model as proposed by Palmer26 was basically sound, there appeared to be two serious deficiencies in his subsequent formulation. The first was the neglect of the important effects of a finite voltage rise time on the depletion offree electrons (preionization electrons) near the cathode during the preavalanche period, and the neglect of those effects on the subsequent development of the electron avalanche process. The second was the absence of any explicit functional relationship between such important physical properties of the gas mixture as the first Townsend coefficient a and the electron mobility fl, and the applied electric field strength to gas density ratio E /n throughout the formulation. In addition, a sizable error was made in relating the electron diffusion radius to the electron mean-free path and the drift distance. These deficiencies not only raised serious questions about the accuracy and range of applicability of Palmer's predictive results, but they also obscured the true potential of his proposed physical model in defining the minimum preionization density required for homogeneous avalanche formation under a more general set of pulsed discharge conditions.

In the present paper we examine the problem of homogeneous volume avalanche formation at high gas pressures between two parallel and properly shaped electrodes27 driven by an electrical circuit of finite voltage rise time. Appropriate nondimensional equations are derived for determining the critical track length and head radius for the individual primary avalanches as statistically averaged functions of the voltage rise time and of various electrochemical properites of the gas mixture. Numerical examples are given for an electrophilic gas mixture used in typical rare-gas halide excimer laser generation. The minimum preionization density and voltage rise time required for homogeneous avalanche formation in such a gas mixture have also been determined using a modified version of the Palmer model,26 and they are compared with some recent experiments.

II. HIGH-PRESSURE BREAKDOWN MECHANISMS

As is well known in the gas discharge literature, there are two types of electrical breakdown which can convert an initially nonconducting high-pressure gas between two parallel electrodes into a highly conducting plasma upon the application of a high-voltage pulse. One is the classical Townsend breakdown28

•29 and the other is the plasma

streamer breakdown. 3o•31 Even though the basic process for

electron multiplication is due to electron avalanche in both types of breakdown, the conditions for occurence and appli-

J.1. Levatter and S.- C. Lin 211


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+ IIIIIIIIIIIIL

7///1///1//7

(0)

+ +

1//77/77/777 !

(b) (c)

FIG. I. Schematic diagrams showing (a) streamer development around a single primary electron avalanche after its space-charge field has grown beyond a certain critical value, (b) continuous backward propagation of the cathode-directed plasma streamer after the arrival of the primary avalanche head at the anode, and (c) complete bridging of the electrode gap by the plasma streamer.

cable ranges offield strength to gas density ratio E /n are quite different, so that it is important to make a distinction between the two.

The well-known Townsend or Paschen breakdown mechanism is characterized by a large number of successive electron avalanches that originate from secondary electron generation. The space-charge field caused by differential motions between the electrons and the positive ions is assumed to be so weak as to be completely negligible. Continuous exponentiation of the electron current within the discharge gap is assumed to be maintained by a positive feedback of the Townsend avalanche process through secondary electron emission at the cathode surface. A self-sustained discharge condition, corresponding to the onset of such positive feedback, is accordingly given by an equation of the type

(a/p)pd = 10g[(1 + y)jy] , (1)

where p is the gas pressure, d is the electrode gap, a is the first Townsend coefficient, which measures the exponentiation rate offree electrons per unit mean drift distance of the electrons under the influence of the constant applied electric field strength E under consideration, and y is the second Townsend coefficient, which measures the total probability of secondary electron emission from all sources associated with a single primary electron emission. If positive ion bombardment at the cathode surface were the main source of secondary emission, the minimum time required for the positive feedback mechanism to become effective after turning on the applied E field at t = 0 would then be some fraction of the ion transit time from anode to cathode,

T,=d/u i , (2)

where U i denotes the mean drift velocity of the positive ions. For He + ions in He at 1 atm pressure, Ui is 5 X 102 m/sec at a typical breakdown field strength of 4 X 105 V / cm. 32 Thus, for a single transit across a 4-cm electrode gap, 7'i - 10-4

sec. For heavier ions across larger gaps, the transit time would be correspondigly longer. On the other hand, if photoelectric effects at the cathode were an important source of secondary emission, the minimum time for positive feedback would then be governed either by the characteristic time for generation of the appropriate excited molecular states dur-


ing the avalanche process or by the radiative lifetime of the excited molecule, whichever is longer. In any case, a relatively long formative time delay (- 10 -6 sec) is generally observed in a Townsend-type breakdown which leads to sparking across the gap.

In contrast to the Townsend breakdown model, "Kanal" or streamer breakdown occurs as the result of a large space-charge field that develops from a single electron avalanche that quickly transforms an otherwise orderly avalanche process into a rapidly propagating plasma streamer. This form of breakdown will therefore allow the sparking phenomenon to begin anywhere inside the discharge gap without relying on the secondary electron generation processes at the cathode surface. The large space-charge fields that develop are due to the relatively low mobility of the positive ions as compared to that of the electrons. On the time scale of interest in a typical short duration pulsed discharge, the electrons are free to move toward the anode while the ions are essentially frozen in space. For simplicity, the propagating avalanche head filled mostly with free electrons can be idealized as a negatively charged sphere, behind which is the positive space-charge [see Fig. lea)]. The shape of the avalanche cone is determined primarily by electron diffusion. 33 At some critical point where the space-charge field of the avalanche head becomes comparable in magnitude to that of the applied electric field E, streamer development begins. At this point, secondary avalanches are initiated by photoionization in front of and behind the head of the primary avalanche. Both an anode- and a cathode-directed streamer develop and move at a much greater velocity than the velocity of the primary avalanche head. The increased velocity of the anode-directed streamer is due to the spacecharge enhanced electric field on the anode side of the avalanche head. The cathode-directed streamer is primarily the result of the positive space charge left behind the avalanche head. In the surrounding gas, photoelectrons are produced which initiate secondary avalanches directed along strong field lines toward the stem of the primary avalanche. The greatest multiplication of these secondary avalanches occurs along the axis of the primary avalanche where the spacecharge field supplements the applied field. As the negatively charged avalanche head propagates toward the anode and exponentiates, it also leaves behind a positively- charged tail which continues to lengthen and intensify at an acceleratin1} pace until the anode and cathode are eventually connected by the self-propagating plasma streamer. In Fig. I, three successive stages of such streamer development are schematically illustrated. Thus, according to this model, breakdown will occur whenever a single primary electron avalanche is allowed to develop to the critical point of streamer initiation anywhere within the electrode gap. A breakdown criterion, attributed to Raether,34 which corresponds to the condition that the critical track length Sc for a primary electron avalanche developed under the influence of a suddenly applied constant electric field is equal to the electrode gap d can accordingly be derived, such that for air, in mks units,

(a/p)pd = 20 + log(d). (3)

According to the early experiments of Townsend,28 the sec-



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ond ionization coefficient r is typically of the order of 0.1, so that the total avalanche gain ad corresponding to the breakdown condition (1) is about 2.4, which is more than a factor of6 smaller than the value of ad corresponding to the breakdown condition (3) for all electrode gaps of the order of a few centimeters or greater. Thus the applied field strength required for observation of a Townsend-type breakdown is generally much weaker than that required for observation of a streamer breakdown. On the other hand, for large-volume high-pressure discharges at high values of E In corresponding to those of general interest in high-power excimer laser excitation as mentioned earlier, the total avalanche gain across the discharge gap often exceeds a numerical value of 20 so that a streamer-type breakdown can be initiated from points far away from the cathode due to the relative shortness of the critical avalanche track length Sc in comparison with the electrode gap d. Furthermore, due to the nonlinear buildup of the space-charge field after the primary avalanche track length has grown beyond Sc within the electrode gap, a streamer-type breakdown can take place in a time scale much shorter than the characteristic time

(4)

for the single transit of a primary electron from cathode to anode at a constant drift velocity Ue • For a free electron in pure He or in a predominantly He gas mixture at 1 atm pressure under the influence of a typical breakdown field strength E = 4 X 105 V 1m (Fig. 4), Ue is about 2 X 104 mlsec, so that 7e - 2 X 10 -6 sec for a 4-cm electrode gap. This explains why streamer breakdown has been observed with formative delay times as short as 10 -9 sec in some fast pulse discharges at high values of E In.

III. INHIBITION OF STREAMER FORMATION BY PREIONIZATION

As one may conclude from the foregoing discussions, breakdown under high E In conditions is essentially caused by the nonlinear deVelopment of the space-charge fields as-

ANODE

~IIIII!IIIIIIII//!J)Y

W-V- W' W-

-- -- -- --- - - -+ + + + + + -+ + + t- + + + ... -+ +

-.1.

r !cri I

1 SN 3 - neo I~ ..... I

/f7III/////7///777//7/I/~

CATHODE FIG. 2. Schematic diagram showing one row of simultaneously formed primary avalanches propagating uniformly toward the anode before their respective space-charge fields reach the streamer development stage. The adjacent avalanche heads will begin to overlap each other when the mean diameter of the avalanche heads becomes equal to the mean distance (Iln.,,)'/.' between the preionization electrons.


sociated with the individual primary electron avalanches randomly initiated within the discharge gap. This is quite unlike the classical Townsend discharge which requires successive generation of secondary electrons at the cathode. For high electron mUltiplication gains (ad;::: 20), breakdown is governed by the streamer mechanism, and the discharge is filamentary from the onset. This is the process by which most overvolted spark gap switches operate. The same high E In, high gain conditions are also typical in high-pressure self-sustained laser discharges. This type of discharge, if allowed to progress naturally in the manner described in Sec. II, will therefore favor the arc mode rather than the desired spatially homogeneous glow mode of operation. The appropriate question to ask then is not why do avalanche-selfsustained discharge lasers arc, but why under certain conditions do they not arc? The answer to this question lies in part in the presence of preionization in the discharges in which a suitable number density of primary electrons is provided uniformly over the whole volume prior to the initiation of the main discharge.

In the streamer breakdown theory, it is assumed that the time development of a single primary electron avalanche does not depend on the simultaneous presence or absence of any other primary electron avalanches within the same discharge gap. Accordingly, there is no explicit dependence of the breakdown criterion (3) on the initial number density of primary electrons. Implicit in the formulation of the streamer breakdown theory, however, is that the space-charge can develop strong local gradients since it is this feature which causes the discharge current to become filamentary. In the case where the space-charge field is due to a single or just a few primary electron avalanches, strong field gradients clearly exist due to the limited diffusive spreading of the individual avalanche patches. As the number of simultaneous primary avalanches is increased beyond a certain level, one would expect the adjacent avalanche patches to overlap. Strong overlapping of the secondary ionization generated by the individual primary avalanches not only will homogenize the resultant plasma density in the discharge, but also may smooth out the local gradients of the spacecharge field to such an extent that streamer formation is completely inhibited. (See Fig. 2.)

During the early phase of avalanche formation, that is, before the development of the individual primary avalanche heads has reached the critical stage described earlier, the space-charge fields associated with each primary avalanche will remain weak in comparison with the applied field. If the collective effects of the space-charge fields from all the simultaneously developing primary avalanches are also negligible, one may then assume that the probability of avalanche initiation is the same for all the primary electrons. Thus the number density of primary avalanches at any given instant can be considered to be identical to the preionization density neO at the beginning of the breakdown process. With identical mean drift velocity Ue for all the electrons, the mean separation distance between centers of adjacent primary avalanche heads will also remain the same as the initial mean separation distance between the primary electrons generated by the preionization process, (l/neO)1/3. By equating this

J.I. Levatter and S.- C. Lin 213


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mean separation distance to the critical radius of the primary I 0 .-------.----~--~--~ avalanche head where the space-charge field becomes com-parable in magnitude to that of the applied electric field E, one can estiamte the minimum preionization density, (nd;)min' required for sufficient smoothing of the local spacecharge field gradients to inhibit streamer formation as Palmer proposed.26

IV. FINITE VOLTAGE RISE TIME EFFECTS

There are numerious writings describing the process of single streamer development,35-37 and a few papers that attempt to extend these ideas to explain the growth of a homogeneous volume avalanche discharge.38

,39 However, almost all of the previous works (including Palmer's26) assumed that the applied electric field is switched on instantly and kept constant at all time, or at least up to the transition of the avalanche process into a spark discharge. This assumption overlooks the effects of the time varying electric field due to the finite voltage rise time across the electrode gap driven by any real high-voltage pulse forming network. Since the electron avalanche rate, electron drift velocity, and other discharge parameters of importance are all sensitive functions of the local E In, an accurate account of avalanche development must include the effects of the time varying field. This is especially true for all high-voltage pulsed discharges where the characteristic time for critical avalanche head formation, or the formative delay time for observation of streamer breakdown in the absence of adequate preionization, becomes comparable to the voltage rise time.

In Figs. 3 and 4, we show some typical examples of the instantaneous electron exponentiation rate, d (logne )/dt,

<:>

\.0 1 _

"'1-~ -0

10 - I ~

106 "--_....L._---1 __ .....L..-_--=:l 103

o 1 2. '3 4 E / n t!0-20V-m2)

.... e

C> -1

FIG. 3. Calculated electron exponentiation (right-hand branch) and electron decay (left-hand branch) rates as functions of the electric field strength to molecular number density E In and of the electron mole fraction [e] in a He : Xe : F, = 200 : 8 : I mixture. For simplicity of presentation, the ionic and metastable atom mole fractions are assumed to be directly proportional to the electron mole fraction in the following ratios: [Xe 7 I = [Xe-] = 2[e]. [F ] = [Xe2 ' ) = 0.25[e]. [He-] = 2[He +] = 0.4[e).

(From Ref. 40.)


8 /

/

IO~ .. ·-;,/ 6 .' v

c...:> • v

Q.)

~ E

10-~ .. -> v 0

4 ./

co.> .;.-:::> .';'

,.;r ,,7

.v

2 ?

"'-- [e) ~ 10-5 J' .Y

.I .'>

0 0 2 3 4

EI n ( 10-2OV-m2 )

FIG. 4. Calculated electron drift velocity u,. as functions of the electric field strength to molecular number density ratio E In and of the electron mole fraction leI, in a He: Xe : F, = 200: 8 : I mixture. Other assumptions are the same as those indicated for Fig. 3. (From Ref. 40.)

and of the electron drift velocity Ue as functions of E /n at various fixed values of the electron mole fraction [e]aone/n for spatially homogeneous discharges in a rare-gaslhalogen-gas mixture. These examples, taken from the recent work ofH,H, LUO,40 are based on non-Maxwellian electron velocity distribution functions obtained from numerical solution of the Boltzmann equation in which the collisional term has taken into account all relevant elastic and inelastic collisions. The inelastic collisions have included, for example, direct electron impact ionization and excitation of the atoms and molecules, two-step ionization via the metastable excited states,13.41 electron attachment, and electron-ion recombination, The particular rare-gas halide mixture under consideration is a helium/xenon/fluorine mixture in mole ratio He : Xe : F2 = 200 : 8 : 1 at a total pressure of 1 atm, For simplicity of presentation, the mole fractions of various ions and metastable atoms have been assumed to be directly proportional to the instantaneous electron mole fraction [e] at the indicated fixed ratios in the figure captions. In any real discharge, the instantaneous mole fractions of the various ionic and excited species would, of course, depend on the detailed history and the discharge, but the indicated mole fraction ratios are quite typical. As can be seen from the semilog plot in Fig. 3, the absolute value of d (logne)/ dt can vary by several orders of magnitUde over the typical range of E /n encountered in an overvolted pulsed discharge. It may also be noted that for each fixed value of [e], there is a corresponding minimum breakdown field strength (E /n)o at which the absolute value of d (logne)ldt collapses to zero. We shall call this value of(E /n)o for small values of[e] (i.e., [e) < 10 -7 as illustrated in Fig. 3) the breakdown threshold, and for larger values of [e), the field strength for maintaining a steady-state discharge at the particular fixed value of [e). For E /n < (E /n)o' the actual rateofchangeofne is negative



148.251.80.252 On: Thu, 03 Apr 2014 15:23:33

so that the plotted absolute value of d (logne)1 dt represents the exponential decay rate of the free electron density in a sub-breakdown plasma. For E In> (E In)o, the actual rate of change of n. is positive and the plotted value of d (logne)1 dt gives the exponentiation rate of free electrons for a homogeneous avalanche process in a frame of reference moving at the mean drift velocity of the electrons, Ue •

Another important feature of the avalanche formation process overlooked by assuming an instantly switched on breakdown field is the influence of the time varying electric field on the redistribution of free electrons prior to the initiation of the electron avalanche process. During this preavalanche period in a discharge driven by any real pulse forming network (PFN) with a finite switch-closing time, the applied voltage would stay below the breakdown threshold over a period of time comparable to the voltage rise time. The corresponding field strength during this period, with E In «E In)o, and hence d (logneldt<O, will not substantially change the total number of primary electrons generated by the preionization process but will merely cause them to drift away from the cathode toward the anode. As we shall see from the analysis that follows, the resultant depletion of primary electrons in the cathode region due to the preavalanche drift does have a direct bearing on the initial distribution of electron avalanches, and hence the subsequent development of the space-charge field.

V. ANALYSIS OF THE FORMATIVE PHASE OF AN AVALANCHE DISCHARGE IN THE PRESENCE OF PREIONIZATION AND FINITE VOLTAGE RISE TIME

When the voltage rise time across the electrode gap is finite, a complete analysis of the discharge development must include the preavalanche phase. If the voltage pulse is such that the electric field strength remains below the breakdown threshold (E In)o for too long a period of time, then a spatial region lacking free electrons will grow near the cathode surface, leading to the development of a nonuniform avalanche discharge for essentially the same reason postulated by Palmer.26

The preavalanche phase can be analyzed by considering the situation in which the voltage, or equivalently the electric field within the electrode gap, rises linearly with time up to a certain maximum value Em before dropping back toward or below the threshold value, Eo = (E In)on, due to the finite output impedance of the driving PFN, such that for O<I<lm.

E=~, ~ where E = constant, and Em = Etm > Eo. This is a reasonbly good approximation to the rising part of the actual time history of the applied electric field for many fast discharge lasers driven by low-impedance transmission lines.

From Figs. 3 and 4, it can be seen that, over a limited range of E In ;;.(E In)o, one may approximate the first Townsend coefficient a by a two-term polynomial such that

d (logne) 1 d (logne ) a= =

dx dt


/ ANODE

~flfllfl/////IJ __

T ue=Jl-E • I _~

2 r --1-

d

p/7/7/77// ///hZ'l//7~CATHODE

FIG. S. Schematic diagram showing the relative positions of primary avalanches originating from various parts of the electron-deficient region x".

(6)

or

(6a)

For the particular example cited, a good fit of the computed values fora at low electron mole fractions [e]< 10 - 7 over the field strength range(E In)o <E In<2(E In)o can be obtained by letting the coefficients

bo = [-ai(Eln)o +a2(Eln)~J= -1.l8·104 m 2,

b l = [a l -2a2 (Eln)o]=8.2x10 23 Y I -m 3

and

b1 = G 2 = 6.5 X 1042 Y - 2 - m' 5,

where (E In)o = 1.3 X 10 -- 20 Y - m2•

During the linear rise of the applied electric field [Eq. (5)J. the electron drift velocity Ue is also increasing linearly with time. From tlJe nearly linear dependence of U e on E In as illustrated in Fig. 4, one can determined an electron mobility fi=uel E which is inversely proportional to the molecular number density n and is relatively insensitive to the electron mole fraction over a wide range of values of[e], such that

nfi = 1.82 X 1024 y-I - m-I - sec-I. (7)

Thus the ensemble-averaged trajectory for the free electrons is govered by the equation

dx . - = U =fiE=fiEt. (8) dt e

By direct integration of Eq. (8), one can readily determine the total thickness of the electron-deficient region near the cathode at th~ end of the preavalanche period, i.e., at t = 10==.EoIE.

ito. . 1 (E) Xo= fiEtd! = ~Et~ = -nfi - to'

o 2 n 0 (9)

Once the value of E In rises above (E In)o, primary electron avalanches will beign to form. As illustrated in Fig. 5,



148.251.80.252 On: Thu, 03 Apr 2014 15:23:33

some of these avalanches can be expected to originate from the electron-deficient region near the cathode. If the number density offree electrons within Xo is very low, the stream wise (i.e., along the direction of avalanche propagation) separation of the individual primary avalanches can be expected to be of the same order of magnitude as Xo . The lateral spread of each avalanche patch, on the other hand, is primarily governed by diffusion of the secondary electrons about the center of the avalanche head. Thus, if we let Ne be the total number of free electrons in the avalanche after an elapsed time Llt = t - to from the instant of initiation, and assume that the diffusion process is spherically symmetric, then the radial distribution of free electron number density will be approximately given by,42

Ne (? ) n,,(r,Llt) = (2rrDLlt)J/2 exp - 2i5Llt

Ne ( ?) = lI' /2R 2 exp - R2 (10)

where i5 denotes some averaged value of the effective diffusion coefficient and dR =(2DLlt )1/2. As long as the spacecharge field remains weak, free diffusion prevails. The appropriate diffusion coefficient to be used in Eq. (10) can accordingly be determined either from simple kinetic theories33

.43 or from the theory of random ftights,44 so that

(11)

whereie and Ce denote, respectively, the electron mean free path and the electron mean thermal speed at some averaged value of the applied electric field over Llt. The avalanche track length 5, defined here as the distance between the instantaneous center of the avalanche head and the point of initiation, is related to the elapsed time through the trajectory Eq. (8), such that

5 = r flEt' dt' = ~E{tl- t6L )r\)

(12)

or, in terms of the e1asped time, Llt = t - to, and the averaged value of the electron mean drift velq.city during the elapsed time,

iIe =,uE (to + (¥it)] , (13)

one can simply write,

(l2a)

Thus, from Eqs. (1O)-(12a), one obtains the following expression for the instantaneous Gaussian radius of the avalanche head for an e-fold decrease of ne from the center value, ne(O, Llt),

(14)

This differs from the avalanche head radius used in Palmer's formulation26 by the factor (2Ce!3iIe )112. At a typical breakdown E In = 1.6 X 10 --20 V m2 in the He : Xe : F2 = 200: 8 : 1 gas mixture cited where Ue = 2x 104 m/sec,

the mean thermal speed for the free electrons is about 1.3 X 106 m/sec.40 The avalanche head radius used by Palmer is therefore smaller than the Gaussian radius [Eq. (14)) by

216 J. AppJ. Phys., Vol. 51, No.1, January 1980

a factor of about 6.6 in this example. The minimum preionization density as estimated by Palmer, which varies inversely as R 3, is accordingly about a factor of 300 too high due to this difference in formulation alone.45

As mentioned earlier, due to the relative immobility of the positive ions, a continuous separation of the positive and negative charge carriers along the direction of the applied electric field will take place during the avalanche process. An accurate determination of the space-charge field will therefore require a detailed accounting of the spatial distribution ofthe positive ions as well as that of the free electrons. However, for the purpose of establishing a simple analytic expression for the instantaneous magnitude of the spacecharge field around the avalanche head, one may ignore the positive ions for the time being46 and consider only the contributions from the spherically symmetric electron distribution [Eq. 10)]. Thus, from Poisson's equation in mks units, one obtains the electrostatic potential due to this distribution,

if! (r, Llt) = - ~ [ r ne(r', Llt )r'2 dr' €or Jo

+ r f'" ne(r',..:it )r' dr'1

eN. - -- erf(rIR) ,

4rrEor

where e = 1.60 X 10 - 19 C is the electronic charge, Eo

(15)

= 8.85 X 10 -12 F/m is the free-space permittivity, and R is related to Llt through Eq. (14). The magnitude of the corresponding space-charge field, which is directed radially inward, is given by

eNe Er = Igrac:kp I = F(rIR) ,

4rr€oR 2

(16)

where

F(rIR) = (UR )2

X [erf(rIR) - _2_ (rIR) exp( - ?IR 2)] (17) rrll2

is a non dimensional function which governs the variation of Er with the radial coordinate r at any given Llt, or R. For small and large values of r I R in comparison with unity, this function can be represented, respectively, by a power series and by an asymptotic series, such that for small rlR,

F(rIR) = _4 ..c. r 1 _ 2. (..c.)2 + _3 (..c.)4 _ J, 37T1!2 R l 5 R 14 R J

and for large rlR,

Thus, progressing from the center of the avalanche head at rlR = 0, the space-charge field first increases linearly with rlR, reaching a maximum, and then decreases with the inverse square of r I R at large distances. The maximum value

J.1. Levatter and S.- C. lin 216


148.251.80.252 On: Thu, 03 Apr 2014 15:23:33

ofF(rlR )isapproximately0.428, located atrlR = 0.955, as one can determine from differentiation of Eq. (17) with respect to rlR and then looking for the positive real root of the transcendental equation, i.e.,

_2 (1+ _1_) 1T J/2 (rl R )2

I Xexp( - rlR 2) - -- erf(rIR) = O. (18)

(riR )3

Starting from a single primary electron at S = 0, the total number of electrons in the avalanche head Ne is related to the instantaneous avalanche track length S through the Townsend formula (6), such that

Ne =exp[f a(S')ds '], (19)

The maximum value of En corresponding to Fmax = 0.428 at rlR = 0.955, is therefore also related to S through Eqs. (14), (16), and (19), so that

(E,)max = 0.0511 elie exp [ (S a(S') dS']. (20) €OAeCeS Jo

By equating this space-charge field to the applied electric field E one then obtains the following equation for determining the critical avalanche track length Sc required for initiation of streamer breakdown, i.e.,

f' a(S') ds' = log(19.6€oEXeCesJeliJ. (21)

The electron mean-free path is inversely proportional to the total molecular number density n so that Xe = UnO, where o is the averaged momentum transfer cross section for electron-molecule collisions.33 In the example of the He: Xe : F2 = 200 : 8 : I mixture cited at a typical breakdown field strength E In = 1.6 X 10 -20 V m2

, Q is about 7 X 10 -20 m2 so that EXe = (E In) (0) = 0.23 V. Upon substitution of the previously cited numerical values for Eo, e, and Celli, = 65 into Eq. (21), one obtains, in mks units,

(5, Jo a(S ')dS' = 23.5 + log(Sc)· (22)

At this point, if we consider a to be a constant and let Sc = d, the above equation becomes

ad = 23.5 + log(d) , (23)

which differs from the Raether breakdown criterion [Eq. (3)] for air only by a somewhat larger numerical constant on the right-hand side which reflects the difference in electrochemical properties of the two gas mixtures under consideration.

When the applied electric field is increasing linearly with time during the formative phase of the electron avalanche [Eq. (5)], the first Townsend coefficient a within the avalanche head is expected to vary rapidly with the instantaneous value of E In in a manner which depends sensitively on the electrochemical properties of the gas mixture. As mentioned earlier, the actual dependence of a on E In can generally be approximated by a polynomial over a limited range of E In. The sensitivity of such dependence can then be measured by the numerical values of the coefficients a l , a2 , ••• , or


bo' b l , b2 , ••• , in Eqs. (6) or (6a). the dependence of E on the avalanche track length S, in turn, is governed by Eqs. (5) and (l2), so that

. ( 2Es) 112 E(S)=Et(S)= E6 + -,;- , (24)

where Eo = Eto is the field strength at the instant of avalanche initiation as defined before. By substitution of Eqs. (6a), (13), and (24) into Eq. (21), and ignoring the weak time dependence of EX. Cellie within the logarithmic term, one then obtains the following equation for the determination of the critical avalanche track length Sc :

(bo + b2n~6 )Sc + ~:~ [ (E6 + 2;< yl2 - E~ ]

(25)

By making use of the electron-deficient layer thickness Xo as defined by Eq. (9), we may recast Eqs. (25) into a more convenient nondimensional form, i.e.,

Here, we define

and

tPc=sJxo,

¢ =bl (E In)oxo = bl (E In)6(nflI2)to ,

A == [bo + b2 (E In)6 ]Ib. (E In)o ,

B = (b2 12b l )(E In)o ,

19.6EoXe Ce C=----

eflb l (E In)o

For the particular example of the He: Xe: F2 = 200: 8 : I mixture cited, the dimensionless coefficientsA, B, and Ctake on the numerical values of A = 1.004; B = 0.0515; and C = 8.81 X 105

• Substitution of these coefficients into Eq. (26) then yields,

10

\0

4>= bl (E/n)o (ueo /2) to

FIG. 6. Nondimensional critical avalanche track length if;, as a function of the nondimensional voltage rise time ifJ obtained from numerical solution of Eq. (27).



148.251.80.252 On: Thu, 03 Apr 2014 15:23:33

10-5 L--~L.....1...J-U..LLL--'-....I-L...LU...uL---'~...J.....1..UJ.JJ I 10 10Z 103

VOLTAGE RISE TIME to (nSEe)

FIG. 7. Calculated values of the critical avalanche track length 5e' the critical avalanche head radius 'c' and the electron-deficient layer thickness x"' as functions of the voltage rise time to and of the initial gas pressurel standard atmospheric pressure ratio pip I for the He: Xe : F2 = 200: 8 : I sample mixture cited in Fig. 3.

H(1 + ¢,)3!'2 -1 ] +0.0515 iflc -1.004 ¢e

1 = - [13.7 + log(q\¢e)] .

q\ (27)

Again, for the particular example cited, h\ (E In)~(nf.l12) = 1.26 X 108 sec -I , so that q\ = 10/(8 nsec). Accordingly, if the voltag~ rise time for reaching the breakdown threshold, to = Eo IE, is in the range of 1 nsec to 1 f.lsec, the numerical value for the nondimensional parameter q\ can be expected to lie in the range 0.1 :S q\:S 100.

In Fig. 6, we show a plot of ¢e as a function of q\ obtained from numerical solution of Eq. (27). From such a numerical solution, and the definition of ¢e and q\ just given, one can determine the critical avalanche track length, 5e = ¢cxo, and also the corresponding critical radius of the avalanche head using Eq. (14),

rc = 0.955Rc = 0.955(2~e 5cie)l12 3u c

~(405cie)112 . (28)

It terms of the cross section Q, the molecular number density n, and the gas pressure p,

rc = (405c lnQ)1/2

= (405c 1n l Q)1I2(Plpl) -\/2 , (28a)

where sUbscript 1 denotes some convenient reference conditions such as the standard atmospheric pressure and temperature. In Fig. 7, we show a plot of 5c and rc(Plpl )112 so determined as functions of to, together with the electrondeficient layer thickness Xo calculated from Eq. (9) for the sample mixture.

VI. CRITERIAL FOR HOMOGENEOUS AVALANCHE FORMATION

With these analytical results, we are now ready to propose and discuss the following criterial for maintaining spa-


tial homogeneity during the formative phase of a rapidly growing electron avalanche.

A. Minimum preionization density required

As mentioned earlier, Palmer26 proposed that ifthe initial number density of free electrons neD withing the discharge gap due to preionization is sufficient at the time when the high-voltage pulse (assumed to be a step function) is turned on, then the high density of primary electron avalanches will automatically smooth out the developing spacecharge field as soon as the neighboring avalanche heads strongly overlap each other and coalesce. If one assumes that one avalanche patch is generated by each preionization electron within the discharge gap, then the minimum value of neD required to inhibit streamer formation during the formative phase of the discharge is simply given by the condition that the initial mean distance between free electrons (lineD )1/3 be smaller than the critical radius ofthe avalanche head r" where rc is defined by Eqs. (28) or (28a). However, due to the finite rise time of the applied voltage pulse, the field-induced electron drift away from the cathode during the preavalanche period will cause the mean electron number density neD within the region Xo (illustrated in Fig. 5) to fall below the preionization density neD in the main part of the discharge gap (d - xo). Accordingly, a more appropriate criterion of adequate preionization density for inhibition of streamer development everywhere within the discharge gap during the formative phase of the avalanche breakdown should be

(29)

or,

(29a)

The actual redistribution of preionization electrons within Xo as well as the mean value neD at the moment of avalanche initiation are, of course, determined by the combined effects of electron generation by the preionization source, electron removal by attachment to gas moleculses and by recombination at the cathode surface (gas-phase recombination is generally too slow to be important at low preionization levels), and electron movement due to drift and diffusion. If the preionization source strength is nearly constant up to the moment of avalanche initiation to the value of neD within Xo

will remian at a level which is generally not more than a factor of 2 or 3 below the value of neD in the undepleted region (d - xo). This can be shown in a more detailed analysis of the space-time variations of the free electron number density within the depleted layer Xo' At any rate, detailed determination of the most appropriate value of neD to be used in Eqs. (29) or (29a) is quite complicated and is beyond the scope of the present paper. In the absence of more detailed information, one may simply let neD = tneD for all cases involving a continuously maintained preionization source up to to'

At this point, it is important to note that no one really knows how closely the adjacent primary avalanche patches must overlap each other in order to inhibit streamer formation. Therefore the choice of rc as the critical mean separa-



148.251.80.252 On: Thu, 03 Apr 2014 15:23:33

>f-en Z LU <=>

:z: ~

I 03 ~~--L..LJu.uL---'--L.L.J...Lw..L---l.--'-""""""'u..J 106

I 10 10 2 103

VOLTAGE RISE T I ME, '0 (nSEC)

FIG. 8. Calculated values of the minimum preionization density (n"')m<n

and of the electron multiplication ratio n,. (tJ/n,. (0) as functions of the voltage rise time to and of the initial gas pressure/standard atmospheric pressureratiop/p, for the He : Xe: F, = 200: 8: I sample mixture cited in Fig. 3.

tion distance between preionization electrons in Palmer's criterion26 as well as in our modified criterion, Eq. (29), has been quite arbitrary. Due to the inverse-cube dependence of neO or fieO on the linear dimension which measures the actual degree of avalanche head overlap required for total inhibition of streamer formation, a factor of 2 uncertainty in this linear dimension would lead to a factor of 8 uncertainty in the required preionization density. Thus an imprecise estimate of the ratio between fieO and neO as suggested above is not likely to introduce any significant error when compared with the basic uncertainty imbedded in the establishment of criterion [Eq. (29)].

Upon substitution ofEq. (28a) into the inequality expression (29a), the criterion of adequate preionization density then becomes

fieO > (neO)min , (30)

where the minimum number of free electrons per unit volume, (ne()min' is defined by

(31)

In Fig. 8 we show a plot of (ne()min (Plpl) -312 as a function of the voltage rise time to, together with the electron multiplication ratio

neCSJlne(O) = NeCSe> = exp [ f' aCS') dt'] . (32)

It is seen that within the range of value of to plotted, the minimum initial number density offree electrons within the depleted layer Xo needed to maintain spatial homogeneity of the avalanche process during the formative phase of the discharge is in the range 104_106 electronsl cm 3 for operation at 1 atm initial gas pressure. For discharges at 6 atm initial pressure, the corresponding range of minimum number density required in the depleted layer becomes lOs_107 electrons/cm3

• The decrease of (ne()min with increasing voltage rise time is obviously due to the increase in diffusive spreading ofthe avalanche head. Within the same range of value of

219 J. Appl. Phys., Vol. 51, NO.1, January 1980

to plotted, the electron multiplication ratio neCSJlne(O) only varies between 5 X 106 and 1 X 108

, and is therefore relatively insensitive to the change in to.

B. Voltage rise time requirement in the absence of any preionization source strength at to

When the preionization source strength is maintained continuously during the avalanche initiation process, the criterion of adequate preionization density as stated by the inequality (30) is quite easily satisfied. This is due to the relatively low value of (ne() )rnin (Fig. 8), and also the fact that fieOln,,,, is generally not very much smaller than unity. However, in situations where the preionization pulse is terminated before the applied electric field is turned on, as with many TEA lasers, there will be no source of volume ionization within the depleted region Xo to replace those free electrons that have been swept away by electron drift during the preavalanche period, 0 < t < to. Accordingly, with the exception of some noise electrons caused by cosmic rays, background radioactivities, field emission, thermionic emission, etc., the free electron number density within Xo will be essentially zero. The free electron number density neO within the main part of the discharge gap (d - xo), on the other hand, need not be zero due to the residual effects of preionization. In such situations, one may expect that the mean distance between free electrons within the depleted region (l/fie()I/3, will generally be much greater than the thickness of the region Xo itself. Due to the uniform drift velocity, the head of any avalanche originating from the noise electrons within the depleted region Xo will be separated from those of the nearest avalanches originating from the residual preionization electrons within the un depleted region (d - xo) by a streamwise distance which is typically of the order of Xo

(Fig. 5). Thus, in order to avoid streamer formation due to inadequate overlapping of these noise electron avalanche heads by their respective nearest neighbors during the critical stage of primary avalanche development, it is necessary to keep

Xo <rc • (33)

This is in addition to the basic requirement

neO > (neO)min . (34)

Here, neO denotes the free electron number density at to within the undepleted region (d - xo) due to the residual effects of pre ionization. In practice, such residual preionization density is expected to be governed by a number of experimental parameters such as the time delay between the termination of the preionization pulse and the onset of the breakdown pulse,47.48 the electron attachment and recombination rates, the strength and duration of the preionization pulse, etc. As can be seen from Fig. 7, inequality (33) requires that to ~ 15 nsec for a I-atm-pressure discharge and to ~ 6 nsec for a 6-atm-pressure discharge in the He : Xe : F2 = 200 : 8 : I gas mixture cited.

VII. COMPARISON BETWEEN THEORY AND EXPERIMENT

Based on the above findings, we have recently con-



148.251.80.252 On: Thu, 03 Apr 2014 15:23:33

(a) (c)

~~~~:!v\kJJ I I I y ,,0

FIG. 9. Polaroid photographs of the time· integrated luminosity and of the voltage waveform across the electrode gap of an x· ray preionized discharge (Ref. 9) in the He: Xe: F, = 200: 8 : I sample mixture cited in Fig. 3 showing (a) a homogeneous discharge when il"" > (n"")mm' and (b) the corresponding nearly constant voltage during the self·sustaining phase of the dis· charge for 20 < t < 90 nsec, (c) an inhomo· geneous discharge when il", < (nO)min' and (d) the corresponding voltage oscillation after the initial collapse at t"" 20 nsec.

TIME (50 nsee/em)

(b)

structed a large volume pulsed avalanche discharge apparatus9 for the purpose of testing the various predictions of the aforementioned theory. The discharge system has been designed to satisfy all of the necessary conditions for homogeneous avalanche formation established in the preceding sections. This system uses an x-ray preionization scheme for creation of an extremely uniform distribution of free electrons at the time of avalanche initiation, and a multiarcchannel rail gap switch5 is used to create a fast voltage rise time (:S 10 nsec). The discharge electrodes have also been properly shaped to avoid strong edge effects. The performance of this apparatus has been examined, along with data of similar fast-discharge lasers that have been reported in the literature, to determine specifically the minimum required preionization density (neO)min, and the maximum allowable voltage rise time for generation of a homogeneous avalanche/self-sustained discharge in rare-gas/halogen mixtures. Discharge homogeneity and stability were judged on the basis of a time-integrated photograph of the discharge plasma luminosity and by the time history of the voltage across the

~ --i>

INHOMOGENEOUS HOMOGENEOUS

:ill 'JI. IT ill n I -C~--~---L--~-C~~LC'LI--•• ~IL-• .-~I-*.--~I • .---

01 104 105 106 107 108 10 9

PREIONIZATION DENSITY lcm-3)

FIG. 10. The dotted line in this graph indicates the minimum preionization density (n"")mm for the He : Xe : F2 = 200 : 8 : I sample mixture cited in Fig. 3 at I atm pressure and for to = 10 nsec. The. points on the log scale indicate the various experimental values of n", at which a homogeneous and stable discharge is observed while the 0 points indicate the varous experimental values of n"" at which an inhomogeneous and unstable discharge is observed, using the same x·ray preionized discharge apparatus described in Ref. 9.


TIME (50 nsee/em)

(d)

discharge electrodes. A stable homogeneous discharge was defined to be one in which the plasma luminosity contains no visible structure other than that of a uniform glow, and one in which the voltage time history appears to be nearly constant after the initial voltage collapse, as illusted in Figs. 9(a) and 9(b).

In order to determine the value of (neO )min in our discharge system, the preionization density was reduced in successive steps by placing stainless steel and lead sheets ofvarious thicknesses between the x-ray source and the discharge chamber. This was done until the discharge began to break up into multiple channels and the voltage waveform across the electrodes became oscillatory, as in Figs. 9(c) and 9(d). These experiments were conducted at 1 atm pressure using the He : Xe : F2 = 200 : 8 : 1 sample mixture cited in the preceding sections. The x-ray source strength, which had been calibrated separately using a pure-argon absorber,9 was maintained continuously up to the moment of avalanche initiation. Thus, according to Fig. 8, (neO)min g;;2 X 105 cm-3

for to g;; 10 nsec and p = 1 atm. As shown in Fig. 10, all of the discharges with a preionization density neO greater than this value were found to be homogeneous and stable. The discharge marked by point V in Fig. 10, with a preionization density closest to (neO)min' appeared to be mostly homogeneous in integrated luminosity although the voltage waveform was not quite constant after the initial collapse. The other discharges with lower preionization densities, marked by points VI and VII in Fig. 10, were definitely filamentary and had an oscillatory voltage waveform as illustrated in Figs. 9(c) and 9(d). The agreement between theory and experiment appears to be good within a factor of2 to 3. This is very reasonable in view of the basic uncertainty imbedded in criterion (29a) as discussed earlier.

In addition to the determination of (neO)min' the significance of the voltage rise time requirement resulting from the necessary condition (33) has also been examined. In Fig. 11, we plot the theoretical ratio of Xo / rc from Fig. 7 corresponding to the known voltage rise time of a number of recent fast

J.1. Levatter and S.· C. Lin 220


148.251.80.252 On: Thu, 03 Apr 2014 15:23:33

10 to-

Xo rc 2

1

® ®@

-

UNSTABLE I - - - - - - - :'(ii')® - - - - - - - - - - - - -

W ® STABLE I © .1-

®CD

VOLTAGE RISE TIME, to (nSEC I

FIG. 11. Calcuaited ratio of the electron-deficient layer thicknessxo to the critical avalanche head radius r, corresponding to the approximately known voltage rise times as indicated for the various fast-discharge-driven rare-gas-halide excimer lasers recently reported in the literature. A-Taylor et al. (Ref. 8); B-Sarjeant et al. (Ref. 50) and Alcock et al. (Ref. 51); C, D. and E-Levatter and Bradford (Ref. 5); F-Wang (Ref. 52); G-Sze et al. (Ref. 53); H-Butcher (Ref. 54); I and l-Levatter and Lin (Ref. 9).

discharge experiments from which rare-gas halide excimer laser generation has been found to be successful to varying degrees. In all of these experiments, a short pulse of ultraviolet radiation has been used as the preionization source. The time delay between the main discharge pulse and the preionization pulse is usually quite distinct and is often adjusted to yield the highest laser output under a given set of otherwise identical experimental conditions.49 We should therefore expect that those experiments which lie below the line xolrc = 1 would have produced a homogeneous discharge pro-

vided that there was also an adequate level of preioniation [condition (34)], and that the electric field within the discharge region was sufficiently uniform. The data points (A) and (B) in Fig. 11 represent the high-pressure (p = 6 atm) excimer laser experiments performed by Sargeant et al.,50 Alcock et al.,51 and Taylor et al. 8 The experimental conditions represented by point (B) are identical to those of point (A) except that the voltage rise time was slowed by approximately a factor of 5. The discharge in (A) was observed to be a uniform volumetric glow, while in the slower rise time discharge of (B), severe streamer formation was reported. The three atmospheric fast discharge KrF experiments of Levatter et al.,5 represented by points (C), (D), and (E) in Fig. II, showed similar results. The faster discharges corresponding to (C) and (D) appeared as uniform volume glows which showed no sign of arcing or tendency for the discharge impedance to collapse. The slower discharge of (E) was, however, constricted and filamentary and occupied approximately one-half the active volume of the faster rise time discharge. The laser output from (E) was also approximately one-half of those from (C) and (D). Although Wang52 [point (F), Fig. 11] reported efficient XeF and KrF laser generation at a point near xolrc = 1, upon close examination, the discharge was always observed to contain a large number of streamers. Notice that the operating point of this experiment was just on the limit of satisfying condition (33). Both Sze53

221 J. Appl. Phys .• Vol. 51. No.1. January 1980

and Butcher4 [points (G) and (H), respectively, in the unstable region indicated in Fig. 11] reported short duration laser action with the rare-gas halides, but noted that visual inspection of the discharge uniformity led them to believe that the discharge always terminated in an arc. The discharge in both of these experiments also showed a nonconstant, collapsing discharge impedance which would be expected if arc channel formation were present. In general, in spite of the diversity in experimental arrangements and variations in gas mixtures, the onset of discharge inhomogeneity and instability observed in these experiments are in good qualitative agreement with our analytical predictions. More quantitative testing of our theory, however, must await more quantitative measurements from the wider set of controlled experiments.

VIII. CONCLUSION AND DISCUSSION

An analytical model of high E In pulsed avalanche discharge has been developed that includes the effects of preionization, finite voltage rise time, and variation of the avalanche rate with the applied electric field strength. Relevant equations are derived and presented in nondimensional forms. From this model and the associated formulation, some criteria or necessary conditions for forming a homogeneous electron avalanche over an extended volume at high gas pressures have been developed. Numerical results have been generated for a typical He/Xe/F2 mixture suitable for electrical discharge excitation of rare-gas halide excimer lasers and compared with experimental observations.

For the situation where the preionization source strength is continuously maintained up to the moment of avalanche initiation. the minimum preioniation density required for homogeneous discharge formation between two parallel and properly shaped electrodes is found to depend on the voltage rise time as well as on the gas pressure and various electrochemical properties of the gas mixture which govern the net rate of change of the first Townsend coefficient with the change in electric field strength above the breakdown threshold. Our predictive results are found to be in good quantitative agreement with some recent controlled experiments carried out in our laboratory.

For the situation where there is a finite time delay between the preionization pulse and the main discharge pulse as is common practice of all previous experiments on fast discharge excitation of rare-gas-halide excimer lasers, we found that there is a short voltage rise time requirement in addition to the basic minimum preionization density requirement (again, assuming that the discharge electrodes are properly shaped and cold, so that there is no significant field emission and thermionic emission from the cathode). Our predictive results here are in good qualitative agreement with experimental observations in spite of the diversity in detailed experimental arrangements and variations in gas mixtures employed.

One important result from our analytical model and nondimensional formulation is that, as long as a sufficiently strong source of preionization is provided uniformly over the entire discharge volume during the formative phase of the electron avalanche, there appears to be no inherent limitation on the maximum volume over which a spatially homo-



148.251.80.252 On: Thu, 03 Apr 2014 15:23:33

geneous high-pressure discharge can be initiated. Once a homogeneous discharge is formed, there also does not appear to be any reason why a spatially homogeneous and temporally stable discharge current distribution cannot be continuously sustained until some relatively slow thermal instability mode begins to develop. Here we assume, of course, that a suitable pulse forming network (PFN) is used to provide a continuous impedance match between the discharge plasma and the driving circuit over the time duration of interest. In fact, in our recent large volume discharge experiments using x-rays as a homogeneous preionization source,9 we did succeed in generating spatially homogeneous avalanche/selfsustained discharges of up to 2.51iter in volume at I atm pressure and a lOO-nsec duration with no sign of streamer formation or arcing. The corresponding XeCI excimer laser output observed from a He: Xe: Helz = 500: 20: 1 discharge mixture was about 5 J in energy per pulse. So far, the discharge volume is only limited by surface breakdown of the insulator within the discharge chamber of finite dimensions. The lOO-nsec duration of the discharge, which is about 10 times the voltage rise time and 5 times the current rise time, is only limited by the electrical length of the waterdielectric transmission line used as a low-impedance PFN for sustaining the quasi-steady-state discharge current within the avalanche-generated plasma. Exploration of the upper limits in discharge volume and in stable discharge duration would certainly be interesting from the point of view of discharge physics as well as in practical high-power laser applications.

'N.W. Harris, F. O'Neill, and W.T. Whitney, App\. Phys. Lett. 25, 148 (1974).

2J.1. Levatter and S.C. Lin, Appl. Phys. Lett. 25, 703 (1974) . .1 A.F. Gibson, T.A. Hall, and C.B. Hatch, IEEEJ. Quantum Electron. QE-13, 801 (1977).

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222 J. Appl. Phys,. Vol. 51. No.1, January 1980

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ena (Oxford U,P., London. 1969), pp. 45. 278. '4H. Raether, Arch, Electrotech, (Berlin) 34, 49 (1940). "j,M, Meek, Phys. Rev. 57, 722 (\940). 'hF. Llewellyn jones and A,B. Parker, Proc, R, Soc. London, Ser. A 213,

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Lett. 32, 705 (1978). 42P.M. Morse and H. Feshbach, Methods a/Theoretical Physics (McGraw

Hill. New York, 1953), p. 1584, It may be noted that the exponential term often appears as exp( - rl4D<lt) instead ofexp( - r/2D<lt) with an appropritae change from 2rr to 4rr in the pre-exponential factor in many standard texts. This only affects the definition of D, but not the solution of the diffusion equation.

4'J, Jeans. An Introduciton to the Kinetic Theory o/Gases (Cambridge University. London, 1940), p. 20 I.

44Chandrasekhar. Rev. Mod, Phys, 15. 1(1943). 4'It may be noted that in the numerical example cited by Palmer in Ref. 26

for the CO, TEA laser breakdown condition, a relatively low value of 10' cm 'was obtained for the minimum preionization density using his Eq, (7). This was probably due to his choice of X"" 10 - , for the electron mean free path, which implied an exceedingly small averaged momentum transfer cross section Q of 4 X 10 17 cm2 for electron molecule collisions in the TEA laser gas mixture at I atm pressure.

""The presence of positive ions within the avalanche head will make the diffusion process somewhat ambipolar in character, and hence will slow down the outward diffusion rate of the free electrons. On the other hand, the presence of a net negative charge within the avalanche head will speed up the outward diffusion rate of the free electrons due to Coulomb repulsion. The two effects therefore tend to nullify each other to some extent.

47J, Hsia, Appl. Phys, Lett. 30,101 (1977). "R.P. Akins, G. Innis. and S.c. Lin, j, Appl. Phys. 49, 2262 (1978). 40 As discussed in Refs. 47 and 48, the existence ofa distinct optimum delay

time for maximum power output as observed in many of these fast discharge driven rare-gas-halide excimer lasers was probably related to the negative ion formation and other electrochemical effects associated with the preionization process. Diffusion times required for smoothing out any preionization inhomogeneities caused by the nonuniform nature of the uv preionization source might also have played a role. In any case, the optimum delay time was often comparable to or much longer than the width (FWHM) of the uv preionization pulse. Accordingly, the source strength at to can be considered as practically zero in most of the uv-preionized fast

discharge experiments reported in the literature. ,oW.J. Sarjeant, A.1. Alcock, and K.E. Leopold, IEEE J_ Quantum Elec

tron, QE-14, 177 (1978), 'IA.J. Alcock, K,E. Leopold, W.J, Sarjeant, and R.S. Taylor, Paper XI2

presented at the 10th International Quantum Electronics Conference, Atlanta, Georgia, 29 May-I June 1978.

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KrF Laser," University of California Los Alamos Scientific Laboratory, Bidders' Conference on Design, Fabrication, and Delivery ofa lOOO-W, lOOO-Hz KrF Laser System, 22 February 1978.

J.L Levatter and S.- C, Lin 222


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Necessary conditions for the homogeneous formation of pulsed avalanche discharges at high gas pressures