Plasma rate equations for an rf discharge in a magnetic fieldSteven G. Sanders Citation: Journal of Applied Physics 49, 2689 (1978); doi: 10.1063/1.325188 View online: http://dx.doi.org/10.1063/1.325188 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/49/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in RF Power Coupling And Plasma Transport Effects In Magnetized Capacitive Discharges AIP Conf. Proc. 787, 437 (2005); 10.1063/1.2098528 Capacitive probing of plasma and surface potentials in a magnetically enhanced rf discharge AIP Conf. Proc. 190, 466 (1989); 10.1063/1.38449 Enhancement of the plasma density and deposition rate in rf discharges Appl. Phys. Lett. 48, 695 (1986); 10.1063/1.96746 Relation between the RF discharge parameters and plasma etch rates, selectivity, and anisotropy J. Vac. Sci. Technol. A 2, 1537 (1984); 10.1116/1.572466 rf electric fields in an inhomogeneous magnetized plasma column J. Appl. Phys. 49, 4366 (1978); 10.1063/1.325478

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

137.149.200.5 On: Sun, 23 Nov 2014 05:26:17

Plasma rate equations for an rf discharge in a magnetic field

Steven G. Sandersa)

Physics Department, Southern Illinois University, Edwardsville, Illinois 62026 (Received 15 April 1977; accepted for pUblication 13 December 1977)

A set of plasma rate equations are obtained which describe the time evolution of the electron density and temperature for an rf discharge in a magnetic field. Numerical solutions to these equations have been obtained for argon at pressures of 0.1-100 mTorr using the classical rf conductivity as the mechanism for transferring energy from the rf field to the electrons. and ambipolar diffusion and disassociative recombination as the dominant loss mechanisms. The typical temperature solution exhibits two well­defined temperature plateaus. The electron-density solution exhibits a monotonic growth to a saturation value which constitutes an appreciable percentage of the initial neutral-particle density. At the higher pressures, such large amounts of power are required to maintain the saturation density that source power may be the principle limitation on the maximum density achievable in the typical rf plasma source.

PACS numbers: 52.S0.Pi, 52.50.0j, 52.25.Dg, 52.50.Dg

I. INTRODUCTION

Rate equations have been employed in fusion-reactor studies,l in afterglow studies, 2 and in the considera­tion of steady-3 state rf discharges. However, they have not generally been applied to the problem of pre­dicting the time evolution of the electron temperature and density which result from the application of rf power. Obviously, in any real rf discharge, even one containing negligible impurities, complex atomic pro­cesses are taking place which defy complete descrip­tion. Furthermore, the distribution function of the electrons is frequently not Maxwellian and a correct solution to the problem can only be obtained by solving the Boltzmann equation. However, even though these considerations might seriously limit the usefulness of the solutions of the rate equations, it is still instruc­tive to determine the behavior of the solutions to these equations.

II. THE RATE EQUATIONS

The first rate equation relates the change in the energy of the plasma to the power absorbed by the plasma and the rate of loss of energy from the plasma.

du dt =P abo - P 1ost ' (1)

We assume the plasma is homogeneous and that u is the plasma energy density, P abo is the absorbed power denSity, and P 108t is the power-loss density. The second equation relates the time rate of change of the density to the ionization source function and the rate of loss of plasma.

dn dt =RI -RL' (2)

where n is the electron density in m-s, RI is the ioniza­tion rate, and RL is the particle-loss rate.

We assume that the plasma energy density can be written as

a)Present address: Northern Arkansas Telephone Company, Flippin, Ark. 72634.

(3)

where n is the electron density and U1 is the energy associated with each electron ion pair. We are con­sidering singly ionized ions of one species. We also assume that the rate of loss of energy from the plasma is related to the particle-loss rate by

(4)

This relationship ties the energy-loss rate to the rate of particle loss, but is not as restrictive as it may appear since radiative loss from the plasma can be treated by carefully defining Up Using these relations we can rewrite Eq. (1) as

ndUI dn _ dt + dt U 1 - P abo - R LUI (5)

or

= (P abo - R;U1)n-1•

We assume that

(6)

U1 =~kTe + ~kTi + 2eVI , (7)

where k is Boltzmann's constant, Te is the electron temperature, T, is the ion temperature, and VI is the ionization potential.

We make use of the observation4 that an energetic electron typically loses two to three times the ioniza­tion energy of the target gas atom for every electron­ion pair it produces. We also neglect ion heating and assume that the ions are at room temperature so

~=~k~ dt 2 dt'

(8)

Equation (1) can be thus replaced by

resulting in two equations, one for the time evolution of the electron density and one for the time evolution of the electron temperature.

(9)

2689 J. Appl. Phys. 49(5), May 1978 0021-8979/78/4905-2689$01.10 @ 1978 American Institute of Physics 2689

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

137.149.200.5 On: Sun, 23 Nov 2014 05:26:17

III. SPECIALIZATION TO AN rf DISCHARGE

We have assumed the absorbed power density is given by

2E2 P _ ne Ven

abIJ - 8m

x(V;n+(W1_Wb)2 + v!n+(~+Wb)2)' (10)

which results from assuming that the plasma exhibits a classical rf conductivity. 5 Here, m is the mass of the electron, e is the charge of the electron, E is the peak electric field intensity, ven is the electron-neutral collision frequency, and Wb = eB / m is the cyclotron resonance frequency for an electron.

We write the ionization rate as

Rj=nv j ,

where Vj is the ionization frequency. We consider three plasma-loss mechanisms, diffusion, recombination, and drift of plasma out of the rf coupler. Thus,

RL =n(l/rd + l/rr + l/Tv)

=n/r, (11)

where rd is the confinement time for diffusion loss, rr is the confinement time for recombination loss, and Tv is the time a drifting plasma would remain within the coupler in the absence of other loss mechanisms,

A. Diffusion

The diffusion time constant is related to the diffusion coefficient D and the diffusion length A by

l/rd =D/N. (12)

For a cylindrical confinement6 vessel (or rf coupler) of radius R and length L in a magnetic field B,

2 = v,/ (2.405) 2 (.!!..)2 A2 v.: + w2 R + L en b

(13)

We assume that the diffusion is ambipolar7 and that T e

»T j so that

DA = kTeiM VIn,

where M is the ion mass and Vip is the ion-neutral collision frequency.

B. Recombination

(14)

(15)

where Ci is the disassociative recombination coefficient reported by Biondi et al. 8

The recombination coefficient was measured for a pressure of 20 Torr and a temperature of 300 OK. The temperature and neutral-particle pressure dependence of Ci was not determined and there is a question as to whether the observed recombination was really due to disassociative recombination. If this is the correct mechanism, one would expect Ci to decrease as T e in­creases and the particle pressure decreases. However, we have not included such effects in our calculations.

2690 J. Appl. Phys., Vol. 49, No.5, May 1978

C. Drift

The confinement time for a drifting plasma is given by

(16)

where Vd is the drift velocity. The drift velocities ob­served in our plasma source are sufficiently small that they were neglected in our computations.

IV. ARGON COLLISION FREQUENCIES A. Ionization frequency

In order to determine the ionization frequency as a function of temperature, we must take cross-section data given as a function of beam energy and integrate it over the electron-distribution function. We have inte­grated Smith's data9 over a Maxwellian distribution and we have obtained results consistent with Lotz'slO values (av) using the relationship

(17)

where (J is the ionization cross section, v is the elec­tron velocity. No is the neutral-particle density in cm-\ and ( ) indicates averages over the distribution function.

We fit argon v/p data, where P is the neutral pressure in Torr, with a tenth-order polynomial which is valid for T < 50 eV.

v/p == 40,767 + 3. 06111 x 106T - 4, 79512

+ O. 560276T7 - 2. 51461 X 1O-3TB

+ 5. 43236 X 1O-6T 9 - 4.36624 X lO-9TlO.

(18)

B. Electron-neutral collision frequency

The beam datal! for these frequencies is given for beam energies up to 35 eV. Our best fit to data inte­grated over a Maxwell-Boltzmann distribution is given by

V,/p :::9. 06854 X 10-7 + 1. 15098 x 109T

+ 4. 10663 X 10sr4 - 1. 6453 X 10 5T 5

+ O. 39179TB - to 21869 X 10-3T9 (19)

and is probably reasonable for T < 50 eV. For energies above 35 eV it becomes difficult to speak of elastic collisi~ns because most collisions will produce excita­tion and ionization. However ,inelastic collisions will also serve to mix the phase of the electron motion so that they can pick up energy from the rf field in accordance with Eq. (10).

Steven G. Sanders 2690

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

137.149.200.5 On: Sun, 23 Nov 2014 05:26:17

C. Ion-neutral collision frequencies

These frequencies can be obtained from the mea­sured line broadening observed in ion cyclotron reso­nance experiments, We have used a value12 of

(20)

V. SATURATION OF ELECTRON DENSITY

In the design of an rf plasma source one frequently wishes to maximize the electron density which the source is capable of producing. However, the actual percentage ionization of most low-pressure rf sources is typically less than 1 %. It is useful to consider the mechanisms which can limit the production of a high­density plasma.

A. Neutral burnout

If we write ilL = l/T, Eq. (2) can be written as

dn dt =(11; - IIL )n. (21)

Since II; and, in fact, all the collision frequencies are proportional to the neutral gas pressure, the plasma­generation process will turn itself off when all of the neutrals become ionized. For a singly ionized plasma, the neutral-particle number density N satisfies the following relation:

N(t) =N(O) - n(t), (22)

or in terms of the neutral partial pressur.e P (Torr) =Po (Torr) - 2.833 x 10-23n (m-3

). Since lien is also pro­portional to pressure, we can expect the transfer of energy from the rf field to the electrons to become less efficient as the neutral population decreases (as long as we are operating away from the cyclotron resonance and collisions are necessary to guarantee energy trans­fer from the field to the electrons). We also note that the ambipolar diffusion coefficient increases as IIln becomes smaller.

10

B. rf shielding

As the electron density builds up, the interior of the plasma will be shielded from the rf fields, Assuming a slab geometry of thickness 2R and purely evanescent rf fields we obtain an expression for the average elec­tric field squared within the plasma

(E~ = (o/2R)[l - exp(- 2R/o)]E~, (23)

where 0 is the skin depth and Eo is the field at the plas­ma surface. We have carried out numerical computa­tions using a purely reactive skin depth

(24)

where nc (= mEow2/ e Z) is the cutoff density, w is the angu­

lar frequency of the rf source, A equals the corre­sponding free-space wavelength, and Eo is the free­space permittivity.

C. Finite power limitation

All real rf sources are limited in their output power, This limitation could be the deciding factor in the maxi­mum attainable plasma density.

VI. SOLUTIONS OF THE RATE EQUATIONS

USing the preceding expressions for the generation and loss rates, our equations take the following form:

(25)

(26)

where Cz and C3 are independent of nand T, and C1

depends upon T and n through the electron-neutral collision frequency. Since the collision frequencies are not strongly dependent upon n until the plasma approaches complete ionization, Eq, (25) is effectively uncoupled from Eq. (26).

E = 10' vIm -------------~--~/

z o III: I­U W ... W

2691

10·

P = lOOm'Torr wb/ w =0.9~

10 • 10 •

TIME (Seconds)

J. Appl. Phys., Vol. 49, No.5, May 1978

10 ' 10 •

E v

10'; .... t/)

z w o z o III: ....

10'~ ... w

FIG. 1. Time evolution of the electron tem­perature and density for a 2. 45-GHz rf dis­charge in argon.

Steven G. Sanders 2691

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

137.149.200.5 On: Sun, 23 Nov 2014 05:26:17

~ m~ Pressure 100 mTorr

I 104 vim 1.10/1.1 94 ,......._._._.-.- _._'_.-.--':'._.

I : i: 5'103 ";'m, ""0/"1.1 .99

o w co

'" o VI co < IOOrriN

1.10,11.1 .94

---._. ./

/ ",

--I I

/ /

/ /

/

1.10,11.1 .99

I

100W I I

I I

; : i : i:

I

I: i: . I

! f I: I: . I

/1 ;/ i: il f

I /

;- J

IMW

100kW

--HI I 1

10mv'~------------~r-----________ ~~ILLa ____ L--r ____________ ~ ____________ ~ 10kVV

109

10 10 10 6 10 10'

TIMESECONDS!-

FIG. 2. The time dependence of the energy-absorption rate for rf discharges in argon.

Figure 1 shows numerical solutions of Eqs. (25) and (26) illustrating the time evolution of an initial burst of ionization having a density of 3.6 x 107 cm-3 and a tem­perature of 1 eV. The fourth-order Runge-Kutta calcu­lations were carried out with the electric field assumed constant. Initially, the energy which the electrons ac­quire from the rf field is thermalized by the electron­neutral collisions. The ionization cross section is very low at these temperatures, so very little energy goes into producing ionization. Only after the temperature rises to about 90% of the first-plateau temperature Tl does the density begin to change.

During the time of major growth in the electron density, the temperature is constant. In the last decade of growth the neutral partial pressure is affected, re­sulting in a decrease in Vi' The right-hand side of Eq. (25) is no longer zero and the temperature must rise so that each electron-ion pair can carry away more energy. This gives rise to the second temperature plateau T 2 • The electron-neutral collision frequency will also drop as the neutral partial pressure drops. Since the power density at cyclotron resonance goes as v;!, this will improve the energy transfer. Off reso­nance, the coupling to the rf field goes as v..Jw2 and the energy transfer will decrease as von decreases.

Figure 2 shows the power absorbed assuming a plas­ma volume of L 528xlO-5m3 for the same conditions as Fig. 1. The plateau early in the discharge can be ex­plained by noting the power absorbed for conditions closer to cyclotron resonance, wblw=0.99. One ob­serves an initial drop in the absorbed power under these conditions. This is because initially, there is little collisional damping of the resonance. As the tempera-

2692 J. Appl. Phys., Vol. 49, No.5, May 1978

ture of the plasma increases, the damping increases and the absorbed power decreases. Eventually, the electron density rises to a point where this decrease in coupling to the rf field is overcome. The saturation power required to maintain the saturation density for a plasma of this volume is quite high. This is because of the high density necessary for neutral burnout. At lower pressure the power requirements are more rea­sonable. For example, at a pressure of 1 mTorr and wbl w = 0.94, 686 Ware necessary to produce a satura­tion plasma density of 2XI013 cm-3

• This plasma has an 11. 4-eV electron temperature and is 57% ionized. At a pressure of 0.1 mTorr, the saturation density of 1. 6 x 1011 cm-3 is achieved with only 6.9 W input power. This plasma is 44% ionized and has an 11. 4-eV elec­tron temperature. These results clearly demonstrate that off resonance the absorbed power scales approxi­mately as the square of the initial neutral pressure. [This.is to be expected since the absorbed power goes as nvon and n will typically saturate at one-half the initial neutral-particle concentration No. Since von

- N(t) - tNo, we have nVen - tN~. ] The high absorbed power at high pressures suggests that our choice of electric field strength in the calculations may have been slightly greater than actually exists in the typical rf source. However, a 104-V/m electric field can be achieved with a 100-W rf source coupled to a microwave cavity having a Q of 1000. If plasma loading reduces the cavity Q to 10, the electric field would drop to 103 Vim unless the rf source power was increased to 10 kW. It should also be noted that close to cyclotron resonance and at low pressures, the electron temperature reaches values in excess of 50 eV, exceeding the validity of our ionization-source functions.

Steven G. Sanders 2692

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

137.149.200.5 On: Sun, 23 Nov 2014 05:26:17

---------Ioo-m-:J,=-o-r-r ---; IdS

E 5· 103'fm

~(J 0.99

-- .... - - - --::~-~- - - -- - ---lOm10;.;---

10

-/'" , , , -,'

/

,,---------/1

___ -" I I I

/

/

I

<' /

E u

100 mTorr , __ ----------~------------~1r

>­l-

z o ct: I­U UJ -' UJ

1 I

10mTorr;,/' ;/ 1 mTorr

,,./JII'

1 '::-::"'~---.-.--.-.-.-.

10'

, / r;-

/. .1 mTorr / I

./ , .// ,

I I , , ,

10mTorr I I , I ,

-100 mTorr p,

i .1 mTorr

106 lOS

10. 3

,-

JO'1

10'

V)

Z w Cl

z 0 0:: I-U W --' w

TIME ISECONDS)---

FIG. 3. The pressure dependence of the time evolution of the electron temperature and density.

In Fig. 3 we illustrate the effects of pressure on the electron heating and density formation in the plasma. We note that as the initial neutral pressure decreases, T2 -T 1 •

In these calculations we have assumed that the aver­age electric field within the plasma is constant. We have also carried out the calculation using the skin­depth correction for a source operating at 2.45 GHz with a cylindrical plasma diameter of 1, 6 cm. Be­cause of the small value of Rl~, the shielding has a minimal effect on the saturation density in our source. 13

VII. VALIDITY OF THE RATE EQUATIONS

Since the rate equations are merely a statement of energy and particle conservation, their validity depends largely upon the use of the correct expressions for the absorbed power and the loss rates. However, there is another limit on the validity of these equations if they are used to describe the time evolution of the electron temperature. The concept of temperature is meaningful only if the electron relaxation time is sufficiently short that the distribution function remains Maxwellian. If the changes in temperature are slow compared to the relaxation time this should be the case. In Fig. 4 we plot ~tl> the time for the solution of the temperature equations to go from 10 to 90% of the temperature dif­ference (Tl - 1.0) and ~t2' the time required to go from 10 to 90% of the temperature difference (T2 - T 1 ),

against the initial neutral gas pressure. We also show the electron-neutral collision time for the correspond­ing temperatures. For the range of electric and mag­netic fields considered in these calculations, at least

2693 J. Appl. Phys., Vol. 49, No.5, May 1978

four collisions take place during the most rapid tem­perature increase, a number probably sufficient to thermalize the electrons.

In Eq. (8) we made the assumption that ion heating could be neglected. Since the solutions of the equations lead to steady-state conditions for which the percent ionization is high, there will be a point where the elec­tron-ion collision frequency Vet' which is proportional to the electron density will become equal to the electron­neutral collision frequency which is decreasing because of the burnout of the neutrals. Though calculations in­cluding ion-heating effects have not yet been undertaken, it is possible to make some observations about the possible effects of this process on the results presented in this paper.

The worse-case situation would be if the ions were so strongly coupled to the electrons that T j -Te • Under these conditions

(27)

and U1 would be approximately 25% larger than the value used in our calculations. Equation (9) would become

(28)

Thus, the rate of change of the electron temperature would be halved and the plateau temperatures would change to satisfy

(29)

Steven G. Sanders 2693

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

137.149.200.5 On: Sun, 23 Nov 2014 05:26:17

---~I,

- -- ~I,

-.,; -0 c 0 u (1)

(/)

(/) lOS UJ

~ f-

UJ (/)

~

UJ c:t<: ::::> f-<X: c:t<: UJ a.. ~

10· UJ f-

107~_-r ___________ ~~~~~_~~

0.1 5 1.0

PRESSURE mTorr

FIG. 4. Comparison of the electron-temperature rise times and electron-distribution-function relaxation times.

where U1 is now the larger value. Since we assume that the power is available, the percent ionization would in­crease so that R; decreases to compensate for the larger Up We estimate the change in the final plateau value of T to be no more than a few percenL

From Eq. (2) we see that the slight adjustment anti­cipated in R j would modify the rate at which the density builds up at the point that ion heating becomes impor­tant. However, this also would only be a small effecL

We conclude that ion heating will have little effect on the character of the curves presented in this paper other than to increase the steady-state absorbed power by an amount necessary to heat the ions. Had we performed the calculations assuming constant input power rather than constant E field, the effects of ion heating may have been more dramatic, perhaps resulting in a lower­ing of the electron temperature.

2694 J. Appl. Phys., Vol. 49, No.5, May 1978

VIII. CONCLUSIONS

The model we consider in this paper was developed to describe microwave discharges operating at rf power densities of less than 100 W / cm3

• Discharges of this type produced by rf sources rated at 10 kW or less will typically yield electron densities in the range lOla to 5 X 1013 particles/ cm3 and electron temperatures in the range 1-20 eV. We have attempted to include all the processes which are generally assumed operative in this type of discharge. Because our source uses a 100-W 2. 45-GHz magnetron, we assumed that ion-heat­ing effects would be negligible. The calculations, how­ever, indicate that the loss mechanism will not produce a saturation of the electron density until the plasma reaches percentage ionization in excess of 25%. Under these conditions one would expect significant ion heating to take place. In Figo 1 this occurs at 3. 5x10-7 s or from Fig. 2 at absorbed powers of about 10 kW (654 W / cm3

). Our 100-W (6054 W / cm3) source would be deliver­

ing its full capability at 2 x 10-7 s, thus leading to a pre­mature density saturation at 2 x lO lD cm-3

• This is typi­cal of the densities measured in our source under the conditions of the calculations.

Although the details of the saturation of the plasma­generation process are somewhat in question because of the neglect of ion heating and the cavalier treatment of the excita·tion energy, the calculations do lead to several important conclusions.

(1) The classical rf conductivity is an effective mecha­nism for transferring energy from the rf field to the electrons. Thus, good impedance matches should be achievable.

(2) For plasmas having diameters less than a free­space wavelength, the shielding effect of the electrons in the pressure ranges considered is negligible.

(3) At the lower pressures, neutral burnout limits the maximum achievable density and electron-ion collisions may be important.

(4) At pressures above 1 mTorr, the achievable den­sity will generally be set by the power capabilities of the source. In this case it is the energy loss rather than the particle loss which limits the maximum achiev­able density.

Our final point has to do with loss mechanisms and the plasma-confinement time. When the neutral partial pressure drops as a result of ionization, the diffusion coefficient DA in Eq. (14) becomes larger because of the decrease in the ion-neutral collision frequency. However, D A cannot increase without limit because this expression is valid only as long as the ions are actually undergoing collisions. If the ion mean free path be­comes greater than the length of the rf coupler, one must estimate the confinement time from the expression3

(30)

with L being the coupler length and

V - ~+-~ (kT 5 kT )1/2

s- M 3 M (31)

Steven G. Sanders 2694

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

137.149.200.5 On: Sun, 23 Nov 2014 05:26:17

10'

<n 0 10 mTorr Z 0 /' v w <f)

w / Oi ffusion ::E LAmbipolar

l-

I-Z w ::E 10 ,; w ~ "-Z 0 V

Ion Sound Speed Flow 10·

10'" ldl 1011 lO"

ELECTRON DENSITY (cmJj

Figure 5 illustrates the electron density dependence of the confinement times appropriate to the loss processes considered in these solutions to the rate equations. Our laboratory plasma source consists of a 7. 6-cm-long microwave cavity located at one end of a 140-cm-long magnetic mirror confinement field. The diffusion velo­city is determined by the length of the entire machine, The confinement time is given by one-half the length of the rf coupler divided by the diffusion velocity. The effect of this longer axial vacuum chamber is to in­crease the diffusion time by a factor of 18 above that obtained for a quartz tube of the same length as the rf coupler. This longer confinement time is the value plotted in Fig. 5. For our source the assumed recom­bination mechanism is the dominant loss mechanism,

lR. F. Post, Rev. Mod. Phys. 28. 338 (1956).

2695 J. Appl. Phys., Vol. 49, No.5, May 1978

FIG. 5. Electron density dependence of the plasma-confinement times,

10'· lotS

2C,J. MacCallum, Plasma Phys, 12, 143 (1970). 3R,W, Motley, S. Bernabei. W.M. Hooke. and D.L, Jassby, J. Appl. Phys. 46. 3286 (1975),

4E, Hinnov, A. S. Bishop. and F. W. Hoffmann, Princeton University Plasma Physics Report No. MATT-270, 1964 (unpublished) •

5W, p. Allis, Handbuch der Physik (Springer, Berlin, 1956), Vol. 21, p. 383.

6S, C. Brown, Introduction to Electrical Discharges in Gases (Wiley, New York, 1966), p. 253,

1F. F, Chen, Introduction to Plasma Physics (Plenum, New York, 1974), p. 140.

8M.A. Biondi and S, C. Brown, Phys. Rev. 76, 1697 (1949). 9p. T. Smith, Phys. Rev. 36, 1293 (1930).

tOW. Lotz, Astrophys. J. Suppl. 14, 207 (1967). l1A.D. McDonald, Microwave Breakdown in Gases (Wiley, New

York, 1966), p, 20. 120, Wobschall, J, Grahain, Jr., and D. Malone, Phys. Rev.

131, 1565 (1963), 13S,G. Sanders and D, Mowery, Trans. Ill. State Acad. Sci.

69, 1 (1976).

Steven G. sanders 2695

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

137.149.200.5 On: Sun, 23 Nov 2014 05:26:17