Nonlinear Electron Acoustic Waves, Part I

Nonlinear Electron Acoustic Waves, Part I RICHARD G . FOWLER Department of Physics, Uniiiersity of Oklahoma, Norman. Oklahoma

I. Introduction

....

I

11. Laboratory Exper A . Breakdown W

B. Secondary Breakdown Waves C. Electric Shock Tube Precursors ......................................................... D. Laser Precursors .................................................. E. Slow Proforce Waves ........................................................................ F. Tertiary Breakdown Waves III. Theories .. A. Early Approaches ... B. Fundamenta Equations in in One Dimension C. Solution of the Electron Equations for Proforce Waves IV. Conclusion to Part 1 List of Symbols .................................................................................... References

4 4 26 31 50 52 52 56 56 58 12 83 84 85

1. INTRODUCTION

In a world with large amounts of quickly disposable energy, a wealth of dynamical phenomena has come to light marked by a very high propagation speed, approaching that of electromagnetic waves. In most cases, the observation is based on a moving luminous domain in a gas, and the domain’s initial boundary is remarkably sharp, suggesting something of a discontinuity. Simultaneous observations of ionization have shown discontinuities to be present which are closely cosputial with those of the luminosity. In a previous article (I) it was hypothesized that these fast waves all involve organized electron motion, and hence all have electron acoustic speed as their reference speed in the Machian sense. In this article the entire subject will be elaborated, and new evidence for the validity of the fluid dynamical viewpoint of this phenomenon will be reviewed. It is probable that the first laboratory observation of visible effects from a wave of the electron acoustic family was by Hauksbee (2) in 1705. He combined the possession of a good vacuum pump and the requisite interest

2

RICHARD G . FOWLER

in electricity needed to bring the phenomenon to visibility. His earliest report to the Royal Society states that “mercury will appear as a shower of fire while descending from the top to the bottom of a long (partially evacuated) tube.” He termed the phenomenon the n?ercuriu/ phosp/iorus, stemming from a demonstration he had just made of the luminous oxidation of the element phosphorus at low pressure. More sophisticated inquiries were possible by 1835, the date which J. J. Thomson credits to Wheatstone as that of the discovery of the electrical breakdown wave. Wheatstone (3),who tried to investigate the propagation of conduction in the onset of Geissler discharges at a time when it was fashionable to wonder about the speed of telegraph signals in wires, truly observed only that the conduction set in through a 2 m long gas-filled tube faster than he could follow it with the rotating mirror he had just invented. In actual fact no phenomenon of this class was definitely detected until I891 when Thomson ( 4 ) himself reported resolution of a fast-moving luminous pulse generated in a long partially evacuated tube. By viewing images of tube segments 15 m apart as superimposed by a rotating mirror, he estimated a speed of about one half of the speed of light and determined that this luminous pulse moved from the anode to the cathode. Von Zahn ( 5 ) had previously asserted that there was no Doppler shift of the emitted radiation comparable with the limiting velocities expected by Wheatstone, indicating that there was either no phenomenon or no mass motion. It is well to emphasize at this point that although this absence of detectable motion in the heavy particles of the gas is assumed to exist in all the situations that will be discussed and is certainly a primary criterion for classification of these waves, it may never have been proved explicitly. Von Zahn was photographing end on what he describes as a Geissler discharge, and as such almost certainly saw light chiefly from the positive column of the steady discharge subsequent to the passage of the various breakdown waves. The breakdown waves themselves are low enough in intensity that one may now properly wonder whether they would have been accessible to nineteenth century spectroscopy at all, even had there been adequate time discrimination in the experiments. In the interval elapsing before the definitive 1930 research of Beams ( 6) , investigators seem to have had their attention diverted to the slower dynamical phenomena of the plasmacoustic class generally known as moving striations, although James (7), using a Kerr cell, reported failure to observe Thomson’s wave. Clearly this often cited negative result arose because again the wave is much too low in intensity to be visible under the severe light losses that handicapped early Kerr cells. Over the next fifteen years Snoddy and his co-workers (8) provided the first significant studies of the electrical breakdown waves or potential w m e s as they came to call them. Beams observed with a rotating mirror that a wave always left his un-

NONLINEAR ELECTRON ACOUSTIC

WAVES,

I

3

grounded electrode, regardless of polarity, and proceeded toward the grounded electrode, where it was succeeded by a much brighter wave (a return stroke) moving at higher speed in the reverse direction. In so doing he evidently became the first to observe a true breakdown wave, an ionizing wave proceeding into a neutral gas. It is almost certain from the observations reported that Thomson’s wave was a return stroke. i.e., an ionizing wave moving into a medium which has been seeded with electrons. usually by a previous wave. Beams observed speeds of 4 x 10’ nijsec with applied voltages of 30 kV in air at pressures of about 0.3 Torr. Although Beams began his research ( 9 ) in an effort to study lags in the order of appearance of spectral lines in spark discharges (a complex effect compounded of the finite radiation lifetimes of quantum states, the molecular transit times to the site of excitation of volatilized electrode materials, and finally of the transit time of the electron waves which initiate the discharges), he soon recognized (1936) that his research on the waves themselves would have an impact on the study o f man’s oldest electrical acquaintance, lightning, a field that was just beginning to have a well-grounded phenomenology. The 1916 techniques of Wilson (/O) provided information on the potentials, currents, and total charge transport of lightning strokes, and the Boys camera ( I 926) revealed the complicated structure of the stroke itself together with rough estimates of the propagation velocity. As first described by Schonland (I/), an initial faint wave called a leader was seen to propagate from the (usually) negative cloud base toward the ground, commonly in a series of 50 m advances at an average speed of greater than lo7 m/sec, interspersed with 50 psec pauses. When the observed potentials of lo8 V were scaled with the pressure, it was clearly probable that a root similarity existed between the phenomena being described. When research began in 1949 with electric shock tubes (12) and some of the other high current, low pressure discharges employed in plasma physics, it was quickly apparent that some form of preheating including preionization took place in (what seemed to be) every case in the expansion chamber of the apparatus well before the advance of the heavy particle shock waves and flows. The causative agent was first clearly identified as a wave by Josephson and Hales (13) who measured its velocity at around 5 x lo5 m/sec, others having tried to explain the effect as photopreionization by ultraviolet radiation. The phenomenon was dubbed a plasma precursor, although this led to a certain amount of confusion with a similar early ionization observed at about the same period in front of strong shock waves in an ordinary pressuredriven shock tube by Jahn ( 1 4 ) and later investigated by Weymann (/5) that proved to be caused by the free diffusion of electrons. Although this effect has had its place in clarifying our overall understanding of the electron waves, it is not sufficiently germane to receive detailed attention in this article.

4

RICHARD G. FOWLER

As phenomena accumulated, it became apparent that the unifying feature of this family of waves, intermediate in speed between electromagnetic waves, and acoustic waves, was the acoustic speed of the electron, (5kTe/3m)”’, and that a number of miscellaneous situations, some of them involving natural phenomena, were probable members of the family. Time lapse movies (16) of the sun have revealed waves of luminosity moving across the face of the sun at these high speeds in conjunction with violent activity. Radar reflection from the sun has shown outward moving waves at similar speeds (17). Satellites have detected shock waves in the interstellar media with speeds of 4 x 10’ m/sec (18). The transit time from the sun and steepness of onset of certain magnetic storms suggest that they involve waves of the same type. Following the 1962 Johnston Island “Starfish” test at high altitude, a worldwide ionspheric effect was observed which may have been simultaneous (19) or may have had a propagation velocity of 2.3 x lo6 m/sec (20). Recently precursors have been observed (21) at speeds of 0.3 to 3.0 x lo6 mjsec from around the explosion generated by focusing a pulsed laser. Since 1962, when Paxton and Fowler (22) first proposed that nonlinear fluid theory applied to the electron component of the gas should describe these waves theoretically, definite progress has been made both experimentally and theoretically toward that goal. 11. LABORATORY EXPERIMENTS

A . Breakdown Waves

Snoddy, Beams, and Dietrich (23) investigated breakdown waves chiefly through and in regard to their electrical properties. They applied the potential of a Marx circuit impulse generator to an electrode in one end of a 15 m long partially evacuated tube (Fig. 1 ) and observed potentials as they appeared along the tube with an oscillograph having about a 20 nsec risetime. Since I IOM

MIt

8 53M

.P

TO CHARGING SYSTEM Gu

io

OSCILLOGRAF&

II I E4

FIG.1. Diagram of Beams’ apparatus showing discharge tube and Marx circuit for supplying potential. From ( 8 ) with permission of Phys. Rev.

5

NONLINEAR ELECTRON ACOUSTIC WAVES. I

they used only a rotary oil pump to achieve vacuum, their gas purity is not free from challenge, but despite this the small difference observed between different gases (Fig. 2) suggests at once that ultrahigh purity would not have been of much import. They found that the principal factors governing the velocity of the wave were the applied potential, the gas pressure, and the radius of the tube in which the wave is confined. The velocity increased c

40

V

: 35 I

w

-2

ln

2

2

c

- -0--

30 25

4

(1

d0

20

0:

a

LL

0

15

P

;10 P v)

0.2 0.4

0.6 0.8 1.0 1.2 PRESSURE (TORR)

1.4

1.6

1.8

FIG.2. Average speeds of antiforce wave propagation for dry air (-), C 0 2 ( . . . . .), and H, (- - - -), applied voltage 125 kV in 5 mrn tube. From (23) with permission of Phys. Reo.

linearly with applied potential (Fig. 3), and varied according to the similarity law product radius x pressure, increasing with this factor for low pressures. They observed that waves from a negative electrode were faster than those from a positive one and that, like the return stroke, a second wave that happened to move into an already ionized medium because it was launched too soon after the first wave passed traveled with extra speed. Examining the wave fronts with a rotating mirror, they found them quite straight.” They measured the input currents supporting the waves and found them to be large (Fig. 4 and Table I). Mitchell and Snoddy (24) subsequently added to this research as part of a general effort to tie the observations into the lightning problem, improving the data on variation of wave speed with gas pressure (Fig. 5), and offering a theory which will be discussed below. In what follows, the unambiguous terminology proforce ” and “ antiforce” will be used to describe the two polarities in which waves may be “

“

6

RICHARD G . FOWLER

TABLE I MAXIMUM CURRENT I N INITIAL WAVEFOR DIFFERENT TUBES Applied voltage

Tube diameter (mm)

Maximum current (A)

(Torr)

+I32

18.0 5.0 5.0 1.7 18.0

230 171 143 91 226

0.60 0.45

(kV)

+ 141 1180

+I32 -I32

Pressure Aim’

9.0 x 10-3

9.0 x lo-’ 7.5 x 1 0 - 2 4.0 x 10-1 9.0 x 10-3

0.08 0.45

0.45

generated. A proforce wave will be one in which the force on the electrons at the immediate front of the wave is in the direction of wave propagation; an antiforce wave is the contrary case. The modern phase of measurement dates from the work of Haberstich (25). To the general layout of a breakdown wave apparatus he added a 30.5 cm diam cylindrical ground shield around the breakdown tube. He experimented with replacing the spark-gap switch, used by Beams, with a vacuum switch but found it unsatisfactory and returned to the spark gap. His great advance was in the diagnostic equipment which he applied to the problem. He introduced photomultipliers to track the waves by their luminosities as well as 40

I

1

1

1

1

I

V

W

v) \

g 35-

-

0

-

2

0 30I-

a 0 a

n

-

0

a 25n

b.

0

-

20W

n cn

15

I

I

I

I

I

7

NONLINEAR ELECTRON ACOUSTIC WAVES, 1 0

-

110

I

I

I

I

I

I

I

voltage

FIG.5 . Spced pressure curves for proforce waves in air, in 14 cm tube. 6 25 kV, 55 kV, A 85 kV, U 115 kV. Froni (24) with permission of Phys. Rro.

8

RICHARD G . FOWLER

electrostatic probes rather similar to those used by Beams to detect them by their accompanying fields. In this fashion he was able to supply the basic evidence missing from the previous work, that the luminosity and charge configuration of the wave were cospatial within the risetime of his equipment which he estimated at better than 10 nsec. Haberstich worked with helium and argon and like Beams did not take any special precautions with gas purity. In his own words, Our unavoidable use of a mechanical pump is admittedly poor experimental practice for the study of electric discharges. But, because of the high speed of the potential waves, it appears reasonable to assume that the collisional phenomena associated with the propagation mechanism are little affected by the purity of the gas. Impurity effects may still be involved in radiative processes, such as photoionization in the gas or at the walls. Unfortunately the role played by these processes in the propagation of the wave is not well understood at present. Therefore, it is difficult to estimate the influence of photo-ionized impurities, for example, on the structure and velocity of the potential waves. In view of the excellent reproducibility of the speed and attenuation observed over a period of operation of our discharge tube of nearly two years, it appears that the results reported here have not been significantly affected by the low purity of the experiments.

His use of gases with well-known atomic constants, even it they were not absolutely pure, was an important advance if comparison with theory is to be made. As will be subsequently seen, the properties of the wave do depend on the ionization potential, ionization cross section, and elastic cross section of the gas, and so gas purity is ultimately desirable; but these are dominated by the mechanical properties of the free electrons to a point that the wave speeds are only moderately sensitive to gas composition. Having demonstrated that the luminosity and electron charge in the wave were cospatial, Haberstich selected the former for his velocity measurements because he noted that electrostatic probes interfered with the motion of proforce waves, especially at low pressures, to falsify the time interval measurements. He calibrated his probe by inserting a metal pipe inside his glass tube and moving it toward the probe under known potential. He arranged light pipes so that a single 1P21 photomultiplier could see four stations at once, and all the successive signals could be recorded in a single triggered sweep of a Tektronix 585 oscilloscope. His results are given in Figs. 6-9; they have been replotted as functions of position at constant pressure rather than vice versa as originally given. They bring out vividly for the first time exponential attenuation of the wave velocity with advance down the tube, a fact obscured in Beams’ apparatus by the lack of symmetry in the ground array and by the limited number of stations used. That the potential at the wave front attenuates was known to Snoddy et al., but again only two stations 8 m apart were employed. Haberstich followed

9

NONLINEAR ELECTRON ACOUSTIC WAVES, I

f

2.5

y to68 6 4

4.5

6.5

2

FIG.6. Wave velocity decrements. Haberstich data for proforce waves in helium. From Haberstich (25) with permission.

61

-

4 -

2 -

:10’ 8 r V \

I

6 :

-6

4 2 :-

g >-

-

--

106 8 6 4 -

4.1 TORR ‘I1

2 -

to1

I

I

-

REVISED II

I

FIG.7. Wave velocity decrements. Haberstich data for proforce waves in argon. From Haberstich (25) with permission.

10

RICHARD G . FOWLER

LOCATION ( M )

FIG.8. Wave velocity decrements. Haberstich data for antiforce waves in helium. Haberstich (25) with permission.

the attenuation of potential at the intervals of 0.5 m, obtaining the results in Figs. 10-13. When these are replotted from the viewpoint of subsequent theory one finds both that attenuation is exponential with either nearly the same or nearly one half the decrement found for the wave attenuation and that there is also an electrode drop in potential for the proforce wave which becomes especially pronounced at low pressures. The attenuations observed are substantial and show at once that the velocities computed by finite differences over such large station separations as 0.5 m require correction by use of the expression

dt

(1)

In the most severe case this amounts to a factor of 1.47. The curve for 11.O Torr in Fig. 7 has been replotted to allow for this factor. Having a measure of local velocities and local potentials, Haberstich plotted one against the other in a few cases and obtained what seemed superficially to be discordant behavior-linearity at some pressures and nonlinearity at others. Replotting all the data as in Fig. 14 indicates that proforce

11

NONLINEAR ELECTRON ACOUSTIC WAVES, I

I o6 8

6 4

t

5

10

2t

4

11.0 TORR

I o4 0

10

2LOCATION 0 3 ( M0I

4 0

50

FIG.9. Wave velocity decrements. Haberstich data for antiforce waves in argon. From Haberstich (25) with permission.

waves in both helium and argon (over the pressure and voltage range studied) together with antiforce waves in argon above 1 Torr when also below 10 kV follow the relation V K ( p 2 / p , while antiforce waves in helium (over the brief pressure range studied) together with antiforce waves in argon below 1 Torr follow the relation V CK (pp except when also below 10 kV. Haberstich also computed what he termed front thicknesses by multiplying the observed risetime of signals on the voltage probes, photomultipliers, and his microwave interferometer by the propagation velocity. A sample of these results is reported in Fig. I5 for proforce waves in argon. Essentially the same thicknesses were observed in helium. Using a microwave interferometer he detected substantial quantities of electrons behind the front, but the response time of the instrument was so slow that it is not possible to relate them i n any sure way to the front itself. A sample measurement was 10l6 electrons per cubic meter at 10 kV and 2.0 Torr He some 50 psec after wave passage. This slowness of microwave response has been a standing handicap in getting useful data about these waves. One of the striking qualitative observations made by Haberstich was that the luminous response of the photomultipliers was more rapid for waves in

'i 5

1.0

1

0

I

I .o

I

I

2 .o 3.0 LOCATION (MI

I

I

4.0

5.0

FIG.10. Wave potential decrements. Haberstich data for proforce waves in helium. Haberstich (25) with permission. 10

,

I

2 -

I.o

I

0.34

I

I

1.9

1

11.0 TORR

-

I

LOCATION ( M I

FIG.1 1 . Wave potential decrements. Haberstich data for proforce waves in argon. Haberstich (25) with permission. 12

FIG.12. Wave potential decrements. Haberstich data for antiforce waves in helium. Haberstich (25) with permission.

.

10

-3

5-

I

I

I

I

311.0 TORR 2

I 0

0

I

I

I

I .o

2.0

3.0

LOCATION ( M I

I

4.0

5.0

FIG.13. Wave potential decrements. Haberstich data for antiforce waves in argon. Haberstich (25) with permission. 13

2

I o6

8 6

+2/P OR + P FIG.14. Haberstich data plotted against an appropriate functional variable $ / p or qV (see text); n = 6,3, respectively. Various symbols distinguish different pressures and initial volt ages.

FIG.15. Apparent thickness d, of the potential rise for proforce wave in argon plotted in terms of the risetime T , at the potential probe,the velocity Vof the wave,and the pressure p of the gas. Waves generated by potentials of 15, 1 1, 7.5, and 5 kV and observed at I = 127 cm. From Haberstich (25) with permission.

NONLINEAR ELECTRON ACOUSTIC WAVES, I

15

argon than waves in helium. This he traced to the presence of A(I1) radiation at the outset of argon waves. The A( 11) radiations are generally much faster in radiation lifetime and thus appear and peak before the A(1) radiations become strong, in response to a pulsed excitation at the wave front. The He(I1) radiation was not observed because its excitation potential was too high for the waves generated, and so in helium only the slower He(1) radiations were seen. Haberstich in fact obtained rough time resolved spectra showing that A(I1) radiations are present at the very front and die out. The research of Haberstich was both satisfying and tantalizing. Haberstich himself, and Burgers (26), offered theories of the phenomenon based on the fluid dynamical approach of Paxton and Fowler. Shelton derived a solution of these equations (27) which predicted that the wave velocities should be mobility controlled and proportional to E/p. The existence of these new theories capable of being tested, their partial discord with the observations of Haberstich, and the general need for more extensive data on the breakdown wave phenomenon prompted the experiments described by Blais (28) and Blais and Fowler (29).Their wave apparatus included the following departures from the design used by Haberstich: (a) The breakdown tube was 7 m long and 51 mm diam. An adjustable spark gap was used to switch the potential as Beams had done. Tests proved that this simplified arrangement gave a more regular and reliable performance than a variety of more modern and sophisticated electronic equipment. The vacuum switches used by Haberstich had limited his voltages to 10 kV because of what he termed bouncing of the relay. Blais found that they broke down at high voltage owing to field emission before closure and that the entire wave propagation phenomenon could take place during the period of arcing and wild voltage oscillation that ensued prior to actual metal-to-metal contact. (b) A standard vacuum system with oil diffusion pump and liquid N, cold traps (to maintain the purity of spectroscopically pure commercial gases) was employed. (c) An improved version of Habersitch’s electrostatic ground was designed of twenty la in. diam aluminium pipes evenly arranged and rigidly mounted parallel to one another as generatrices of an open lattice cylinder 0.27 m i. d. This style of construction provided unlimited access ports to the wave tube. (d) Wave speeds were measured by compiling in a multichannel analyzer the time intervals for a large number of successive waves between given stations. With one photomultiplier at a reference station and a second at a downstream station, start and stop inputs were delivered to a time-to-pulseheight converter. The output pulse was then stored in the analyzer until a statistically significant number of events was obtained. The entire apparatus could be calibrated against measured delay lines to better than 2 nsec. The principal advantage of the method lay i n the time-to-pulse-height converter

16

RICHARD G . FOWLER

that has a higher time resolution, can deal with smaller time intervals, and can be calibrated more accurately than the time base of an oscilloscope. Lacking an analyzer, one might display the output of the converter on an oscilloscope and use the ability to calibrate its vertical amplifier with known voltages to achieve about the same accuracy. Care must be taken to be sure that the triggering waveforms remain constant in profile and amplitude, otherwise a falsification of the velocities can easily enter because the converter responds upon passing a given threshhold voltage of the trigger pulse. (e) The diagnostic methods they employed were chiefly optical, based on the observation by Haberstich that metallic probes, even external to the wave, could interfere with the wave propagation. From their optical observations they derived electron temperatures and electron densities in addition to wave speeds. Blais and Fowler noted for the first time that the attenuation of the wave velocity as measured by Haberstich was exponential; one purpose of their research was to investigate this matter in detail in order to determine whether the exponential behavior was exact or approximate. Because of the attenuation, as has already been mentioned, it is not possible to obtain true velocities from the finite ratios Az/At measured between stations. Rough plots of this finite ratio suggested, however, that dV/dz = -yV"

(2)

where 0.5 < n < 2.5. The cases of n = 1 and 2 could be treated by finite differences to yield linear plots of At vs. t for n = 1 and At vs. z for n = 2. After putting the two cases to as many tests as possible, they concluded that n = 1 (the exponential decay) was the more likely and probably was an exact result. It was found that y is strongly dependent on (po, the initial applied voltage, and on the pressure. The best fit was obtained by forming the quantity y ( p 0 2 i 3 , which then exhibits a linear change with pressure at low pressure and a saturation behavior above 5 Torr as shown in Fig. 16. The data of Haberstich can be placed on the same curve plot with an adjustment by a factor of 4for the relative tube sizes ( 2 in,/l in.). The general behavior suggests electron loss under diffusion as the controlling factor present in the constant y. The constancy with pressure at high pressures indicates that the column is selfconstricted in the radial direction as observed by Winn (see below), and as expected above an crp of about I0 for an electron-dominated column at high pressure (30). Since concern over the possible perturbation of the wave speed apparatus by any probes with which the local potentials might have been measured had led Blais and Fowler to avoid their use, they were compelled to adopt a different approach which is itself not without objection. The exponentially

9"

7l

NONLINEAR ELECTRON ACOUSTIC WAVES. I

17

5

TORR HE

FIG. 16. Decrement of wave velocity vs. gas pressure in helium 0 Antiforce, Proforce, a Haberstich proforce, A Haberstich antiforce. From Blais and Fowler (29) with permission of Phy.7. Fluids and Haberstich (25) with permission.

decreasing observed velocities at each point were extrapolated back, using the

Ar,r plot of the data points, to the velocity that would have been observed at z = 0, i.e., in the vicinity of the electrode. This was then treated as the velocity

corresponding to a wave moving at the initial electrode potential as measured by a voltage divider connected to the electrode. The data they obtained for proforce waves are given in Fig. 17, plotted against cp/ap, the natural variable. The data for intermediate pressure seem to cluster around a linear dependence of Vupon cp/ap, but data for high pressures and low pressures are offset and at high pressures and weak fields the dependence seems to be going over into the cp2 dependence found in the Haberstich data. A t high pressures the shifted position of the data again indicates a column which is of smaller radius than the tube, 8 mm, in fact. At low pressures, the shift of the data to low velocities may be because of the ignored cathode fall in potential. The tacit assumption exists that there is no cathode drop in potential. Subsequent restudy of the Haberstich data has shown that this is a good approximation for antiforce waves and proforce waves at pressures above lmm, but not below.

18

RICHARD G . FOWLER

A

*

.

- 301 - 98 - 098 - 295 - 0.30 0 - 163

0

A A m

0 - HI45

e- 24

8 - 4 5 0 - 6 5

0

I

1

1

1

1

I

l

l

I

FIG. 17. Comparison with theory of observed initial velocities vs. applied potential for proforce waves in helium. Data symbols below H are from Haberstich, above are from Blais. From (29) with permission of Phy.~.Fluids and (25) with permission.

The data for antiforce waves are plotted in Fig. 18. As yet there is no theory with which to compare this data. It must be noted that these data do not show that c p / q is the simple controlling variable as clearly as do the proforce wave data. A striking prediction of the Shelton theory is that breakdown waves will not run into a neutral gas beIow a critical velocity of (2eqi/rn)'/'. In helium this is 2.99 x lo6 m/sec. The prediction was found to be borne out by experiment reasonably well in two different situations: (a) In the experiments from 1-3 Torr no starting discharges were observed below about 2.5 x lo6 m/sec, a fact that is in accord also with the observations of Haberstich. (b) In experiments where the velocity of the decelerating wave passed below 3 x lo6 at some point in the apparatus, no wave was detectable beyond this point. The waves at 30 Torr were an exception to this, showing starting and running values to below 8 x lo5 m/sec. The method used for determining electron temperatures and densities was an adaption of a spectroscopic method proposed by Sovie (31) and perfected

NONLINEAR ELECTRON ACOUSTIC

WAVES,

7t

19

I

5

A h

a d

. 16 L 7

@,

lo4

..

A

0

A

1

3

5

7

105

+

3

1

5

7

L . i

1

3

106

(V/rn/Torr)

FIG. 18. Antiforce wave velocities vs. applied potential in helium. I:', 2.95, A I .63, 0 0.98. From Blais (29) with permission of P/?)P.Y. Fhids.

30. .1,

9.8,

by Latimer. Mills, and Day (32) in a situation similar to this one, but quasistatic. It depends on taking the ratio of the intensities of two emission lines. When the excitation of gas atoms is due solely to random electron collisions, and multiple processes such as excitation transfer or excitation from the metastable states are negligible, the intensity ratio of two spectral lines, as measured by the signal S from a photomultiplier tube, is given by or

SjklSim = ( Q j r l l > I ( SjklSlm

Qimzl>

== < . ~ k ( l ' ) L ' > . j h / < f i m ( ~ ~ ) Z 1 ) ~ l w i

(3)

(4)

where ojkand g l m ,respectively, are the values of the optical cross section of the two lines at maximum. The optical cross section Q.jk is defined as Q j k= Bj, Pi'. Bjk is the branching ratio for t h e j + k transition and Qj'is the apparent cross section for theJth level, i.e., the cross section of thejth level uncorrected for cascade from higher levels. The functions f ( v ) describe the shapes of the excitation functions with electron velocity, i.e.,

20

RICHARD G . FOWLER

The angle brackets mean that the cross section as a function of velocity has been multiplied by velocity and averaged over the plasma electron velocity distribution which is generally taken to be Maxwellian. Thus if the two cross sections are different functions of electron velocity then the ratio will be a variable function of electron temperature. The best pairs of helium lines to use are those which originate from the n'S and n3S levels, since the excitation cross sections of these lines have been shown by Miller (33) to be relatively free from pressure effects. These lines also have the advantage that the radiation from the electron beam cross section measurement experiments is unpolarized (34) and hence the available measured cross sections will be more accurate since no corrections for anisotropy of the emission of radiation are necessary. TABLE 11 MEASURED OPTICAL CROSS SECTIONS QIk FOR THE HELIUM SPECTRAL LINES

Optical cross section (10-l' m2) Line

(A,

Transition ~

5048 4713 4438 4141 4686

Electron energy (eV)

Jobe and St. John (35)

Present data

Branching ratio B,,

Lifetime (nsec)

0.59 0.62 0.47 I 0.477 -

89 62 110 140 2.1

~

2IP - 41s 2 3 -~ 43s 2lP - 5's 2 3 -~ 53s He+34

33.5 26.5 33.7 27.7 205

13.80 20.6 5.14 7.64 1.07

13.60 19.80 5.19 7.64 -

The cross section data (Table II) used are the recent data of Jobe and St. John (35)up to 400 eV. They were measured at pressures and currents low enough to ensure the absence of multiple effects. The data of Moussa (36) normalized to that of Jobe and St. John at 400 eV was used to achieve extrapolation to 1000 eV. A computer integration routine suggested by Elste (37) was applied 'to calculate the Maxwellian averages of the cross sections. When a Maxwellian velocity distribution of electrons is chosen ( Q j k u ) is given in terms of energy by u,,,.,

ju

( Q j Au ) = 3.583 x 107(l/kTe)3/2

"7

I

"

Qjk

(U)Ue-"lkTedU.

(7)

Here U is the energy in eV, kT, is the electron temperature in eV (i.e., 1 eV = 7737"K), Urninis the threshold for excitation and Urn,,is the maximum energy to which the integration was carried out, in this case generally IOkT,. These

NONLINEAR ELECTRON ACOUSTIC WAVES, I

21

averages for several lines and ratios for these line pair combinations are given in Table 111. When the 4686 A Hef line is visible, it can be used along with one of the atomic helium lines to check on the reliability of the plasma temperature measurements. If good agreement is obtained with that shown by pairs of atomic lines, one has a test of the validity of the Maxwellian distribution since the ionized helium line is excited by electrons in the tail of the distribution while the atomic helium lines are excited by the low energy electrons. In a time changing excitation, where changes take place more quickly than atomic systems can respond to them, the method must be modified. Thus, in practice the highest temperature in a breakdown wave is always observed at the leading edge of the wave even though the intensity of the optical signal there is zero. The fact that the quantum states providing the radiation have natural lifetimes and do not respond instantly to the excitation makes it very difficult to obtain early information about the structure of fast waves, a fact that is often overlooked in interpreting oscilloscope traces of luminous profiles. The slow linear rise usually present at the leading edge of a wave does not indicate a corresponding slow rise of the controlling variables in the wave, but rather the normal belated response of a linear system to a step function of excitation. In fact, until the atomic response is unfolded from the data, the observed profile is wholly deceptive and hidden wave structures may easily pass unnoticed. The differential equation governing the excitation process for the kth state is dNk/dt = - N J r I ( ~ ~ v ) n N (8) where Nk is concentration of excited states, T k their lifetime, and ( u ku> is the Maxwellian average given in Table 111. The electron concentration is n, and the neutral atomic concentration N . The photon flux at the photomultiplier is K(Nk/Tl),where K is an apparatus constant that varies with wavelength only as the cosine of the diffraction angle of the monochromator grating. The photomultiplier produces an output srgnal ski which is proportional to the input photon flux according to an efficiency factor hkj which is wavelength dependent. Substituting the observed photomultiplier signal skj for Nk in the differential equation yields

+

If the time history of a second line is observed as well, the ratio of these histories gives (on the right-hand side) the function tabulated by Latimer et. al. (32) and instantaneous temperatures can thus be determined

TABLE 111 MAXWELLIAN AVERAGES OF OPTICAL CROSS SECTIONS FOR HELIUM< Q l k v : A ~ CALCULATED D RATIOSAS OF ELECTRON TEMPERATURE"

5048

A

4113A ~

1.333 1.666 2.000 2.333 2.666 3.000 3.333 3.666 4.000 4.333 4.666 5.000 5.333 5.666 6.000 6.333 6.666

8.64E-5 8.83E-4 4.09E-3 1.24E-2 2.8 1 E-2 5.29E-2 8.76E-2 1.32E-1 1.86E-1 2.48E-1 3.16E-1 3.91E-1 4.70E-1 5.52E-1 6.37E-1 7.23E-1 8.10E-1

_

1.66E-4 1.62E-3 7.2953 2. I I E-2 4.63E-2 8.46E-2 1.36E-I I .99€-1 2.73E-1 3.55E-1 4.42E-1 5.33E-1 6.26E-1 7.19E-1 8.12E- I 9.03E-1 9.91E-I

4438 _

A

4121

A

4686 8,

5048 4713

7.93E-18 1.71E-I4 2.90E-I 2 1.15E-10 1.82E-9 1.58E-8 8.89E-8 3.68E-7 1.21E-6 3.30E-6 7.84E-6 1.66E-5 3.22E-5 5.77E-5 9.71 E-5 I S5E-4 2.36E-4

0.521 0.544 0.565 0.587 0.605 0.624 0.643 0.660 0.679 0.697 0.715 0.731 0.749 0.765 0.784 0.800 0.816

_

2.70E-5 2.85E-4 1.37E-3 4.16E-3 9.578-3 1X2E-2 3.05E-2 4.63E-2 6.55E-2 8.77E-2 1. I3E-I 1.40E-I 1.68E- I 1.98E-1 2.29E-1 2.61E-1 2.93E-1

4438 4121

A

FUNCTION 4121 4686

~~

4.59E-5 4.75E-4 2.22E-3 6.6OE-3 I .48E-2 2.75E-2 4.47E-2 6.63E-2 9.1 5E-2 1.2OE-1 I SOE-1 1.80E-1 2.15E-I 2.47E-1 2.80E-1 3.1 2E-l 3.44E-1

0.588 0.6 0.61 5 0.631 0.647 0.665 0.682 0.698 0.716 0.733 0.75 0.167 0.784 0.802 0.819 0.836 0.853

76000 36400 19100 I0960 6670 4280 2880 2070 1456

8.OOo 9.333 10.66 12.00 13.33 14.66 16.66 20.00 23.33 26.66 30.00 33.33 36.66 40.00 43.33 46.66 50.00 53.33 56.66 60.00 63.33

1.1 6E-0

1.49E-0 1.79E-0 2.06E-0 2.31E-0 2.52E-0 2.81E-0 3.19E-0 3.49E-0 3.73E-0 3.92E-0 4.07E-0 4.2OE-0 4.3 I E-0 4.41 E-0 4.48E-0 4.55E-0 4.61 E-0 4.66E-0 4.70E-0 4.73E-0

1.3lE-0 1.56E-0 I .76E-0 I .90E-0 2.01 E-0 2.08E-0 2.14E-0 2. I5E-0 2. I2E-0 2.06E-0 1.98E-0 I .90E-0 1.83E-0 I .75E-0 I .68E-0 1.62E-0 1 .SSE-O 1.50E-0 1.44E-0 1.39E-0 1.35E-0

4.21 E-l 5.43E-1 6.54E- I 7.55E-1 8.46E-1 9.26E- I I .03E-0 1. I7E-0 1.28E-0 1.37E-0 1.44E-0 1.49E-0 I .54E-@ 1.56E-0 1.61E-0 I .64E-0 1.66E-0 I .68E-0 I .70E-0 I .7l E-0 1.73E-0

4.57E-1 5.49E-1 6.19E-1 6.71E-1 7.08E-1 7.33E-1 7.55E-1 7.62E- I 7.48E- I 7.26E- I 6.99E-1 6.71E- 1 6.43E- 1 6. I6E- I 5.92E-1 5.68E-1 5.46E-I 5.26E-1 5.07E-1 4.89E- I 4.73E-1

9.04E-4 2.38E-3 4.96E-3 8.82E-3 1.40E-2 2.06E-2 3.28E-2 5.82E-2 8.83E-2 1.21E-1 1.55E-1 1.89E-I 2.23E-1 2.56E-1 2.87E-1 3.17E-1 3.46E-1 3.73E-1 3.98E-1 4.22E- I 4.44E-1

0.882 0.955 1.017 1.084 1.15 1.21 I 1.31 1 1.477 1.642 I .8l 1.971 2.135 2.30 2.46 2.61 8 2.77 2.935 3.07 3.22 3.36 3.505

0.921 0.989 1.057 1.13 1.195 I .26 1.366 I .54 1.71

1.885 2.056 2.226 2.394 2.56 2.72 2.884 3.042 3.2 3.35 3.5 3.65

506 230 I25 76.0 50.4 35.6 23.03 13.1 8.48 5.99 4.51 3.54 2.88 2.41 2.06 1.79 1.58 1.41 1.274 1.16 1.066

’ Multiply all averages by lo-” m3 sec-’.

h)

w

24

RICHARD G. FOWLER

Blais and Fowler applied this method to obtain time history temperature data for both proforce and antiforce waves, at 2.0 m and at 5.0 m flight distances (see Fig. 19). They used a Jarrell-Ash m monochromator. Mounted at the exit slit was a 7746 photomultiplier. The signal was sent to a Tektronix 519 oscilloscope, an instrument with a real time risetime of 0.28 nsec.

+

0

"

1

A

0

10

I

20

P 30

?

40

x 50

60

TIME (nsecl

FIG.19. Temperature profile for 3.0 Torr, 42 kV proforce and antiforce waves at 2.0 m 0 anti) and at 5.0 m (0 pro, anti). From (29) with permission of Phjs. Fhids.

(0 pro,

Although their data are for relativelyearly times ( 10 nsec) after the wave front arrival, it is not early enough to detect the temperatures believed by the theory to exist in the sheath layer and evidently applies to what Shelton called the quasineutral, or thermal region behind the front. As the data show and theory expects, the thermal region is relatively insensitive to the polarity of the wave. Examining the discrepancy between theory and experiment, however, suggests the desirability of repeating and improving the measurements; it also suggests that the thermal region, being one in which the electron gas is stationary, must be affected strongly by diffusion to the walls and by the presence of the conduction currents found in a real three-dimensional geometry, and that these must be ultimately added to the theoretical analysis. In Fig. 20, the data are present for the initial values of temperatureobserved at various stations for both proforce and antiforce waves. If the constants Kand h,, can be determined by absolute calibration against a standard lamp, then one can measure n, the concentration of electrons in the N

25

NONLINEAR ELECTRON ACOUSTIC WAVES, 1

W

W

a

W

I

0

,

2

.~ ~

__-1

4 STATION LOCATION (rn)

6

1

.-

FIG. 20. Peak (earliest measurable) temperature observed for proforce and antiforce waves at various initial voltages and locations. From (29) with permission of Phys. Fluids.

wave, as a function of time by direct application of the differential equation. Referred to the photomultiplier signal from a standard lamp S,, through the same monochromator optics, with the lamp ribbon and experimental tube only partially filling the monochromator aperture,

where W is the width of the observation slot on the experimental tube, and R , is the tube center distance to the monochromator, s is the slit width of the monochromator, ,f is its focal length. N , is its grating line number, O1 is the angle of diffraction, A , is the standard lamp ribbon area, R , is its distance, S, is the lamp signal, and P, is its monochromatic photon radiance (photons/ unit wavelength/sec/cm2/sr). I n Fig. 21 are given some electron density measurements i n the thermal region of the wave at the two stations together with proforce wave theory calculations. Both the general trend and final values of 1 7 , are encouraging. It is evident, however, at the conclusion of this review of the existing experimental work, that much remains to be done on careful measurement of velocity and structure of the wave.

26

RICHARD G . FOWLER

2

I

I

I

I

1

I

,2.0

1

5.0

21

-

n I

I

I

1

I

I

B. Secondary Breakdown Waves In 1959, Westburg (39,proposing to study moving striations in glow discharges, found that he had developed an apparatus for the study of secondary breakdown waves-those in which an electron space charge moves in an already ionized gas, augmenting that ionization. In lightning discharges, and in the work of Beams. these waves were called return strokes. The glow discharge offers controllable degrees of preionization at almost exactly the right level to simulate conditions in a normal return stroke. Using a 40 mm diam Pyrex tube 1.5 m long and gas purity procedures of the highest quality (which he later found to be unnecessary-the results being insensitive to intentional impurities, even mercury), he began each run with an 0.05 pF (estimated) capacitor held by a power supply at 1-3 kV across a 100 kR resistance and the experimental tube in series. This produced a glow current that, depending on the gas and its pressure, could be 1-20 mA. The resistor (which was in the anode side of the circuit) was then shunted with a mercury

27

NONLINEAR ELECTRON ACOUSTIC WAVES, I

relay leaving a residual resistance of 100 R. The circuitry was coaxial, and the experimental tube was run inside a metal shield. Waves were then followed from cathode to anode and return, by means of five probes through the tube wall and a pair of movable photomultipliers. N o details are given on the risetime of the equipment. At some brief instant after the shorting of the 100 kR resistor, a glowing spot appeared on the cathode and the luminosity moved as a sharp-fronted wave toward the anode with an initial velocity in the 2 x lo5 m/sec range which slowed to about half of the initial value as the wave reached the anode, after which a wave with a less steep front returned to the cathode at about the same velocity as the first wave had initially. The former should be called a secondary proforce wave, the latter a secondary antiforce wave. The currents delivered by the cathode in support of the first wave were 1-10 A , and were a slight function of cathode material but a strong function of gas and gas pressure. Typical voltage distributions reported by the probe stations as a function of time are given in Fig. 22. In Fig. 23 the derivative of these

0 Cothode

1.00

0.50 M

I .46 Anode

Fici. 22. Westburg’s potential distribution plots for argon at 97 p and 1700 V during first 160 nsec of secondary wave propagation with A1 cathode in tube 146 cm long. Asterisks indicate zero time which is the instant the potential begins to change a t the first plasma probe. From (38) with permission of PItys. Rev.

28

RICHARD G . FOWLER 7.50

-

5.00 P9

0

0 0

-

\

n

X

2

X

z

I >

\

>

v

W

2.50

0

Co thode

0.50

M

1.00

0

I .46

Anode

FIG.23. Electrical field and reduced field (Ejp) distributions during the first 100 nsec of secondary wave propagation as derived from curves in Fig. 22. From (38)with permission of Phys. Rev.

gives some measure of the electric field as the wave progresses. They both show the progressive character of the phenomenon, but the spacing of probes is so great that the profile and magnitude of the electric field is largely a function of the French curve used to connect the points. In Fig. 24 the velocity data read from these figures is plotted against the reduced field. Westburg gave an analysis of the phenomenon as follows: There is a brief delay while the negative charge which has been applied to the cathode attracts ions and repels electrons. The high concentration of ions collecting on the cathode face breaks down an oxide layer to permit a large current to be released which augments the electron space charge in front of the cathode. A zone of very high field now forms between this cloud of negative charge and the plasma of the glow discharge containing very fast electrons, mainly at the front of the cloud. Owing to their high velocity and the low pressure these electrons only lose energy over distances of centimeters. The velocity of the wave will relate to the mobility of the electrons in this front. Losses of fast electrons to the walls must be compensated by acceleration of new electrons by the field in the front, but is probably never complete, and so the front decelerates.

29 I O ' ~I L

I

1

I

I

1

I

1

1

1

1

,

I

8 -

6 4 -

2 0 W

m

-I \

k

7

10

--

:8 -

-1 W

'

6 -

4 -

2 -

lo6

I'

IO*

I

2

4 P

I

6

I

o

8

n

1

1

lo5

2

(V/M/TORR)

FIG.24. Westburg's velocity data vs. reduced field. From (38) with permission of Phys.

Reis.

Many important questions were left unanswered by this research. Could the waves equally well have been originated at the anode end by pulsing the anode? How did they vary with the degree of ionization of the glow discharge? Winn (39) undertook to obtain answers. He found the waves of both species could be made to originate at the anode end of the glow column by applying the voltage pulse to that electrode, although he found that a proforce wave could only be originated there if there was a neighboring temporary cathode available. He formed this by painting a silver coating on the wall around the anode.

30

RICHARD G. FOWLER

5x10~ W

v) \

3 i-

u0

4x10’ 3x10~

0

10

0

20

30

40

50

GLOW DISCHARGE CURRENT

60

70

80

(mA)

FIG.25. Secondary wave velocities in nitrogen as a function of shielding diameter and glow dischargecurrent __ f 2 4 kV, - - - -24 kV, pressure 0.5 Torr. After Winn (39) with permission of Phys. Rev.

6x1O7

1

O L

0

I

10

I

1

I

I

I

I

I

I

I

I

50 60 GLOW DISCHARGE CURRENT ( m A )

20

30

40

I

1

70

I

80

FIG.26. Secondary wave velocities in nitrogen as a function of pressure and glow discharge current 24 kV, - - - -24 kV. After Winn (39) with permission of Phys. Rev. ~

+

31

NONLINEAR ELECTRON ACOUSTIC WAVES. I

Winn’s apparatus was a Pyrex tube 0.06 m diam and 1 m long. optionally surrounded by a 0.1 m diam copper shield. He found that the presence of the shield decreased the velocity of both species of wave, but not in exactly the same way (Fig. 25). Thus it must be concluded that in addition tothe strengthening of the electric field at the front which is introduced by the induced charge on the nearby shield, there is an adverse structural effect on the proforce waves. The effect of glow discharge current on the two species is shown in Fig. 26. If the electron density in the glow discharge is calculated, a curve like that in Fig. 27 can be obtained for the velocity as a function of degree of ionization. SPEED OF LIGHT POINTS FOR FALL-IN-POTENTIAL WAVES L I E IN THIS REGION

V

k-“ +24 kV

1014

lod5

1Ol6

10”

lOl8

ELECTRON DENSITY AHEAD OF THE WAVE FRONT (M-’)

FIG.27. Secondary wave velocities as a function of electron density ahead of the wave front, 0, 0.5 Torr; 0 , m 0.25 Torr. After Winn (39) with permission of Phys. Rev.

C. Electric Shock Tube Precursors As the electric shock tube became an increasingly interesting research tool in plasma physics, the question of the discrepancy between Rankine-Hugoniot predictions and the excessive velocities observed (40) became a matter of increasing concern. Voorhies and Scott (41) called attention to the high intensity of the radiation which could be observed in advance of the plasmacoustic shock wave and its singular features. It showed small Doppler broadening and so was radiated by cold atoms. It would pass through a 0.4 W/m2 field seemingly without hindrance. It was not noticed in a sidearm off the main expansion chamber. Polarity reversal of the electric driver current had no effect. These motional results must be construed in the light of their apparatus’ minimum resolving time which was certainly very low, probably no better than 50 nsec.

32

RICHARD G . FOWLER

The research of Voorhies and Scott was done with a Josephson conical driver shock tube of 0.075 m diam, and shortly thereafter, Josephson and Hales conducted a detailed study of this device under higher resolution. They observed the precursor with both photostations and probe stations. The precursor was clearly distinguishable on the probes as a progressive wave of negative charge in both polarities of the driver current, with 24 kV on the capacitor, and showed up as a more slowly rising intensity of light on the photomultipliers. Sample figures from their work are shown in Fig. 28. A

C

FIG.28. Josephson and Hales’ observations of 24 kV precursor waves in deuterium. (A,B) probe, (C,D) phototube; (A,C) end electrode positive, (B,D) end negative. Numbers are distances from ring electrode in centimeters. From (13) with permission of Phys. Fluids.

They operated with the ring electrode of the shock tube grounded and observed that when the end electrode was positive the wave was of simpler structure and about tenfold higher velocity than when it was of negative polarity. The velocities they obtained were in the 106-107 mjsec range. Among other striking observations was the fact that the front of the plasmaacoustic shock wave, owing to the loss of negative charge into the precursor, bore a residual positive charge which accorded well with the energies available in the shock, Fig. 29, and that since the expansion tube in which the electrostatic probes were placed was made of copper rather than glass, it is apparent that the precursor can be propagated through a metal-walled tube. Finally, the parabolic shape of the probe potentials implies a slab of electrons passing the probe, and if one applies Poisson’s equation to the probe curves in Fig. 28, he can estimate the electron concentration behind the wave at 1.5 x 1013 mP3.Typical gas pressures ranged upward from 100 p.

33

NONLINEAR ELECTRON ACOUSTIC WAVES, I

1 1

to4

/

,

2

1

1

1

1

1

l

4 6 8 lo5 V =VELOCITY OF SHOCK FRONT ( M I S E C )

2

FIG.29. Potential jump across plasmacoustic shock vs. velocity. From Josephson and Hales (13) with permsision of Phys. Fluids.

McLean, Kolb, and Griem (42),recognizing that many of the peculiarities of the electromagnetic (T) shock tube derived from the precursor, examined the radiation with essentially the same results as Voorhies and Scott but worked with a 0.03 m diam quartz tube. Fowler and Hood (43), testing a new design of shock driver that was intended to eliminate the blast wave behavior of electric shock tubes having short drivers, discovered an intense wave precursor on their mirrorgrams of the shock tube flow. The wave had a speed in the lo6 m/sec range which seemed to increase linearly with capacitor potential, Fig. 30, and be unaffected by gas pressure at low pressure, decreasing as perhaps p-”’ above 0.5 Torr of H, . The wave did not seem to be affected by the electrical arrangements of surrounding conductors, or by a transverse magnetic field of 0.1 W/m2, and was of higher velocity in argon than i n hydrogen. When the recombination afterglow of a I A pulsed glow discharge was present in the region through which the wave was moving there seemed to be a suppression of the afterglow just in advance of the wave. They hypothesized that heat conduction from the

34

RICHARD G . FOWLER 6-

-

I

I

-

1

G 4W

I n -

\

I

g

-

I

I

-

i

0

-

0

2 -

0

0

-1 W

>

-

lo6

0

0

-

0

0

-

ff-

I

2

3

4

5

6

7

CAPACITOR POTENTIAL

8

9

10

II

(kV)

FIG.30. Wave velocity vs. capacitor potential. From Fowler and Hood ( 4 3 ) with perniission of Pliys. Rev.

plasma driver somehow supported the advance of this self-sustaining space charge wave. Pugh (44), using a simple single segment electric shock tube of 0.075 m diam, observed the precursor in hydrogen at essentially tlie same speeds as Josephson and Hales had found in deuterium, thus negating their explanation based on a beam of deuterium ions. He observed two additional effects that shed new light on the phenomenon. He observed that a Lucite endplate reflected the precursor wave with increased intensity that decreased somewhat when the plate was a metallic one, and that a round-ended rod inserted centrally into the expansion chamber from the downstream end developed a luminous sheath around itself of a thickness that increased linearly with distance along the rod. These effects resembled a reflected shock wave and a fully developed boundary layer, respectively. Pugh calculated the flow velocity from the wedge angle if the gas were hydrogen at room temperature as 7 x lo3 mjsec. This was very nearly the velocity of the plasma-acoustic wave, but that could not yet have reached the rod at this epoch. Unless one invokes some process like that seen by Schreffier and Christian (45),it seems more consistent with the rest of the picture to treat the flowing gas as composed of electrons. If then one introduces the electron mass in tlie boundary layer formula and a temperature of perhaps 3 x 104"K, the indicated fow velocity becomes 3 x lo6 m/sec, a result quite consistent with those of other observers for the precursor wave. Haberstich (46) began research on the precursor of the electromagnetic T shock tube and quickly decided it was a potential wave deriving from the high voltages present, so that he revised his apparatus and directed his attention to the research already reported above.

NONLINEAR ELECTRON ACOUSTIC WAVES, I

35

Gerardo, Hendricks, and Goldstein (47) observed the precursor in the course of microwave studies of an electric shock tube. They found that there was a background electron density in the first 130 p e c of 10l6 m-’. They believed that pliotopreionization accounted for the ionization observed. lsler and Kerr (48) embarked on a study of the electric shock tube as a spectral source in local thermodynamic equilibrium. They observed a luminosity front traveling faster than lo6 m/sec, but only when the ring electrode was positive. They observed also a second precursor that had an intensity maximum at about 1 p e c after discharge initiation and independent of polarity. They assigned the latter to scattered light and fluorescence, believing that it coincided temporally with a pinch in the driver gas. Lubin and Resler (49) injected shock waves from an electromagnetic T tube into a glass-lined wave guide and examined the behavior of microwaves entering at the other end of the wave guide and reflected from the advancing plasma. They found clear evidence for a wave precursor for which they measured speeds up to 6 x lo6 m/sec, as shown in Fig. 31. They used a transverse microwave cavity to acquire data on the electron concentration (Fig. 32) and the riselength to 900.; of final electron concentration in the front (Fig. 33). From their work they concluded that the precursor exhibits a wavelike structure with a definite electron density front; that the rise length of electron concentration as well as the average electron concentration behind the front are dependent on both the discharge field and ambient pressure; that the rise length and precursor velocity varied monotonically with the parameter E/p; that the precursor begins propagating down the tube prior to the main discharge, immediately (time resolution unspecified) upon application of the high voltage to the electrodes; and that currents of the order of one ampere were found to be distributed throughout the precursor electron density distribution. Much of the confusion involved in understanding the nature of precursor waves in the electrical shock tubes has been due to the difficulties in resolving possible photon and electron contributions. It was therefore a step forward when Russell (50), in spite of the negative results of Voorhies and Scott and McLean et at., found it possible to detect a clear-cut wave precursor in a 50 mm sidearm off from a 50 mni expansion chamber on a single segment collinear electric shock tube. I n a sidearm, arranged perpendicular to the main expansion tube, the luminous plasma in the shock tube can only be seen by double scattering, but a wave was still observed in this sidearm which propagated with the electron acoustical speeds predicted by theory. Mills, Naraghi, and Fowler (51) took up the research where Russell left off. During the course of their investigation, the apparatus, formed around a linear electric shock tube, evolved through several stages until the causes of the various precursor phenomena were finally established. In the

36

RICHARD G. FOWLER 10’

-u W

u)

- o6 \

I t

cu

I

0

-I W

>

t

I

los

I

I

I 1 1 1 1 1

Io6

I

I

I

I

I Ill1

10’

FIG.31. Precursor velocity dependence on E / p for (a) hydrogen and for (b) argon. Experimental data: 11 kV, 0 8 kV, A 6 kV, x Fowler and Hood. From Lubin and Resler (49) with permission of Phys. Fluids.

first stage, the multiple 3egment linear electric shock tube constructed by Hamberger and J o h n s r , ~(52) was used, as in Fig. 34. Failing to observe recognizable common phase points in the signals at two photomultiplier stations, Russell had introduced the sidearm expansion chamber, and in the process of reconstruction had simplified the driver to a single segment enclosed in a grounded metal box intended to reduce stray electrostatic fields which might generate breakdown waves. He observed a well-defined low intensity wave precursor in the sidearm, while the gross early luminosity in the main tube remained largely unchanged. He was able to measure the speed

r21:-: 10’’ 8 :

I

I

I

-

6-

I

-

--

I

200 M TORR

4 -

-

n

-Ia C

100 M TORR

10’6

8 6

-

42 -

I 0‘’

1

-I

I

I

I

IIV

Flu. 32. Average electron density determination of (a) hydrogen and (b) argon by nanosecond pulse technique and transverse cavity technique. 0.: j Microwave pulse technique; 0.. microwave cavity technique. From Lubin and Resler ( 4 9 ) with permission of Phys. Fluids.

I

I

Fit,. 33. Experiinental E?iperimenlal precursor precursor front front thickness thickness dependence dependence on on E/p Eip for for hydrogen hydrogen and and Fici. argon. 0 I 1I kV, kV, 0 C 88 kV, kV. .',.66 kV. kV. From From Lubin I.ubin and and Resler Kesler (49) (4Y)with with permission pernlission of ofPhys. Phys. for argon. Flit ds. N i li id$.

38

t

1IICHAKD C . FOWLER ~.

.. 70 ern MOLYBDENUM E L E C T R O D ES ~~~~

CONSTANT FLOW G A S INLET

% 0-20KV de

-

FIG. 34. Multisegmented electric shock tube. From (52) with permission of J . Qirant. Spertrosc. Radiat. Transfer.

of advance of the former at 3 x lo6 m/sec and that of the latter at 3 x lo8 mjsec, confirming the general opinion that the bulk of the early luminosity and ionization associated with the normal configuration of the electric shock tube can arise from energy transferred at light speed away from the driver plasma, yet isolating a wave phenomenon of the type reported by many observers, under circumstances where radiant energy support was clearly lacking. That there was also a wave precursor in the main expansion tube, which could not be perceived because of scattered light, was quickly shown by introducing a second sidearm downstream from the first and determining the time delay between the waves seen in the two sidearms. I t corresponded exactly to flights from a common point. Moreover, using a photostation near the head of a sidearm as a reference point, Liou (53)now found it possible to follow (with some difficulty) the precursor along the main expansion tube even in the presence of the scattered light, which proved to have little or no effect upon the wave. At this point in the research, replacement of the 99.95 % helium employed by Russell with 99.9995% helium resulted in such a great reduction in the scattered luminosity that the wave could be clearly followed even in the main expansion chamber, suggesting impurity as one more of the many causes of

NONLINEAR ELECTRON ACOUSTIC WAVES, I

39

discrepancy between various observers. In fact it was observed that the intensity and structure of the wave precursor were so sensitive to impurities that, for reliable results, it was necessary to flush out the apparatus after each discharge to remove the gaseous contaminants formed by wall disintegration in the preceding discharge. It is perhaps noteworthy that Hales and Josephson who first saw the precursor so clearly, had used a flowing deuterium system of gas filling. Russell had looked for any influence of electrical geometry on the wave precursor by introducing various configurations of shields around the sidearm and connecting them in various ways to the active elements of the driver, and had found no effect. Mills, Naraghi, and Fowler carried this idea further, showing that a metal section could be used in place of the Pyrex pipe with the wave reappearing at the far end after a delay corresponding exactly to the transit time through the tunnel. It was also possible to place a metal screen across the throat of the metal tube, with only a modest reduction in the intensity of the reappearing wave, and again, apparent normal transmission. It began to seem that the wave was independent of all support from the electrical circuits. Finally compelled to undertake a complete rebuilding of the apparatus in which the modifications i n Fig. 35 were made, however, Mills et a/. found that the wave was no longer observed at all. It then became clear that the wave is a breakdown wave, although not an electrostatic breakdown wave driven by the potential applied to the capacitor as is commonly the case with breakdown waves. It occurs when maximum rate of rise of current sets in in the driver, which then generates very large potentials across small inductances in the circuit of the driver discharge. Anything other than SHIELDING CHAMBER

/

ALUMINA TUBE

IGNITRON SWITCH

b-H

CAPACITOR

Fiti. 3 5 . Precursor-free electric shock tube arrangement. From (51) wilh permission of

P/I.I.S. Fluids.

40

RICHARD G . FOWLER

the most careful grounding and symmetrizing of return circuits can result in induced voltages of lo4 to lo5 V for about Q cycle of the discharge between some point in the driver and infinity, when currents change at 101’-10’2 A/sec across inductances of 10-7-10-8 H ; during this interval a breakdown wave can move as much as 3 m. Not only self-inductance in the ground circuit but also mutual inductance to it must be avoided. When mutual inductance is involved, the driving voltage can be larger than the capacitor voltage. Whereas Russell had taken adequate precautions against a breakdown wave of electrostatic origin (i.e., proportional to the instantaneous voltage on the capacitor) and against self-inductance, he had not considered the possibility of mutually induced potentials. One striking observation is that if a sufficiently long run is provided the precursor, it comes to a sudden halt as the current in the driver nears a maximum. SHIELDING HIGH POTENTIAL DRIVING ELECTROD:

CAGE

2,in. I D EXPANSION CHAMBER

IGNITRON SWITCH

1 -

FIG.36. Precursor-producing electric shock tube arrangement. From (51) with permission of Phys. Fluids.

Russell’s actual arrangement is shown in Fig. 36. The inductance labeled L is formed by the very short heavy connection from electrode No. 2 to the capacitor (including any interior wiring of the capacitor). The inductance L , was a heavy strap 8 in. long and 3 in. wide running from the capacitor to the shield can. The mutual inductance between circuit A and circuit B developed the potential difference between the shield can and electrode No. 2 that initiated the precursor. No one can determine from published work alone, without examination of the actual apparatus structures used, whether the necessary emf’s in the ground circuit were present in any given experiment. It seems evident, however, that Hales and Josephson had them, that Fowler and Hood most certainly did, and since it is probably never possible to be free from such a

NONLINEAR ELECTRON ACOUSTIC WAVES. I

41

wave in the Kolb T-configuration, it seems reasonable to look for a similar cause in the experiments of Lubin and Resler. It is probable also that the apparatus of Hamberger and Johnson was free from this effect, since they in fact (and Russell, subsequently, in that same apparatus) looked for it and apparently could not find it although one cannot be certain for they did not look at sidearm flow. Finally, the observations reported by Isler and Kerr of a precursor wave under only one discharge polarity may well have been a result of shifting a small amount of inductance to or from the ground circuit while achieving the polarity reversal. It is worthy of remark that the observation of a precursor by Hood and Fowler, but none by Hamberger and Johnson, although superficially in the same style of apparatus (a multisegment electric shock tube) is reasonable because Hamberger and Johnson, in the hope of increasing the efficiency of the shock tube for higher power operation, carefully tailored the return circuit of each segment into a parallel-plate, double-coaxial structure which minimized ground circuit inductances. Mills et al. found, however, that although the connection to an electrical emf is necessary for the existence of this precursor, it is not sufficient to account for all of its peculiarities. They proved that the precursor derives the bulk of its vitality from the adjacent driver plasma by secondary processes and it is this which gives it its intense luminosity, insensitivity to intromittent electrodes, insensitivity to electrostatic grounds, etc. They then designed an apparatus which would produce the precursor intentionally. They used a linear constant bore shock tube with a 2 in. expansion chamber of Pyrex pipe. The driver chamber was made from a special high purity alumina tube. The electrodes were either of platinum or molybdenum with indistinguishable results. They used fast rise photomultipliers (low sensitivity with 1 nsec risetime, high sensitivity with 2.5 nsec) and observed the luminosity associated with the precursor to obtain velocities, temperatures, and electron densities. Each photomultiplier was tested against the inverse square law to set linearity limits on usable signal size. Photomultiplier signals were fed through terminated 50 R cables to an oscilloscope of risetime commensurate with the photomultiplier used. Wave-speed studies of the precursor were made of both proforce and antiforce waves utilizing the same techniques. By applying Lenz' law to the circuit of Fig. 35, it can be seen that if the floating electrode of the driver section is pulsed positively, then the induced emf is such that electrode No. 2. becomes positive with respect to ground, thus generating antiforce waves. When electrode No.1 is pulsed negatively, proforce waves are propagated. The driving voltage on the electrode was measured in relation to the capacitor voltage and found to be 0.80 of the latter. Wave speeds were obtained by measuring the time of flight of the wave for viewing slots 0. I m apart along the discharge tube. The widths and effective

42

RICHARD G. FOWLER

heights of the slots were adjusted to insure signals of near constant amplitude for accurate velocity measurements. The wave-speed data were plotted as a function of distance and the results for waves of both polarities appear in Figs. 37 and 38. The attenuation that is a well-known characteristic of breakdown waves (23,25) is clearly visible. Blais and Fowler had found the attenuation to be generally exponential with a decrement that was both pressure a n d voltage dependent. Mills

0 85 0 77

I

I O ~ L A - L - - .

I L - L _ . POSITION ( m )

FIG.37. Velocity vs. position curve, proforce waves. Numbers on each curve are attenuation i n n i - ' . 17 kV, 0.2 T o r r ; 1 3 16 kV, 0.2 T o r r ; 0 14 kV, 0.2 Torr; 0 I 7 kV, 0.65 Torr; 16 kV, 0.65 Torr; A 16 kV, 1.25 Torr. After Mills, Naraghi a n d Fowler (51) with permission of Phjs. FINids.

et at. found that precursor breakdown is under a n entirely new regime, with the attenuation not only as much as 30 times greater but increasing slightly with applied voltage rather than decreasing as i n the electrostatic apparatus. The increase was not marked over the small range of variation studied and might have been either linear with voltage or merely constant at 0.95 m-' within the experimental error. N o clear distinction was found between proand antiforce precursors in this respect.

43

NONLINEAR ELECTRON ACOUSTIC WAVES, I

,"

08

10

I2

14

I6

18

20

22

POSITION ( m )

FIG.38. Velocity vs. position curve, antiforce waves A 17 kV, 0.2 Torr; n 16 kV, 0.2 Torr; V 14 kV, 0.2 Torr; 0 17 kV, 0.65 Torr; 17 kV, 1.2 Torr; A 14 kV, 0.65 Torr. Numbers on each curve are attenuation in m - ' . From Mills er a / . ( 5 l ) with permission . o f P h ~ s Flitiris.

When the technique of Blais and Fowler was used, and thevelocity data were extrapolated back to the electrode, it was found that they then fell somewhere near where corresponding data taken in the electrostatic breakdown apparatus would have fallen. However, their trend more nearly indicated that the velocity has a q 2 p - " 2 dependence than the 'pp-' dependence found in the breakdown wave apparatus and expected by theory. In Fig. 39 they are plotted against this new parameter. Figure 40 shows the measured peak electron temperature in a proforce wave as a function of wave advance. Figure 41 shows the measured electron temperature through the wave. The temperatures were obtained by using the line ratios at the points on the optical profiles corresponding to 10 nsec intervals after the onset of luminosity. Figure 42 shows the measured peak electron temperature of an antiforce wave as a function of gas pressure at a fixed position along the sidearm. When a single spectral line was used and the system calibrated absolutely against a standard lamp, the electron densities in Fig. 43 were obtained.

44

RICHARD G . FOWLER

I

105

3

lo3

C2

7

5

lo4

6‘”(orb units)

FIG.39. Extrapolated initial velocity vs. q*p-’’’, From (51) with permission of Phys. Fluids.

They are quite similar to those reported by Blais and Fowler for breakdown waves. Although the precursor cannot exist without a driving voltage, it is apparent that given that voltage there is another and more important mechanism aiding. Mills et al. had satisfied themselves that this could not be radiation from the driver. Fowler and Hood, believing that there was no field at all present, had postulated that the very hot electron gas existing in the shock tube driver after the initiation of the discharge might provide an energy

0

0

t“ 4

12

14

16

18

20

DISTANCE (rnl

Fro. 40. Peak electron temperatures in an advancing proforce wave at 17 kV and 1.2 Torr. After Mills, Naraghi, and Fowler (51) with permission of Phys. F1uid.r.

NONLINEAR ELECTRON ACOUSTIC WAVES. I

oLLLu_i 1

0

40

80

I20

-

45

1

200

I60

TIME (nsecl

FIG.41. Temperature profile in time of a 17 k V proforce wave at 1.5 m from the electrode, 1.2 Torr. After Mills, Naraghi, and Fowler (51) with permission of Phjss. F/uids.

source for precursive effects. Energy could then be transported to the wave front via electron-electron collisions. To test this mechanism, which seemed to be the only remaining one for energy transport to the wave, several experiments were conducted. The first was to examine the breakdown wave which could be produced by the apparatus without plasma contact. A brass plate was pressed tightly into the ring electrode of the driver. Once in place, the driver was completely closed while the discharge circuitry _

w [

3

U

w

Q

L

_

~

-

500

+

? I L

0

0 50

I 00

I50

PRESSURE (Torr)

FIG.42. Peak temperatures vs. gas pressure for an antiforce wave at 17 kV, 1.5 m from the electrode. After Mills, Naraghi, and Fowler ( 5 / ) with permission of Phj7s. Fluids.

46

RICHARD G. FOWLER

I

I

I

I

I

I

I

I

I

1

I

I

I

I

1

1

TIME (nSECI

FIG.43. Electron density vs. time for a proforce wave. After Mills, Naraghi, and Fowler

(51) with permission of Phys. Fluids.

of the shock tube was not affected. In this way none of the hot driver gas could escape into the expansion chamber, but a breakdown wave precursor might still be launched and propagated. All measurements were made in the sidearm. The plugged driver resulted in a decrease in the velocity and a marked decrease in the intensity of the wave, but the wave still existed. N o further tests were made to see if the attenuation and E / p dependences returned to normal. Instead the hypothesis that there might be another major energy transfer mechanism operating in the precursor prompted the initiation of investigations with magnetic fields. It is well known that a transverse magnetic field has the effect of reducing the flow of heat in a plasma, to a first approximation, by a factor (1 + m t e c r 2 ) - * ,where w,, is the electron cyclotron

N O N L I N E A R E L E C T R O N ACOUSTIC' WAVLS, I

47

frequency and T is the electron collision time. To the extent that the precursor relies on transport processes, the establishment of a transverse magnetic field along the shock tube should increase the heat insulation properties of the gas and thus impede the propagation of the precursor. I n the experiments discussed below, both ;I transverse and axial magnetic field were utilized. The latter was invariably found to have no first order effect upon the Now. Magnetic fields of up to 0.27 W/mZ were employed. Figure 44 shows the temperature profile observed for the wave at a point downstream from a transverse magnetic field. The temperature was measured as a function of the axial distance behind the onset of the wave. The reduction in the temperature of the supporting column is striking. -

7--0 ~

. "

1

o l ' L ' . 0

0.2

04 06 DISTANCE (rn)

08

10

FIG.44. Electron temperature along the wave profile at a point downstream from a transverse magnetic field : B 0 G, .B = 800 G B = 1400 G , I ,B =~ 2400 G. After Mills. Naraghi, and Fowler ( 5 1 ) with perinissioii of' Phj..~.Nuids.

Accompanying the reduction in downstream wave temperature is a reduction i n wave velocity. The velocity was measured over a 0.27 m span at a distance of 0.16 m from the magnetic field. Using the Ramsauer value of 5 of 0.95 x sec for 10 eV electrons, the theoretical curve in Fig. 4.5 was plotted. These results certainly indicate an energy transport process, but

48

RICHARD G . FOWLER I .8 x

lo-'

I

I

I

I

I

1

I .6

c

1.4

\

u

W

v)

-I>

1.2

I .o

0.8

0.6 0

1

I

1

I

I

I

2

3

4

5

0'

I

6I

(w'/ M 4 )

FIG.45. Wave velocity after passage through a transverse magnetic field. After Mills, Naraghi, and Fowler (51) with permission of Phys. Fluids.

0

I

2

3

4

5

6

7

8x162

B2h2/m4)

FIG.46. Wave velocity during transit of a transverse magnetic field. After Mills, Naraghi, and Fowler (51) with permission of Phys. Fluids.

NONLINEAR ELECTRON ACOUSTIC WAVES. I

49

whether it is conventional heat transport as envisioned by Hood and Fowler or some sort of Thomson effect has not been established. Although the wave speed downstream is slowed by a magnetic field, the wave speed in the field may have been enhanced (Fig. 46). This result is not unreasonable because the field can produce a rise in the local electron pressure, both via T, and n,. Using the two probes opposite each other and passing through the walls of the confining tube, Mills et al. determined the emf set up by the electron flow in the magnetic field. The region of electron motion in the ionizing wave was found to be about 0.154.3 m long and of fairly constant speed. The electron speed calculated from this experiment was found to be less than the value found by previous methods, indicating that it is smaller than the velocity of advance of the luminous wave front, as it should be. By introducing an electric field probe into the flow, Mills et al. were able to determine the critical value of the B-field at which the electric force on the electrons becomes equal to the magnetic force to be about 0.16 W/m2, with u = 8 x 10' m/sec, giving an EZ in the wave front of 1.28 x lo5 V/m. Using an antenna probe calibrated at 1.5 M H , they searched for rf fields around the apparatus. At a point 0.3 m down the sidearm, signals for E, and EB were detected as having approximately one fourth the magnitude of the signals for EZ. The approach of the electric field E , was sensed some 100 nsec before the arrival of the luminosity of the front, and Er underwent a sharp lowering of rate of increase at the onset of luminosity. The frequency spectrum of Ez contained at least three components: the base frequency of the capacitor discharge 3 x 10' H, a second frequency of 2.5 x 10" H, and a third of 2.5 x lo7 H. Applying the probe calibration they estimated the amplitudes of these fields as 8 x lo4 V/m, lo3 Vim and 10' V/m. The probe was probably insensitive above lo8 H. In an earlier version of this apparatus, Latimer et a/. reported finding two 6 x IO5"K peak excursions of temperatures at 30 and 55 cm from the driver electrode which may have indicated standing waves of 5 x lo8 H, whose amplitude would then have been 2 x lo4 Vjm. These peaks were found to build up late in the history of the flow (10 psec) and no peaks were found further down the tube than 30 crn nor were they observed subsequently in the Mills et a/. apparatus. A search for microwaves in the latter apparatus was made without success. Mills et a/. therefore concluded that the delivery of energy to the precursor primarily related to the base (quasi-dc) frequency of the capacitor discharge. The work reported by Latimer et a/. is of additional interest because it bears on conditions in the photopreionized gas in the main expansion chamber, which is seen to range in temperature from a minimum of 20 eV to a maximum of 70 eV as shown in Fig. 47.

50

RICHARD G . FOWLER

90

I

I

I

I

I

1

I

I

I

-> 70 80

u

W

5

60

l-

a

4

50 40

z

0 [L

I0 W

30 20 10

0 DISTANCE FROM DRIVER (M)

Fro. 47. Measurements using three ratios, of the electron temperature in the precursor as a function of distance from the driver at a time 10 psec after the driver is triggered. The driver voltage was I2 k V and the helium pressure was 0.65 Torr. After L a t h e r , Mills, and Day ( 3 2 ) with permission of J. Qunnr. Spec/rosc.Rarlint. Tramfir.

D. Laser Precursors At the outset of the research of Mills, Naraghi, and Fowler it was believed that a component of the electric shock tube precursor would be found, caused purely by energy transfer processes from the extremely hot driver plasma, sans electric field support. This proved to be an illusory belief. There remained the untested possibility that the driver plasma was not hot enough, and that something in the thermonuclear range would still display an energy transfer supported precursor. Koopman (21) has now demonstrated that an electron precursor does accompany the generation of a focused laser plasma. He observed a spherical electron flow at distances up to 0.3 m from a carbon target on which a 6 J , 30 nsec laser pulse at 6943 A was focused by a 0.1 m lens. He observed the plasma with Langmuir probes and X-band microwave interferometry. The probes indicated the near instantaneous arrival of a small negative charge excess, followed by a fairly thin strong double layer, first negative then positive, followed by a relatively long zone

NONLINEAR ELECTRON ACOUSTIC

WAVES,

51

I

of negative charge excess, and finally positively charged plasma-acoustic flow of the laser plasma itself. The microwave probes indicated that the electron concentration itself increased abruptly near time zero and inverse square law tests proved this to be ultraviolet photopreionization. The concentration then underwent a sudden step-up as the double layer passed, followed by a plateau during the passing of the long zone of negative charge, and an enormous increase when the plasma-acoustic wave arrived. These results are summarized in Figs. 48 and 49. Observations at larger distances resulted only 0.50

I

1

I

I

1

I

I

TIME ( p S E C )

FIG.48. Position vs. time graph of features observed on probe and microwave diagnostic devices. 0 Front, x plateau, 0 laser plasma. From Koopnian (21) with permission of Phys. Fluids.

in a dispersion of the features above. In Fig. 50 the effect of gas pressure was investigated, and the results fitted to a theory that the momentum loss rate of the electrons is dominated by ionization. This yields the correct density dependence of P ' ' ~ for the empirical constant CI describing an exponential growth of the velocity of the wave front. Koopman concluded that he indeed had an electron ionization wave for the propagation of which no high voltage or electric current is required. He felt it could be described as a self-consistent balance of electron diffusion and space charge coupled to collisional ionization processes. This raises once again the question of whether the ordinary shock tube diffusion precursor observed by Weyrnann and analyzed by Pipkin (54) is capable of becoming a shock-fronted electron ionization wave at extreme energies.

52

RICHARD G . FOWLER

"""'"

+ 60

- 3

?

5

LD

-0 2

+ 40

u

c

I

+20:

0

O

-> A

?z F

0

n

- 20 LASER PLASMA I

0

I

I

I

2

3

I

4

1

5

I

6

- 40 7

- 6_ 0 _

TIME ( p S E C )

FIG.49. Electron density n, (A), plasma potential V , ( O ) ,and floating potential Vf (0) as functions of time at a location 14.9 cm from the plasma origin, as measured by a cylindrical Langmuir probe. After Koopman (21) with permission of Phys. Fluids. 2.5

-

2.0

1.5 u 1

w

cn

r-

E

1.0

a 0.5 0

0

I 2

3

4

5

6 7 8 9 1011 1 2 1 3 1 4 (MTORR)"~

FIG.50. Plot of velocity proportionality constant A vs. square root of gas pressure, using data on average velocity of ionizing fronts obtained from microwave apparatus. After Koopman (21) with permission of Phys. Fluids.

NONLINEAR ELECTRON ACOUSTIC WAVES, I

53

E. Slow Proforce Waves Although the broad assumption is made that primary breakdown waves are always fast waves and secondary waves are even faster, some evidence has always existed dating even from Mitchell and Snoddy (1947) that supposedly primary waves could travel at speeds below the ionization minimum (2eq1,/m)”~. Scott (55) has investigated this matter and found that inherent in it lies the possibility of further understanding of the entire phenomenon. Antiforce waves, he finds, are unique in structure for all pressures, gases, and voltages and exist without any apparent lower bound of velocity at ( 2 e q 1 J m ) ~ / ~ . Proforce waves, on the other hand, are found to exist in two styles, not one, as supposed. One style shows velocities below the ionization velocity minimum and has a slowly rising intensity at its front. The other does not exist below the velocity minimum and shows a discontinuity in structure at its front. Since, as Shelton showed, the velocity minimum arises from the shock conditions at the breakdown front, it is apparent that the slow proforce wave need not meet this shock condition and that the theory must be reexamined in search of other initial conditions. F. Tertiary Breakdowti Waves

Barach and Sivinski (56), working with a collinear electric shock tube with the rear driver electrode grounded and spaced a few centimeters from the ring electrode, introduced a second grounded electrode downstream in the expansion chamber at a distance of 14 to 30 cm. They worked in argon and observed the existence of a primary breakdown wave with speeds greater than lo6 m/sec moving away from the ring electrode, which was followed (probably after an unobserved secondary return wave) by a slow-moving wave having a large current (Fig. 51) at its front. They called the latter a precursor arc and observed that it was present only in the proforce mode (i.e., when the ring electrode was negative). They found that it produced substantial preheating of the expansion chamber gas so that the speed of the plasma-acoustic shock wave was strikingly enhanced. Figure 52 gives this increase as a function of initial pressure; Fig. 53 gives it as a function of capacitor voltage. The passage of the waves was detected on Langmuir double probes flush with the wall in small side tube recesses and with a photomultiplier. Barach and Sivinski determined that the velocity of the front fitted well with the equation V = E/pne or, in other words, was primarily governed by electron drift. This is illustrated by Fig. 54 where they plotted V VS. conductivity ( I / p ) at constant E. The electron concentration 11, they found to be nearly constant at 1.75 x l O I 9 m-j. It seems certain that we have here the first clear-cut information on tertiary breakdown waves.

FIG.51. Precursor current vs. capacitor voltage. Initial pressure 0.55 Torr, 0 magnetic . probe, 0 single probe. After Barach and Sivinski ( 5 6 ) with permission of P ~ J ' sFluids. I

0.018

50

0.016 0.014

0

40

0.012

W

P

ln

W

a.

m

' zi 0.010

-

30

P

;0.000

52

I

V

a

a

z

v)

20

0.006

0.004 10

0.002 I

I

0 0

1000 BANK VOLTAGE

500

I

1500 (V)

0 2000

FIG.52. Shock wave speed vs. capacitor voltage for an initial gas pressure of 0.55 Torr,

0 with arc, 0 without arc. After Barach and Sivinski (56)with permission of PIiys. Fluids 54

I

0 014

-

0 012

-

0002

-

I

I

I

50

I

I

-

40

- 10 1

I

I

I

I

FIG.53. Shock velocity vs. initial pressure. Capacitor voltage 1000 V, 0 with arc, 0 without arc. After Barach and Sivinski (56) with permission of Phys. Fluids.

VELOCITY

(M/SEC)

FIG.54. Electron front velocity vs. gas conductivity for constant voltage gradient 0 0.345 V/m. After Barach and Sivinski (56) with permission of

0 0.12 Vim, A 0.16 Vim, Phys. Fluids.

55

56

RICHARD G . FOWLER

111. THEORIES A . Early Approaches The breakdown of gases between closely spaced plane parallel electrodes led in 1889 (57) to Paschen’s law that for a given gas and electrode material, the breakdown potential is a simple concave-upwards function of pd. Thereafter it is a strong function of the nature of the gas and a weaker function of the electrode material. Townsend (58) undertook to explain this by the concept of an electron avalanche in which electrons with energies acquired in random flights in the electric field produced ionization ( a processes). He envisioned a single electron as the initiator of each avalanche and the avalanche as connecting the electrodes in establishing conduction. Since a single avalanche might initiate a discharge but could not sustain one because the electrons in it, however multiplied, would eventually all be swept to the anode, he supplemented the a process with an in-gas process in which the positive ions produced by the electrons replaced the initial electron by their collisions with the neutral gas (p processes). Subsequently this process was shown to be quite improbable and Townsend advanced an electrode process in which the positive ions or excited atoms released new electrons from the cathode (y processes). With the discovery of the photoelectric effect, the promptness of the y processes could be better understood, and new life was given to the p processes, although they remained less probable sources of electrons than they processes, Much investigation of the Townsend hypothesis has been made. Paschen curves taken under these conditions can certainly be explained by these processes, of which the first and third are preponderant with closely spaced electrodes (59). The beauty of the concept of the Townsend avalanche dominated efforts to explain spark discharges in all their configurations for a half century. The overly brief transit and current growth times observed by Rogowski (60) were explained with superficial adequacy by von Hippel and Franck (61). The existence of long sparks with both pro- and antiforce directions of advance was regarded by Loeb and Meek (62) as rationalizable by a concatenation of Townsend avalanches aided by photoelectric fi processes. Even Mitchell and Snoddy, at the culmination of the research begun by Beams on low pressure breakdown in tubes, invoked the avalanche as a fundamental building unit in the wave advance although it led to a velocity dependence on field strength of El’’ which was not the dependence found by them or by any observer since. As time passed it became evident that although the Townsend avalanche

NONLINEAR ELECTRON ACOUSTIC WAVES, I

57

theory does indeed describe the growth of one single electron's ionization swarm until the Debye length in the swarm becomes less than the dimensions of the swarm, thereafter a general fluid response of the system must be considered. Raether's cloud chamber studies of the avalanche revealed this dramatically (63). He found departures from avalanche behavior at electron densities of I O l 5 m-3 with electron temperatures of -1O'"K and swarm dimensions of 1 mm. Penning (64) made the strong case that space charge and multiple processes must be considered before a solution of the long spark problem could be expected. Paxton and Fowler (22) advanced the hypothesis that when the ionization has transcended the swarm theory limits, the electron gas can be described by fluid dynamical equations. They assumed that the leading edge of the ionization wave was amenable to treatment as a shock front. In their view, the principal drive force for the wave came from the electron pressure generated by the high temperatures of the electrons in the electric field. Thus no great distinction was to be expected between proforce and antiforce waves. They called attention to the importance of the zero current condition at the leading edge of primary waves, but neglected to include ionization energy losses in their equations (which will subsequently be seen to make their results apply to secondary or tertiary waves rather than primary waves). Their model was that of an infinite plane wave, while the experimental situation is always strongly three-dimensional, and they had very scanty data with which to make comparison ; nonetheless, the results were encouraging. The chief advantage of the photoionization-avalanche (streamer) theory had been its ability to permit the existence of antiforce as well as proforce waves. The fluid theory could also do this. The chief disadvantage of the streamer theory was that it used a spherically symmetrical energy transfer mechanism to transport ionization forward to new sites while the streamer is in fact strongly filamentary and grows always at its end, in a direction roughly parallel to the cylindrical sides of the filament. The fluid theory expects growth at this tip because the electric field will be strongest there. It is probable that the budding" of branches onto a spark streamer is caused by and therefore is a measure of the frequency and importance of photoionizing processes as in lightning, or in the Lichtenberg branching studied by Nasser and Loeb (6.5). Burgers (26) attacked the problem of the structure of a one-dimensional electron wave by formulating the collision operators which Paxton and Fowler had left undefined and attempting an overall solution of the equations by integrating them through the wave from -co to +a.He found that the velocity of the wave must be of the order of electron sound speed (kT,/rn)1'2, but his attempts to relate the velocity to the impressed electric field did not

-

"

58

KICHARD G. FOWLER

agree with the experimental results of Haberstich. He found V K E"'. The difficulty may have lain in the form he chose for the electron ionization coefficient, postulated as

dnldt = aiNn(v - V ) . He chose this form because it made the integration simple, but there is no reason to expect that ionization will cease when the electrons are at rest in the frame of the heavy particles, i.e., when u = V. Thermal ionization may continue for some time afterward if T, is large. The formulation of the zeroorder approximation should have been

Lubin (66) approached the problem of the electric shock tube precursor on the assumption that it was electromagnetically coupled to a transmission line formed by the plasma-filled tube and moved at the group velocity for that line. The idea was an appealing one, since experimenters on all types of these electron waves have expressed the feeling that there is a resemblance between the wave's behavior and that of a transmission line. The analysis provided by Lubin begs the question on two essential points, however. ( 1 ) He chose an ad hoc equation for electron production which has no theoretical basis and was justified only in that it produced velocity dependence agreement with the observations

and ( 2 ) he selected the velocity of his unloaded line as that value which gave scale agreement with the observations despite the fact that the magnetic coupling is less than that of the electric coupling. The agreement can thus only be regarded as contrived. B. Rinu'anienial Equatioris it1 One Diiiietuion The fundamental equations are those of a three-component fluid (I). It is in this respect that the formulation differs from the problem so commonly treated in plasma physics, the fully ionized two-component case. Only small differences exist between the individual equations in the two cases, but they are crucial. The principal one is that the charged particles interact with each other only by way of the electric field, but both species interact with the

59

NONLINEAR ELECTRON ACOUSTIC WAVES. I

preponderating neutral gas viscously. The interaction of the positive ions with the neutral molecules is so great that it has been found necessary to separate the equations of momentum and energy for heavy particles. In one dimension, the system of equations descriptive of the above model is

etrE -

1

W)V = eN, VE

+ A,

(i 1 -it7iu2

-

(: 1

Ai -niv2

.

(21)

In the above equations, the ion iind atom velocities and temperatures are set equal because of the strong collisional interaction between the two species. The symbols t 7 , z i , p and MIdenote electron density, velocity, pressure and internal energy, respectively; capital letters denote corresponding quantities for heavy particles, with a subscript i if special reference to ions is needed. E is the electric field (applied field plus space charge field) and q is the heat conduction vector. The ionization frequency is denoted by /I while A, denotes an elastic transfer operator for the indicated quantity and A i is a similar inelastic transfer operator. These will receive detailed discussion later.

60

RICHARD G . FOWLER

That the basic equations already imply Kirchhoff’s fundamental statement about the electric current can be shown as follows. Subtracting Eq. (15) from (16) and multiplying by the electron charge, one has

(ajar) [e(N, - n)] + (a/dz) [e(Ni V - nu)] = 0.

(22)

Employing Poisson’s equation, this becomes

(a/&)

+

[EO(dE/dt) e(Ni V - nu)] = 0,

(23)

or E,

(aE/at)+ e ( N , V - nu)

=

i, ( t ) .

This is Kirchhoff’s law and says that in one dimension the total current, convection plus displacement, is independent of position. The application of this result will vary from one set of conditions to another, but in the important case of a wave moving into a neutral gas, one can generally evaluate io( f ) as zero on the argument that N and n are zero ahead of the wave, while E must be constant somewhere sufficiently in advance of the wave. This result is known as the zero current condition. Its corollary is that if one places himself in the wave frame of a steady profile wave, he observes that N i V = nu.

(25)

By summing the system of individual species equations, the ordinary global equations of fluid dynamics can be obtained (68). It was first noted by Paxton and Fowler (22) that these equations lead to the description of an electron fluid wave if and only if one is meticulously careful not to ignore the small difference between ion mass and neutral mass. Adding Eqs. (16) and (17), one obtains a second theorem

mi+N)+ -(Ni a + N ) V = 0, at aZ

(26)

which can be fittingly termed, in nuclear jargon, the continuity of baryons. In the hypothetical steady profile rest frame, one can integrate across the wave to obtain (Ni

+ N)V = N o Vo ,

(27)

where N = N o , N i = 0, and V = V, in the neutral gas ahead of the wave, Vo being the frame velocity. One can express this result in the words “ baryon flux is conserved in a steady profile wave.”

61

NONIJNEAR ELECTRON ACOUSTIC WAVES, I

1. The Collision Operators and the Electron Equations

Before one can proceed to investigate the detailed structure of electronacoustic waves it is necessary to ascertain as Burgers did the forms of the collision operators. The elastic collision operators are well known. The momentum operator for m e M is

Ae(mu) = K , n ~ ( t 7 V).

(28)

In a n approximation which has a maximum inaccuracy of about 40//,, the coefficient K , takes the form (with Maxwell-Boltzmann statistics)

where n is the experimentally determined total elastic collision cross section for momentum transfer. The elastic energy operator, also for m < A4 is

Ae (+mu2)= (2m/M) nK, [$kT,

+ (m/2)(c

-

V j 2 ]-tK,mn(z’- V )V . (30)

At high temperatures in helium ( >5 x 1 0 ° K ) . K , is temperature independent and K , i p = 2.41 x lO’/sec-Torr. The essential feature of this operator is that, n o matter what the value of n , the well-known factor 2 m / M is present in two of the three terms. Consequently they may be neglected when V. The average K , is one of a generally useful family of averages over distribution functions which occur in these problems. In the serious studies of swarms made in 1930-1950 it was frequently noted that the distribution function could be non-Maxwellian. A complete theory of such distributions was devised by the efforts of many workers [Morse et a(. (68), Smit (69), Druyvestyn (70)l. The non-Maxwellian character usually took the form of a n enhancement or depletion of the number of fast electrons relative to the mean velocity and therefore would have more effect on the ionization rates than on mechanical constants such as K , . Accordingly, as a first approximation it is still useful to have Maxwell-Boltzmann averages for this and similar quantities. These were given in a previous article in this series but have been revised for the new data of Golden and Bandel ( 7 / ) and recomputed on the assumption that all cross sections decrease reciprocally with velocity (Born approximation) at large velocities. The new results are given in tables in the next section of this article It is difficult to impossible to approach the inelastic collision operators from first principles because the more interesting ones involve three-body collisions. Shelton has shown how the vector terms which are involved can be rl#

62

RICHARD G . FOWLER

identified by requiring that the basic equations be invariant under a Galilean transformation. Then the essential new terms are

Ai (mu) = pnin V

(31)

and

Ai(fmu2) = +pmnV2 - pnecp,, where we must add the term -jnecpi ad hoc, to recognize a portion of the inelastic energy loss which eludes the transformation test because it is a scalar. Physically speaking, the inelastic terms state that the nascent electrons bring to the electron system only the momentum and energy they possessed when they were attached to the heavy particle system. The transformation technique presents additional terms of the form K ( P- V ) in the momentum operator and K(n - V ) V in the energy operator which can be recognized as the exchange to be expected even if the collision were elastic and which can be regarded as included already in the elastic coefficient K , by choosing the total cross section for the cross section 0 . There is likewise a term K ( v - V)' presented in the energy operator that must be of the same character as the similar term occurring in A e ( ~ m t l z )and hence must bear a factor of order 2m/M making it negligible. Including these terms, the steady profile equations for the electron component of the fluid can now be written

dni)/dz= /In,

(33)

+ nkTJ = -enE - K,mn(o - V ) , ( 34) (d/dz)[rnn(u2 - V2)u+ nc(SkT, + 2e(pi)+ q] = -2envE - 2K,mn(u - V ) V . b

(d/dz)[mn(u- V ) v

(35) In reaching this result, the new inelastic terms have been transferred into the divergence term by using the first equation to eliminate p.

2 . Collision Probabilify Averages as

The averaged collision probability (P:.") (see Tables IV and V) is defined

and is expressed in collisions per meter per torr. The abscissas are given in square root of electron volts, with one electron volt being equal to 7737°K.

NONLINEAR ELECTRON ACOUSTIC WAVES. I

63

If one wishes t o use these averages to compute the collision rate per unit volume of B quantity A which is a power function of such as 11,

then

Averages suitable for calculation of the transport coefficients have also been computed. They (see Tables VI-IX) are defined as

((k,"") ~ J p , o ' 1

Pn

df

=

(39)

For (P,"') and (P,'"). which are rarely needed, the values in Volume 20 of Adiwnces in Electronics arid Electron Plijisics may be consulted. Averages for

cesium and methane are also given there.

3. ltiitiul Conditions .for Steady Profile Wares

The only complete solutions of the equations which have been obtained so far have been developed on the assumption that a wave frame exists and that viewed from this frame the dependent variables change with position only. Then in one dimension the global differential equations can be integrated (67) to yield t1P = N i v, (40) (Ni t ~ n / tt' (

mnr(

t j 2

+ N ) V = No vo,

- V ) + M N , Vo ( V - V,) + N o k(T - 7'0) - V2)

+ MN,

+ nkT, + ( ~ , 1 2 ) ( E , -~ E 2 ) = 0 V , ( V 2 - V o z )+ 5 N , Vo k(T - To) + 5kT, + (2eyi)nr+q + iy = 0.

(41) (42) (43)

Equations (42) and (43) have been simplified by the use of Eqs. (40) and (41). We can use these equations to determine the leading edge condition on the wave. The arrival of the wave is signifed by the existence of electrons with some velocity r1 and some temperature ( T e ) l while , El = E, since there can be no surface charge singularity on the interface; V , = V0 since there is n o time to accelerate the ions across the interface: and (Ti), = To. Moreover i and q must be zero at the front. Hence, n,[z1,(v, ),'b

and

+ k(Te),/n7]= 0,

(44)

+

(45)

n1z1,[z112- Vo2i- (5k(Te),,h7i) ( 2 e y i / m ) ] = 0.

TABLE IV


Argon

CO

7.7 x 1 0 3 1.3 x lo4 1.7 2.2 3.1 4.1 4.8 5.8 7.0 9.5 1.2 x 1 0 5 1.6 I .9 2.8 3.8 5.0 7.7

6.475 11.64 15.65 20.5 28.3 34.8 37.9 41 .O 43.25 44.4 43.5 41.0 39.0 33.9 29.8 26.4 21.5

55.6 64.7 65.3 62.5 56.6 51.5 49.0 46.4 44.5 41.7 40.2 38.0 36.6 20.7 30.4 27.9 23.6

Helium 17.9 17.6 17.3 16.8 15.8 14.85 14.2 13.4 12.58 11.17 10.15 8.85 8.128 6.76 5.8 5.07 4.04

Hydrogen

Mercury

42.3 45.1 45.5 44.7 42.2 39.1 37.0 34.5 43.0 27.8 24.85 21.60 19.8 16.2 13.9 12.15 9.68

202.0 201 .o 189.0 172.0 146.0 124.2 113.0 100.8 90.5 77.2 70.0 63.4 60.4 55.1

51.0 47.4 40.75 ~~

'Multiply tabular values by lo2 to obtain collisions per meter at 1 Torr.

Neon 5.37 6.48 7.02 7.63 8.47 9.14 9.50 9.91 10.29 10.75 1I .02 11.05 1I .05 10.68 10.15 9.52 8.35

Nitrogen 4.375 5.04 5.06 4.9 4.54 4.25 4.12 3.99 3.89 3.75 3.66

3.50

3.40 3.11 2.84 2.595 2.18

Oxygen 17.9 19.35 20.4 21.5 23.6 25.4 26.2 27.6 28.7 30.0 30.5 30.4 30.1 28.6 26.8 24.8 21.2

Thallium 27.1 29.6 32.1 34.5 36.4 37.0 36.5 36.3 35.6 34.0 32.6 30.65 29.4 26.6 24.2 22.1 18.5

Xenon 23.05 43.9 58.3 72.2 87.2 94.6 96.2 97.0 95.5 88.9 81.7 71.4 65.0 51.8 42.6 35.8 27.2

-w

F6

P

8

e

!i

T"

Argon

CO

Helium

Hydrogen

7.7 x 103 1.3 x 104 1.7

13.5 24.0 32.2

90.5 90.9 83.0

23.9 22.7 21.9

61.6 61.75 60.1

294.0 245.0 213.5

8.35 9.94 10.93

2.2 3.1 4.1 4.8 5.8 7.0 9.5 1.2 x 105 1.6 1.9 2.8 3.8 5.0 7.7

41.35 53.75 61.1 63.7 64.25 63.3 59.0 53.8 48.5 44.0 35.8 30.8 27.0 21.7

13.9 63.4 57.7 55.9 53.7 52.6 50.8 48.5 46.6 43.7 38.2 33.9 30.6 25.0

20.85 19.03 17.3 16.3 14.9 13.75 11.85 10.45 9.2 8.275 6.18 5.72 5.03 4.02

56.5 49.9 43.9 40.8 36.8 33.4 28.6 25.15 22.1 19.8 16.05 13.7 12.05 9.55

181.5 142.0 117.0 106.0 94.75 87.2 79.4 75.2 73.6 70.4 64.8 58.8 53.7 44.5

I I .83 12.97 13.85 14.3 14.55 14.9 15.25 15.1 15.0 14.4 13.2 12.05 1 1.05 9.19

a

Mercury

Multiply tabular values by lo2 to obtain collisions per meter at 1 Torr.

Neon

Nitrogen

Oxygen

Thallium

Xenon

1.07 7.02 6.50

25.5 27.9 29.9

35.75 45.2 49.4

49.3 91.8 114.2

5.97 5.36 5.08 5.025 4.92 4.86 4.76 4.57 4.375 4.10 3.55 3.14 2.83 2.30

32.3 35.8 38.6 40.0 40.8 41.8 42.25 41.4 40.7 38.7 34.4 30.8 21.9 22.8

51.5 51.3 49.6 48.3 46.2 44.6 41.75 39.2 37.1 34.7 30.0 26.6 23.9 19.5

130.4 140.5 138.5 135.0 127.0 118.0 100.5 86.7 72.4 63.0 46.9 38.3 32.2 24.9

0

5

5, * m

G 4

$ 2

c, m

2 c,

<

9

<

,c?.

"

TABLE VI <(l/Pc)(-9 T" 7.7 x lo3 1.3 x lo4 1.7 2.2 3.1 4.1 4.8 5.8 7.0 9.5 1.2 x IO' 1.6 1.9 2.8 3.8 5.0 7.7

Argon

co

Helium

Hydrogen

0.986 0.5675

0.0297 ,0.02065 0.0266 0.02595 0.0257 0.0262 0.0266 0.0271 0.0278 0.0288 0.0297 0.0308 0.0315 0.0337 0.0359 0.0388 0.0487

0.0703 0.0534 0.07175 0.0727 0.07475 0.0774 0.0795 0.0829 0.0867 0.0953 0.104 0.117 0.126 0.151 0.1745 0.188 0.272

0.0371 0.0257 0.0329 0.0318 0.0317 0.0321 0.0327 0.0339 0.0355 0.03915 0.0428 0.0485 0.0522 0.06275 0.0727 0.0836 0.1 14

0.644

0.539 0.426 0.3425 0.304 0.262 0.226 0.1 785 0.1495 0.121 0.108 0.0862 0.076 0.0706 0.0745

~

~

~

~

~~

Mercury 0.593 0.298 0.317 0.255 0.188

0.148

0.129 0.1105 0.0952 0.0755 0.0645 0.0540 0.0488 0.0410 0.0369 0.0348 0.0369

~~~

Multiply tabular values by 10' to obtain collisions per meter at I Torr.

Neon 0.306 0.208 0.259 0.244 0.222 0.205 0.1965 0.186 0.177 0.163

0.154

0.1455 0.1405 0.134 0.131 0.132 0.153

Nitrogen 0.392 0.363 0.347 0.334 0.328 0.3265 0.327 0.3285 0.331 0.3345 0.339 0.346 0.351 0.370 0.392 0.421 0.529

Oxygen 0.0758 0.0736 0.0715 0.0695 0.0663 0.06325 0.0618 0.0594 0.0573 0,0549 0.0518 0.0497 0.0484 0.0472 0.0473 0.0487 0.0580

Thallium 0.0416 0.0472 0.0476 0.0472 0.0454 0.0435 0.04275 0.0418 0.04125 0.0402 0.04015 0.0407 0.0411 0.0433 0.0460 0.0495 0.0624

Xenon 0.I063 0.0991 0.0904 0.08075 0.0677 0.0574 0.0525 0.04665 0.0420 0.0355 0.0321 0.0294 0.02845 0.0287 0.0307 0.0340 0.0450

9 6 ?

4G

55

TABLE VII <(1/P<)"' )

T" 7.7 x lo3 1.3 x 104 1.7 2.2 3.1 4.1 4.8 5.8 1.0 9.5 1.2 l o 5 1.6 1.9 2.8 3.8 5.0 1.7 8

Argon

co

Helium

Hydrogen

Mercury

Neon

0.372 0.205 0. I 50 0.120 0.0738 0.054 0.0456 0.0379 0.0325 0.027 1 0.0253 0.025 1 0.02575 0.029 0.0329 0.0374 0.0457

0.01 14 0.01 53 0.0154 0.0174 0.01 76 0.019 0.0195 0.0201 0.0207 0.021 5 0.0222 0.0232 0.0241 0.0267 0.0296 0.033 0.0396

0.0473 0.048 1 0.0496 0.05125 0.0549 0.0597 0.0628 0.0677 0.0737 0.0852 0.0958 0.1 I 1 0.121 0.148 0.173 0.199 0.246

0.0206 0.0191 0.01905 0.021 1 0.021 1 0.0235 0.0252 0.0275 0.0302 0.0354 0.0400 0.0464 0.0505 0.0617 0.0722 0.0834 0. I035

0.1085 0.0466 0.0308 0.0236 0.01 575 0.01 38 0.01 33 0.01 34 0.01 365 0.01 42 0.0147 0.0151 0.0155 0.01 64 0.0176 0.01 92 0.0225

0.1675 0.1417 0.130 0.1295 0. I065 0.0984 0.0928 0.0894 0.0859 0.0812 0.0789 0.0777 0.0774 0.0810 0.0862 0.0933 0.1090

Mulitply tabular values by 10' to obtain collisions per meter at 1 Torr.

Nitrogen 0.225 0.198 0.196 0.216 0.209 0.218 0.21 8 0.225 0.229 0.233 0.2385 0.249 0.257 0.2865 0.3195 0.357 0.434

Oxygen

Thallium

Xenon

0.0477 0.04425 0.0427 0.0404 0.0374 0.0349 0.0330 0.0319 0.0307 0.0292 0.0285 0.0283 0.0286 0.03055 0.03325 0.03665 0.0438

0.0375 0.0346 0.03 16 0.0289 0.0260 0.0250 0.02425 0.0246 0.0249 0.0259 0.02705 0.0289 0.0299 0.0339 0.0376 0.0422 0.051I

0.0717 0.0472 0.0365 0.0257 0.01975 0.01 55 0.01 37 0.01255 0.01365 0.01205 0.01 305 0.0152 0.0169 0.0219 0.0266 0.0314 0.0400

TABLE VIII < I lP,'3'>

T"

Argon

co

Helium

Hydrogen

Mercury

Neon

Nitrogen

Oxygen

Thallium

Xenon

7.7 x lo3 1.3 x lo4 1.7 2.2 3.1 4. I 4.8 5.8 7.0 9.5 1.2 x 10'

0.226 0.113 0.0817 0.0604 0.0417 0.0327 0.0298 0.0276 0.0271 0.0286 0.0308 0.0358 0.0382 0.0462 0.0532 0.0616 0.0753

0.0195 0.0193 0.0215 0.0239 0.0267 0.0281 0.0288 0.029 0.0295 0.0306 0.0317 0.0346 0.03585 0.0413 0.0463 0.053 0.0643

0.0634 0.0663 0.0697 0.074 0.0825 0.0922 0.0998 0.1093 0.121 0.143 0.1605 0.190 0.2045 0.248 0.286 0.331 0.407

0.02485 0.0245 0.0258 0.0280 0.0327 0.0378

0.0290 0.01255 0.01 11 0.01163 0.01 39 0.01605 0.0172 0.0183 0.0193 0.02035 0.0207 0.0218 0.02205 0.0241 0.0264 0.0300 0.0360

0.190 0.158 0.1445 0.133 0.119 0.1 125 0.109 0.1045 0.1027 0.101 0.1005 0.1052 0.106 0.1176 0.129 0.1495 0.1735

0.2555 0.247 0.2635 0.282 0.301 0.3085 0.3125 0.312 0.315 0.3265 0.336 0.37 0.3835 0.445 0.504 0.578 0.705

0.0598 0.0544 0.0515 0.04825 0.0438

0.0505 0.0390 0.0345 0.0320 0.0309 0.03145 0.0325 0.0334 0.0349 0.0375 0.0395 0.0437 0.0454 0.0528 0.0595 0.0682 0.0830

0.0652 0.0336 0.0241 0.0180 0.01377 0.0217 0.0218 0.01335 0.01476 0.01795 0.0213 0.0269 0.0300 0.0389 0.0460 0.0543 0.0670

1.6

1.9 2.8 3.8 5.0 7.7

0.0415

0.0457 0.05099 0.0600 0.0670 0.0793 0.0852 0.1035 0.120 0.1 395 0.171

Multiply tabular values by 10' to obtain collisions per meter at 1 Torr.

0.0405

0.0393 0.0376 0.0369 0.0367 0.0368 0.0392 0.04025 0.0456 0.051 0.05825 0.0708

9

3 G 73

TABLE IX

< 1IP:

T"

Argon

co

Helium

7.7 x lo3 1.3 x lo4 1.7 2.2 3.1 4.1 4.8 5.8 7.0 9.5 1.2 x lo5 I .6 1.9 2.8 3.8 5.0 7.7

0.373 0.198 0.1415 0.1403 0.0840 0.0746 0.0745 0.0763 0.081 6 0.0940 0.105 0.1245 0.133 0.1615 0.186 0.220 0.258

0.0525 0.06375 0.0737 0.145 0.08675 0.0880 0.0887 0.0889 0.09075 0.0961 0.1015 0.1137 0.1195 0.1405 0.1595 0.184 0.22

0.194 0.208 0.223 0.242 0.278 0.319 0.348 0.384 0.426 0.503 0.565 0.669 0.716 0.872 1.004 1.16 1.40

Hydrogen 0.0716 0.0755 0.0837 0.1915 0.1 I45 0.134 0.147 0.162 0.1795 0.211 0.2335 0.278 0.2985 0.365 0.423 0.731 0.588

5'>

Mercury 0.0265 0.0265 0.0324 0.0395 0.0493 0.0557 0.0586 0.0605 0.0622 0.0634 0.0642 0.0683 0.06975 0.0796 0.0893 0.1027 0.1115

Neon 0.503 0.418 0.385 0.359 0.328 0.31 1 0.306 0.298 0.2965 0.301 0.306 0.33 0.339 0.386 0.4325 0.740 0.592

Nitrogen

Oxygen

Thallium

Xenon

0.692 0.781 0.860 1.58 0.945 0.94 0.95 0.947 0.964 1.02 1.08 1.22 1.284 1.525 1.74 2.01 2.30

0.1 70 0.1525 0. I43 0.1985 0.1187 0.1 117 0.109 0.1067 0.107 0.1 103 0.1 142 0.1255 0.131 0. I54 0.175 0.2025 0.242

0.133 0.0968 0.0900 0.1555 0.0929 0.0985 0.104 0.1073 0.1 123 0.1213 0.1285 0.1443 0.1515 0.1 80 0.205 0.354 0.284

0.121 0.0574 0.0428 0.0359 0.0340 0.0370 0.0403 0.04525 0.0522 0.06675 0.0798 0.1007 0.1115 0.141 0.165 0.195 0.231

z 0 z

c

3> 7J

m el 7J 0

8e vl

=!

5

0

5

M

H

Multiply tabular values by lo2 to obtain collisions per meter at 1 Torr.

QI

W

70

RICHARD G. FOWLER

This condition can be satisfied in three ways. The first is n, = 0 with u, and ( T J , undetermined. The second is u1 = 0, T, = 0 with n , undetermined. The third case is n , # 0. It results in true shock solutions, although the shock is recognizable only in the electron gas. Solving for v l and (Te)lgives

5V0 [9Vo2+ 16(2ecpi/m)]”2

u,=-+

8

8

(46)

Since ( T J , > 0 and since the zero-current condition requires Vo and u to be of the same sign, V, < u1 5 0 . It then follows that dE/dx must be negative at the front, so E always increases as one passes from the undisturbed region in front of the wave toward - 03. Thus there is always a negative charge at the front, regardless of the polarity of the potential driving the wave. The actual lower limit on til is, however, considerably higher than V , . As Vo goes to infinity, 2ecpi/m becomes negligible and u, approaches a limit of V0/4,the usual strong shock result. Thus in general,

0. (48) It is perhaps easier to understand this condition when written in absolute values (49) 0 < IU, I < I v0/41. From the upper limit, v1 5 0, one concludes by substitution in Eq. (46) that + m v O 2>ecp, (50) and that there is a lower limit on wave speed for the shock fronted wave. The wave must also be bounded on its backside with certain conditions. Since there can be no current in this one-dimensional case, while there is a conducting plasma present, the electric field must approach zero. Nor can there be any relative motion of the charges, so u must approach V . The wave thus replaces a neutral gas with a partially ionized quiescent plasma and is basically analogous to the N wave in shock theory except that the discontinuity on the leading edge where the flow abuts on neutral gas is a strong discontinuity, while that on the backside may be and usually is a weak discontinuity.

Vo/45 01

4. The Heavy Particle Equations, a Priori Approximations

The basic approximation under which a solution of the problem is possible was actually first suggested by Thomson and Thomson (72). It is that the heavy particles are mechanically undisturbed by the wave passage. One can

NONLINEAR ELECTRON ACOUSTIC WAVES, I

71

verify this assumption empirically before using it to solve structural problems, and afterwards can check the fact that a given solution continues to fulfill the assumption. We write the heavy particle equations for a steady profile wave as follows

( d / d z ) [ M N V 2 + Mi N i V 2

+

dN V / d z = pn,

(51)

dN V / ~=Z -/It?,

(52)

+ (Ni + N)kT]

= eN,E

+ K,inrz(v - V ) - PmnV,

(didz)[MNV3 MiN, V 3 + 5(Ni + N ) V k T ] = 2eNi V E + 2VK,mn(v - V ) - flmriV2.

(53) (54)

By carrying out the indicated differentiation of these equations and eliminating dT/dz, one can obtain, by use of baryon conservation

[ M N o Vo

- ntNi V - 3 M N o Vo ( k T i M V ' ) l ( d V , ' d ~ )

= eNi E

+ K , m n ( ~- V ) .

(55)

Now on the left side, n7Ni is certainly less than M N , and k T / M V 2 is much less than unity, so only M N o Vo remains. Of the two terms on the right-hand side of the equation the first is the electric force and the second is the viscous damping, which must always be the smaller of the two, so we retain only the first, convert the equation to an inequality, and set E = -dcp/Jz. Integrating through the wave

AViV I (cci/MVo')Acp.

(56)

Empirically, at 40 kV the wave velocity is 2 x 10" cmjsec and so A V / V I a i , ai being the fractional ionization. In practice cci < as measured with

microwaves by Haberstich and optically by Mills. Thus A V / V - l o - * and the change in velocity of the heavy particles seems superficially to be so small as to be completely negligible. Shelton (73) has examined the consequences of a rigid application of the assumption that any changes in velocity of the heavy particlescan be neglected. He found that the structure of the wave cannot be consistently developed under this assumption. The relative velocity of the electrons and heavy particles cannot be brought to zero behind the wave and dE/dz therefore remains nonzero so that E becomes singular eventually, a result which is physically unacceptable. Further examination of the basic equations shows that constant V is not really a good approximation for universal application. The forces acting on one fluid are essentially the same as those acting on the other, so that the total changes of momentum and energy of the two fluids are similar in magnitude. Thus even though the velocity change of the heavy particles is small,

72

RICHARD G. FOWLER

the vastly greater density of the heavy particle fluid gives it comparable changes in total fluid quantities. This does not mean, however, that the nearly constant value of V is a useless fact. If one returns to the electron fluid equations, he notes that nowhere in them is the velocity V multiplied by the heavy particle density so that it may be treated as constant there. This decouples the equations and makes a solution possible. C . Solution of the Electron Equations f o r Proforce Wazies

Since Vis only slightly varying with respect to u, Shelton and Fowler used this quantity to form dimensionless variables in which to express the electron equations. Let these variables be v = (2etpi/e, Eo2)n, ’I = EIEo

$

0 = kTJm V 2 ,

= u/V,

< = (eE,/mV)z

9

and let a set of dimensionless parameters also be introduced, p

= (mV/eE,)P,

CL

and

= 2eqi/inVZ,

K = (mV/eE,)K.

(57)

The electron equations (including Poisson’s equation) then become

(4&)(v$) (d/ds)[v$($- 1)

+Vo]

(58)

= pv9 = -Vq

(d/d<)[V$($’ - 1) -kV$(50 -k E)]

=

- K V ( $ - I),

(59)

-2V$)7 - 2KV($ - I),

(dq/d<)= (via)($ - 1).

(60) (61)

It is important to note that for a proforce wave, E, is intrinsically negative, so that for a wave facing in the positive z direction, with the shock discontinuity at z = 0, the description of the wave profile involves negative values of z , and hence positive values of 5. In terms of the new variables, the initial conditions at the discontinuity become v 1 # 0;

$1

=

[5 - ( 9

+ 16~)”~]/8;

8, = $ 1 ( 1 - $,);

q I = 1.

(62)

As a first step, the four differential equations can be reduced to three with the formation of an exact differential by subtracting twice the momentum equation (59) from the energy equation (60) and introducing the Poisson equation in the result. After being integrated across the shock discontinuity, with the constant of integration evaluated in the undisturbed gas, this yields the algebraic relation v$($

-

+v(5$ - 2)O +v$a

+ a(#

- 1) = 0.

(63)

NONLINEAR ELECTRON ACOUSTIC WAVES, I

73

From this we can deduce a relation across the entire wave from the neutral gas in front to the quiescent plasma lying somewhere sufficiently far behind. Since at the latter point we have I(/ --t I , q -+ 0, we obtain Vf(LY

+ 3 4 ) = a,

(64)

where the subscript f denotes this “final” state. Because the electrons finally equilibrate thermally with the heavy particles, Of will be much smaller than a and the major meaning of this result is that Vf ZY

1.

From the definition of v , the implication of vf = 1 is that the electrostatic energy density present in the field in front of the wave has been converted exactly into ionization energy density behind the wave. Figure 64 gives these predictions for helium. Naturally this exact balance is achieved only because we have neglected all other inelastic processes, and treated the heating of the gas as negligible. Nevertheless the result proves to be a good first approximation and justifies the not wholly facetious description of these breakdown waves as detonations in the mixture gas-plus-electrostatic-field. Attempts to integrate the equations numerically by successive approximation bring the immediate observation that the dependent variables ( I - I(/) and q fall into one class and v and O into another as far as rapidity of variation with ( is concerned, with the former undergoing their entire variation over a span of 5 which is at least an order of magnitude smaller that that required by the latter. The rapidity of variation of q can easily be understood from physical reasoning. Any conductor quickly readjusts its surface charge distribution so that its interior is shielded from external electric fields. When the conductor is a metal the layer in which this occurs can be treated as mathematically zero in thickness. This is not possible in a gas, although the layer may be very thin. For a quiescent plasma the layer has a well-known thickness called the Debye length. In the present case the layer is somewhat thicker than a Debye length because it is continually being formed in dynamical balance but it is still very thin. The variable ( 1 - I(/) shows a companion degree of sensitivity with respect to q because electrons are the ultimate in responsiveness to the electric field and their drift velocity is the mechanical measure of the rate of adjustment of the charge distribution in the layer. ( 1 - $) begins to assume values at the point where the electrons first experience the field and it must vanish as the electric field vanishes. The thin dynamical region in which q and J/ undergo adjustment is succeeded by a much thicker thermal region in which the electron gas which

74

RICHARD G. FOWLER

has been heated in the dynamical layer cools while continuing to produce ionization. It is thus natural to divide the attack into two parts. the shock layer or Debye sheath, and the thermal layer or quasineutral region. In his original attack on the problem Shelton believed that the boundary between these regions was a vague one that he could choose at a definite point in 5 merely as a convenient approximation. Further study, however, has led to the opinion that this boundary is actually a second discontinuity and that I) = I and 9 = 0 identically, rather than approximately, at points beyond this surface. The equations treated here are not precisely the set of which it has been proved ( 7 4 ) that an initial discontinuity must break up into more than one discontinuity as time passes, but bear a sufficient similarity to them to be suggestive. In any event a solution can be found by assuming the interface between the sheath and thermal region to be a weak discontinuityin thecase of the proforce wave and a strong discontinuity (shock) in the case of the antiforce wave, as will be discussed later. In the quasineutral region, since I) is nearly unity, it can be set at this value everywhere except where I) - I occurs. Then two equations emerge free of I) and q , the electron production equation dv/d[ = L

(66)

~ S

and the algebraic relation (64)previously derived in terminal form only v(a

+ 30) = M.

(67)

Although Eq. (66) seems free of 8, in fact it is not, since p, which determines p, is a strong function of 0. The function p has been calculated by averaging the ionization cross section over a Maxwell distribution. The result for helium is given in Fig. 55, employing Smith's (75) measurements. Also displayed arethreeempirical fitting functions with various ranges of usefulness. These are

fl

/I= 1.13 x lO9p(5.O5 x 10'/T)1i2[-Ei(-5.05 x 105/T)] 5 x 10' < T < 2 x lo7, = 6.1 x lO*p[l - e - 1 . 6 5 x 1 0 - h ( T - 1 0 s ) ] 1.5 x 10' < T < I x lo',

o

9, = 5.17 x IO8p(T/2.87 x I 0 5 ) 6 ' 1 0 ~ - 2 ~ 8 7 x 1 0 5 < / TT < 2

(68) (69)

lo5. (70)

Herep is gas particle density expressed in units of 3.54 x 10'6/cm3,the number of molecules per cubic centimeter at I Torr pressure, 0°C.

The appropriate form for the quasineutral region is certainly the last one, based on the temperature studies of Blais and of Mills. Corresponding empirical formulas for neon, argon, and molecular nitrogen and hydrogen can also be obtained from Smith's data by fitting it to the formula 0 = ao(Wi/W)[(W/Wi) -

I]"

(71)

75 I

I 1 Illll

I

I 1 1 1 1 1 1 1

I

I

I

r

m

4 -

a

2 -

0

-

-

2 lo6

-U’l

I

I

I I 1 1111

I

I

F w . 55. Maxwellian ionization rate in helium Fluids.

at low energies, or

0 = oo’(Wi/ W

I I

I

ILIII

I

I I I I I I

(PI. From (75) with permission of PhJ,s.

) In ( Wi W , )

(72) at high energies, where W and W , are the electron kinetic energy and the ionization energy, respectively. I n Table X values of these empirical constants are given. TABLE X ~~~

Helium 00

n pOip

4.81

\

lo-”

1.10 5.17 y lo8

Eliminating obtains where

Neon

1’

6.05

1

lo-’’

1.25 6.50 toR

Argon 4.63 1 00 4 9 5 x 109

~~-

~

Hydrogen 1.34 10-16 1.10 1.44 109 ~

Nitrogen 2.17

i 10-16 1.55 2 33 x 109

between the equations for the quasineutral region, one

i = (K,/h.fio)[f(cc/20) - f(r120,)l.

is given in Table X and

(73)

76

RICHARD G . FOWLER

For helium f ( u ) is represented in Fig. 56. One must of course invert the functionfto find 8, and then v also follows from Eq. (67). The constants ti and 0, take their values by connection of the quasineutral region with the sheath.

1ci3

10‘2

10’

100

lo‘

I02

Io3

Id

lo5

flu)

FIG.56. Functional relation between position 6 and temperature 0 in the quasi-neutral (thermal) region.

At the interface between the sheath and quasineutral region, I) --t I and q -+ 0. It also follows directly from Poisson’s equation that as II/ -, 1, dq/d< = 0, since v will remain finite and nonzero. Therefore as one approaches the interface from the sheath, 1 - I) K ( 1 - ( i s ) and q K ( I - (is)’ or q K ( 1 - I))~. Shelton’s original solution differed at this point by the added assumption that on the interface dII//d< = d2q/@ = 0 as well. This amounts to using the relation q K ( 1 - 11/)3’2 to approach the interface. By postulating that this known behavior near the interface would give an adequate extrapolation across the very thin sheath region, Shelton found an approximation to a solution for the sheath problem for proforce waves. Subsequent study has shown that one can form a solution in the case of the proforce wave using any monotonically decreasing function which connects the points ( I ,I),)and (0,l) in the (q,$) plane. Actually to obtain an exact solution one should probably use an empirical or theoretical microscopic relation between terminal electron drift velocity and electric field. Unfortunately, even the most recent data (76) give no clue as to the relation in this region of high E’p and one can 2 nearly resembles the usual only surmise that the expression q = ( I - ~ ) most

NONLINEAR ELECTRON ACOUSTIC WAVES. I

77

form for a situation in which elastic energy losses probably once more predominate. Since the differences between solutions were found to be slight, Shelton and Fowler (73) also chose an easily integrable third relation as a model + [(I - *)/(I - *])I2 = 1. (75) (9 It too has the necessary parabolic properties near II/ = 1. In order of mention the models are referred to as I, 11, and ill. The fact that q = 0 at the front of an infinite plane wave with no ionization ahead of its actually determines the slope dq,d$ at the leading edge of the It follows that sheath because it implies that dO/d( = 0 at $ = (d9/d*), = -\‘I[(*, - 1)/(4$1 - 1)l(76) Although this condition will help to choose the proper path, Shelton and Fowler did not make use of it because it is fulfilled on some path between the parabolic and the elliptical extremes and these two actually display such small differences in their corresponding solutions. Equations (58-61) must be satisfied by the solution but one of these can be replaced by the algebraic integral [Eq. (63)], which is regarded as giving 0 when v, 9, and $ are known. A further simplification can be effected by substituting Eq. (61) in the second right-hand term of Eq. (59) to obtain

- 1)

(d!&) [ V $ ( $

-k V o $. KEY] =

Replacing v with the electron convection current j canonical form for solution.

+

j $ ( $ - I ) ~j ( 5 $

- 218 + . N u

( d / W [ j ( $ - 1)

- \’q. =\$

(77) reduces the set to a

+ all/(r12- I ) = 0 ,

+ ( j O / $ ) + ~v1= - j M 7

diicjt = p j N , drlidt =.i($- I)/a$.

(78)

(79) (80) (81)

One divides Eq. (80) by Eq. (81) to obtain a r-free equation rljidv = ~/l/(lc/ - I),

(82)

which is now integrable using relation (75) under one minor assumption. For temperatures above the 106”K expected in the sheath p is found to be essentially constant, even for variable 0, and so for a first approximation p is assumed to be constant over the sheath thickness. This assumption could be iterated away in an orderly fashion if it were found invalid. Then

The constant of integration is evaluated at the shock discontinuity where 9 = 1.

78

RICHARD G . FOWLER

One next divides Eq. (79) by Eq. (81) and the result can also be integrated using relation (75) t o obtain

Evaluating the integration constant first at point (1, and then point (0, I ) in the (y1, $) plane, the expression will be a solution if K is chosen as

0 can now be found by introducing these expressions for j and ti into Eq. (78) except that 0 will have a singularity at $ = 2/5 unless another condition is met, i.e., that the balance of the terms in the energy equation are zero at $ = 2 / 5 . This results in the determination of j , and of j , via the intermediate valuej*

i" = where j

=j *

at $

=

I -{I

- [I

I

-3'/5'(1

- $1)2]"2}2

+ (32/52u)

(86)

9

2 / 5 , so that for example with model 111,

Since j is a function of p, which in turn contains ti as it is defined, Eq. (85) is not the final form of the expression for t i . Solving, one finds explicitly

The correction implied by the second factor is usually small. In Fig. 57 plots of the critical parameters of the electron wave ($,, ti, 0,. and t2) are given as functions of E,, for theory 111. Although the natural independent variable is E o , the implicit nature of the calculation is such that the order

V,, -+ x

-+

$,

--$

j"

+ 1; + Eo -+

p

+,i2-+v2

--f

0,

must be followed. The profile of a wave is found most easily by integratingj as a function of 5. It is

79

NONLINEAR ELECTRON ACOUSTIC WAVES, I

K

IO0 W

_1

W

2

m

08

v,

a w

06-

t U

$ 04-

(2

0

0

& !P

iV/rn/Torr)

FIG.57. Critical parameters of the electron wave: $, , K , 6,, a n d for theory 111.

t 2a s functions of

Eo

A profile for ti can now be had by use of Eq. (83), as can a profile for II/ by using the assumed relation between q and I). after which the 0 profile follows from either Eq. (84) o r the algebraic relation. A similar calculation has been carried through for the other II/>q relations and the results differ only slightly. as can be seen in Figs. 58 and 59 in which they have been compared for four initial fields. Attention i s called to the extremely long tail on the 0, v curves found in the quasineutral region (Figs. 60-63). It is evident that under most experimental situations this i s all that the time resolution available permits one to observe. Referring to Fig. 16 in the section on experiment, the theoretical curves given by models I, 11, and Ill have been plotted against the data of Blais and Haberstich. The actual placement of the theoretical curves with respect to the data involves many complicated unsolved questions of factors of order unity that must be considered in a true calculation from first principles. These relate to the electrostatic geometry of the apparatus plus wave which fully described a t any instant i s that of a charged partially conducting rod with a cloud of charge of arbitrary shape at the end inserted into a glass sleeve and surrounded by a larger conducting cylinder. Some discussion of this will be given in Section 1V. The relation between the central field and potential at the

80

RICHARD G . FOWLER

c FIG.58. Sheath profiles for

4 and 17 vs. (, 4, = 0.01.

end of a fully conducting plane-ended rod is E = 0.502 $/a, but the geometrical constant is increased by the presence of a glass tube. If the conductor is a poor one with a somewhat extended volume of charge at its tip, the constant is also increased. There is added uncertainty as to a possible factor that comparison of the one-dimensional fluid theory to three-dimensional experiments may introduce. Therefore, in plotting the theoretical curve, these geometrical factors have been arbitrarily set at unity. The agreement then shown between

10

08

06

04

02

'0

01

02

03

04

05

06

07

08

E

FIG.59. Sheath profiles for (CI and 71 vs. 6,4%= 0.238.

09

07

06

05

04 0. I

03

02

01

'0

02

04

06

08

I0

12

14

16

I

18

E

FIG.60. Sheath profiles for electron temperature 0 and concentration v as function of position 5, Initial velocity I,!J, = 0.01. (Multiply ordinate scale by 80 to obtain v.)

20

I

\

\

\

04L2-L2 '0

01

02

03

04

05

0'6

0;

08

09

0

E

FIG.61. Sheath profiles for electron temperature 0 and concentration v as function or position 5. Initial velocity 4, = 0.238.

82

RICHARD G . FOWLER

.20

9

10

0 lo-'

Io3

lo'

I00

lo4

E

FIG.62. Quasi-neutral (thermal) region profiles for 8 with

'

4, = 0.01 and 0.238.

'I

0'

,

I

1

U

L

1

1

-

1

_

l

i

L

E

Fic. 63. Quasi-neutral (thermal) region profiles for v with

I/,= 0.01 and 0.238.

83

NONLINEAR ELECTRON ACOUSTIC WAVES, I

theory and experiment seems adequate in the present state of both theory and experiment. Scale factors for the conversion of 0, v, and ( to physical variables T,, n, and 2 are given in Fig. 64 for theory 111. '

0

I

6

3

7

5 7 1

-~~ ~~~~

3 5 7

I

ABSCISSAS

3 AS

5 7 1

3

5 7 1

3

5 7 1

LABELLED

FIG.64. Scale factors for conversion of 6, v, and 6 to T, ,n,and z for the third (elliptical relation) theory.

IV. CONCLUSION TO PARTI It seems apparent that a theoretical resolution of the various aspects of the breakdown wave is now at hand. Following the direction of the Shelton solution for the shock-fronted proforce wave in one dimension, solutions for the other waves now seem possible within the overall requirement that they exhibit only small differences in dependence of velocity on reduced field. Moreover, Turcotte and Ong (77) have indicated solutions for the cylindrically restricted thermal region, and Albright and Tidman (78) have discussed the time dependent case, if it should prove to be needed to understand any of these waves. It is now of importance to ascertain to what extent these theories describe various natural phenomena such as lightning and to perform improved experiments on all these waves both in nature and in the laboratory. It is proposed to deal with these questions as far as possible in a continuation of this article at a subsequent date.

84

RICHARD G . FOWLER

LISTOF Tube radius Dimensionless ionization constant Ion fraction Ionization frequency Wave velocity decrement Speed of Light Electron charge Electric intensity mks permittivity Dimensionless electric field Dimensionless electron temperature Total electric current density, conduction plus convection plus displacement Dimensionless electron flux Boltzmann’s constant, also angular wave number Momentum constant (proportional to elastic collision frequency) Dimensionless collision frequency Electron mass Neutral particle mass Ionic mass Dimensionless ionization rate Electron concentration, also an arbitrary integer Neutral concentration Initial neutral concentration lon concentration

SYMBOLS

Dimensionless electron concentration Dimensionless position variable Electron pressure, also ambient gas density in Torr Heavy particle pressure Collision probability Electric resistivity Heat conduction vector Cross section, with appropriate subscripts Time Heavy particle temperature Electron temperature Electron collision interval Radiation lifetime of state k Electron velocity either in a general frame or special (as specified) fluid frame of reference Heavy particle velocity either in a general frame or special (as specified) fluid frame of reference Initial heavy particle velocity Electron internal energy Coordinate along flow direction Dimensionless electron velocity Electric potential Ionization potential Capacitor potential Cyclotron frequency

REFERENCES 1. R. G. Fowler, Aduan. Electron. Elecfron Phys. 20, 1 (1964). 2. F. Hauksbee, Phil. Trans. Roy. SOC. London 24, 2129 (1705) 3. C. Wheatstone, Phil. Trans. Roy. SOC.London, 124, 583, (1834). 4 . J. J . Thomson, Proc. Roy. SOC. London 49, 84 (1891). 5. W. von Zahn, Wied. Ann. 8, 675, ( I 879). 6. J. W. Beams, Phys. Rev. 36,997 (1930). 7. J. James, Ann. Phys. Phys. Chem. 15,954 (1904). 8. L. B. Snoddy, J. W. Beams, and J. R. Dietrich, Phys. Rev. 50,469,1094 (1936); 51, lo08 (1937). 9. J. W. Beams, Phys. Rev. 28,475 (1926).

NONLINEAR ELECTRON ACOLISTIC WAVES, I

85

10. C . T. R. Wilson, Pruc. Roy. Soc., Ser. A 92, 555 (1916). 11. B. F. J. Schonland, fror. Roy. Soc., Ser. A 164, 132 (1938). I Z . R. G . Fowler, Electrically Energized Shock Tubes. Oklahoma University Research Institute, Norman, Oklahoma, 1963. 13. V. Josephson and R. H. Hales, Space Tech. Lab. Rep. STL/TR-60-0000-19313; Phys. Fluids 4, 373 (1961). 14. R . G. Jahn and F. A. Grosse, f h . w . Fluids 2,469 f 1959). 15. H . D. Weymann, Phys. Fluids 3 , 545 (1960). 16. G . E. Moreton, Sky Telescope p. 145 (March, 1961). 17. J. P. Wild, J . fhys. Soc. Jap. 17, Suppl. A-I 1, 249 (1962). 18. L. H. Burlaga and N. F. Ness, Solar Phys. 9, 467 (1969). 19. J. Roquet, R. Schlich, and E. Selzer, J . Ceophys. Res. 68, 373 I ( 1 963). 20. R. G. Fowler, J . Ceophys. Res. (submitted for publication). 21. D. Koopman, Phys. Fluids 15, 56 (1972). 22. G . W. Paxton and R. G . Fowler, fhys. Rec. 128, 993 (1962). 23. L. B. Snoddy, J. W. Beams, and J. R. Dietrich, Phys. Rec. 52, 739 (1937). 24. F. H . Mitchell and L. B. Snoddy, Phys. Re!).72, 1202 (,I 947). 25. A. Haberstich, Ph.D. dissertation, University of Maryland, College Park, Maryland, 1964. 26. J. M. Burgers, see Appendix to Haberstich dissertation (25). 27. G . A. Shelton, Ph.D. dissertation, University of Oklahoma, Norman, Oklahoma, 1967; G. A. Shelton and R. G . Fowler, f h y s . Fluids 11, 740 (1968). 28. R. N. Blais, Ph.D. dissertation, University o f Oklahoma, Norman, Oklahoma, 1971. 29. R. N. Blais and R. G . Fowler, fhys. Fluids, in press. 30. R. G . Fowler, Proc. f h y s . Soc., London 68, 130 (1955). 31. R. J. Sovie, Phjss. Flitids 7,613 (1964). 32. I . D. Latimer, J. I . Mills, and R. A . Day, J . Quant. Spectrusc. Radiat. Transfhr 10, 629 ( I 970). 33. F. L. Miller, Ph.D. dissertation, University of Oklahoma, Norman, Oklahoma, 1964. 34. B. L. Moiseivitsch and S. L. Smith, Rru. Mod. f h y s . 40, 1 (1968). 35. J. D. Jobe and R. M. St. John, Phys. R ~ L 164, . . 117 (1967). 36. H. R. Moustafa Moussa, F. J. DeHeer, and J. Schlutter, fhysica (Ufrecht)40, 517 (,I 969). 37. G . Elste, J . Qimnt. Spectrusc. Radiat. Tramfer 3, 209 ( I 963). 38. R. G . Westburg, Phys. Rev. 114, 1 (1959). 39. W . P. Winn, J . Appl. Phys. 38, 783 (1967). 40. W. R. Atkinson, Ph.D. dissertation, University of Oklahoma, Norman, Oklahoma, 1953. 41. H. G . Voorhies and F. R. Scott, Phys. Flitids 2, 576 (1959). 42. E. A. Maclean, A. C. Kolb, and H. R. Griem, Phys. Fluids 4, 1055 (1961). 43. R. G. Fowler and J. D. Hood, Pltys. Rec. 128, 991 (1962). 44. E. R. Pugh, Ph.D. dissertation, Cornell University, Ithaca, New York, 1962. 45. R. S. Schreffler and R. H. Christian, J. Appl. fhys. 25, 324 (1954). 46. A. Haberstich, Bull. Atner. fhys. Soc. 9, 585 (1963). 47. J. B. Gerardo, C. D. Hendricks, and L. Goldstein, Phj)s. Fluids 6, 1222 (1963). 48. R. C. Isler and D. E. Kerr, Phys. Fluids 8, I176 (1965). 49. M . J . tubin and E. J. Resler, fhys. Fluids 10, 1 (1967). 50. G . R. Russell, fhys. Fluids 12, 1216 (1969).

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RICHARD G . FOWLER

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