1. Plasma Waves and Echoes

I. PLASMA WAVES AND ECHOES*

.

I. Plasma Waves In the physics of any medium, waves occupy an important position. As the response of the medium to disturbances, their characteristics give considerable insight into the properties of the medium. Conversely, the medium must be thoroughly understood to predict the character of its waves. The complexity of plasma physics is reflected in the waves that exist in plasmas. At any frequency, two transverse modes corresponding to electromagnetic waves and one longitudinal mode can in general be present. The longitudinal waves, which exist only in matter, are most sensitive to the properties of the plasma and are the prime topic of this part. An accurate description of the plasma must be used to calculate the properties of these waves, and consequently observations of these waves serve as a powerful diagnostic technique for the plasma. Both density and temperature can be measured without affecting the plasma. I n some cases, details of the distribution function and fluctuation spectrum can be inferred from wave measurements.? I. I . I. Equations for a Plasma

T h e equations for a collisionless plasma comprise Vlasov equations for the electron and ion distribution functions and Maxwell’s equations for the fields. For longitudinal modes, the only nontrivial field equation is the Poisson equation. I. I.1. I. Infinite, One-Dimensional Model. T o simplify the theory, we shall consider an infinite plasma with only one dimension. This avoids consideration of a magnetic field. Special results for other cases will be quoted, but all the essential physical processes appear in this model. T h e equations are

?f + v a!f - (;)Eat

* Part

dx

1 by K. W. Gentle.

t See also Parts 2 and 3. 1

af

av

=

0,

(1.1.1)

2

1.

PLASMA WAVES AND ECHOES

+ v -ap ax + ( & ) E

aF

at

dF

a71 - = 0,

( 1.1.2) (1.1.3)

where thc distribution functions are denoted by j for electrons and P for ions. Considering small amplitude perturbations about an electrically neutral, field-free equilibrium, one obtains a set of linear partial differential equations for the perturbed field and distribution functions, fi and F,. These can be solved directly by representing the perturbations as Fourier transforms, e.g.,

dkf(71, k , w ) exp[i(kx - w t ) ] . (1.1.4)

do

fi(x, v , t ) = (11277)

To obtain a particular solution, thc boundary conditions must be specified; we shall assume that conditions at t = 0 arc given and that all perturbations are zero before that. This specification has the important effect of fixing the contours along which the integrals of (1.1.4) must be taken. Here, k is strictly real, but w may be complex. To ensure that j1(v,x, t < 0) = 0, it suffices to choose the contour to go above the poles off(v, k, w ) . Then for t < 0, the contour can be completed in the upper half plane and not contain any poles. This prescription introduces the boundary condition, but rctains all the usual properties of the Fourier transform for the t > 0 solution. (It is mathematically cquivalent to the Laplace transform approach used originally by Landau.') I. I. I.2. Linear Wave Equations. The equations for the perturbed quantities become equations for the transforms

+ ikvf, - (eE/m)df,/dv

=

0,

(1.1.5)

+ ikvF, + (eEjM) dF,/dv

=

0,

(1.1.6)

-iwfi -iwF,

ikE

=

4ne

J:m

(F,

- fi) dv.

(1.1.7)

This set of linear equations is easily solved. I , I . I .3. The Plasma Dielectric Function. Eliminating the perturbed distribution functions, one finds

[I -

'$Jym[{(b)2+ (A)%}/.:

- F)]dv]E(k, w)

=

0.

(1.1.8) L. L). Landau, J. Phys. (USSR)10,25 (1946).

1.1. PLASMA

3

WAVES

The coefficient of E is the plasma dielectric function, ~ ( ka,), and this equation states the requirement that a nontrivial longitudinal wave can exist only when the dielectric constant is zero. Strictly speaking, u is a real variable, but the integral in (1.1.8) is well defined as a complex integral for analytic fo. Since the integral of (1.1.4) is taken above all poles, e.g., Im w > 0, the integral of (1.1.8) along the real axis may be analytically continued as a complex integral with a contour chosen to go under all poles. For this prescription and the choice of Maxwellian distribution functions of equal density, m fo(v) =

Fo(u) =

exp( - ~ ’ / Z V , ~ )

(2rc)1’2ue

M (~n)1/2gi

)

( 1.1.9)

exp( - ~ / Z V ~ ~ ) ,

the integrals of (1.1.8) are tabulated functions. In terms of the well-known W-functions,” the dielectric constant is

(1.1.10) where ,ID = DJw, and cop2 = 4rcne2/m are the Debye length and electron plasma frequency. The real and imaginary parts of the W-function for real argument are sketched in Fig. 1. The function has simple asymptotic

FIG.1. The real and imaginary parts of the W-function for real argument.

expansions for large and small values of the argument.

* T h e W function is related to the 2 function of

r1

(51 42) Z( - J2)= +Z(- 61 J2).

Fried and Conte* by W ( { )= -1

+

B. D. Fried and S. D. Conte, “The Plasma Dispersion Function.” Academic Press, New York, 1961.

1. PLASMA

4

WAVES AND ECHOES

I . I .2. Electron Plasma Waves ‘l’here are two frequency bands in which thc dielectric function (1.1 .lo) has zeros for w with small imaginary parts, At all other frequencies, the roots occur for o with a large imaginary part, implying rapid damping. Onc of these bands occurs for frequencies at and slightly above the plasma frequency. At these high frequencies, the ions cannot follow the oscillations. T h e argument of the W-function for ions, w/kui, is very large compared with one and W is approximately zero. Even the argument of the Wfunction for clcctrons, wlkv,, is large enough to justify use of the expansion (1.1.1 1). Assuming the imaginary part of w is small, thc dispersion relation can be easily calculated. I t reduces to w N mae[l i- +k2i1,’ - i(7~/8)’”(1/kA,~) exp(-($ i- l/2k2J.,>’))] (1.1.13) in thc limit that ki, G 1. (Then Q may be replaced by apein the small terms.) This dispersion relation is sketched in Fig. 2. These are the

3.05

W

5 a z

a 0 3.025

0

01

0 2

03

04

0.5

k b FIG.2. ‘I’he dispcrsion relation for clectron plasma wavcs. l’he frequency and wavenumhcr arc scaled as w/w,. and k l , . ‘I’he real part of ~ o / m pips given by the left scale, and the imaginary part by the right scale.

well-known electron plasma oscillations and a dispersion relation of this form was obtained by 1,andau.’ At long wavelengths, the waves propagate

1.I.

PLASMA WAVES

5

with high phase velocity, v J k l , , but with low group velocity, 3vekA,, and negligible damping, thus satisfying the assumptions of the derivation. As the wavelength becomes short, k l D approaching one, the phase velocity approaches the electron thermal velocity and damping becomes appreciable. This is the collisionless damping of plasma waves first predicted by Landau.’ I . I .2. I . Landau Damping. T h e Landau damping of the wave is attributable to resonant electrons, the electrons on the tail of the distribution having a velocity equal to the wave phase velocity. These electrons are moving with the wave and see a constant electric field, not an oscillating field. They can therefore extract energy from the field and damp the wave. T h e details of the process are more subtle. An electron traveling with the wave will be either accelerated or decelerated, depending upon where it is on the waveform. If the energy transfer is averaged over electrons at various places on the waveform, one finds that the electrons moving slightly slower than the wave receive an acceleration, while the average velocity of electrons moving slightly faster than the wave decreases. T h e average effect is therefore for slower particles to extract energy from the wave, while faster particles feed the wave. T h e net result depends upon whether there are more electrons moving slightly faster or slightly slower than the wave phase velocity. T h e derivative of the distribution function at the phase velocity thus determines whether the wave grows or damps. For the Maxwellian distribution treated here, the derivative is always negative and the waves are damped. If a beam were injected into the plasma to produce a bump in the tail of the distribution, there would be a range of phase velocities for which the derivative of the distribution function would be positive, implying growing waves. I . I .2.2. Large Amplitude Waves. This analysis fails if the wave amplitude becomes too great. We have considered only the initial motion of the electron caused by the field. As the electron accelerates, it moves to a different part of the wave, where its acceleration changes. From this point on, a different analysis is required. The time scale for this effect is the “trapping time.” The resonant electrons have so little kinetic energy in the wave frame that they may be trapped in the periodic potential of the wave. This trapping is tnanifest in a time equal to the period of the o~cillation,~ z N (m/ekE)’’2, ( 1.1.14) the time an electron takes to move back and forth in the wave potential. If linear theory is to be applied, the wave amplitude must be made so small that this time is longer than the time during which the observations are T. Stix, “The Theory of Plasma Wavcs.” McGraw-Hill, New York, 1962.

6

1.

PLASMA WAVES AND ECHOES

made. T o use linear theory for times up to N plasma periods, the wave energy density, E2/Xn, must be less than the plasma therrnal energy density, ;nmve2, in the ratio u,/ulh

5 1/(Nk21,2)2.

(1.1.15)

1.1.3. Ion Acoustic Waves

T h e second lightly damped wave occurs at low f r e q ~ e n c i e s Equation .~ (1.1.10) can bc solved for the root under any conditions, but a general analytic approximation can be found only if Te % T i . Otherwise thc wave is strongly Landau damped by the ions and w has a large imaginary part at all low frequencies. For hot electrons and cold ions, the wave solutions have phase velocities much below the electron thermal velocity but above the ion thermal velocity. Hence the small argument approxiniation is applicable to the electron W-function and the large argument approximation to ions. T h e equation can then be solved directly for (11 to give

2 . dispersion relation for T J T i = 10 which isuseful for k l , << ( 7 ; / T i ) 1 / The is sketched in Fig. 3. At low frequencies, the wave behaves like a sound wave with speed ai(T,/Ti)'i2.'l'his is the plasma analog of an acoustic wave because the electrons and ions move together as a single fluid. I n contrast with electron plasma waves, ion acoustic waves are always somewhat damped, even in the long wavelength (small k) limit. The first contribution to the imaginary part of (1.1.16) comes from electrons and is usually small. T h e second contribution is ion Landau damping. It is the same for all long wavelengths btXduSe these waves all have the same phase velocity and resonate with the same ions. As the frequency approaches the ion plasma frequency, and the phasc velocity drops, the wave rcsonatcs with morc of the ions and is more heavily damped. For plasma with T, not much greater than T i , even the least damped waves are strongly damped and (1.1.1 6) is inapplicable. €3. D. Fried and R. W. Gould, Phys. Fiuids4,13Y (1961).

1.1. PLASMA

7

WAVES

I

04

0.2

0

0.2

0.4

0.6

0.0

1.0

k AD

FIG.3. T h e dispersion relation for ion acoustic waves with T,= IOT,.

I. I .4. Finite Geometries

T h e infinite, one-dimensional plasma is a useful model because it is the simplest model that demonstrates the characteristic behavior of a plasma. However, only a few experiments produce conditions that even approximate infinite geometry; for most experiments the theory must be recast for the experimental configuration. Among various possible experimental geometries, one will be examined here as an illustration of the effects; it is appropriate to several experiments on plasma waves." I . I .4. I . Equations for a Plasma Column. We consider a column of plasma in a uniform magnetic field along the axis. T h e column is bounded by a conducting cylinder at radius R. This problem has been treated by Trivelpiece and Gould' for cold plasma having uniform density from the center to a radius r' 2 R. They obtain simple expressions for the dispersion A. W. Trivelpiece and R. W. Gould, J. Afipl. Phys. 30,1784 (1959).

* We shall confine the discussion to electron plasma waves, the case treated extensively in the literature. Since the phase velocity of ion acoustic waves is independent of density for the lightly damped waves, the density gradients of finite geometries are often unimportant.

8

1. PLASMA

WAVES AND ECIlOES

relation of (electron) waves in this case. T h e more general case of finite temperature and a density distribution n(r) is Jiscusscd by Ihok" ; numerical integration is required using the n(r) and T,(r) of the experimental plasma. The cquations are simply the three dimensional analogs of (1.1.1) and (1.1.3), but with the forcc in (1.1.1) becoming ( - e/m)(E + {v x Blc)). Providing the Larmor radius arid Debye length arc both much shorter than the scale length for density variation, the Vlasov equation can be integrated along unperturbed orbits to give the perturbed clectron density as a function of E and zero-order density. For these longitudinal waves, E can be derived from a potential, and (1.1.3) then becomes V . € * V V = 0. (1.1.17) Complete details of the integration (1.1.l) to obtain the dielectric tensor may be found in S t i ~Equation .~ (1.1.17) is tractable only in the limits that the plasma frequency is much larger or much smaller than the cyclotron frequency, w,, . The latter case usually obtains in experimental studies of plasma waves. For strong magnetic fields, the dispersion relation separates into a series of modes. The very high frequency modes near the cyclotron frequency and its harmonics are discussed in Part 7 . We are concerned here only with the somewhat lower frequency mode near the plasma frequency, thc mode corresponding to the electron plasma wave in the one-dimensional case. Assuming the usual form of solution for cylindrical geometry, $(r) exp(im0) exp(i(kx - at)),Eq. (1.1.17) reduces to

The components of the dielectric tensor are

&,

=

1 - (w;c/k25e)w(- fB/kUe),

=

1-

w i e / ( 0 2 - w:,J.

(1.1.19)

In the limit (of, 9 mie, cL may be taken as onc, but higher order corrections for m,, approaching apemay be found in Hook.' The dielectric constant depends upon position through the plasma frequency, which is a function of the local density, n(r). 'This is an eigenvalue problem: w must be chosen so that $(r) meets homogeneous boundary conditions at both r = 0 and r = H. At the center, $' = 0 for m even, and li/ = 0 for m odd. At the wall, $ ( H ) = 0 to give zero tangential field at the conductor. L). L.Book, Phys. Pluids 10, 198 (1967).

1.1. PLASMA

9

WAVES

1.1.4.2. Finite Plasma Dispersion Relation. Solving (1.1.18) will produce roots w(k, nz, n), where n is the number of zeros that $ ( r ) has between the center and wall. No analytical solution is possible in general, but the problem is a simple one for a computer to solve using the density distribution i n ' a n experimental plasma. For a plasma with a radius of many Debye lengths in a strong magnetic field, a satisfactory dispersion relation o ( k ) for m = 0 may be obtained by solving the following equation:

The solutions of this equation for two illustrative cases are shown in Fig. 4."In contrast with the infinite geometry case, waves with frequencies below the plasma frequency are possible ; they occur for wavelengths comparable with and longer than the tube radius. As the wavelength

0.1

0.2

0.3 ll

0.4

AD

FIG. 4. The dispersion relation for electron plasma waves in a bounded, cylindrical column. The upper curve is the result for a plasma of uniform density and a radius of 100 Debye lengths. T h e lower curve applies to a plasma with the same central density but with a profile n = n o [ l - sin2(nr/2ro)].The arrow indicates the wave with wavelength 1 = ro.

* Equation (1.1.20) is satisfactory for klD < 0.3. For larger k , the W function of complex argument must be used in E~~ of (1.1.18).

10

1. PLASMA

WAVES AND ECILOES

becomes shorter than the radius, the dispersion curve approaches that of the infinite plasma. Frequencies above the plasma frequency are possible at wavelengths approaching the Debye length. If the plasma radius is many Debye lengths, this portion of the curve is little affected by finite geometry. As the figure illustrates, use of the correct density distribution n(r) is required only if the dispersion curve must be computed to good accuracy. The central density can be inferred within 200/, from an experimental dispersion curve without detailed calculations. Damping becomes important as the wavelength approaches the Debye length. Any plasma large enough to be described by a dielectric constant (a radius of many Debye lengths) behaves almost like an infinite plasma at these wavelengths. Since the damping is caused by resonant electrons, it depends primarily on the wave phase velocity, independent of geometry. Drummond and Malmberg’ have shown that for plasmas at least a few Debye lengths across in a strong magnetic field, (1.1.21)

which is correct for ki/kr < 0.1, thc usual range of interest. (‘l‘hey consider the case of complex k instead of complex w . See next section.) This form of the result is independent of geometry. For a given k,, o and the group velocity v g will depend on geometry, but if the relation among them is known from either the experimental or theoretical dispersion curve, the damping can be obtained. The effect of finite geometry is of crucial importance for electron plasma waves because it makes wave propagation possible over a broad range of frequencies. T h e dispersion relation is qualitatively different, and much easier to work with experimentally. 1.1.5. Space and T i m e Damping

The foregoing theoretical analysis has assumed that the plasma is perturbed with wavenumber k at t = 0 and thereafter left free. It then oscillates at a frequency v = w,/2n, damping at a rate w i . Experiments are rarely done this way. Usually, a wave is continuously excited at w by a source at some position in the plasma. Instead of assuming real k and finding complex o,one excites a real o and observes a complex k , the wave oscillating and damping away from the source. If the damping

’

J. H. Malmberg, C. B. Wharton and W. E. Drummond, Plasma Phys. Contr. Nucl. Fusion Res., Proc. Conf., Culham, Engl., 1965 1,465. IAEA, Vienna, 1966.

1.2. OBSERVATION

OF PLASMA WAVES

11

is small, the relation between damping in time and damping in space is i y ( y 4 wr)>one obtains simple. Writing w(k,) = w,

+

K

= k,(w,)

- iy/(dw/dk) = k, - i y / ~ , .

( 1.122)

The damping rates are related by the group velocity of the wave. When the wave damps by a significant fraction in a period, the dispersion relation must be resolved for complex k with real o. Reference to theoretical calculations for specific experiments will be found in the next section with the experiments. When the damping becomes strong, the problem becomes quite complex. Two effects complicate the calculation. T h e first arises from our method of We kept only the root of E(k, w ) = 0 that had small damping, solving (1.1 3). and never inverted E(k, w ) to find E(x, t ) explicitly. T h e justification was implicit in the fact that other roots were all strongly damped and after a short time only the lightly damped root would remain to contribute to E(x, t ) . However, when all the roots are strongly damped, no contribution will remain at long times, and all roots must be kept for the short times when electric fields still appear. Even worse, roots with larger imaginary w may not have larger imaginary k. For the important case of ion acoustic Gould' finds that the roots of increasing wi for k waves with T, = Ti, real correspond to roots of decreasing k i for w real. Some additional complications also arise in changing from time to space calculations. Detailed theory must be done for each case. A second problem has been studied by Hirshfield and Jacob.' In the usual calculation, one assumes that the antenna excites a normal mode of the plasma, one in which the electric field and distribution function are coupled by Eqs. (1.1.1) and (1.1.3). However, at the antenna, the electric field in the Vlasov equation should include the applied field produced by the source and not just the field produced by the plasma. T h e perturbation of the distribution by the applied field is generally ignored because it dies away within a few Debye lengths. In a strongly damped wave, this distance can be comparable with the damping length and neglect of this effect is no longer justified. Hirshfield and Jacob suggest the criterion that k i 5 O.4kr for neglect of the effect.

I .2. Observation of Plasma Waves To date, most experimental work on plasma waves has been devoted to observing them and verifying that the theoretical description fits the R. W. Gould, Phys. Rev. 136, A991 (1964). J. L. Hirshfield and J. H. Jacob, Phys. Fluids 11,411 (1968).

12

1.

PLASMA WAVES AND ECHOES

phenomena. Although Landau predicted collisionless damping of plasma waves over twenty years ago, convincing experimcntal evidence of this has only begun to accumulate within the last five years. There is not space here to give a complete historical review of the many workers who have identified or produced these waves in plasmas. Only recent work will be cited, and the discussion will concentrate on experiments and techniques that supply complete information on the waves, i.e., both wavelength and damping as a function of frequency. This work has been most decisive in confirming the theory and in developing techniques that make plasma waves a useful diagnostic. 1.2. I. General Requirements

There are several requirements for any apparatus intended for plasma wave observation. T h e first and most important is that the plasma be collisionless in an appropriate sense. For electron plasma waves, this means that all the electron collision frequencies be less than the plasma frequency. For ion acoustic waves, all ion collision frequencies must bc less than the applied frequency, and dissipation through electron collisions must be negligible. Corrections to the theory can be made for collisions, but if the collision frequency is as large as the applied frequency, the wave will be damped too rapidly for useful observations. For collisions to be ignored completely, the mean free path for all collisions must be long compared with the distances over which the measurements are made. A second requirement is effective isolation of the receiver from the transmitter. Various techniques will be illustrated in the experiments discussed here, but some means must be used to discriminate between signal reaching the receiver through the plasma and signal appearing by direct electromagnetic coupling. A third requirement concerns the noise lcvcl in the plasma. T h e signal-to-noise ratio must be good at the receiver, and techniques to enhance this are available. Furthermore, the plasma must be stable in the sense of having no major density or temperature fluctuations in thc time the measurements are made. Otherwise the dispersion relation will be a function of time and the result is uninterpretable. I .2.2. Ion Waves

Ion waves are often observed in low-pressure discharge plasmas. With hot electrons and moderately low collision frequencies, the waves can propagate nearly undamped. 'l'hc first observations werc of naturally

1.2. OBSERVATION

OF PLASMA WAVES

13

occurring oscillations, often under conditions where the discharge tube formed a resonant cavity.' O , Ioa More recently, methods for externally exciting the waves have been developed. The complete dispersion relation, including damping, can now be empirically determined. I.2.2. I. Experimental Techniques

1.2.2.1.1. TRANSMiTTINC ANTENNAS. The wave is usually impressed on the plasma by plane, parallel grids used singly or as a closely spaced pair; the grid generates a plane wave moving perpendicular to itself.' I t is made of fine wire to intercept as little plasma as possible, and the wire spacing is several Debye lengths. With a single grid, a negative bias is applied to collect ions. Modulation of the probe voltage modulates the ion current drawn and hence the ion density. As an example, Wong et al." used a grid spanning their plasma column made from 0.0025-cm wire with 0.075-cm spacing in a plasma with a 0.001-cm Debye length. A 5-V modulation of the -20-V bias produced a 10% density modulation. When double grids are used, their spacing should be less than half of the shortest wavelength excited, but they require no bias. If plasma losses on the exciter would not be too great, as in some B = 0 plasmas, a plane plate can be used. T h e large perturbation in density near the plate does reduce coupling efficiency somewhat. 1.2.2.1.2. RECEIVERS.Several techniques have been used to detect the ion waves. In some experiments, the receiver and transmitter are identical, and the ion current drawn by the biased grid is observed. However, other methods have advantages in some cases. Alexeff and Jones12 discuss the use of hot-wire electron emitting probes biased negatively and of cold probes positively biased. They report increased sensitivity with such probes, although care must be taken that the electron current drawn to the positively biased probe is not sufficient to perturb the plasma. W ~ n g used ' ~ a standard microwave interferometer as a detector. A microwave beam narrower than the acoustic wavelength crosses the plasma l o I. Alexeff and R. V. Neidigh, Phys. Rev. 129, 516 (1963). (Contains extensive references to earlier work.) "'H. Tanaca, M. Koganei and A. Hirose, Phys. Rev. Letters 16,1079 (1966). A. Y. Wong, R. W. Motley and N. D'Angelo, Phys. Rev. 133, A436 (1964). l 2 I. Alexeff and W. D. Jones, Phys. Letters 20,269 (1966). A. Y. Wong, Phys. Rev. Letters 14,252 (1965).

_____ * The plane wave will have the same diameter, d, as the grid (or plasma, if the grid

spans the plasma column). If the grid is embedded in a large plasma, the plane wave will diverge by diffraction with an angle d/d.

14

1.

PLASMA WAVES AND ECHOES

perpendicular to the direction of wave propagation. I t is combined with a reference signal from the same source, and the sum is detected. T h e density variations of the acoustic wave modulate the index of refraction for the electromagnetic wave. The consequent phase modulation of the microwave beam appears as an amplitude modulation of the sum of beam plus reference. Although this method of detection is more complex, it is sensitive only to the density of plasma in the microwave beam and is free of the experimental difficulties discussed below. 1.2.2.1.3. WAVEDETECTION. A major problem in all experiments designed to excite and detect ion acoustic waves is the separation of direct-coupled and plasma-coupled signals at the receiver probe. A transmitted signal will always appear at the receiver, but it can reach the probe by capacitive coupling and be no indication of a plasma wave. T h e separation of capacitive and wave signals is often accomplished by timeof-arrival measurement. T h e transmitter is driven by a gated, sine-wave oscillator. The direct-coupled signal disappears immediately when the oscillator is turned off, but the wave signal continues for the propagation time. Since these are slow waves, the times are usually many microseconds and easily observed on an oscilloscope. T o measure wavelength, at least one of the probes should be moveable." T h e wave signal can be distinguished because its phase relative to the transmitted signal changes as the distance between transmitter and receiver is changed, whereas the direct signal has constant phase, T h e change of phase with distance gives the wavelength. Another common problem in detecting the waves is obtaining an adequate signal-to-noise ratio.? Even in relatively quiet, stable plasmas, a probe will pick up oscillating signals. In many experiments, the combination of this noise and the damping of the waves has limited measurements to within a few wavelengths of the transmitter. Beyond this distance, the wave signal becomes comparable with the noise, and its time of arrival cannot be ascertained. The simplest technique for extracting signals from noise is synchronous detection. A synchronous detector (lock-in amplifier) compares the received signal with a reference and supplies an output proportional to the component of received signal that has the same frequency as the reference and a chosen phase relation. In this case, the reference signal would be the frequency transmitted ; commercial instruments are available for frequencies from 10 Hz to beyond 100 kHz. T h e detector output would register the component of receivcd signal at the transmitted frequency and determine its phase relative to the transmitter.

* A less desirable possibility is a number of fixed probes. t See also Volume 2, Part 12.

1.2.

ORSERVATION OF PLASMA WAVES

15

Unfortunately, the received component would include both direct and plasma coupled signal, inseparably combined. 'I'he detection technique o f Wongl circumvents this difficulty. T h e microwtaw interferometer responds only to density variations in the plasma; there is no open probe to pick up stray signals. Wong is therefore able to synchronously detect the wave signal in the output from the interferometer and follow the bvavcs over much longer distances in the plasma. T h e wave may be recovered from the noise of other density fluctuations in the plasma. I'or cases in which time of arrival must be determined to eliminate direct-coupled signal, some of the new signal averagers would prove useful.14 'I'he transmitter is gated in the usual manner, and the received signal is fed into one of these devices. I t averages the signals received at a chosen time after each gate pulse. T h e noise averages out and the signal due to the wave at that time remains. Ijy taking an average at each of a number of times, the complete waveform can be found, and the delayed portion representing the plasma wave can be selected. I .2.2.2. Results in Q-Machines. \Vong et al.' I made the first measurements of both \vavelength and damping length of acoustic waves using a plasma produced by contact ionization. Cesium and potassium plasmas were produced on a hot tungsten plate at one end of the 2-machine, forming a meter-long plasma column 3 cm in diameter. T h e strong axial magnetic fields gave ion cyclotron frequencies much greater than the applied frequencies, rendering the system effectively one-dimensional. T h e background pressure of lo-' 'l'orr was sufficiently low that ion-neutral collisions could be ignored, and the plasma density was kept low enough to neglect ion-ion collisions. 'I'he question of collisions is thoroughly discussed in the original article." T h e ion and electron temperatures in these plasmas are approximately equal to the plate temperature, 2500 "I<. When T , = T , , the wa\e is strongly Landau damped at all frequencies, giving rise to the difficulties discussed in Section 1.1.5. GouldMhas done the calculations and finds that in a region several Debye lengths from the transmitter but \vithin two wavelengths, a disturbance propagates that behaves like the simple root" of the dispersion relation for Te = T , . ( T h e disturbance propagates beyond this region and has been followed by Wong,' but it looses its wavelike character.) 'I'he experimental observations were confined to this region because Landau damping reduced the amplitude below the noise level in that distance. An additional experimental complication is caused by plasma streaming. T h e plasma is formed l4

1. Xlcxeff and W.D. Jones,Phys. Rev. Letters 15, 286 (1965).

* The simple root of (1.1.10)

to :I root with

(r) reill

is thc m for k rc;il that is Ie;ist damped. This corresponds and k comnlex for the sn;iti;ilproblem of the experiment.

at one end and streams down tlie niachine. A correction must he niade f o r the streanling elocity. I’igure 5 sho\vs the e\perinicntal results for k, and k , ;is a function of frequency for clo\vnstre;irn propagation i n ;I potassium plasma. T h e points include results from densities between 2 and 15 x 10” ~ m - the ~ ; lines are the theoretical predictions, based on the assumed temperatures and the experimental value of the streaming velocity. ‘I’he agreement is well within csperirnental error and clearly shows the adequacy of collisionless theory and thc existence of Imidau damping. kl

05

10

15

2 0

k , cm-‘

I .2.2.3. Results in Hot-Electron Plasma. ‘I’he most favorable conditions for observation of ion acoustic waves obtain in plasmas with an electron temperature much above the ion temperature. ‘I’he most extensive measurements in such plasmas have been obtained in experinients by illexeff and Jones.“ - I 7 A plasma with a density near 10’ c n - ’ is produced l5

I’

I . Xlexeff, W. I). Jones and 1). hIontpomcry, f%j’s.l i e a . L e t l e t s 19,422 (1067). I . Xlexetf, W. I>. Jones and I). hIontgornery, f%j,s. I;/uids 11, 167 (1968). I<. I.onngren, I1.C. illontgoniery, I . ;\lcxef m d \V. I ) . Jones, f%jps. I , P / ~ P25A, ~S

629 (1967).

1.2. OBSERVATION

17

OF PL4SMA WAVES

by an electron gun in a spherical vessel that contains gas at a pressure of a millitorr. There is no magnetic field. T h e electron temperature is a few volts, but the ions are close to room temperature. A flat plate excited by a gated sine wave serves as an antenna in the essentially infinite plasma. T h e wave phase velocity was measured in several gases at frequencies well below the ion plasma frequency (kAD 4 1) and found to agree with Eq. (1.1.16).14 Because the plate does not launch a plane wave, a decrease in wave amplitude with distance cannot be quantitatively associated with damping. T o test the theory without requiring an absolute measurement of the damping length, they have varied certain plasma parameters that have an important effect on the damping and observed the results.

THEORETICAL GAS DAMPING

THEORETICAL L A N D A U DAMPING

i

I

1 tL ' 2

4

10

20

40

Te/ T i

FIG.6. Relative damping of ion acoustic waves as a function of temperature ratio.

In one case, they added a small fraction of helium to a xenon p l a ~ m a . ' ~ This does not affect the wave phase velocity, because that is determined by the average ion mass. Therefore, k, is insensitive to a small impurity. However, the Landau damping is determined by the density of particles resonant with the wave. The light impurity ions have a high thermal velocity, and they make a large contribution to the number of resonant ions. The great increase in damping with only a small fraction of helium has been found in quantitative agreement with theoretical predictions. In another experiment, Alexeff, Jones and Montgomery'6 observe the

18

1. PLASMA

WAVES AND ECHOES

change in phase velocity and damping as the ratio T J T , is changed, keeping other plasma parameters fixed. They place a hot, electronemitting tungsten plate in the plasma. By varying the emission current, they control the degree to which relatively cold electrons from the plate replace hot plasma electrons. The Tell2dependence of phase velocity is confirmed, and Fig. 6 shows the agreement for damping. T h e plasma is not truly collisionless, a fact that becomes evident when the Landau damping is small. Because only relative damping can be measured, the observations confirm only the functional dependence of the damping but this is a significant dependence. on TJT,, Sessler18-'sband Lonngren et al." have also observed the acoustic wave at frequencies approaching the ion plasma frequency. Observations have been difficult because the waves are always strongly damped there and they may become confused with signals arising from ions accelerated by the transmitter that stream directly to the receiver" (cf. 1.1.5). T h e signal from streaming ions has been called a pseudo-wave. I .2.2.4. Diagnostic Usefulness. Ion acoustic wave propagation can easily be used to measure electron and ion temperatures." The electron temperature follows from the wave propagation velocity, which can be found with any configuration of transmitter and receiver. If a transmitter that launches plane waves can be used, the temperature ratio can be inferred from measurements of wave damping. By varying the frequency, the Landau damping can be separated from other damping or loss terms. Even if plane waves cannot be generated, qualitative estimates of damping sufficient to distinguish between T , = T iand T , % T i can be made. If T, % Ti,raising the frequency until the damping rises steeply will give a useful estimate of the ion plasma frequency. This is the only way of deducing density from ion acoustic wave observations. Collision frequencies can be estimated by lowering the wave frequency until additional, non-Landau, damping appears, which can be ascribed to collisions, primarily ion-neutral and ion-ion. I .2.3.Electron Plasma Waves

Historically, electron plasma waves have been hard to observe and identify. The high frequencies make detection difficult, and the fastest G. M. Sessler, Phys. Rev.Letters 17,243 (1966). IsnG. M. Sessler, J. Acoust. SOC.Am.42,360 (1967). '*'G. M. Sessler and G . A. Pearson, Phys. Rev. 162,108 (1967). l 9 I. Alexeff, W. D. Jones and K. Lonngren, Phys. Rev. Letters21,878 (1968).

* See also Volume 7B,Section 10.3.

1.2.

OBSERVATION OF PLASMA WAVES

19

collision process in most plasmas, electron-neutral, is very effective in damping the wave. I .2.3.I. Experimental Techniques. In principle, techniques for observing electron waves are the same as those for ion waves, but the higher frequencies of electron plasma oscillations necessitate more elaborate apparatus. 1.2.3.1.1. ANTENNAS.In designing transmitting and receiving probes, the entire experimental apparatus must be considered. At the electron plasma frequency, the vacuum electromagnetic wavelength is generally comparable with the apparatus size and radiative effects are important. Since the coupling to the plasma is rarely efficient, good discrimination against radiative coupling must obtain to find the plasma signal. With the faster characteristic speeds of these waves, time-of-arrival techniques appropriate for acoustic waves fail. Two general approaches to this problem have been devised. For experiments in an effectively infinite plasma, the antennas must be shielded. Pairs of grids on the scale of those used in ion wave studies may be One element of the pair is at rf ground and the other is connected to the center of a matched, coaxial transmission line. (At these frequencies, impedance matching of all components is necessary.) Grounding the facing elements of the transmitting and receiving pair minimizes radiative coupling. I t may be necessary to completely surround the center element of the transmitter with an rf ground mesh. Plasma passing through the cage is modulated, but little rf field escapes. T o minimize plasma losses on these grids, they are generally connected to be floating in dc potential or biased to the plasma potential to eliminate the electron sheath. Plane grids have been used to produce plane waves, but other geometries are possible if the waves need not be plane waves. T h e essential feature is the use of a shielding grid to reduce radiation. For experiments in a finite plasma, another possibility exists. A plasma column in a magnetic field can be bounded by a cylindrical conductor. If the density is below 10'' emp3, the cylinder radius can easily be chosen so small that the cylinder is a waveguide beyond cutoff at the plasma frequency and somewhat above. Then a simple antenna formed by extending the center conductor of coaxial cable into the plasma can be entirely satisfactory. No electromagnetic radiation can propagate from the transmitter to receiver through the cylinder. Use of miniature coaxial Iine and a small center conductor is recommended to minimize the plasma perturbation. Insulating the elements or allowing them to float in dc potential is desirable. Long slits made in the cylinder for moveable probe 2o 21

G. Van Hoven, Phys. Rea. Letters 17,169 (1966). H. Derfler and T. C. Simonen, Phys. Rev. Letters 17,172 (1966).

20

1.

PLASMA WAVES AND ECHOES

access must be closed where the probe is not present to maintain the electrical integrity of the cylinder. The electronic equipment generally used with the 1.2.3.1.2. RECEIVERS. transmitter and receiver is shown in Fig. 7. Noise from the plasma is often so severe and the signal so weak that synchronous detection is necessary. T h e high-frequency signal source is modulated or chopped with a lowfrequency square wave. The received signal enters an optional rf amplifier and then a mixer-preamplifier where it is converted to the I F frequency and amplified in a high-gain IF strip. Video and dc outputs are available from this and are fed to the synchronous detector to extract the component derived from the transmitter. Alternatively, the synchronous detector may be driven from the video output of a high-gain rf amplifier connected

SO. WAVE

- GENERATOR -

SYNCHRONOUS DETECTOR

x-Y

IF AMP VIDEO DETECTOR

R E C E I V E R POSITION

RECORDER

directly to the receiver. The tuning implicit in conversion to IF is accomplished by a tuneable filter or amplifier at some stage of the amplification. For convenience in obtaining plots of the wave, a position transducer attached to the receiver drives the X-axis of the X - Y recorder. This gives the best output record for measurement of wavelength and damping length. One element of the rf system, often the mixer-preamp, may have variable bandwidth, but the system should not be extremely narrow-band. T h e need for a wide-band receiver can be understood by considering the

1.2.

OBSERVATION OF PLASMA WAVES

21

behavior of the received signal. T h e phase of the received signal relative to the transmitted signal is (1.2.1) the integral running between antennas. In simple theory, (1.2.2)

and is therefore sensitive to the plasma density along the path. Noise, which is associated with density fluctuations, will cause modulation of the phase of the received signal. Over long path lengths, these modulations can greatly exceed 277 and are effectively a frequency modulation. Under good conditions, Aw/w may be as small as but the relative bandwith required increases when the signal is converted to a lower IF frequency. This problem is exacerbated when using 1.2.3.1.3. INTERFEROMETERS. the interferometer loop of Fig. 7. I n this mode, the sum of the received signal and a reference signal (somewhat larger) from the transmitter is recorded. As the receiver moves, the phase of the received wave signal changes and its interference with the reference signal evinces the change." This allows one to measure wavelength directly, but after a few wavelengths, the phase difference becomes large. T h e modulation effects of density variations cause fluctuations in the phase of the received signal. When these fluctuations reach 2n in amplitude, the received signal no longer has an average phase relative to the reference, and the interferometer output becomes independent of position. T h e interferometer pattern damps within a few wavelengths of the source, although the wave is still present and detectable by the receiver without interferometer loop. One must not confuse this with wave damping, for all the original energy is still in the wave. T h e apparent damping distance is merely the wave phase correlation distance. Beyond this distance, the plasma signal and any undesired radiative signal at the receiver behave differently. T h e radiative signal has a well-defined phase that is almost independent of position, whereas the plasma signal has random phase. T h e radiative contribution can therefore be cancelled by the addition of a signal of equal amplitude and opposite phase. In contrast, the sum of plasma and nulling signal is unaffected by changes in the phase of the nulling signal and cannot be made zero.

* T h e presence of unwanted radiatively coupled signal at the receiver does not adversely affect this measurement. Since the plasma wave length is much shorter than the electromagnetic wave length, the radiative contribution appears with nearly constant phase. It becomes part of the fixed reference signal.

1. PLASMA WAVES AND

22

ECHOES

When using synchronous detection with the interferometer, it is best to modulate or gate the transmitter signal after the reference signal for the interferometer is extracted. Then the reference amplitude is constant and the signal from the synchronous detector is derived from the wave only. If the reference channel is also modulated, the synchronous detector registers a large, constant output even in the absence of a wave signal. These techniques have been employed by several workers,Zo-z2but we shall only discuss the results for two cases where both wavelength and damping were quantitativcly measured in agreement with theory.

/

FIG. 8. Observation of the dispersion relation for electron plasma waves in one dimension.

I .2.3.2. Results in “Infinite” Geometry. By restricting observations to a small region of plasma, one can obtain results that coincide with the theory for an infinite plasma. Derfler and Simonen” used a small sodium plasma (of Q-machine type) with no magnetic field. Using plane grid antennas that were larger across than their separation distance, they approximated a one-dimensional plasma. T h e experimental results for the real and imaginary parts of k are shown in Fig. 8. The curves are theoretical predictions based on the measured density of 2 x lo7 cmW3and temperature of 2000°K and using tabulated values of the W-function to evaluate the 22

J. H. Malmberg and C. B. Wharton, Phys. Rev. Letters 13, 184 (1964);17, 175

(1966).

1.2.

OBSERVATION OF PLASMA WAVES

23

dispersion relation. (The plasma frequency is 30 MHz and the Debye length is 0.7 mm; the plasma is completely collisionless.) Agreement is well within experimental error. However, the theory is inaccurate at high damping because only the first root of the dispersion relation was kept. Very long wavelengths and damping lengths could not be observed within the confines of the small plasma. 1.2.3.3. Results in Finite Geometry. The most complete and elegant experiments on plasma waves have been done bj Malmberg and Wharton.22 The experiments cover a greater range of parameters and the accuracy of the results provides a more sensitive test of theory than any other experiment. The general arrangement of the apparatus is shown in Fig. 9.

FIG.9. The experimental apparatus of Malmberg and Wharton.

The Duoplasmatron source produces a hydrogen plasma with a density somewhat above 10’ cm-3 and an adjustable temperature of 5-10 eV. The plasma is bounded by a conducting cylinder 10 cm in diameter and two meters long. T h e plasma terminates on a grid and conducting plate. I n a magnetic field of 200 G, the electron cyclotron frequency is sufficiently above the plasma frequency to satisfy the theoretical requirement for an “infinite” magnetic field. With a background pressure of lo-’ Torr, the shortest mean free path is that for electron-neutral collisions: 40 m. The plasma is collisionless. T h e end plate is equipped with an electron velocity analyzer. Using this, they have measured the electron temperature of the plasma and verified that the electron distribution is Maxwellian in the region of the tail important for damping. For wave measurements, the interferometer configuration gives a plot of wavelength, and a direct measurement of received signal strength gives the damping. Damping can be observed over several orders of magnitude and is found to be exponential.’ T h e observations of wavelength are shown in Fig. 10. The solid line is calculated by computer integration of Eq. (1.1.18),

24

1. PLASMA

WAVES AND ECHOES

k AD

FIG.10. Observations of the real part of the dispersion relation for electron plasma waves in finite geometry.

keeping the W-function in the dielectric tensor." The temperature used was that determined from the electron analyzer, but the density was chosen for best fit. (Probes are quite unsatisfactory for density measurements in this sort of plasma.) Agreement is excellent over the broad range of experimental points. Thc results for damping are shown in Fig. 11, using coordinates especially suitable for presenting Landau damping. T h e abscissa is the square of v,/ce, the exponent of e in the expression for the number of resonant electrons in a Maxwellian distribution, These are the electrons that cause the wave damping, T h e straight line represents the theoretical result that the damping is just proportional to the density of resonant electrons. [The line is a plot of (1.1.21) for the parameters of this experiment.] T h e experimental data are in good agreement with this prediction for two different temperatures and over a decade range of damping lengths. Further evidence that the damping arises from the electrons traveling with the same velocity as the wave is provided by use of the plate at the end of the machine. Ions from the source stream to the plate, striking it and recombining. A sheath forms at the plate that repels most electrons. The electron distribution is therefore essentially symmetric, the net drift at the ion speed being negligible on the scale of V,. The symmetry is proved by the observation of points on Fig. 11 for waves propagating in either

* Only the m = 0 lowest radial mode is used. Higher modes have lower phase velocities and are rapidly damped.

1.2. OBSERVATION OF

25

PLASMA WAVES

0.10

0.05 k,/k,

0.02

0.01

T6.5; K T - 9.6ev

0.005 0

5

10

\ 15

,

20

.

25

( V p /ve 12

FIG.1 1 . Observation of damping of electron plasma waves in finite geometry.

direction along the column. By changing the bias on the end plate, the potential barrier can be lowered to the point that sufficiently energetic electrons can reach the plate and be lost instead of. reflected. T h e velocity distribution will then be that of Fig. 12, with no high-energy tail in the direction of the source. An abrupt drop in damping for that direction

E L E C T R O N VELOCITY. V

FIG.12. The electron distribution function modified by application of a potential to the end plate.

1. PLASMA

26

WAVES AND ECHOES

occurs when the rcsonant electrons are removed by biasing, while the wave traveling toward the plate remains damped as before. T h e wavelength is unchanged. This providcs the most elegant experimental confirmation of the physical picture of resonant electrons receiving the wave energy from what appears to them to be a dc electric field. Only the resonant electrons contribute to the damping, but those electrons do not affect the wave otherwise. I .2.3.4. Large Amplitude Waves. All the experiments discussed have used wave amplitudes sufficiently small that linear theory applies. Application of criterion (1.1.15) is difficult in borderline cases because knowledge of electric field strength in the plasma is hard to obtain. If the transmitter and receiver are identical, the principle of reciprocity provides a satisfactory estimate. By calibrating the receiver, one can determine that the received signal is N dR below the transmitted power. Ry reciprocity, coupling in at the transmitter occurs with the same efficiency as coupling out at the receiver. Therefore the power coupled into the plasma is N / 2 dB below the transmitter power." The total power in the plasma is the product of energy density times group velocity, integrated over the plasma cross section. This provides the basis for an estimate of the wave energy density required for (1.1.15). Malmberg and Wharton' have produced large amplitude waves that can be observed for times violating (1.1.15) within their machine. T h e resonant particles responsible for the damping are the ones trapped in the wave, and they found large deviations from Landau damping for these waves. The observed slow oscillations in wave amplitude agree qualitatively with theoretical work for similar case^.'^'^^ They have also directly observed the distortion in the background distribution function caused by the acceleration of the resonant electrons. 2 6 I.2.3.5. Diagnostic Usefulness. Electron plasma waves can serve as an excellent diagnostic for plasma density and temperature. From the known experimental geometry and relative density distribution, a family of dispersion curves can be calculated for various densities and temperatures. T h e T , and n giving the best fit to the experimental curve are chosen. J. 11. Malmberg and C . R. Wharton, Phys. Rev. Letters 19,775 (1967). T. M. O'Neil, Phys. Fluids 8,2255 (1965). 2 5 L. M. Al'tshul' and V. I. Karpman, Zh. Eksperinr. i Teor. R z . 49, 515 (1965). Soviet Phys. J E W (Erzglish Transl.) 22, 361 (1966). 2 6 J. H. Malmberg and T. 13. Jensen, Plasma Phys. Contr. Nucl. Fusion. Res., 3rd, Novosabirsk, IJSSR, 196'8. IAEA, Vienna, 1969. 23

24

.~

* This is subject to the restriction that the N

dB decoupling include no losses from plasma dissipation and that the coupling he due to a wave mode in which the transmitter and receiver are similarly placed.

1.3.

PLASMA WAVE ECHOES

27

(Since the density most affects the finite geometry behavior at long wavelengths while the temperature affects behavior as kAD approaches one, the two parameters are determined nearly independently.) T h e density and temperature can be determined to the same percentage accuracy as the wavelength, and the goodness of the best two-parameter fit gives an index of the reliability of the method. Like all wave measurements, the result is unaffected by sheath and probe effects. The indicated temperature is a good estimate of 2 even if the electron distribution is not Maxwellian. Measurements of the damping length as a function of frequency can indicate the shape of the distribution function in the tail. T h e strength of the damping at each phase velocity determines df,/dv at that velocity, and a graph of the distribution can be inferred from a series of such measurements. This does require absolute measurement of the damping. I n the experiments discussed here, the geometries were chosen to give simple electron plasma modes for which calculations would be relatively easy. The experimental accuracy in these cases is remarkable and establishes the validity of the theory beyond doubt. In more complex geometries, the difficulty of computing a dispersion relation may preclude accurate temperature and density determinations. Experimentally, Landau damping may be inextricably mixed with spreading of the wave front. At worst however, a reasonable estimate of density and temperature should be possible.

1.3. Plasma Wave Echoes Plasma wave echoes share a common phenomenology with spin echoes, which have been known for some time. A macroscopic disturbance excited in the medium decays, but not through a collisional or other truly randomizing process. A second disturbance likewise dies away, but at a later time an echo of the disturbance appears. Since at least two pulses are required to excite an echo, these are often called two-pulse echoes. They are possible in many-body systems when a macroscopic disturbance decays by a process that does not obliterate all trace of the disturbance. If a “memory” of the disturbance remains in the microscopic motions of the many bodies, some of the information can be recovered in a macroscopic manifestation under the proper conditions. I .3. I. Types of Echoes

T h e echo can be separated from the primary disturbances in either space or time. The first echoes observed were temporal echoes. Pulses were applied to a sample at t = 0 and t = z.Echoes appeared at intervals of 7 after the second pulse. For spatial echoes, disturbances are excited con-

1. PLASMA

28

WAVES AND ECHOES

tinuously at two different places, and the “echo” appears at a third place, one where no signal would be found if either of the sources were present alone. Spin echoes are always temporal echoes because the spins are fixed in space and can be affected only by disturbances applied at their position. In contrast, plasma wave echoes have been observed as spatial echoes only. Various types of disturbance have been used to excite echoes in plasmas. Herrmann et aL2’ made the first observation of echoes by exciting the plasma with pulses of radiation at the electron cyclotron frequency. After the first pulse, the plasma briefly continues to radiate strongly at the cyclotron frequency, but this decays after a few periods because the magnetic field is inhomogeneous. Electrons with different frequencies are soon out of phase and the radiation does not add coherently. After the second pulse, the radiation again decays quickly. However, the phase of each electron (starting from the first pulse) is well defined at the time of the second pulse, and the total energy acquired from the two pulses depends on this phase. (The second pulse can either add to or subtract from the contribution of the first pulse.) Any mechanism that affects electrons of different energy differently, e.g., an energy dependent collision cross section, selectively removes electrons that had a certain phase, 8, after the separation time t. At a time t later, these electrons, the ones that would then have had phase 28, are missing. Electrons of all phases are therefore not present and coherent radiation appears, thc echo. Considerable theoretical work has been published for cyclotron echoesJ* When the density of the plasma becomes moderate, one cannot use a single-particle model to treat the cyclotron echo ; the electrons interact and the plasma modes must be used. The echoes then appear at the upper hybrid frequency, (oze-i- wpZ,)’12, a case examined by Bauer et a1.35 G. F. Herrmann, R. M. Hill and D. E. Kaplan, Phys. Rev. 156,118 (1967). F. W. Crawford and R. S . Harp, Phys. Letters 21, 292 (1966); J. Appt. Phys. 37, 4405 (1966). 2 9 D. E. Kaplan, R. M. Hill and A. Y. Wong, Phys. Letters22,585 (1966). 3 0 R. S. Harp, R. L. Bruce and F. W. Crawford, J. Appl. Phys. 38,3385 (1967). 31 R. W. Gould, Phys, Letters 19,477 (1965). 32 W. H. Kegel and R. W. Gould, Phys. Letters 19, 531 (1965). 3 3 W. H. Kegel, Phys. Letters 23, 317 (1966). 33a W. H. Kegel, Plusma Phys. 9,23,339 (1967). 3 4 G. F. Herrmann and K. F. Whitmer, Phys. Rev. 143,122 (1966). 3 s L. 0. Bauer, F. A. Blum and R. W . Gould, Phys. Rev. Letters 20,435 (1968). 27

* Echoes acising from energy-dependence cross sections arc discussed by Herrmann et ~ l . , ~Crawford ’ and Harp,’* Kaplan et al.,29and Harp et aL30 Other possible mechanisms are treated by G o ~ l d , ~ Kegel ’ and G o ~ l d , ~ Kege1,33-33a * and Herrmann and Whitmer. 34

1.3.

29

PLASMA WAVE ECHOES

Echoes can also be obtained if the plasma is excited with electron plasma waves or ion acoustic waves. Both experiment and theory are best developed for electron plasma wave echoes, and we shall discuss them in greatest detail. 1.3.2. Mechanism for Electron Plasma Echoes

A simplified theory of plasma wave echoes can be constructed following ~ 13 is appropriate to spatial echoes in one dimension. Gould et u Z . ~Figure TRANSMITTERS

w2= 2 9

PLASMA

5 (2.3)

FIG.13. The appearance of spatial echoes.

The antennas excite fields at frequencies w1 and w 2 continuously. To predict echoes, it suffices to consider only the effect of these applied fields on the particles and ignore the self-consistent plasma response to the fields, i.e., the plasma waves. In this idealized case, a transmitting antenna is a pair of closely spaced grids that influences an electron only during the short time the electron is between the grids. T h e electron perturbation is proportional to 4 exp( - i d ) , the value of the field between the grids at the time the electron passed through them. Thereafter, it streams with velocity a. For electrons traveling to the right, the distribution function , t ) = Fl(v)q5, between antennas includes a perturbation f l ( ~ x, exp( -io,(t- XI.)).T h e function F(u) is slowly varying in v, similar in character to the unperturbed distribution function. T h e density perturbation at any point is n(x, t ) = jf(u, x, t ) du, and this is the amplitude 36

R. W. Gould, T . M. O’Neil and J. H. Malmberg, Phys. Rev. Letters 19,219(1967).

30

1. PLASMA

WAVES AND ECHOES

of the disturbance at the point. At distances from the antenna such that o x % Ce, the term exp(io,x/u) in f, is a rapidly oscillating function integrated over the distribution function, leaving the net disturbance n(x, t ) small. For plasma waves with w 2 w p e , this implies that the macroscopic evidence of the disturbance disappears within a few Debye lengths of the source, although the perturbations in the distribution function remain. When the electrons pass through the second grid, they suffer a perturbation f 2 ( v ,x, t ) = F 2 ( v ) 4 2expi- i w 2 [ t - (x - Z)/v]}. T h e field from this antenna also dies away within a few Debye lengths. However, the second grid affects not only the equilibrium distribution, but also the perturbed distribution from the first grid. This second-order contribution in $ is f12(2', x, t ) = F12(v)41+zexp -i{wl(t - x/v) - w 2 [t - (x - Z)/v]). The velocity dependence of this exponent is [ w 2 ( x - 1) - q x ] / v . At x = ~ 2 Z / ( w 2 wl), the v dependence disappears, and the integral for the density perturbation lacks the oscillating term to make it zero. An echo appears at a point. x = w21/(w2- w1)

(1.3.1)

if w 2 > wl. (If w 2 < ol,a similar argument would show the presence of an echo at the corresponding point to the left of the first antenna.) T h e amplitude of the echo is proportional to and is therefore second order, The echo appears at a frequency w 2 - wl. By considering higher order perturbations from each antenna, one can infer the position of higher order echoes. An echo of order m + n with amplitude proportional to $lm$ z" occurs at a position and frequency given by x = nw2Z/(no2- mul), w = n u 2 - mw,,

(1.3.2)

providing nw2 > mwl . Some of these are depicted in Fig. 13. A graphical picture due to Ahern, Baker and W ~ n shows g ~ ~how the velocity spread of electrons blurs out a perturbation, but two such perturbations interact to refocus the electrons farther along their paths. Figure 14 indicates the trajectories of electrons that are admitted to the system by the first antenna during short pulses. Only those electrons that arrive at the second grid during the times it is open are permitted through to continue their unperturbed orbits. T h e figure is drawn for w 2 = ZW, . At positions somewhat beyond the second gate, the density of trajectories is constant in time; there are no macroscopic disturbances there. However, at the position indicated, there are times when no electrons are present; the density depends on time, manifesting the echo. 37

D. R.Baker, N. R. Ahernand A. Y. Wong,Phys. Rev. Lettevs20,318 (1968).

1.3.

PLASMA WAVE ECHOES

31

These are grossly simplified pictures of echoes that suffice only to predict the positions and frequencies. More sophisticated theory is required to predict the characteristics of the echo waveform.

x=p

x =0

0

r

2r

3r

4r

5r

6r

TIME

FIG.14. A simple picture of the formation of spatial echoes. The lines represent sample electron trajectories as affected by the two antennas. The diagrams at the right indicate the resulting electron density n ( t ) at various points in space. The blurring of the direct antenna perturbations and the subsequent echo formation are apparent.

I .3.3. Theory of Echoes

T h e theory of echoes appeared in the original note on plasma wave echoes,36 and simplified treatments have appeared elsewhere. * O’Neil and G o ~ l have d ~ published ~ the most thorough and detailed treatment to date. They present a complete calculation for second-order temporal and spatial echoes and supply results for higher order echoes that are correct under specified restrictions and will be a useful guide generally. 38

39

R. W. Gould, Phys. Letters25A, 559 (1967). T.M.O’Neil and R. W. Gould, Phys. Fluids 11,134(1968).

1. PLASMA

32

WAVES AND ECHOES

The calculation is quite complex; the details scarcely belong in a chapter of this nature. I n outline, the work follows the discussion of the preceding section, using the Vlasov equation to calculate the perturbation in the distribution function and the electric fields in the plasma. Because the applied fields do not merely perturb individual particles but excite coherent plasma waves, the primary effect of the antenna is to excite an electron plasma wave. The wave propagates away from the antenna, decaying by Landau damping. Although the electric fields and density variations of the wave disappear, the perturbations in the distribution function do not and are strongest for those electrons that were resonant with the wave and absorbed its energy. The second transmitter likewise excites a plasma wave that Landau damps, leaving its imprint on the distribution function. T h e effect of the second wave on the residual perturbation from the first produces the echo, T h e strongest second-order echo occurs when w1 and the echo o3 excite only slightly damped waves, i.e., wl, o3 ope. This requires that w2 2mne. Tn the simple picture, the echo occurs at x = I’ = o,l / ( 0 2- wl) = w21/w3.O’Neil and Gould3’ quote expressions for the large contributions to the echo field near x = 1’. For x < E’,

-

-

E3 =

kI2lo,

- 4142(%)(F) ,~ - 2 i k , 1 - ~ 4exp[i(w3/w,)(k, 3k,2&2[1

For x > I’,

E, = -

4142

-(

+ iT1)(l’ - x)]

o ~ ~ /-o12ik,r,1,2] ~ ~ )

*

(1.3.3)

(+)(“‘) 4

2 i k , 1 - , 1 ~ exp[i(l’ ~ - x ) ( k 3 - z3)] 3kZ2,lD2[1- ( w ~ ~ / -w 12ik3r,&,2]l’ ~ ~ )

where k, and Tn are the real and imaginary parts of the wavenumber that is the root of the dispersion relation at on.At the front of the echo, x < l‘, the field grows at a rate (03/01)r,,proportional to the Landau damping constant of wl, and it oscillates with a wavenumber ( ~ 0 3 / w l ) k 1 , At the back however, the wave oscillates and decays as would an ordinary plasma wave at 0 3 . I.3.4. Effect of Collisions on Echoes

Echoes are interesting phenomena and a sensitive test of the accuracy of the Vlasov equation, but they are more than fascinating curiosities. Echoes are generally employed to measure relaxation effects in a medium. In a plasma, this relaxation or loss of memory could arise from collisions

1.3.

33

PLASMA WAVE ECHOES

or microturbulence.* The echo depends on the existence of perturbations of the form exp(iwx/v) in the distribution. Even a very small-angle collision can change v enough to change the phase of the exponential appreciably. When the phase becomes random, memory of the perturbation at w is lost and the echo vanishes. Su and Oberman4’ and O’Nei141 have added a Fokker-Planck operator to the Vlasov equation to describe the effect of collisions on the distribution function.

af af eE af at + v - ax + m av

(-)-

= -

( i ) D l f + ($)D,f.

(1.3.4)

I n general, the D’s depend upon both position and velocity, and D , is most important for perturbations of the type considered. T h e diffusion in velocity space introduced by this operator could be caused by either classical electron collisions or effective collisions produced by a turbulent wave spectrum in the plasma. Experimentally, the third-order echo with w1 = w 2 = o3 is strong and easy to produce. There are also theoretical reasons favoring this echo for study of the effects of collisions. O’Nei14’ derives explicit results for this echo with diffusion arising from two sources. T h e first is Coulomb collisions. Then D depends only on velocity, and the collisions reduce the echo by a factor exp( -D,(w/k)2Z3k5/3w3), (1.3.5) where Z is the distance between transmitters. Since the perturbation in the distribution function is strongest for resonant electrons, diffusion at that velocity is most effective in destroying the echo. By observing peak echo amplitude as a function of transmitter separation, one could determine D,(w/k). One could thus measure D 2 ( v )throughout most of the tail of the distribution. Asecond assumed D , treats the case of diffusion produced by turbulence. A third transmitter at a position Zl between the two echo transmitters launches waves with a power spectrum E2(w) and random phases. This simulated turbulence reduces the echo amplitude by a factor exp[ -(e/m)2EZ(w)Z12k5/ I ki 1

OJ~],

(1.3.6)

where k j is the damping length for waves at w , the frequency of the echo. Experiments have confirmed this result, thereby providing perhaps the best evidence for the oft-quoted theoretical prediction of diffusion pro40

41

-

C. H. Su and C. Oberman, Phys. Rev. Letters 20,427 (1968). T. M. O’Neil, Phys. Fluids 11, 2420 (1968).

See also Parts 2 , 3 , 9 and 10.

34

1. PLASMA

WAVES AND ECHOES

portional to the power spectrum E Z ( w )of t ~ r b u l e n c e . ~T’h e echo is thus established as a powerful diagnostic for plasma turbulence.

I .3.5. Ion Wave Echoes A.detailed theory comparable to that available for electron wave echoes does not yet exist. However, the same considerations that lead to the prediction of electron echoes apply to ion acoustic waves. Antennas can excite ion waves that Landau damp and leave a residue of perturbations in the distribution function to produce echoes. Equation (1.3.2) correctly gives the positions and frequencies of echoes.

I.4. Observations of Echoes The observation of echoes is inherently no more difficult than observing the primary waves themselves, although some requirements are somewhat more stringent. T h e plasma must be large and truly collisionless. T o observe the primary wave, a plasma only a few damping lengths long suffices. The scale must be increased by an order of magnitude to observe echoes. The great sensitivity of the echo to collisions requires that the mean free path really be longer than the device. If collisional damping is detectable in the primary waves, it will probably destroy the echo. I .4. I, Experimental Techniques

Only a few additions to Sections 1.2.2.1 and 1.2.3.1 are required here. T h e transmitters required for echoes are identical to those discussed earlier. The receivers are also identical, but signal averaging or coherent detection to improve the signal-to-noise ratio is imperative for echoes. T h e echoes are second or higher order in the wave amplitude and are therefore much weaker than the primary waves. The interferometer techniques to measure wavelength also require modification. T h e echo occurs at w 3 = no, - mo, , and a constant phase reference signal at o3 is required for the interferometer. For electron plasma wave echoes, this is obtained by putting o1and ozinto a crystal mixer. A narrowband filter tuned to w 3 and an rf amplifier provide the desired output. For the low frequencies of ion acoustic waves, a simple, sensitive interferometer can be made with a synchronous detector. A mixer and filter-amplifier produce a signal at o3that is used directly as the reference signal for the detector. Since w g # ol, w z , there is no direct-coupled 4f

T. H. Jensen, J. H. Malmberg and T.M. O’Neil, Phys. Fluids 12,1728 (1969).

1.4. OBSERVATION

35

OF ECHOES

signal at w 3 that must be discriminated against, and this method is quite effective. I .4.2. Observation of Electron Plasma Wave Echoes

Malmberg el ~ 2 1 first . ~ ~reported observations of these echoes. Using the apparatus discussed in Section 1.2.3.3, they have found spatial echoes of orders 2, 3, 4, and 5.44 For each order, the echoes occur at the position and frequency predicted by Eq. (1.3.2), and only when the frequency inequalities are satisfied. Furthermore, the echo amplitude depends on the amplitude of the primary waves in the expected way. For a third-order echo, Fig. 15 shows typical interferometer traces at the applied and echo

f 3 = 140 M H z

RECIEVER GAIN INCREASED 20 d 0

-20

0

20

40

60

80

100

120

RECEIVER POSITION, C M

FIG.15. Interferometer traces for a third-order echo. The uppermost trace shows the wave excited by the first antenna, the middle trace shows the second wave, and the bottom trace shows the appearance of the echo atf3 = 2f2 =fi in the expected place.

frequencies. T h e echo wavelength is apparently that of w 3 , but that is experimentally indistinguishable from (1.3.3). 4 3 J. H. Malmberg, C. B. Wharton, R. W. Gould and T. M. O’Neil, Phys. Rev. Letters 20,95 (1968). 44 J. H. Malmberg, C. B. Wharton, R. W. Gould and T. M. O’Neil, Phys. Fluids 11, 1147 (1968).

1. PLASMA

36

WAVES AND ECHOES

A rather unusual sort of third-order echo was found in this machine. For the case o1 = o2 = co3, an echo can be produced by a single transmitter. If the transmitter is placcd a distance 1 from the sheath at the end of the machine, electrons perturbed by the transmitter can be reflected by the sheath and pass the transmitter for a second perturbation. T h e reflection at the shcath gives the effect of a virtual transmitter at a position 1 behind the sheath. The experiments are compatible with the theory, but they are not yet sufficiently precise to verify the theoretical predictions of the internal structure in the echo: its wavelength variations and its rate of rise and fall. I .4.3. Observations of Ion Acoustic Echoes

Ion wave echoes have been observed in Q-machines by Ikezi and T a k a h a ~ h and i ~ ~by Baker et d3'Ahern, Baker, and Wong have examined second-order echoes, and find that Eq. (1.3.2) correctly predicts their location. (Corrections are required when the plasma has a net drift down the machine.) They have not used an interferometer to measure the wavelength in the echo, but they qualitatively investigated the effect of collisions in damping the echo, They increased ion-ion collisions by increasing the plasma density and increased ion-neutral collisions by adding an inert gas background. Both procedures reduced the echo amplitude, but ion-ion collisions were much more effective because even when the mean free path for 90" deflection is long, there are frequent small-angle deflections that suffice to destroy the echo. Echoes show the expected sensitivity to Coulomb interaction. Ikezi and T a k a h a ~ h ilikewise ~~ find echoes in agreement with (1,3,2), and from an interferometer they conclude that the echo has the same wavelength as the normal acoustic wave at 03.

45

H. Ikezi and N. Takahashi, Phys. Rev. Letters20,140 (1968).