J. Nucl. Energy, Part C: Plasma Physics. 1960, Vol. 1. pp. 190 10 198. Pergamon Press Ltd. Printed in Northern Ireland

LONGITUDINAL PLASMA OSCILLATIONS IN AN ELECTRIC FIELD B. D. FRIED, M. GELL-MANN,* J. D. JACKSON? and H. W. WYLD? Space Technology Laboratories, Inc., Los Angeles, Calif.

(Received 27 June 1959)

Abstract-The properties of longitudinal plasma oscillations in an external electric field are investigated. I n a completely linear approximation, it is found that the direct-current electric field introduces essentially no new effects. A quasi-linear approximation is also considered, in which couplings between different plasma modes are neglected while the space-averaged distribution functions are assumed to be approximately independent of time. In this case, a Maxwellian distribution function is found to be always unstable against the growth of very long wavelength oscillations.

If E, = 0, the linearized form of these equations can readily be solved. The resulting dispersion equation (JACKSON, 1958) predicts Landau damping (LANDAU, 1946) if the unperturbed distributions have no relative mean velocity and gives growing waves if the mean velocities differ by more than E times the electron thermal velocity (for Ti= T?),where E is a number of order one whose exact value (JACKSON, 1958) depends upon the form assumed for the unperturbed velocity distributions. It is the aim of the present paper to generalize these field-free results and to examine the effect of an external electric field upon the plasma waves. With the usual separation off into a space averaged part, f o , and the fluctuations, fl, around that, we find that in astrictly linear theoryJ,must be time-dependent. Consequently, the equation for fi does not have harmonic solutions and there is no dispersion equation in the usual sense. This is discussed in Section 2. In Section 3 we consider briefly the consequences of assuming foto be time-independent, asmight be appropriate in a quasi-linear theory which takes account of the effects of the fluctuations upon fo but neglects the coupling among the fluctuation modes. In this case a dispersion equation of the usual sort can be derived and leads to growing waves with a Maxwellian foeven in absence of a relative electron-ion drift. We The ion distribution function, F, satisfies the same conclude that either the quasi-linear approximation equation with elm + -e/M. The external electrical with time-independent f,,is inherently inconsistent or field E,(t) is a given function of time, while the self- else that it demands a special form for f o , different consistent plasma field, E, is determined from Poisson’s in character from a Gaussian. equation 2. THE L I N E A R T H E O R Y 1. I N T R O D U C T I O N

IN the course of an attempt to understand in more detail the possibility, mggested by BUNEMAN (1958), that long-range co-operative effects in the forni of growing plasma waves may provide a new mechanism for plasma resistivity, we have studied the dispersion equation for longitudinal plasma waves in the presence of an external electric field. While we have not, as yet, succeeded in achieving a quantitative understanding of BUNEMAN’S mechanism, the results concerning the effect of an electric field on plasma waves are self-contained and may be of value also in other investigations. We consider a plasma composed of electrons and ions and assume that the distribution function (in phase space) for each species obeys a collisionless Boltzmann equation, with electromagnetic fields whose sources are the plasma charge and current density. Since the two-stream instability which BUNEMAN considers involves only longitudinal plasma waves, we neglect the magnetic field due to the plasma current. We also assume that no external magnetic field is present. The problem is then essentially onedimensional, and we have for the electron distribution function, f ( x , U , t ) ,

_ aE - 4rre ax

*

t

s

dc(F -f ) .

(2)

It is convenient to make a Fourier expansion of the x dependence of the distribution functions,

Present address: California Institute of Technology. Present address: University of Illinois.

f ( x , 0, 190

r)

= n,fo(u, 1 )

+ 2.hC(v,t ) exp ikx k

Longitudinal plasma oscillations in an electric field

191

where the bar denotes a Fourier transform with respect to U,

wit11 similar expansions for F and for

E(x, t ) = IE,(t) exp (ikx).

m

fO(o)=J- m

h‘.

Reality of f,E requires fk* =f-];, etc. The space averaged density of both ions and electrons is indicated by no, and fo is normalized to 1. (The k spectrum is Inade discrete by using periodic boundary conditions with a period L SO that the allowed k values are multiples of 2n/L.) The equations for the Fourier aiiiplitudes are then

nu exp {-iue)fo(u)

=I

(10)

cc

-&.(e,

0)

du exp {-iuO)f,(u,

0),

-m

and a n integration by parts has been used to transfer (d/du) from fo to exp [iku(t‘ - t ) ] . Substituting equation (9) and a similar expression for ion density into Poisson’s equation, equation (9,we obtain finally a n integral equation for €,(f).

+ ru,2~dt’Ek(r’)(t-t’)

Ek(t)

ikE,

= 4ne

i

dc(F, - fk),

(5)

In the linearized approximation we drop the righthand sides of equations (3) and (4). Then equation (3) is solved by takingfo to be an arbitrary function of I / = 1;

+ (e/7v)~Ec(tr)

(6)

Introducing u and t as independent variables in place of 1‘ and I, we can write equation (4) as

with a similar equation for F,. Since the coefficients are time-dependent, the solutions of equation ( 7 ) are not plane waves and we cannot find a dispersion equation in the usual sense. However, we can solve equation (7) by using an integrating factor.

where

s,”

4(t) = ( e / m ) dt’(t

- r’)&(t’).

The electron density is then

x

l

tlf’E,(t’)(t - t’) . f ” [ k ( t - t’)]

x e x p [-ikm(+ - +’)/MI = ( 4 ~ e i / k [exp ) (ik$)f;,(kt, 0)

- exp (-ikni+/M) Fk(kt, O)]

(11)

where cop2 = 4nn,e2/nz is the electron plasma frequency. I n absence of t h e external electric field, 4 = 0 and the integral equation is of the convolution type. A solution is readily obtained by means of Laplace or one-sided Fourier transforms, E,(w) =

No)

1

+ D(to)

where R(w) is the transform of the right-hand side of equation (1 1) and D((o) is the transform of to,2t[fo(kt)

+ (~u/M)Fo(kr)l.

The necessary and sufficient condition for stability of the oscillations is that the denominator of equation (12) have no roots in the upper half-w-plane. This problem and the properties of D(w) have been carefully discussed by JACKSON (1958). The integral equation is also simple if only electron fluctuations are considered. In the limit m/M-+ 0 we have again a convolution equation, this time for the quantity E exp (-ik4). Since $ is real, the stability properties are identical with those in absence of a n external field. For the case where neither m / M nor E, vanishes, equation (11) is rather formidable. For any given initial conditions, the right-hand side of equation (1 1) is known and one could at least obtain a numerical solution. To determine the stability properties, however, it is necessary to decide whether equation

192

B. D. FRIED, M . GELL-MANN, J. D. JACKSON and H. W. WYLD

(1 1) has solutions with unbounded E for any initial equation (16) is complicated, for the asymptotic form conditions. This information is readily obtained from of Z’ is (see Appendix) t h e usual dispersion equation but we do not know a z’(x) -20id.irx exp (-x2) + x-2 for 1x1 CO, general technique for extracting it from the integral equation. Some progress can be made by rewriting (19) equation (11) in terms Of a ‘perator repre- where = 0, 1, or 2 according to whether Jm(,y) is -+

sentation, as follOws. We equation (7) by formally inverting the differential operator,

h=

noe

a

+ ik(u -

[%

positive, zero, or negative. For the corresponding g it follows that

g(t)

The density is then

--f

-

6-2

- 2iod&/n3) exp ( - ~ / n 2 ) for ( 1

a. (20)

---f

Instead, we shall use the simpler function, g(5) = ( 5

+

(21)

which corresponds to the choice of a resonance shape distribution function

where the function g is defined by

,fo(u) = r-1u(u*

the singularity in the integrand for real x being defined in the manner appropriate to a n initial-value problem. Substitution of equation (14) and the analogous expression for ion density into Poisson’s equation gives the operator form of equation (ll),

4-a y .

(22)

For the case where the two species have equal velocity spreads and equal masses ( i n = M , n = A ) , the equation for Et is then

(23)

where G is defined as in equation (15) with the ion distribution, F,,, in place of f o . l f f , is Maxwellian, exp (- u2/a2) f O ( 4

=

The ia term in the denominators, which represents Landau damping for our particular fo, can be eliminated by the substitution

E,z r - 1 1 2

(24)

(-02)

exp ( -q2) clq

Rationalizing the denominators in equation (25) and setting Y = P2r (27) we have finally a fourth-order equation for 7, (z2 -k P2)q = (k2/co,~)u2,92f7

id; exp ( - x 2 ) - 2x Y(x),

(28)

We now specialize to the case of a constant external field. Since electrostatic instabilities tend to be more exp (q2)dq. (18) serious for the longer wavelengths, we first study Y(x) = exp (-x2)x-l equation (28) in the limit of very small k . An explicit (For some useful properties of Y and Z , see JACKSOX definition of the ‘small k’ regime can be obtained by (1958)). Even in the low-temperature limit ( a -+ 0) imagining that the external field is switched off at Y ( x ) being real for real x,

L:

Longitudinal plasma oscillations in an electric field

time t , leaving the two species with velocities V= -. --.e/$//ii. The differential equation (28) can then be solved with an exponential exp (iukt), where u is the root of 2 ( ~ ? V 2 )= (k2//ttj,,2) (U' - V')'.

+

The correction to the k = 0 solution, u2 = - V 2 , is sinall provided kV/to,< 1. Thus, we consider k as 'small' if k iiico,,/eE,t. (29) ~f we define s = k t and 7 = eE,/km


r

= 2(9's2 - a2/as*)r = 0

whose general solution is 7

=~l'~.Z~~~(ips~/2)

(30)

where Z,,, denotes any Bessel function of order 114. The character of the small k solution is now clear. For some choice of initial conditions, the Bessel function i n equation (30) will involve at least some of the Hankel function of second kind, so that q ( s ) will grow exponentially 77(s)

w exp ( 9 ~ ~ 1 2 )

for p*/2

(31)

>I .

I t follows from equations (27) and (24) that J' will have the same growth character as 17, while Ek will grow only when the increasing exponential in equation (31) exceeds the Landau damping, i.e. p2/2

> CIS.

(32)

These results can most conveniently be summarized i n ternis of three characteristic times : T = ii?wJeE,,k, the time for the field to produce particle velocities of W I , / k; __ tu = d T / w l ,= di?&ecX ~

the time at which the Hankel function begins its exponential growth (corresponding to p2= 1 ) ; t, = 2ma/eE, = 2(kn/w,,)T,

the time at which equation (32) is satisfied and also the time required for the field to produce a relative drift velocity of order a. For given k , it follows from equation (29) that the solution of equation (30) is valid only for t < T. Thus, there are three possibilities.

193

(a) If the values of n and E, are such that t,

< td < T

or eE,/kina2 < 1 < (kU/2k)'

(33)

(where k , = oJ,/n is the Debye wave number), then the Hankel function growth starts at a time (t,) when its rate is less than the Landau damping. Later on, (at t,) but still before t = T, the relative drift velocity exceeds a and E, begins a n exponential growth which continues a t least until time T. (b) If t, < t, < T, or 1 < eE/mka2 < (kD/k)' (34) then even though the relative drift velocity exceeds a at time t,, growth of E is postponed until the later time (t,) when the Hankel function attains its asymptotic character. This result is at first surprising; in the case E, = 0 a drift velocity greater than a leads to growth, so that one would here expect growth a t a time of order td. However, the energy exchange between particles and wave which constitutes the physical reason for growth of the wave (JACKSON, 1958) cannot occur in a time less than that required for a particle to traverse one wavelength, and this time is just t,." Hence we have the double condition for growth in presence of a n electric field: t must be great enough for the external field to produce a relative drift velocity greater than the thermal speed and also to accelerate the particles through a distance of at least one wavelength. (c) If t, > T or t D > T that is, if k > k , or eE/mka2 > (kD/k)?(35) then we can only conclude that no growth of Ek occurs before a time T. Whether it occurs subsequently can only be determined by dropping the restriction to small k o r small t. In the opposite limit of large k o r large t, we expect that an approximate solution should follow from setting the right-hand side of equation (28) equal to zero. Noting equation (27) we then have x'y = 0

(36)

+

whose general solution is y = (cls c2) exp ( - i y s 2 / 2 ) where c1 and c2 are constants. Thus, y has no exponential growth and the Landau damping, exp (-as), prevails. The physical reason for the absence of growth is simply that at times greater than T the * The time for a particle t o go a distance Ilk in virtue of its thermal velocity alone is greater than I,, when the inequality equation (34) holds.

194

B.D. FRIED,M. GELL-MA”, J. D. JACKSONand H. W. WYLD

electric field has accelerated all particles to velocities greater than the phase velocity of plasma waves, co,lk, leaving no particles to be trapped by the waves. We see that the general characteristics are just those to be expected from consideration of the field-free case, the only new features being the requirement that growing waves occur only if there is time to accelerate a particle through one wavelength, and that after long times ( t > T ) waves of a given k stop growing and decay by Landau damping. I t seems reasonable to expect a similar behaviour in the case n? # M and also for other choices of f,,but we have not explicitly demonstrated this.

example, by straightforward application of the method of characteristics) is then

where T

=

2

t - t’,

= eE,/nt.

The electron density is 3. A QUASI-LINEAR APPROXIMATION

We now adopt a different point of view. Instead of assuming the fluctuations to have an amplitude small enough to permit complete linearization, we suppose that as a consequence of BUNEMAN’S mechanism a kind of quasi-equilibrium is established in which fo and F, are nearly time-independent. This can come about only if the amplitudes of the fluctuations have increased to a point where the right-hand side of equation (3) approximately balances the term containing E,. In fact, we would require f, and Fa to have such shapes as to lead to little growth of the fk, while also demanding that the fk have a velocity dependence which enables the nonlinear term in equation (3) to cancel the E, term. It is far from clear whether the equations have any self-consistent solution of this character. As a first step in studying this, however, we have examined the consequences of assuming that (a) fois independent of time, (b) the nonlinear terms in equation (4) can be neglected, (random phase approximation). At worst, this can be regarded as an approximation to the problem discussed in the previous section, valid over times short compared to that in which fa changes appreciably t

/eEe.

Ek = 4ve ~ V ( F-AL). ,~

im~t~7c(r)

= (-i/k2)

(38)

We shall assume that E, is independent of time. The general solution of equation (37) (obtained, for

j

a‘ mdOJo(0) exp (i(u0 -1- A02/2k)} du o

(45)

with U

= w/k.

Longitudinal plasma oscillations in an electric field

195

The dimensionless parameter 2/(2i./ka2) is just the ratio of the velocity increment produced by the field in a distance l / k to the thermal velocity. I n the limit ;I= 0, equation (51) reduces, as it should, t o 1 ( U D 2 [R(tu) + r(w)] the dispersion relation given by JACKSON (1958). the integral is to be carried out along a contour which For 1. # 0 but 2/kal2 1 the properties are qualipasses above all of the singularities of the integrand. tatively similar to the zero field case. However, Aside from poles of X(to), which depend upon the for A/kaI2 1, the character is quite different. particular initial conditions chosen, t h e poles of the I n particular, we find that growing waves occur for integrand will occur at points where the denominator arbitrarily small values of the drift velocity V, and, vanishes. in fact, even in the limit M / m 4 00 where the ions are D(to) = 1 to,:[I'(cU) R(W)] = 0. (47) very heavy and d o not participate in the oscillations. Consider the latter case, i.e. a n electron plasma If equation (47), which is just the dispersion relation with a background of heavy positive ions to provide for this system, has roots in the upper half-plane, charge neutrality. We want to know whether the then E,(r) will grow exponentially at large times, the dispersion equation, which now simplifies to i.e. the oscillations will be unstable. To gain some familiarity with the dispersion equation (47), we investigate its properties for the particular case of Maxwellian distributions for fo has any roots with Im(u) > 0. The use of a Nyquist and F,. We choose a frame in which the drift velocidiagram, as described by JACKSON (1958), enables ties are & I/ and we assume both species to have the us to answer this without the necessity of evaluating same temperature, equation (52) for complex U. Unfortunately, even if exp [ - - ( c - V)2/a,2} U is real, the argument of Z' is complex because of '0 = d/.rral p, and the separation of 2 into real and imaginary parts is simple only when the argument is real, pure exp (-(U v)*n,2) aZ2= (in/M)a12. (48) imaginary, o r proportional to 4; We therefore Fo = 2&, exploit the fact that in the large field limit, ?,/ka2 1, The Fourier transform offo is p2 is nearly pure imaginary. Introducing the velocity

R ( ~is~defined ) in a n analogous fashion. I n inverting equation (43), exp (-iwt) X ( w ) E,(t) =J^_mLdro (46)

+

>

+

+

>

f o ( 6 ) = l-du exp (-irO)fo(c) = exp (-[nl2O*/4

+ iYO]}

(53)

(49)

(we shall assume that both k and E, are positive) we have P = 2/-4iy2/n2 1

and the function I' required for the dispersion equation is

= 2y/a.\/-(l

+ + in2/8y2+ . . . ).

(54)

If we neglect the n2/y2term, then where ,MI = 2/(1

- 2ilL/kaI2),

U =

co/k

and Z is the 'plasma dispersion function' defined in equation (18). (The reduction of the integral in equation (50) to the Z function requires just some completions of the square in the exponent.) The dispersion equation (47) is then (55) where C and S are the Fresnel integrals C(9)

+ iS(x3) = Jf Joaexp (it21 dr,

x

> 0.

(56)

196

B. D. FRIED, M. GELL-MA”, J. D. JACKSONand H. W. WYLO

For small or intermediate values of x = u/2y, t h e representation equation (55) is a good approximation for large ria. However, in the asymptotic region (x -1) it is not correct; the real part of p 2 causes a damping of the linearly divergent, oscillatory character predicted by equation (55). To show this, we use the large argument asymptotic form* of Z ,




1 3.1 5.3.1 +-+t+-+.... x2 2x 22x6 where the term in brackets is to be included if, and only if, Jm(x) < 0. Including the a2/y2correction to p i n equation (54) we then find

FIG. ].-Real

and imaginary parts of‘ ,u-2Z’(u/oj() vs. X

=

ll/2>J.

origin. The dispersion equation (52) will have roots in the upper half-plane (leading to growing waves) if this spiral includes (at least once) the point k2a2/coP2. This will happen if

y 2 a(kD/k>’, k, G w,/n. (58) In order for the large field approximation to be valid, a. This, combined we must simultaneously have y (57) with equation (58), gives as a condition for instability (k/k&ma2 E,2 nina2(k,/k)2, (59) We see that if -U/? is large compared to 1 but still small compared to y / a , then the first term of equation a condition which can always be satisfied, for non(57) dominates. It is just the asymptotic form of vanishing Ec, at a sufficiently large wavelength. equation (55). For - u / y large compared to ria, This is not a physically reasonable result, since it however, the first term becomes exponentially sniall predicts that an electron plasma with a Maxwellian (due to the fact that p2 is not pure imaginary) and the distribution will have some exponentially growing second term (which is itself tending towards zero) waves n o matter how small the applied electric field. dominates. The first term in the brace of equation When the ions are assumed to have a finite mass, it is (57) is proportional, in magnitude t o not surprising that the same disease manifests itself and one finds growing waves for an arbitrarily small d g 2 x exp (--azx2/4y2) relative drift velocity. The reason for this difficulty which has a maximum at

>>

<
0. For Im(=) < 0, Z ( z ) is defined to be the analytic continuation of the function from the upper half-plane of = t o the lower half-plane. By expressing (0 - z)-l as a Fourier integral,* equation ( A I ) can be transformed to

Z ( r ) = 2i exp (-?)

Lm

(A2)

exp ( - t 2 ) dt

a representation valid for both signs of Im (2). For real z, it is convenient to separate the real and imaginary parts of equation (A2), writing it as

-

Z(x) = i

dn-exp (-x2)

- 2xY(x)

(A3)

exp ( t 2 ) dr

(A4)

where Y ( x ) = x-l exp (-xz)

s,”

is real. JACKSON’S small and large argument expressions for Y , which we reproduce here,

[

CONCLUSIONS

(2xyy- 1)” n!/(2!2

+ i)!

x
0.

Similarly,

(A12)

1

}

,

('413)

z2'1+1 I;+~ O for IiI -+ CO, x < 0. To establish the lemma, we note that from equations (A9) and (A13), -+

1;

=h

n

exp ( z 2 - t2) dt

E

for x > 0, where P denotes an asymptotic expansion, while for x < 0 we write

=s

t2) dt

Z ( z ) N [2i 4 : exp (-z2)]-C

> 0, then x < s in the integrand for I$ and so

1 1 1 :

-

t2n

A , (-1)"/~2'~+~ (A16) where the bracketed term is included if Re (z/i) = I m (z) < 0, and is omitted if Re (z/i) = I m ( z ) > 0. For I m ( z ) = 0 it would appear that both forms of (A13) are satisfied and that it is equally correct to include the bracketed term in (A15) o r to omit it. This is in fact the case, for with z real and z + CO, the error committed in stopping at any term I / Z ~ " + ~ is greater than exp (-?). Nevertheless, it is important for dispersion applications to know the correct form of the imaginary part of Z(x), even if it is negligible compared to the real part. We see from equation (A3) that for z real, Im ( Z ( x ) )= i d ; exp (-x2) while comparison of equations (A3) and (A16) shows that R e ( Z ) = -2xY(x) N -&l,(--l)"/~~"+~ giving the asymptotic expansion for Y(x) quoted in equation (A5). To summarize, for I=/ -+m, Z has the asymptotic expansion

'

where cr = 2 if Ini (i)< 0, cr = 1 if 1111(=) = 0, and cr = 0 if Im ( z ) > 0. --5 Of course, it should not be inferred from (A17) 1" that Z(z) has a discontinuity a t the real axis, since it is = 6exp(9) 1; N d; exp ( z ) - 2 A ~ / Z ~ ~ +byI definition a n analytic function. The significance of the equation is rather as follows. For -y 2 x 1, the term cTid/TT exp (-z2) may be quite large and must be included. For y 2 x > 1 n o such exponential term should be included. For IyI x things are more complicated. Right on the axis ( y = 0) the entire (A 15) imaginary part of Z comes from the exponential. where In a sufficiently close neighbourhood of the axis, the A , = ( - l ) ' L ( / ~- &)!I(.&)! exponential will still provide a larger contribution and the bracketed term is included if x < 0, omitted to Iiii (2)than will the first N terms or the series, but if x > 0. This completes the proof, since (A6) follows the size of this neighbourhood shrinks as x increases.

1;

n

-m

exp (z2 - t2) dt

+

+J

exp (12 - t2) dt

+

>