Download longitudinal plasma oscillations in an electric field

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,<(t)= exp (--nkt)y(t).
Then equation (23) becomes
k2
df2a
to,2Y
then
g(0
jcu
= 2i
exp
_~/o(o - x - iEl-1 exp
(-x2)
-J:
=
= (v.-2
+ 8-2)1)
where
= a-ZZ’(t/a)
where Z(x) is the ‘plasma dispersion function’ which
is always encountered in an analysis of plasma
oscillations linearized about a Maxwellian distribution
z ( x >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
<
then the only explicit occurrence of k is in the factor
k2 on the right side of equation (28). In the limit of
small k we then have
( 2+ /?)>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
<m
s
exp (ikILr2/2)-I- .f;,(kt,0) exp (ikjbt2/2)] (40)
where the bar denotes the Fourier transform, defined
as in equation (10). Upon substituting this and an
analogous expression for ion density into Poisson’s
equation, equation (5), we obtain again an integral
equation for €,(t),
Ek(t)
+ c o l F r dt’E,(t’) [f,( k ~t’), exp ( i k i . ~ ~ / 2 )
+ p o ( k ~t‘), exp (-iki727b~z/2M]= X ( t ) (41)
T
0
where
x(t)= (4ve/ik) [Fk(kt,0 ) exp (--ikh2n?/2M)
- &(/ct, 0) exp (ikj.t2/2)] (42)
depends on the initial conditions.
When fa is independent of time, the integral
equation (41) is of convolution type, and the solution
by Laplace transform is immediate. With
~ , ( w )=
Ek(w) = X(wj {I
(43)
+
CO,*[V(U)
+ R(w)]}-’
(44)
where r and R are transforms of the kernels of
equation (41),
Thus, we study the linear system
with a similar equation for Fk and with
s
exp (icot)
and a similar definition for X ( O J )we have
(du ufao>/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 ,
<
z’(x) = (-4 d,x
exp <-x2)>>
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
>>
< <
x = d&ja,
the maximum amplitude being
--
2(y/a) d277je.
We can now sketch the form of the real and imaginary parts of I/p22’(u/pa) as functions of ti (Fig. 1)
and hence the form of the Nyquist plot, i.e. the map
of the real u-axis in the plane of p-zZ’(u/pa) (Fig. 2 ) .
As we go from U = + a towards U = -cc along the
real u-axis (the opposite direction from that indicated
by the arrows in Fig. 2), the image point starts from
the origin, moves outward in a gradually-widening,
clockwise spiral until it reaches a radius of order
(air) x (27~/e)l/~,
then quickly spirals back into the
*
A discussion of the asymptotic form of
Appendix.
Z is given in the
FIG.2.-Schematic
Nyquist diazram for the dispersion
equation, equation (52). (The arrows correspond t o traversal
of the rr-axis from - m t o -7- CO.)
Longitudinal plasma oscillations in an electric field
may be that the original hypothesis is inconsistent;
t ~ l e r eis no solution for which tlie random phase
and the approximation of nearly
constant,fo are both valid. A t any rate, if a solution
of tile indicated character does exist, then the present
results show that f, and F, must have forms very
different from a Maxwellian distribution.
A S a final point, we recall tlie remark, made at the
beginning of this section, that the present analysis
sl1ould describe the completely linear problem of
Section 2, at least during a time in which fo does not
change appreciably. The results found Iiere-instability for any external field-will agree with those
of Section 2 only if we can show that the gro\vth mte
is small compared to eE,./rnn, the rate a t which f,
is changing. Such a demonstration can, in fact,
be given so that the results of the two sections are not
inconsistent.
197
for Im(=)> 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<
1
On the basis of the linear analysis of Section 2,
we conclude that at least for the special distributions
treated there, and probably for more general ones
as well, the presence of an external electric field then give corresponding expressions for Z ( x ) .
causes no significant changes in the stability character
However, a little care must be exercized in obtaining
of tlie linearized plasma waves. If the field is very asymptotic expressions for Z ( z ) when z is not real.
strong, then i t may produce a separation of electron We require the following result, due to Stokes (WHITand ion mean velocities greater than the electron TAKER and WATSON, 1952): Len?ma. The asymptotic
thermal speed (thus satisfying the field-free condition expansion of the error function has the form
for growing waves) before it has carried a particle
exp (-9)
through one wavelength of the oscillations. In that
l e x p (- t2) dt N & 1/;/2
2z
case, growth is delayed until the particles have gone a
1
1.3
1.3.5
distance of order Ilk and thus had chance to exchange
23~6
. (A6)
I--+--2_,2 22=4
energy with the plasma waves.
From tlie results of Section 3 it appears that if a
where the upper sign is to be taken if - ~ / 2< argz <
solution of the complete equations in which .f, and
n-12, the lower sign if 4 2 < nrgz < 3x12.
Fo are nearly constant in time exists, it must involve
either an f, and F, with non-Maxwellian shapes or Proof: Integration by parts shows that the two
else must be affected in an important way by the integrals
nonlinear terms in equation (4) which couple one
mode to another.
REFERENCES
satisfy the recursion relation
+ . .)
0. (1958) P h j s . Reu. (Letters) 1 , 8.
J . D. (19%) P/mnic/ Oscil/Lc,ions, Physical Research
Laboratory Report; J . N i d . Eiie,;;ry Part C: Plasnia Phys.
This issue, p. 171.
L A N D A U J . (1946) J . Plijs. Moscow 10, 25.
W H l T T A K E l l and WArsoN (1952) Modern Aflcllysis. p. 152,
Cclnibridge University Press.
BUSEM4N
JACKSOW
From equation (A8) it follows that
APPENDIX
The plasma dispersion function is defined by
where
A,, = (- 1)’L
* This was suggested by
(12
-
;)!Id;.
E. S. WEIBEL.
B. D. FRIED,
M. GELL-MA", J. D. JACKSON
and H. W. WYLD
198
To establish the validity of equation (A9) as an from (A15) by using
asymptotic expansion, we must show (WHITTAKER
l e x p (z2 - t 2 ) dt = (2/;/2) exp ( z 2 )
and WATSON,1952) that
zZn+l r,+l -+ 0 as IzI
a3
(A 10)
- mexp (z2 - t2) dt.
We choose a path of integration for In* in equation
(A7) which starts at t = z = x
iy and runs parallel However, the form equation (A17) is actually more
to the real axis to i CO
iy. (The contribution from convenient for our purposes.
Thus, with z = i t we have from equations (A2)
the vertical strip of length y at fCO vanishes.) With
and (A14)
t =s
iy we have
J:'" exp (z2 - (s iy)2}
Z ( z ) = 2i exp ( E2 )
exp ( - t 2 ) dt
h
---f
+
+
+
+
I,'(z) =
p1L
=i".
exp {x2 - s2
If x
::1
ds
+ 2iy(x - s)} ds.
= 2iLmexp( E 2
From equation (AS) we have
which with equation (A1 1) gives
<
1
for x > 0.
1=211+1/
Therefore,
z2n+l '1?iL1
-+o for
Izi
-+
CO,
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
+
>
<