Kursunoglu, B.

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I ,  ' 
LANDAU DAMPING IN A RELATIVISTIC PLASMA 
Behram Kursunoglu 
Center for Theoretical Studies 
University of Miami 
Coral Gables, Florida 3 3 124 
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, 
1 
LANDAU DAMPING IN A RELATIVISTIC PLASMA* 
Behram Kurpnoglu 
Center for Theoretical Studies 
University of Miami, Coral Gables, Florida 
It is shown that over a time interval T = ,q-q the phenomenon 
of Landau damping, for electron mass small compared to KT, can 
induce a cooling mechanism in a relativistic plasma by an 
"amplification" of longitudinal as well as of transverse waves. 
*This research is supported by United States Air Force Contract 
No. AF 49-(638)-1260 and National Aeronautics and Space 
Administration Contract No. NASA NGR 10-007-010. 
. 
2 
Introduction 
In a plasma,energy losses by a beam of particles traversing 
it can be put into two categories: (i) Incoherent losses or 
microscopic losses arising from close collision of ions and 
electrons. Such losses occur 
rD = (7) whose axis of 
& particle; (ii) coherent or 
KT 1/2 
47me 
inside a cylinder (of Debye radius 
symmetry coincides with the path of 
macroscopic losses arising from 
an interplay of collective oscillations and particles outside 
the Debye cylinder. 
For a relativistic (rarefied) plasma, the cross-section 
for the direct Coulomb interaction of charged particles 
and coherent energy losses eventually take over. For a non- 
magnetized plasma the excitation of longitudinal plasma waves 
decreases 
constitutesthe main mechanism for almost all coherent losses 
suffered by a beam of particles. 
The phase velocity of a longitudinal plasma wave, with a 
first order relativistic correction, is 
where 
k = -  A 2rr ' =SJ(~?~), fig) = Vr = mean thermal velocity 
of the plasma electrons. 
3 
The basic restriction on the emission of a plasma wave 
is contained in the "plasma propagator" defined by 
where 
1 
2 - 5  E = y m c ,  2 p - y m x ,  y = ( l - = )  V 0 
C M 
The second term generates the Landau damping of the plasma 
waves when they are emitted, v i z . ,  
which can be written as 
W = ~ V C O S ~  . 
Thus the necessary condition for the emission of plasma waves 
is 
v > o ,  U u = E  03 
If the particle velocity v is less than d 3  Vr then the 
necessary condition (1.4) cannot be satisfied,and in this case 
4 
most of the energy losses experienced by the particles are 
due to close collisions (incoherent losses). If particle 
velocity v exceeds& VT the condition (1.4) is satisfied 
for certain range of wave lengths k, and coherent losses can 
occur. However,we shall see presently that the nature of 
these losses,i,e. from particles to waves or from waves to 
particles will, for relativistic plasmas, depend on the energy 
of the particles. 
2n 
11. Relativistic Landau Damping; 
Landau damping is a property of the collisionless plasma 
and to that extent it is rather sensitive to the form of velocity- 
distribution function. More importantly the damping seems to 
assume a basic role in differentiating between short and long 
time processes in the plasma. 
in terms of time does, naturally, necessitate a careful analysis of 
the linear and nonlinear nature of plasma oscillations. 
reasonable limit on the time interval, during which a plasma behaves 
linearly, can be obtained, 
equation for a relativistic plasma(’) (in k space), 
A discussion of plasma phenomena 
A 
Let us consider the nonlinear integral 
(11.1) 
B. Kursunoglu, Nuclear Fusion 1, 213-223 (1961). 
c 
5 
where 
4nie kbpv-kvP~ p(krp) d4p 
k ka a 
mc 
is the Fourier transform of the electromagnetic field in the 
plasma and 
is the relativistic angular momentum operator. 
We are using relativistic &vectors according to 
ka = g ab kb, kaka = k4 2 - kl 2 - k2 - k3 2 , 
and arguments of f and p are ‘+-dimensional vectors. 
In (11.1) a definite direction of time flow is chosen 
2 2 2  by assuming that E = cp4 = cJ(p +m c ) has a small positive 
imaginary part (see 1.2) . However, in the case of nonlinear 
theory, in view of an interaction between plasma oscillations, 
the relationship of particle velocities to phase velocity of 
plasma oscillations is not well defined. The two cannot be ex- 
pected to be in phase and hence flow of energy from one to the 
other has a random behaviour in a nonlinear state. Therefore, 
on this basis Landau damping cannot occur. However for shorter 
time intervals, namely j u s t  before the appearance of nonlinear 
(11.2) 
(11.3) 
. 
effects ,  the plasma as a l inear  system can experience a Landau 
damping . 
The equation (11.1) can be writ ten as 
where - M and 3 are  angular momentum vector operators corresponding 
t o  (II03),and CD = ck4. 
(11.4) (or the l inearized r e l a t i v i s t i c  co l l i s ion less  Boltzmann 
equation) provided i n  the r e s t  frame of the pa r t i c l e  (i .e.  rsI) p = 0, 
3 = 0) we impose the condition 
One can apply perturbation theory t o  
- 
1 Putting p4 = mc we may write f o r  the time in t e rva l  7 = c u ,  
over which the  l i nea r  behaviour can be assumed, the expression 
T h i s  provides, f o r  a r e l a t i v i s t i c  plasma, a reasonable c r i t e r i o n  
f o r  the existence of favourable s t a t e s  i n  which Landau damping 
can occur. 
t he  direct ion of the damping, i .e.  energy absorption by the  
p a r t i c l e  or  by the wave. 
The condition (11.6) does not, of course, determine 
(11.4) 
(11.6) 
7 
For large electric fields the electrons have available a 
much smaller time interval than ions to absorb from or to lose 
energy to the wave. 
Now to calculate the actual damping we resort to linearized 
theory (as was already done in reference 1) and obtain the dis- 
persion relations 
2 2 
9 
2 bI 
refering to transverse and longitudinal waves, respectively. 
The dispersion integrals are given by 
I o = I  + i I L ,  P 
where 
2 2  
IP du, Z = aJ(l+q -u ), 
2 mc . = E =  phase velocity of the wave, a = ~ ' p  , 
(11.7) 
(11.8) 
8 
The term IL is the cause of Landau damping. 
IL in (11.7) is given by 
The integral 
1 
where 
co The damping of a plasma wave of velocity 2) = by a 
relativistic electron with velocity v depends on the form of 
IL given by (11.9). It can be written as 
where 
(11.10) 
2 E = ye mc 
9 
Thus the damping of a plasma wave i s  proportional t o  the 
density of plasma electrons, possessing a t  u = v (or p = l), 
according t o  a r e l a t i v i s t i c  Maxwell dis t r ibut ion,  an energy 
E = ye mc‘ and velocity equal t o  the velocity of the wave. 
T h i s  i s  i n  exact correspondence with the absorption mechanisms 
pointed out above. 
absorption of energy by the electron from a plasma wave decreases 
w i t h  an increase of electron energy. 
However, it i s  important t o  note that 
A s  an i l l u s t r a t i o n ,  l e t  us consider the  extreme case 
where the r e s t  mass of the electron i s  small compared t o  KT. 
this case the r e l a t i v i s t i c  dispersion in t eg ra l  (see r e f .  1) 
f o r  longitudinal waves becomes 
In 
where the imaginary p a r t  refers  t o  Landau damping i n  t h i s  
extreme case. 
we obtain the r e su l t  
Using (11.11) i n  the  dispersion re la t ion  (11.8) 
2 = coth [t (1 + $411 . 
u 2u C 
P 
(11.11) 
(11.12) 
From (11.12) it follows that longitudinal wave velocity exceeds 
the  velocity of l i gh t .  
reverse of Landau damping occurs: 
from the electron and reaches phase ve loc i t ies  exceeding that 
Therefore, i n  this extreme case, the 
the wave absorbs energy 
10 
of light. T h i s  means that Landau damping in the extreme 
relativistic state provides a mechanism for cooling the plasma 
as contrasted with the heating effect that it provides in a 
warm plasma. If the imaginary part of (11.11) were omitted 
then one would obtain the relation (II.l2),with tanh instead 
of coth. In this case 2) is always less than c .  
On the basis of this extreme relativistic case the dis- 
persion relation for transverse waves has a similar structure 
to that of longitudinal waves. The transverse dispersion 
integral I1 together with transverse dispersion relation leads 
to 
1 
1 
one of its solutions being 2) = c. For all other solutions, 2) > c .  
The energy absorbed by the waves may be dissipated amongst 
the waves themselves, enhancing the mechanism of waves scattering 
from one another. 
which does not allow Landau damping ( 2 )  after the time 7 .  
The plasma thus assumes its nonlinear behaviour 
For a detailed discussion of Landau damping in nonrelativistic 
plasma,see Chapter 7 
H. Stix, 1962, McGraw-Hill. 
of THE THEORY OF PLASMA WAVES, by Thomas 


Landau damping in a relativistic plasma

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