Plasma Physics II

CHAPTER 0: WAVES IN PLASMA II

In the following short chapter, we will conclude briefly the kinetic description of longitudinal modes in a warm

plasma we started in the first part of the course. This will be done in order to fully understand the physical

meaning of Landau damping.

.Closure of the kinetic description of collective modes in a warm plasma

In the first part of the course, we studied the linear(small amplitude), collision-less description of collective

modes in a warm plasma in the absence of any external field. We especially saw a possible solution of the

Vlasov linearized equation, through the adiabatic switching on of an initial perturbation and the subsequent

application of a Fourier in space-Laplace in time transform. This gave us the electric potential,

+∞ G (v )

∫

q dv

a //

Σ

4 i

π a //

v p/ k

−i

−∞

a

k , p)=− //

Φ( d F

3

k 0, a

2

q

4 π +∞ dv

a ∫

1− d v

Σ // //

2 m v – i p/ k

k −∞

a a //

and the perturbation of the distribution function q f



1 a 0, a

f ' k , p)= k

( (g +i Φ ⋅ )

a a

p+i k⋅v m v

a

Then we saw that the problem was about the evaluation of the poles of the potential

N p)→ or D( p) 0

( ∞ →

leading to two contributions: one from the initial perturbation

G v

( )

+∞

∫ a //

N p)=Σ q dv

( a //

p

−∞ v

a −i

// k

and on from the unperturbed properties d F

2 0,a

q

4 π +∞ d v

∫

a

D( p)=1− d v

Σ // //

2 pk

m

k −∞ v – i

a a //

Thus, we want to conclude the discussion about these aspects, before trying to give a physical interpretation of

Landau damping. Let us consider the evolution in time of the perturbation of the distribution function;

antitransforming its Laplace transform, we get such evolution:

g q f

 p)

1 i Φ (

∫ ∫

a pt a 0,a pt

f ' , t e dp+ k e dp

(k )=

a 2 i p+i kv 2 i m v p+i k v

π π 

// a // //

The first contribution, related to the initial perturbation g , can be solved using the residual method of complex

a

analysis, giving us that such integral is proportional to kt v

−i

g e //

a

which, as a function of the parallel component of the velocity, is an oscillating function limited by g (see

a

figure #1) with a time dependent frequency ω=kt. Figure #1: contribution of the oscillating behavior of the perturbation of

the distribution function for two different times

This oscillation is never damped and acts as a memory of the initial perturbation,as we would have expected

from a collision-less model. This contribution is also called ballistic contribution. Notice that this

contribution is not present in the scalar potential, since this integral contribution dies as time increases, leaving

no effect on the potential; this show furthermore the self-consistent character of the dynamics in a plasma.

Let us see now the solution to the problem of determining the zeroes of the denominator of the scalar potential,

that is d F

2 0, a

ω

1 +∞ dv

∫

p , a

z d v , with z=ip

ε ( )=1+ Σ =0

//

// //

k n z – k v

−∞

a 0,a //

This equation can lead to approximated solution, assuming some particular values of the wave vector; for

example, we can study the limiting case in which k v // ≪1

z

that is for small wave vectors. Notice that this approximation prevent us from studying the resonances, which

are the real purpose of this equation. However, this case allows us to analytically solve the integral, writing the

binomial expansion of the denominator (up to first order): −1

kv kv

1 1 1

// //

–

= (1 ) ≈ (1+ )

z−kv z z z z

//

Considering now the contribution of the electrons only -dominating because of the presence of the plasma

frequency squared-, we can write the resonance condition as

2 kv d F

ω +∞

∫

p , e // 0,e d v

(1+ ) =−1

//

k z n z d v

−∞

0,e //

However, the contribution d F

+∞

∫ 0,e d v =0

//

d v

−∞ //

since the distribution function must tend to zero at infinity, otherwise we would have infinitely many particles.

The other integral, on the other hand, can be written, though an integration by part, as

kv d F kn

k

∫ ∫

// 0,e 0,e

d v F d v

=− =−

// 0,e //

z d v z z

//

Thus, substituting these results, we get that for small wave vectors,

2 2

z =ω p ,e

which is the result for the dispersion relation we got in the cold plasma approximation. Using the second order

of the denominator expansion, we would get 3 2 2

k

ω=ω (1+ λ )

p ,e D , e

2

that is the dispersion relation of collective modes in a warm plasma (Vlasov dispersion relation), we found

through a multifluid approach.

Consider now the dispersion relation in the case in which the imaginary part of z is much less than the real part

of it: γ≪ω

In this case, we can write an expansion for the dielectric tensor as ε(ω)

ε (z )=ε (ω+i γ)≈ε(ω)+i γ

// // ω

stopping at first order with respect to γ. Now, making explicit the real and imaginary parts of the dielectric

tensor, we obtain  ε ( ω) ε (ω)

R I

ε (z )≈ε (ω)+iε (ω)+i γ −γ

// R I ω ω

and the dispersion condition requires both the real part and the imaginary to be equal to zero, that is

{ ε I

ε −γ =0

R ω

ε R

ε +i γ =0

ω

I

However, being 2

ε ∝γ

R

we can also write { ε =0

R ε I

γ=− ε / ω

R

All evaluated in real frequencies, we can use the Plemelj formula and write the dielectric tensor as

2 2 ( )

q k d F d v q d F

/

4 4

π π

∫

a 0, a // a 0,a

P d v

ε (ω)=1+ Σ +i(− Σ )

// //

2 2

m – k v m d v

ω

k k

ℝ

a a k

ω/

a // a //

giving us the real and imaginary parts. Using these results one can find ω and γ as functions of k and

eventually verify the γ is actually small. One can also show that for sufficiently long times, in the

approximation of γ<<ω, the electrostatic potential becomes proportional to

i(k⋅x−ω(k )t ) γ(k )t

k , t)≈e e

ϕ(

Suppose now to have a particular initial distribution function, such as a Maxwell-Boltzmann one:

2 2

−3/

T m v

0, a a

f exp(−

(v)=n (2 π ) )

0,a 0, a m 2 T

a 0,a

thus resulting in 2 2

−1/

T m v

∫ 0, a a //

F f dv dv exp(−

(v)= =n (2 π ) )

0,a 0, a x y 0, a m 2 T

a 0,a

given which we can explicitly calculate the dielectric tensor:

{ }

[ ]

1 z

, z 1+D(

ε (k )=1+Σ )

// 2 2 √ 2 k v

k λ

a th , a

D ,a

in which we defined 2

−u

x e

∫

D( x)= du

1 2 u−x

/

π

In particular, if x is real we can use the Plemelj' formula: 2

−u

x e 2

∫ 1/ 2 −x

D( x)= P du+i x e

π

1 2 u−x

/

π

Notice that the properties of the dielectric tensor depend on those of D; in particular, then, the ratio D(x)/x is of

relevance and it is called plasma dispersion function.

In the particular case in which x is real and much larger than unity, we can write an asymptotic expression for

D(x): 1 3 2

1 /2 −x

D( x)≈−1− – x e

+i π

2 4

2x 4x

in our case, this limiting case correspond to take phase velocities that are much larger than the thermal velocity

z ≫v th ,a

k

Substituting in the dielectric tensor the expression we obtained for D(x), considering the contribution of the

electrons only and setting the real part equal to zero we obtain the following dispersion relation:

T

2 2 2 0, e

k

ω =ω +3 ( )

p ,e m e

which is the Bohm-Gross formula with polytropic index equal to 3; moreover, we obtain from the imaginary

part ω 3 1

π

√ p ,e exp –

γ=− (− )

8 2

3 3 2 2

k 2 k

λ λ

D ,e D , e

which is always negative, so that the mode is always damped in time. Notice that γ is much less than one if

−1

k , otherwise it becomes large and the description doesn't hold anymore. Thus, we may conclude that

≪λ D , e

the mode found by Vlasov are characterized by small and negative imaginary part of the frequency,

corresponding to damped modes in time.

Before moving to the physical justification of Landau damping, we can talk some about the possibility of

solving the dispersion relation in a general way. First, we can write

2

q m

a e

F F

=F + Σ

0 0,e 0,e 2 m

e

a e a

and obtain a dispersion condition as 2

m k

e

G( z /k )= 2

4 e

π

in which we defined the function F '

∫ 0

G(u)= dv //

v −u

//

This means that the condition ε (z)=0 is satisfied if G(u), complex-value function of complex variable, takes

//

real and positive values. However, we could also study the behavior of G for complex values of the u variable.

For example, for Im (u)>0

G will be an analytic function if F is such; furthermore, this would correspond to unstable behavior in the

potential. In the other hand, negative value of the imaginary part of u will correspond to stable behavior

(damped) of the potential. Thus, finding the mapping of real u will provide us with the zone of stable and

unstable behaviors.

Without going into much detail, we can see a couple of properties. We can say that G is finite for all values of

u so that the imaginary part is negative, thus corresponding to a finite region of the G complex plane; on the

other hand, for real values of u going to infinity, G will tend to zero. If F has a stationary point, that is a local

0

maximum of local minimum, G will have imaginary part equal to zero; in particular, a necessary but not

sufficient condition to have an unstable behavior is for F to have a minimum.

0

.Physical meaning of Landau damping

Rather than considering a collective behavior, we can consider a single charged particle, say an electron, in

motion with a velocity v , directed along the z direction,so that its motion is

0 z t

=v

0 0

Then, let us consider a longitudinal, small amplitude, adiabatically switched on electric field:

[ ]

i t)

(kz−ω αt

E z ,t lim Re E e e e

( )= ̂

0 z

+

α →0

We are interested in evaluation the perturbation induced on the electron motion, by such electric field . In order

to compute it, we can assume linear treatment and write

z= z z and v=v v

+δ +δ

0 0

In particular, from the linearized Newton's law, we can write

d v

δ

m E z ,t e E z t

=−e ( )≈– ( )

0,

dt

in which we supposed that the electric field acts on the unperturbed trajectory of the particle. Substituting then

the expression of this field, we obtain ω

d v [ ]

δ ikt (v − ) t

0 α

k

m Re E e e

=−e 0

dt

Notice that the round brackets suggest some kind of resonance at v =ω/k. Through this expression, we obtain

0

by integration: [ ]

[ ]

E z t E z t

( ) ( )

e e

0, 0,

v Re and z t)=− Re

δ (t )=− δ ( 2

m ikt k t m

(v −ω/ )+α t)

(ikt (v −ω/k )+α

0 0

Rather interesting is now to consider the average power exchanged over an oscillation period, between the

electric field and the particle. This is given by

⟨ ⟩

W F v e E z , t v

⟨ ⟩

= =− ( ) (t )

Assuming again a small amplitude for the field, we can expand it around the unperturbed position of the

particle and write ⟨ ⟩

E



W v)[ E z z

≈−e (v +δ ( )+( ) δ ]

0 0 z

 z 0

and further elaborating this expression, by taking its real part, and recalling that

1 *

Re(a) Re( b)= Re(ab )

2

we get [ ]

2 v α

e d

2 0

W E

∣ ∣

= 2 2 2

2m dv α +k (v −ω/k )

0 0

Notice that here too a resonance at v =ω/k is present.

0

Now, consider a population of electron with a given distribution function, we can compute the mean power

exchanged between the longitudinal wave and such population as

∫

W W f

̄ = (v )dv

e 0 0

and, substituting the equation we found for W and integrating by part, we obtain

df v

2 α

e 2 ∫ e 0

̄

W E

∣ ∣

=− ( )dv 0

2 2 2

2m dv k

α +k (v −ω/ )

0 0

Recalling now the adiabatic switching on of the perturbation, so that we can make α tend to zero, and

exploiting Plemelj formula, we obtain ( )

df

2

e π ω

2 e

̄

W E

∣ ∣

=− 2

2m dv

k v k

=ω/

0 0

Notice that this is the same result we got from the description Vlasov gave for waves in plasma. The advantage

here is the simpler physical interpretation: the particle that will have a velocity lower than the one of the wave

will take energy from the wave, while, vice versa, the ones with higher velocity will give energy to it

(Cherenkov effect). Actually, after a characteristic time, the distribution function will flatten around ω/k, due to

the change in energy of the particles. CHAPTER 1: RECALLS OF

SPECIAL RELATIVITY

One of the most important results obtained from the theory of electromagnetism, proposed by Maxwell, is the

fact that the velocity of an electromagnetic wave in vacuum (i.e. light) is constant and equal to

8

c=2,99792⋅10 m/ s

This was not at all a trivial aspect of this theory, since it was completely wrong from the point of view of

Galileo's relativity principle, which states that a body moving with constant velocity, with respect to a fixed

frame of reference, will be seen, by a moving inertial observer, with a different velocity. Experimentally, this

was not true for a light wave.

In 1905 Albert Einstein proposed, then, a different theory for the relative motion of bodies, substituting (and

embodying the Galilean one), starting from two postulates:

all laws of physics must be invariant in form in all inertial frames (i.e. moving with constant velocity);

• there exist a maximum finite speed in every inertial frame, equal to the speed of light in vacuum.

•

As we already said, assuming the second postulate to be true, Galilean transformation cannot be true and must

be changed with another set. Moreover, it can neither be true that time is absolute and distances are too, that is

in general 2 2 2 2 2 2

tt ' and x y x ' y ' z '

+ +z  + +

while the equality one is accustomed to are direct consequences of Galilean transformations. Evidently, thou,

changing these aspects will have a direct impact on the kinetic and dynamic treatment of a moving body; on

the other hand, relevant physical aspects, such as the momentum and energy conservation, must still be

present, due to the first postulate.

1.1 LORENTZ (OR BOOST) TRANSFORMATIONS

The introduction of the new set of transformation can be done in many different ways; here we will follow the

most concrete one, the one Einstein proposed. We assume that at times t=t'=0 (notice the plural), two cartesian

orthonormal frames of reference, K and K', are set with the same origin with parallel axes; at those same

instants, a spherical light wave is emitted from K's origin, and the frame K' starts moving with constant

velocity v, directed along the x axis of K, as in the sketch below.

The light front is identified in K by the sphere equation

2 2 2 2 2

x y t

+ +z =c

while in frame K', because we assume c to be constant, by

2 2 2 2 2

x ' y ' ' t '

+ +z =c

since,for simplicity, we assume v along the x axes, we can write, subtracting these two equations side by side,

2 2 2 2 2 2

c t – x t ' – x '

=c

Such equality is satisfied by the following linear transformations:

{ x ' Chφ – ct Sh

=x φ

ct ' Ch – x Sh

=ct φ φ

in which we introduce an angle phi which depends on the relative velocity between the two inertial frames

(this is why this set is called boost transformations). In fact, exploiting the coordinates of O' in K and K', we

can write that v

Th φ= =β

c

which is bounded between 0 and 1 -assuming that c is the maximum possible velocity.

Now, using the fundamental relation between the hyperbolic sine and cosine,

2 2

Ch φ−Sh φ=1

we can obtain that 2 2 2

−1/ −1 /2

Ch and Sh

φ=(1−β ) =γ φ=β(1−β ) =β γ

Thus, introducing this new notation, the boost transformation become

{ 0

x ' x−x

=γ( β)

0 0

x ' x

=γ ( −x β) 0

having introduced also a new notation for the time coordinate, x =ct. Notice that is it is the gamma factor to

make the difference with respect to Galilean transformations; in fact, if we assume β~0, that is for low

velocities compared to the speed of light, we obtain &gam