lecture15: plasma physics 1sites.apam.columbia.edu/courses/apph6101x/plasma1-lecture-15-b… ·...

Lecture15:

Plasma Physics 1APPH E6101x

Columbia University

Last Lecture

• Langmuir probes

• Langmuir probes in magnetized plasma

This Lecture

• Beam-plasma instability

Instabilities

• “Micro-instabilities”

• “Macro-instabilities”

8.1 Beam-Plasma Instability 199

Fig. 8.2 (a) Electron velocity distribution f (v) for the beam-plasma system. (b) Analogy to pop-ulation inversion in an atomic laser system. The solid line gives the thermal population accordingto a Boltzmann factor. The dashed box indicates the overpopulation of level c w.r.t. the thermalpopulation in level b

Consider the production rate of photons, which is given by the (negative) rate atwhich the upper laser niveau c is net depopulated by the competition of stimulatedemission Bbcnphnc and absorption Bcbnphnb,

dnph

dt= −dnc

dt= Bbcnphnc − Bcbnphnb . (8.1)

Spontaneous emission processes between levels c and b can be neglected at highphoton density. For non-degenerate levels, the Einstein coefficients are identical,Bcb = Bbc. The number of photons grows exponentially in time when nc > nb.The associated exponential growth in the photon density is the laser process. Thisanalogy shows that instability in terms of exponentially growing waves is a conse-quence of the strong deviation from thermal equilibrium.

For the beam-plasma situation, let us denote the total electron density by ne0 witha correponding electron plasma frequency ωpe. The beam population represents afraction nb = αbne0 of the total electron population, and we will assume αb ≪ 1.The beam velocity is v0. The problem is considered as one-dimensional with thebeam propagating in x-direction.

8.1.2 Dispersion of the Beam-Plasma Modes

In Sect. 6.4 we had used first-order perturbation theory to derive the dielectric func-tion of a cold plasma as

ε(ω) = 1 −ω2

pe

ω2 = 1 + χp . (8.2)

The dielectric function is the sum of the permittivity of the vacuum (“1”) and thesusceptibility χp(ω) of the plasma electrons. χp can also be interpreted as the ratioof the electron conduction current to the vacuum displacement current, at a given

8.4 Macroscopic Instabilities 215

8.4.2 Pinch Instabilities

The pinch effect was already introduced in Sect. 5.3.4. The pinch effect is not nec-essarily a homogeneous mechanism. When we assume that the plasma cross sectionis reduced at some point, the magnetic pressure at the plasma surface will increase,because Bϕ = µ0 I (2πa)−1, as shown in Fig. 8.13a. This increased magnetic pres-sure further reduces the plasma radius at this point, and the plasma column developsa sausage instability.

The magnetic pressure can also deviate from its equilibrium value, when theplasma column is curved, see Fig. 8.13b. Because the magnetic field lines are per-pendicular to the local current direction, the field line density, and the associatedmagnetic pressure, is higher on the inner side and lower on the outer side of thecurved plasma column. Hence, the imbalance of magnetic pressure will further dis-place the column forming a kink.

The sausage and the kink instability can be stabilized by a superimposed longi-tudinal magnetic field, which is frozen in the plasma. The magnetic field lines havea tension T = B2/µ0 that tends to straighten the field lines, see Sect. 5.2.2. Thisgives a net restoring force that counteracts the instablity from the magnetic pressureimbalance of the azimuthal magnetic field component, as shown in Fig. 8.13c.

(a) (b) (c)

pmag pmag pmag pmag

Frest

Tmag

Tmag

I I Bz

Fig. 8.13 (a) Sausage instability, (b) kink instability of a pinch plasma. The magnetic pressureincreases when the cross-section shrinks or becomes asymmetric when the plasma column iscurved. (c) The magnetic tension of a superimposed longitudinal magnetic field counteracts theinstability

8.4.3 Rayleigh–Taylor Instability

A cartoon of the Rayleigh-Taylor instability for the equatorial ionosphere is shownin Fig. 8.14. The unpurturbed plasma boundary is shown by the horizontal dashedline. The plasma fills the upper halfspace. The magnetic field is perpendicular to

8.3 Beam Instability in Finite Systems 209

Different from the vacuum diode, both electrodes in the classical Pierce diode areconnected to each other and held at ground potential Φ = 0 (Fig. 8.8). The classicalPierce diode is completely characterized by a single parameter αP = ωpeL/v0, aswe will see below.

8.3.2 The Dispersion Relation for a Free Electron Beam

The dielectric function for an electron beam traversing an immobile ion backgroundis the same as for the Buneman instability (8.21), except for ωpi = 0 due to theassumed immobility of the ions that can be achieved by assigning an infinite ionmass. Then, the oscillations in the electron beam follow from

0 = ε(ω, k) = 1 −ω2

pe

(ω − kv0)2 . (8.25)

These are waves travelling with the beam or against the beam with wavenumbers

k+ = ω + ωpe

v0and k− = ω − ωpe

v0. (8.26)

8.3.3 The Influence of the Boundaries

Intuitively, we could assume that the influence of the boundaries is the formation ofstanding waves. However, the situation is more complex, because the forward andbackward propagating waves have different wavelength. Moreover, we must accountfor electric charges on the surface of the metallic boundaries, which also produce anelectric field inside the diode. This is why Pierce composed the oscillating electricwave potential inside the diode from four ingredients:

Φ(x, t) =!

Ax + Be−ik+x + Ce−ik−x + D"

e−iωt . (8.27)

Ax + D is the solution for the potential in a parallel plate capacitor in vacuum. Theother terms represent the two counterpropagating waves. The following boundaryconditions must be met by the resulting full wave field:

0 = Φ(0, t) = Φ(L , t) = ne(0, t) = ve(0, t) . (8.28)

The first two conditions require the vanishing of the electric potential at the groundedelectrodes, the other two conditions state that the electron beam is not modulatedwhen it enters the diode. The potential condition at x = 0 then gives

0 = Φ(0, t) = B + C + D . (8.29)

8.3 Beam Instability in Finite Systems 209

Different from the vacuum diode, both electrodes in the classical Pierce diode areconnected to each other and held at ground potential Φ = 0 (Fig. 8.8). The classicalPierce diode is completely characterized by a single parameter αP = ωpeL/v0, aswe will see below.

8.3.2 The Dispersion Relation for a Free Electron Beam

The dielectric function for an electron beam traversing an immobile ion backgroundis the same as for the Buneman instability (8.21), except for ωpi = 0 due to theassumed immobility of the ions that can be achieved by assigning an infinite ionmass. Then, the oscillations in the electron beam follow from

0 = ε(ω, k) = 1 −ω2

pe

(ω − kv0)2 . (8.25)

These are waves travelling with the beam or against the beam with wavenumbers

k+ = ω + ωpe

v0and k− = ω − ωpe

v0. (8.26)

8.3.3 The Influence of the Boundaries

Intuitively, we could assume that the influence of the boundaries is the formation ofstanding waves. However, the situation is more complex, because the forward andbackward propagating waves have different wavelength. Moreover, we must accountfor electric charges on the surface of the metallic boundaries, which also produce anelectric field inside the diode. This is why Pierce composed the oscillating electricwave potential inside the diode from four ingredients:

Φ(x, t) =!

Ax + Be−ik+x + Ce−ik−x + D"

e−iωt . (8.27)

Ax + D is the solution for the potential in a parallel plate capacitor in vacuum. Theother terms represent the two counterpropagating waves. The following boundaryconditions must be met by the resulting full wave field:

0 = Φ(0, t) = Φ(L , t) = ne(0, t) = ve(0, t) . (8.28)

The first two conditions require the vanishing of the electric potential at the groundedelectrodes, the other two conditions state that the electron beam is not modulatedwhen it enters the diode. The potential condition at x = 0 then gives

0 = Φ(0, t) = B + C + D . (8.29)




dnph

dt= −dnc






ε(ω) = 1 −ω2

pe

ω2 = 1 + χp . (8.2)


200 8 Instabilities

frequency ω. From this example, we can immediately write down the total dielectricfunction for the beam-plasma system

ε(ω, k) = 1 + χp + χb = 1 −(1 − αb)ω

2pe

ω2 −αbω

2pe

(ω − kv0)2 . (8.3)

Here, we have adjusted the electron densities by the factors (1 − αb) and αb andused the Doppler-shifted frequency in the denominator of χb. The reader can seethat, for electrostatic waves, the dielectric constant of a plasma can be simply com-posed by adding up the susceptibilities of the participating plasma constituents. Forcompleteness, a derivation of the Doppler shift in the beam susceptibility will begiven below in Sect. 8.1.4.

Electrostatic (longitudinal) waves are characterized by ε = 0, which definesthe dispersion relation ω(k). Inspecting (8.3), we see that determining the zeroes ofthis equation is mathematically equivalent to solving a fourth-order polynomial withreal coefficients. As stated by the fundamental theorem of algebra, such polynomialshave either real roots or pairs of complex-conjugate roots. Therefore, we expect fourindependent branches of wave dispersion ω(k). If a pair of complex-conjugate rootsappears, we are finished because one of these complex roots will be exponentiallygrowing in time and defines instability.

Let us first inspect the functional dependence of the dielectric function ε(ω, k)

(8.3) on the wave frequency ω, which is plotted in Fig. 8.3 for various valuesof αb. The dielectric function has (negative) singularities for ω = 0 and ω = kv0,as expected from the vanishing denominators of (8.3). There is one real root,ω/ωpe ≈ −1 that is nearly unaffected by the presence of the beam. The roots withω/ωpe > 0 show a different pattern. For the weakest beam fraction (αb = 0.001),there is one root close to ω/ωpe = +1. We identify these two roots as the plasmamodes. A second pair of roots is found symmetric about ω/ωpe = kV0/ωpe, whichwe call beam modes. With increasing αb, the separation between the beam rootsincreases until the left beam root merges with the right plasma root. For even

Fig. 8.3 The dielectricfunction of the beam-plasmasystem (8.3) forkv0/ωpe = 1.5. Forαb = 0.001 and αb = 0.01there are four real roots. Apair of complex roots isexpected for αb = 0.01




dnph

dt= −dnc






ε(ω) = 1 −ω2

pe

ω2 = 1 + χp . (8.2)


200 8 Instabilities


ε(ω, k) = 1 + χp + χb = 1 −(1 − αb)ω

2pe

ω2 −αbω

2pe

(ω − kv0)2 . (8.3)






200 8 Instabilities


ε(ω, k) = 1 + χp + χb = 1 −(1 − αb)ω

2pe

ω2 −αbω

2pe

(ω − kv0)2 . (8.3)







Fig. 8.4 The dispersionrelation for the beam-plasmamodes at αb = 0.01. Thedotted lines mark theasymptotes ω = ωpe andω = kv0. The plasma modedevelops into the fastspace-charge wave whichthen approaches the fastbeam mode. Forkv0/ωpe < 1.3 the beammode is a complex conjugateslow space-charge wave. Atthe triple point it splits intostable modes, a slow beammode and a plasma mode.The second plasma modewith negative ω remainsunaffected by the beam

higher values of αb, the dielectric function stays negative in the entire interval0 < ω < kv0. However, if ω is taken as a complex quantity, there will be a pairof conjugate complex roots with the real part of ω lying in this interval.

The dispersion relation for the beam-plasma modes consists of different branchesω(k), see Fig. 8.4. According to our wave perturbation ∝ exp[i(kx − ωt)] andallowing for a complex ω = ωr + iωi , there will be growing waves ∝ exp[i(kx −ωr t)] exp[ωi t], when ωi > 0. In the limit αb ≪ 1, the plasma modes at ω =±ωpe and the (degenerate) beam mode ω = kv0 are uncoupled (see dotted linesin Fig. 8.4). For non-vanishing αb, the positive plasma mode connects to thebeam mode and becomes the fast space-charge wave, which has ω/k > v0. Forkv0/ωpe < 1.3 the beam modes form a conjugate pair. These waves are propagatingmore slowly than the beam and are called slow space-charge waves. The one withωI > 0 is exponentially growing in time. The growth rate takes a maximum valuenear the intersection ωpe = kv0. At the triple point, the slow space-charge wavesbecome real and form the slow beam mode and the plasma mode. The second plasmamode with negative ω remains unaffected by the beam.

8.1.3 Growth Rate for a Weak Beam

For small values of αb, the slow space-charge wave that has the maximum growthrate γ = ωI, is found close to kv0/ωpe = 1. This means that the phase velocity of thewave is nearly resonant with the electron beam, vϕ ≈ v0. Therefore, it is reasonableto seek an approximate solution for ε(ω, k) = 0 in the vicinity of the resonancepoint ωpe = kv0. Introducing ω = ωpe +&ω, we can rewrite the dielectric functionin this regime as







202 8 Instabilities

0 = ε = 1 −ω2

pe

(ωpe +#ω)2 −αbω

2pe

(ωpe +#ω − kv0)2 (8.4)

≈ 1 −ω2

pe

ω2pe

+ 2#ωω3

pe−αbω

2pe

(#ω)2 . (8.5)

In the second line, we have used a Taylor expansion of the plasma dielectric functionfor#ω ≪ ωpe, and applied the resonance condition in the last term. Solving for#ω,we obtain

#ω =!αb

2

"1/3ωpe en2π i/3 with : n = 0, 1, 2 . (8.6)

Here, we have noticed that there are three roots in this region, two of which arecomplex. (A fourth root at ω = −ωpe is non-resonant). This gives, for n = 0, thefast space charge wave as

ω = ωpe

#1 +

!αb

2

"1/3$

. (8.7)

The slow space-charge wave has

ωR = ωpe

#1 − 1

2

!αb

2

"1/3$

(8.8)

ωI = ±31/2

2

!αb

2

"1/3ωpe . (8.9)

The most spectacular result for the unstable mode is the fact, that the growth ratedepends on the third root of the beam fraction αb. Therefore, a beam fraction of α =0.002 generates a wave that has an e-folding after only ten wave periods (ωR/ωI ≈10). The real part of the frequency is close to ωpe and this wave can therefore beidentified as the Langmuir wave. The high growth rate explains why self-excitedLangmuir oscillations are so ubiquitous in dc discharges.

8.1.4 Why is the Slow Space-Charge Wave Unstable?

Let us first consider the motion of a single electron in an oscillating electric field,which is described by (6.30) and results in an an oscillatory velocity of amplitude

ve = eiωme

E . (8.10)







202 8 Instabilities

0 = ε = 1 −ω2

pe


2pe

(ωpe +#ω − kv0)2 (8.4)

≈ 1 −ω2

pe

ω2pe

+ 2#ωω3

pe−αbω

2pe

(#ω)2 . (8.5)


#ω =!αb

2

"1/3ωpe en2π i/3 with : n = 0, 1, 2 . (8.6)


ω = ωpe

#1 +

!αb

2

"1/3$

. (8.7)


ωR = ωpe

#1 − 1

2

!αb

2

"1/3$

(8.8)

ωI = ±31/2

2

!αb

2

"1/3ωpe . (8.9)




ve = eiωme

E . (8.10)

202 8 Instabilities

0 = ε = 1 −ω2

pe


2pe

(ωpe +#ω − kv0)2 (8.4)

≈ 1 −ω2

pe

ω2pe

+ 2#ωω3

pe−αbω

2pe

(#ω)2 . (8.5)


#ω =!αb

2

"1/3ωpe en2π i/3 with : n = 0, 1, 2 . (8.6)


ω = ωpe

#1 +

!αb

2

"1/3$

. (8.7)


ωR = ωpe

#1 − 1

2

!αb

2

"1/3$

(8.8)

ωI = ±31/2

2

!αb

2

"1/3ωpe . (8.9)




ve = eiωme

E . (8.10)

202 8 Instabilities

0 = ε = 1 −ω2

pe


2pe

(ωpe +#ω − kv0)2 (8.4)

≈ 1 −ω2

pe

ω2pe

+ 2#ωω3

pe−αbω

2pe

(#ω)2 . (8.5)


#ω =!αb

2

"1/3ωpe en2π i/3 with : n = 0, 1, 2 . (8.6)


ω = ωpe

#1 +

!αb

2

"1/3$

. (8.7)


ωR = ωpe

#1 − 1

2

!αb

2

"1/3$

(8.8)

ωI = ±31/2

2

!αb

2

"1/3ωpe . (8.9)




ve = eiωme

E . (8.10)

202 8 Instabilities

0 = ε = 1 −ω2

pe


2pe

(ωpe +#ω − kv0)2 (8.4)

≈ 1 −ω2

pe

ω2pe

+ 2#ωω3

pe−αbω

2pe

(#ω)2 . (8.5)


#ω =!αb

2

"1/3ωpe en2π i/3 with : n = 0, 1, 2 . (8.6)


ω = ωpe

#1 +

!αb

2

"1/3$

. (8.7)


ωR = ωpe

#1 − 1

2

!αb

2

"1/3$

(8.8)

ωI = ±31/2

2

!αb

2

"1/3ωpe . (8.9)




ve = eiωme

E . (8.10)

202 8 Instabilities

0 = ε = 1 −ω2

pe


2pe

(ωpe +#ω − kv0)2 (8.4)

≈ 1 −ω2

pe

ω2pe

+ 2#ωω3

pe−αbω

2pe

(#ω)2 . (8.5)


#ω =!αb

2

"1/3ωpe en2π i/3 with : n = 0, 1, 2 . (8.6)


ω = ωpe

#1 +

!αb

2

"1/3$

. (8.7)


ωR = ωpe

#1 − 1

2

!αb

2

"1/3$

(8.8)

ωI = ±31/2

2

!αb

2

"1/3ωpe . (8.9)




ve = eiωme

E . (8.10)







202 8 Instabilities

0 = ε = 1 −ω2

pe


2pe

(ωpe +#ω − kv0)2 (8.4)

≈ 1 −ω2

pe

ω2pe

+ 2#ωω3

pe−αbω

2pe

(#ω)2 . (8.5)


#ω =!αb

2

"1/3ωpe en2π i/3 with : n = 0, 1, 2 . (8.6)


ω = ωpe

#1 +

!αb

2

"1/3$

. (8.7)


ωR = ωpe

#1 − 1

2

!αb

2

"1/3$

(8.8)

ωI = ±31/2

2

!αb

2

"1/3ωpe . (8.9)




ve = eiωme

E . (8.10)

202 8 Instabilities

0 = ε = 1 −ω2

pe


2pe

(ωpe +#ω − kv0)2 (8.4)

≈ 1 −ω2

pe

ω2pe

+ 2#ωω3

pe−αbω

2pe

(#ω)2 . (8.5)


#ω =!αb

2

"1/3ωpe en2π i/3 with : n = 0, 1, 2 . (8.6)


ω = ωpe

#1 +

!αb

2

"1/3$

. (8.7)


ωR = ωpe

#1 − 1

2

!αb

2

"1/3$

(8.8)

ωI = ±31/2

2

!αb

2

"1/3ωpe . (8.9)




ve = eiωme

E . (8.10)


Because of the inertia of the electron, the oscillation velocity ve lags behind theelectric force −eE by a phase angle of 90◦. In the language of electronics, theelectric “current” −eve lags behind the “voltage” E . Hence, an electron behaveslike an inductor.

Consider now an electron with velocity v0 in a wave field. This beam electronobeys the linearized equation of motion

∂vb

∂t+ v0

∂vb

∂x= − e

meE exp[i(kx − ωt)] . (8.11)

In Fourier notation, and solving for the oscillation velocity, this becomes

vb = ei(ω − kv0)me

E . (8.12)

Again, the electron behaves as an inductor, as long as ω−kv0 > 0, which is realizedfor the fast space-charge wave. But, for the slow space-charge wave, the oppositecase is realized withω−kv0 < 0. Therefore, beam electrons show a strange behaviorwhen they interact with the slow wave. Their motion in the wave field is such as ifthey had a “negative mass”. In the language of electronics, the electron now behaveslike a capacitor for which the current leads the voltage by a phase shift of 90◦.

We can easily see that the concept of a “negative mass” has a physical meaning.We will discuss this effect in terms of the average kinetic energy of the wave. For thispurpose, we must calculate the density fluctuations in the beam, which are relatedto the velocity modulation by the continuity of the flow

∂nb

∂t+ ∂(nbvb)

∂x= 0 . (8.13)

Linearizing nb = nb0 + nb1, vb = v0 + vb1, and setting the first-order quantities∝ exp[i(kx − ωt)], we obtain for the perturbed quantities

(−iω + kv0)n + iknb0vb = 0 (8.14)

and

nb = nb0kω − kv0

vb . (8.15)

The density fluctuations are in phase with the velocity fluctuations for the fast waveω − kv0 > 0 but opposite to the velocity fluctuations for the slow wave. Using thevelocity fluctuations from (8.12), we find the density fluctuations as

nb = enb0ki(ω − kv0)2 E . (8.16)




∂vb

∂t+ v0

∂vb

∂x= − e

meE exp[i(kx − ωt)] . (8.11)



E . (8.12)



∂nb

∂t+ ∂(nbvb)

∂x= 0 . (8.13)


(−iω + kv0)n + iknb0vb = 0 (8.14)

and

nb = nb0kω − kv0

vb . (8.15)


nb = enb0ki(ω − kv0)2 E . (8.16)




∂vb

∂t+ v0

∂vb

∂x= − e

meE exp[i(kx − ωt)] . (8.11)



E . (8.12)



∂nb

∂t+ ∂(nbvb)

∂x= 0 . (8.13)


(−iω + kv0)n + iknb0vb = 0 (8.14)

and

nb = nb0kω − kv0

vb . (8.15)


nb = enb0ki(ω − kv0)2 E . (8.16)




∂vb

∂t+ v0

∂vb

∂x= − e

meE exp[i(kx − ωt)] . (8.11)



E . (8.12)



∂nb

∂t+ ∂(nbvb)

∂x= 0 . (8.13)


(−iω + kv0)n + iknb0vb = 0 (8.14)

and

nb = nb0kω − kv0

vb . (8.15)


nb = enb0ki(ω − kv0)2 E . (8.16)




∂vb

∂t+ v0

∂vb

∂x= − e

meE exp[i(kx − ωt)] . (8.11)



E . (8.12)



∂nb

∂t+ ∂(nbvb)

∂x= 0 . (8.13)


(−iω + kv0)n + iknb0vb = 0 (8.14)

and

nb = nb0kω − kv0

vb . (8.15)


nb = enb0ki(ω − kv0)2 E . (8.16)




∂vb

∂t+ v0

∂vb

∂x= − e

meE exp[i(kx − ωt)] . (8.11)



E . (8.12)



∂nb

∂t+ ∂(nbvb)

∂x= 0 . (8.13)


(−iω + kv0)n + iknb0vb = 0 (8.14)

and

nb = nb0kω − kv0

vb . (8.15)


nb = enb0ki(ω − kv0)2 E . (8.16)




∂vb

∂t+ v0

∂vb

∂x= − e

meE exp[i(kx − ωt)] . (8.11)



E . (8.12)



∂nb

∂t+ ∂(nbvb)

∂x= 0 . (8.13)


(−iω + kv0)n + iknb0vb = 0 (8.14)

and

nb = nb0kω − kv0

vb . (8.15)


nb = enb0ki(ω − kv0)2 E . (8.16)




∂vb

∂t+ v0

∂vb

∂x= − e

meE exp[i(kx − ωt)] . (8.11)



E . (8.12)



∂nb

∂t+ ∂(nbvb)

∂x= 0 . (8.13)


(−iω + kv0)n + iknb0vb = 0 (8.14)

and

nb = nb0kω − kv0

vb . (8.15)


nb = enb0ki(ω − kv0)2 E . (8.16)




∂vb

∂t+ v0

∂vb

∂x= − e

meE exp[i(kx − ωt)] . (8.11)



E . (8.12)



∂nb

∂t+ ∂(nbvb)

∂x= 0 . (8.13)


(−iω + kv0)n + iknb0vb = 0 (8.14)

and

nb = nb0kω − kv0

vb . (8.15)


nb = enb0ki(ω − kv0)2 E . (8.16)

204 8 Instabilities

The mean kinetic energy of the beam can be written as

⟨Wkin⟩ = 12

me!(nb0 + nb)(v0 + vb)

2"

= 12

me!nb0v

20 + 2nb0v0vb + nb0v

2b + nbv

20 + 2nbv0vb + nbv

2b", (8.17)

where ⟨· · · ⟩ denotes the average over one wavelength. In the sum on the r.h.s.,all odd powers of fluctuating quantities vanish in the average. Therefore, the onlyremaining terms are the zero-order beam energy and the second-order corrections

⟨Wkin⟩ = 12

nb0mev20 + 1

2me!nb0v

2b + 2nbv0vb

". (8.18)

Using (8.15), we find that for nearly resonant particles, |ω/k −v0| ≪ v0, the secondterm in the angle brackets is much larger than the first. Moreover, this contributionof the wave to the beam energy has different signs for the slow wave and the fastwave.

For deriving the energy relations, Fig. 8.5 shows the electric field, the velocityfluctuation, the density fluctuation of the beam, and the resulting beam energy fromthe second-order contribution (8.18). On average, the beam energy is lowered by thepresence of the wave. This justifies the terminology of a negative energy wave [161].

Fig. 8.5 The slowspace-charge wave. (a)Electric field, (b) beamvelocity modulation, (c)beam density modulation, (d)resulting beam kinetic energy.The mean beam energy isreduced by the presence ofthe wave

204 8 Instabilities

The mean kinetic energy of the beam can be written as

⟨Wkin⟩ = 12

me!(nb0 + nb)(v0 + vb)

2"

= 12

me!nb0v

20 + 2nb0v0vb + nb0v

2b + nbv

20 + 2nbv0vb + nbv

2b", (8.17)

where ⟨· · · ⟩ denotes the average over one wavelength. In the sum on the r.h.s.,all odd powers of fluctuating quantities vanish in the average. Therefore, the onlyremaining terms are the zero-order beam energy and the second-order corrections

⟨Wkin⟩ = 12

nb0mev20 + 1

2me!nb0v

2b + 2nbv0vb

". (8.18)

Using (8.15), we find that for nearly resonant particles, |ω/k −v0| ≪ v0, the secondterm in the angle brackets is much larger than the first. Moreover, this contributionof the wave to the beam energy has different signs for the slow wave and the fastwave.

For deriving the energy relations, Fig. 8.5 shows the electric field, the velocityfluctuation, the density fluctuation of the beam, and the resulting beam energy fromthe second-order contribution (8.18). On average, the beam energy is lowered by thepresence of the wave. This justifies the terminology of a negative energy wave [161].

Fig. 8.5 The slowspace-charge wave. (a)Electric field, (b) beamvelocity modulation, (c)beam density modulation, (d)resulting beam kinetic energy.The mean beam energy isreduced by the presence ofthe wave


The coupling of this negative energy wave with the plasma fluctuations (which rep-resent a positive energy wave) is the reason for wave growth. The loss in kineticenergy appears as gain for the wave potential energy. In this way, the wave potentialgrows in time at the expense of the beam velocity. In Sect. 9.4 we will trace thisevolution into the non-linear regime by means of computer simulations.

Returning to the concept of “negative mass”, for the slow wave, the densityclumps are growing by decelerating the original beam whereas, for the fast wave,the clumps are filled up by accelerating slower electrons. In the first case, energy istransfered from the beam to the wave, and the wave grows, in the second case. thewave accelerates electrons, thereby losing energy.

8.1.5 Temporal or Spatial Growth

In the last paragraph it was assumed that the wave is characterized by an imaginarypart ωi of the wave frequency. This means that everywhere the wave amplitudegrows at the same rate. In the beam-plasma system, one can also imagine an unmod-ulated beam that enters a finite plasma. While the beam propagates through theplasma, any small wave-like perturbation will grow in space. For this case, we haveto solve the dispersion relation ε(ω, k) = 0 with the dielectric function from (8.3)for real frequency ω and complex wavenumber k. Simple algebraic manipulationgives

kv0 = ω + i

!αbω

2peω

2

ω2pe − ω2

"1/2

, (8.19)

Fig. 8.6 The spatial growthrate in the beam-plasmasystem for αb = 0.1 as afunction of the real wavefrequency ω. Note that thegrowth rate even becomesinfinite at ω = ωpe






kv0 = ω + i

!αbω

2peω

2

ω2pe − ω2

"1/2

, (8.19)


206 8 Instabilities

which yields the imaginary part of the wavenumber (as shown in Fig. 8.6)

kI = ωpe

v0

α1/2b ω

(ω2pe − ω2)1/2 . (8.20)

The subtle differences between spatial and temporal growth are discussed in [162].

8.2 Buneman Instability

A second, related example for the instability of counter-streaming charged particlesis found in a current carrying plasma, in which a dc electric field leads to a flow ofall plasma electrons relative to the ions. This instability was first discussed by OscarBuneman (1913–1993) [163]. Again we neglect collisions and assume that the driftvelocity of the plasma ions is much smaller than the electron beam velocity. There-fore, we describe the instability in the rest frame of the ions. Further, we assume thatthe electron beam velocity v0 is much higher than the thermal spread of the electrondistribution (cold beam approximation).

8.2.1 Dielectric Function

The dielectric function for this system contains the susceptibilities of stationary ionsand beam electrons

ε(ω, k) = 1 + χi + χe = 1 −ω2

pi

ω2 −ω2

pe

(ω − kv0)2 . (8.21)

Here, the ion contribution is determined by the ion plasma frequency ωpi, whichaccounts for the higher ion mass. Obviously, the mathematical structure of the prob-lem is similar to the beam-plasma instability in (8.3) when we recognize that, forequal electron and ion density, ω2

pi/ω2pe = me/mi is a small quantity. Different from

the beam-plasma system, it is now the ion term that represents a small perturbationof the streaming electrons. Therefore, we can expect that the unstable waves havefrequencies that are small compared to the electron plasma frequency. These low-frequency ion fluctuations couple with the Doppler-shifted electron plasma oscilla-tions in the beam.

The dispersion branches for the Buneman instability are shown in Fig. 8.7. Theinstability is generated by the coupling of the slow beam mode ω4, which is a neg-ative energy wave, to the ion plasma fluctuations near ωpi, resulting in the unstablebranch ω1,2.






kv0 = ω + i

!αbω

2peω

2

ω2pe − ω2

"1/2

, (8.19)


Next Lecture(Monday)

• Numerical Simulation of two-stream

• Ch. 8: MHD Instabilities (Part 1)

lecture15: plasma physics 1sites.apam.columbia.edu/courses/apph6101x/plasma1-lecture-15-b… ·...

Documents