lecture5: plasma physics 1 - columbia...

Lecture5:

Plasma Physics 1APPH E6101x

Columbia University Fall, 2015

More Consequences of Large ND

• Most interactions are “distant encounters” with small-angle collisions (very few “close encounters”, very few large-angle collisions, and very very few three-body collisions)

• Mean potential energy is much less than mean kinetic energy

• Energy density of electrostatic fluctuations are much less than the plasma’s kinetic energy density

“Close” and “Distant” Encounters

Potential and Kinetic Energy

Fluctuation and Kinetic Energy

Outline (Piel, Ch. 4)

• Distribution function

• Electron ionization cross-sections

• Other cross sections

• Electron/Ion neutral collisions

• Coulomb Collisions (next week)

• Mobility and electrical conductivity

Distribution FunctionIsotropic and Maxwellian

Flow, Temp, Pressure

Distribution FunctionBi-Isotropic

f(μ,E)

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NIST Home > PML > Physical Reference Data > Electron-Impact Cross Section Database

Version History | Disclaimer

Y.-K. Kim,1 K.K. Irikura,2 M.E. Rudd,3 M.A. Ali,4 and P.M. Stone1

J. Chang,1 J.S. Coursey,1 R.A. Dragoset,1 A.R. Kishore,1 K.J. Olsen,1 A.M. Sansonetti,1

G.G. Wiersma,1 D.S. Zucker,1 and M.A. Zucker,1

1NIST, Physical Measurement Laboratory2NIST, Material Measurement Laboratory3University of Nebraska-Lincoln, Department of Physics and Astronomy, Lincoln, NE 68588-01114Howard University, Department of Chemistry, Washington, DC 20059

This is a database primarily of total ionization cross sections of molecules by electronimpact. The database also includes cross sections for some atoms and energy distributionsof ejected electrons for H, He, and H2. The cross sections were calculated using the Binary-

Encounter-Bethe (BEB) model, which combines the Mott cross section with the high-incidentenergy behavior of the Bethe cross section. Selected experimental data are included.Electron-impact excitation cross sections are also included for some selected atoms.

Introduction and References

Contributions of the following colleagues are gratefully acknowledged:W. M. Huo, NASA Ames Research Center, Moffet Field, CA 94035-1000W. Hwang, Samsung Electronics, Suwon, Korea

© Orla/2010 Shutterstock.com

Access the Database:

Atoms | Molecules

NIST Standard Reference Database 107

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Online: August 1997 - Last update: August 2005

ContactDr. Karl Irikura

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http://www.nist.gov/pml/data/ionization/index.cfm

http://www.nrl.navy.mil/ppd/content/nrl-plasma-formulary

84 4 Stochastic Processes in a Plasma

loss per unit time. τcoll is the mean free time between two collisions defined inEq. (4.15).

The elastic scattering of electrons on atoms is almost isotropic [68]. Therefore, onaverage, the electron loses its mean momentum mev̄e and we can write the equationof motion for an average electron

m ˙̄v = −eE − mv̄νm . (4.25)

This average electron now moves in −E-direction. The quantity νm = 1/τcollis the effective collision frequency for momentum transfer. Because of the one-dimensional motion, the vector symbol was dropped. The solution of this equationof motion

v̄(t) = − eEmνm

!1 − e−νmt "+ v(0)e−νmt (4.26)

has two parts: the first describes the approach to a terminal velocity

vd = − emνm

E = −µe E , (4.27)

the second the loss of memory on the initial velocity v0. The terminal velocity vd iscalled the drift velocity, which is established when the electric field force is balancedby the friction force. The mobilities of electrons and ions are defined as

µe = emeνm,e

; µi = emiνm,i

. (4.28)

4.3.2 Electrical Conductivity

The drift velocity of electrons and ions can be used to define the electric currentdensity

j = je + ji = n[(−e)vde + evdi] = ne(µe + µi)E = σ E . (4.29)

The linear relation between current density and electric field is the equivalent toOhm’s law. The quantity σ is the total conductivity1 of the gas discharge. Likewisewe define the conductivity of the electron and ion gas

σe,i = neµe,i = ne2

me,iνm. (4.30)

1In the literature, the same symbol σ is used for the conductivity and the collision cross section,or µ for the mobility and the magnetic moment, but confusion is unlikely because of the differentcontext

thermal velocities, i.e., u2 ≪ kTα/mα, kTβ/mβ ; and (6) anomalous transportprocesses owing to microinstabilities are negligible.

Weakly Ionized Plasmas

Collision frequency for scattering of charged particles of species α byneutrals is

να = n0σα|0s (kTα/mα)1/2,

where n0 is the neutral density and σα\0s is the cross section, typically ∼

5 × 10−15 cm2 and weakly dependent on temperature.When the system is small compared with a Debye length, L ≪ λD , the

charged particle diffusion coefficients are

Dα = kTα/mανα,

In the opposite limit, both species diffuse at the ambipolar rate

DA =µiDe − µeDi

µi − µe=

(Ti + Te)DiDe

TiDe + TeDi,

where µα = eα/mανα is the mobility. The conductivity σα satisfies σα =nαeαµα.

In the presence of a magnetic field B the scalars µ and σ become tensors,

Jα = σσα · E = σα∥ E∥ + σα

⊥E⊥ + σα∧E × b,

where b = B/B and

σα∥ = nαeα

2/mανα;

σα⊥ = σα

∥ να2/(να

2 + ω 2cα);

σα∧ = σα

∥ ναωcα/(να2 + ω 2

cα).

Here σ⊥ and σ∧ are the Pedersen and Hall conductivities, respectively.

39




m ˙̄v = −eE − mv̄νm . (4.25)


v̄(t) = − eEmνm

!1 − e−νmt "+ v(0)e−νmt (4.26)


vd = − emνm

E = −µe E , (4.27)


µe = emeνm,e

; µi = emiνm,i

. (4.28)






me,iνm. (4.30)





m ˙̄v = −eE − mv̄νm . (4.25)


v̄(t) = − eEmνm

!1 − e−νmt "+ v(0)e−νmt (4.26)


vd = − emνm

E = −µe E , (4.27)


µe = emeνm,e

; µi = emiνm,i

. (4.28)






me,iνm. (4.30)


Electrical Conductivity



The elastic scattering of electrons on atoms is almost isotropic [68]. Therefore, onaverage, the electron loses its mean momentum me Nve and we can write the equationof motion for an average electron

m PNv D −eE − m Nvνm : (4.25)

This average electron now moves in −E-direction. The quantity νm D 1= τcollis the effective collision frequency for momentum transfer. Because of the one-dimensional motion, the vector symbol was dropped. The solution of this equationof motion

Nv. t/ D − eEmνm

!1 − e−νmt " C v.0/e−νmt (4.26)


vd D − emνm

E D −� e E ; (4.27)


� e D emeνm; e

I � i D emiνm; i

: (4.28)



j D je C ji D n T .−e/vde C evdi U D ne. � e C � i/E D σ E : (4.29)


σe; i D ne � e; i D ne2

me; iνm: (4.30)

1In the literature, the same symbol σ is used for the conductivity and the collision cross section,or � for the mobility and the magnetic moment, but confusion is unlikely because of the differentcontext






Nv. t/ D − eEmνm

!1 − e−νmt " C v.0/e−νmt (4.26)


vd D − emνm

E D −� e E ; (4.27)


� e D emeνm; e

I � i D emiνm; i

: (4.28)






me; iνm: (4.30)


4.3 Transport 83

4.3 Transport

Transport processes in plasmas comprise mobility-limited motion, electric currentsdescribed by conductivity, and free or ambipolar diffusion. Here, only steady-stateprocesses will be discussed. The section ends with a discussion of the influence ofa magnetic field on mobility.

4.3.1 Mobility and Drift Velocity

In a gas discharge with a low degree of ionization, the motion of electrons and ionsis governed by the applied electric field and collisions with the atoms of the back-ground gas. Most of the electron collisions are elastic. Therefore, we will neglectionizing collisions in the calculation of friction forces. Because of the equal massof positive ions and atoms of the parent gas, the momentum exchange between theheavy particles is very efficient. Besides elastic scattering, the process of chargeexchange plays an important role, in which a moving ion captures an electron andleaves a slow ion behind. In the momentum balance this process is equivalent to ahead-on collision in a billiards game.

A cartoon of electron motion in a gas background is shown in Fig. 4.9. In thecollision between a light electron with a heavy atom the momentum transfer is small.Rather, the incoming electrons experience a random redirection of their momentum.The trajectory is a sequence of parabolic segments. Since we have no diagnostic tofollow the trajectories of individual electrons, we must be content with evaluatingthe average motion of an ensemble of electrons.

The equation of motion for an individual electron can be written as

me Pve D −eE C!

k

me!vkδ. t − tk / : (4.24)

Here me!vk is the momentum loss in the kth collision. By averaging this equationover many collisions we obtain the mean drift velocity ⟨ve⟩. Then, the sum on ther.h.s. of Eq. (4.24) becomes me⟨!ve⟩τ−1

coll, which represents the average momentum

Fig. 4.9 Cartoon of anelectron trajectory in ahomogeneous electric field.The trajectory is interruptedby elastic collisions withneutral atoms

vd

E

Collisional Diffusion

4.3 Transport 85

This concept for the conductivity of a gas discharge is applicable at a low degreeof ionization ne = na. For a typical gas pressure of 1 mbar and room temperature theatom density is na D 2: 8 × 1022 m−3 whereas a typical electron density can bene ≤ 1019 m−3.

4.3.3 Diffusion

The average electron and ion motion in a gas discharge is determined by electricfield forces and by density gradients. The latter type of net motion is called diffusion.What is diffusion on a microscopic scale? Let us consider a situation with a hot elec-tron gas that has initially a density gradient in negative x-direction (see Fig. 4.10).

Since the electron thermal speed is much higher than that of the gas atoms weassume that the gas atoms are at rest. Because of the density gradient it is evidentthat per unit time more electrons will move to the right than to the left. This gives riseto a net down-hill motion. However, this electron motion is inhibited by collisionswith the gas atoms. This combination of electron thermal motion and friction withthe neutral gas is described by a diffusion coefficient D. Again, we can only describethe average motion in terms of a relation between the density gradient and a resultingparticle flux Γ e; i

Γ e; i D ne; i Nve; i D −D∇ne; i ; (4.31)

which is known as Fick’s law. Such relations were originally developed for neu-tral gases, in which the motion of particles is determined by collisions with otherparticles of the same species. In that situation, diffusion is the result from a greaternumber of collision with neighboring particles of the same kind from the left thanfrom the right, which on average gives a net force directed down-hill.

Fig. 4.10 Cartoon of electrondiffusion in an electrondensity gradient

Γ

n

4.3 Transport 85

This concept for the conductivity of a gas discharge is applicable at a low degreeof ionization ne = na. For a typical gas pressure of 1 mbar and room temperature theatom density is na D 2: 8 × 1022 m−3 whereas a typical electron density can bene ≤ 1019 m−3.

4.3.3 Diffusion

The average electron and ion motion in a gas discharge is determined by electricfield forces and by density gradients. The latter type of net motion is called diffusion.What is diffusion on a microscopic scale? Let us consider a situation with a hot elec-tron gas that has initially a density gradient in negative x-direction (see Fig. 4.10).

Since the electron thermal speed is much higher than that of the gas atoms weassume that the gas atoms are at rest. Because of the density gradient it is evidentthat per unit time more electrons will move to the right than to the left. This gives riseto a net down-hill motion. However, this electron motion is inhibited by collisionswith the gas atoms. This combination of electron thermal motion and friction withthe neutral gas is described by a diffusion coefficient D. Again, we can only describethe average motion in terms of a relation between the density gradient and a resultingparticle flux Γ e; i

Γ e; i D ne; i Nve; i D −D∇ne; i ; (4.31)

which is known as Fick’s law. Such relations were originally developed for neu-tral gases, in which the motion of particles is determined by collisions with otherparticles of the same species. In that situation, diffusion is the result from a greaternumber of collision with neighboring particles of the same kind from the left thanfrom the right, which on average gives a net force directed down-hill.

Fig. 4.10 Cartoon of electrondiffusion in an electrondensity gradient

Γ

n


4.3.3.1 Ambipolar Diffusion

The situation for plasma electrons is quite different, because diffusion does notmean that the electrons collide with other electrons. This effect can be neglectedfor weakly coupled plasmas. Rather, as described above, the net motion is onlydetermined by their thermal motion and the inhomogeneous density distribution. Inthis way, electron diffusion is similar to drift motion with the electron temperature—together with the density gradient—providing the driving force. The same consid-erations can be applied to ions.

Einstein had shown that diffusion coefficient and mobility are related by thetemperature of the gas

D�

D kBTe

: (4.32)

This relation quantifies the arguments given above that electron diffusion in a neutralgas background with a density gradient is driven by the temperature and inhibitedby electron-neutral collisions. Because the diffusion of electrons and ions leads todifferent values of the particle fluxes, which would lead to unequal densities ofelectrons and ions, the plasma reacts by forming a space charge electric field E .This field reduces the electron diffusion and accelerates the ion diffusion until thetwo fluxes reach a common value and the plasma remains macroscopically neutral.This final state is called ambipolar diffusion when electrons and ions are lost at thesame rate and E is called the ambipolar electric field.

Figure 4.11 shows schematically how electron and ion density profiles in aplasma look like under the influence of ambipolar diffusion. The difference betweenthe two profiles is exaggerated, for clarity. In the plasma center, a surplus of ions isexpected that generates a positive plasma potential in the plasma center becauseelectrons have the tendency to leave the system faster than ions. Therefore, aslight surplus of electrons is found in the outer plasma region. The correspond-ing space charge field E that accelerates the ions but slows down the electrons,is indicated.

The particle fluxes for this diffusion process are given by

Γ e; i D � n � e; iE − De; i∇n : (4.33)

Fig. 4.11 Cartoon of ion andelectron density profile forambipolar diffusion. Theplasma is bounded by walls atx D � a

+++ +++

---

-- -

----

ne

n

E E

x

ni

–a a0






Nv. t/ D − eEmνm

!1 − e−νmt " C v.0/e−νmt (4.26)


vd D − emνm

E D −� e E ; (4.27)


� e D emeνm; e

I � i D emiνm; i

: (4.28)






me; iνm: (4.30)


Next Lecture

• More Piel / Chapter 4: Transport

• Fusion confinement and ignition criteria

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