Collisions

Basic Plasma Physics

3.6 Diffusion in Partially Ionized Gases

3.6.1 Collisions

Charged particles in a plasma interact with each other primarily by coulomb collisions and also can collide with neutral atoms present in the plasma. These collisions are very important when describing diffusion, mobility, and resistivity in the plasma.

When a charged particle collides with a neutral atom, it can undergo an elastic or an inelastic collision. The probability that such a collision will occur can be expressed in terms of an effective cross-sectional area. Consider a thin slice of neutral gas with an area A and a thickness dx containing essentially stationary neutral gas atoms with a density n,

.

Assume that the atoms are simple spheres of cross-sectional area

a,

The number of atoms in the slice is given by n,Adx

.

The fraction of the slice area that is occupied by the spheres is

(3.6-2) n,Aadx

-- A - n,odx.

If the incident flux of particles is T o , then the flux that emerges without making a collision after passing through the slice is

r (

^{x )} =

ro (

1 - nu o h ) ,

The change in the flux as the particles pass through the slice is

(3.6-3)

- dT = -Tn,a

.

dr

The solution to Eq. (3.6-4) is

(3.6-4)

(3.6-5) where

A

is defined as the mean free path for collisions and describes the distance in which the particle flux would decrease to lie of its initial value. The mean free path is given by

A=- 1

n,o

’

^(3.6-6)

which represents the mean distance that a relatively fast-moving particle, such as an electron or ion, will travel in a stationary density of neutral particles.

The mean time between collisions for this case is given by the mean free path divided by the charged particle velocity:

1 n,ov

z=-.

_(3.6-7)

Averaging over all of the Maxwellian velocities of the charged particles, the collision frequency is then

1 ^-

v = - = n , o v .

z

^(3.6-8)

In the event that a relatively slowly moving particle, such as a neutral atom, is incident on a density of fast-moving electrons, the mean free path for the neutral particle to experience a collision is given by

(3.6-9) where v, is the neutral particle velocity and the reaction rate coefficient in the denominator is averaged over all the relevant collision cross sections.

Equation (3.6-9) can be used to describe the distance that a neutral gas atom travels in a plasma before ionization occurs, which is sometimes called the penetration distance.

Other collisions are also very important in ion and Hall thrusters. The presence of inelastic collisions between electrons and neutrals can result in either ionization or excitation of the neutral particle. The ion production rate per unit volume is given by

%

^{= nun,} (ope),

dt (3.6- 1 0)

where

oj

is the ionization cross section, v, is the electron velocity, and the term in the brackets is the reaction rate coefficient, which is the ionization cross section averaged over the electron velocity distribution function.

Likewise, the production rate per unit volume of excited neutrals, n* , is (3.6-1 1)

where

o*

is the excitation cross section and the reaction rate coefficient is averaged over the electron distribution function and summed over all possible excited states j . A complete listing of the ionization and excitation cross sections for xenon is given in Appendix D, and the reaction rate coefficients for ionization and excitation averaged over a Maxwellian electron distribution are given in Appendix E.

Charge exchange [2,6] in ion and Hall thrusters usually describes the resonant charge transfer between like atoms and ions in which essentially no kinetic energy is exchanged during the collision. Because this is a resonant process, it can occur at large distances, and the charge exchange (CEX) cross section is very large [ 2 ] . For example, the charge exchange cross section for xenon is about m2 [7], which is significantly larger than the ionization and excitation cross sections for this atom. Since the ions in the thruster are often energetic due to acceleration by the electric fields in the plasma or acceleration in ion thruster grid structures, charge exchange results in the production of energetic neutrals and relatively cold ions. Charge exchange collisions are often a dominant factor in the heating of cathode structures, the mobility and diffusion of ions in the thruster plasma, and the erosion of grid structures and surfaces.

While the details of classical collision physics are interesting, they are well described in several other textbooks [1,2,5] and are not critically important to understanding ion and Hall thrusters. However, the various collision frequencies and cross sections are of interest for use in modeling the thruster discharge and performance.

The frequency of collisions between electrons and neutrals is sometimes written 181 as

(3.6- 12) where the effective electron-neutral scattering cross section o(T,,) for xenon can be found from a numerical fit to the electron-neutral scattering cross section averaged over a Maxwellian electron distribution [8]:

where Tev is in electron volts. The electron-ion collision frequency for coulomb collisions [5] is given in SI units by

n lnA v,i = 2.9 x

T2b2

’

where 1nA is the coulomb logarithm given in a familiar form [ 5 ] by lnA=23--In

:

^{) ;;[l}

-

The electron-electron collision frequency [ 5 ] is given by n lnll

v,,

= 5 x

T:b2

’

(3.6- 14)

(3.6- 15)

(3.6- 16) While the values of the electron-ion and the electron-electron collision frequencies in Eqs. (3.6-14) and (3.6-1 6) are clearly comparable, the electron- electron thermalization time is much shorter than the electron-ion thermalization time due to the large mass difference between the electrons and ions reducing the energy transferred in each collision. This is a major reason that electrons thermalize rapidly into a population with Maxwellian distribution, but do not thermalize rapidly with the ions.

In addition, the ion-ion collision frequency [5] is given by

(3.6- 17)

where 2 is the ion charge number.

Collisions between like particles and between separate species tend to equilibrate the energy and distribution functions of the particles. This effect was analyzed in detail by Spitzer [9] in his classic book. In thrusters, there are several equilibration time constants of interest. First, the characteristic collision times between the different charged particles is just one over the average collision frequencies given above. Second, equilibration times between the species and between different populations of the same species were calculated by Spitzer. The time for a monoenergetic electron (sometimes called a primary electron) to equilibrate with the Maxwellian population of the plasma electrons is called the slowing time, 7 , . Finally, the time for one Maxwellian population to equilibrate with another Maxwellian population is called the equilibration time, zeq. Expressions for these equilibration times, and a comparison of the rates of equilibration by these two effects, are found in Appendix F.

Collisions of electrons with other species in the plasma lead to resistivity and provide a mechanism for heating. This mechanism is often called ohmic heating or joule heating. In steady state and neglecting electron inertia, the electron momentum equation, taking into account electron-ion collisions and electron- neutral collisions, is

The electron velocity is very large with respect to the neutral velocity, and Eq. (3.6-18) can be written as

0 = - e n

( :)

E + d -enve ~ B - r n n ( ~ , i + v ~ ~ ) v , + r n n v ~ i v i . (3.6-19) Since charged particle current density is given by J = qnv, Eq. (3.6-19) can be written as

(3.6-20) where

J,

is the electron current density, Ji is the ion current density, and qei is the plasma resistivity. Equation (3.6-20) is commonly known as Ohm's law for partially ionized plasmas and is a variant of the well-known generalized Ohm's law, which usually is expressed in terms of the total current density, J = en(vi - v e ) , and the ion fluid velocity, vi

.

If there are no collisions or net

current in the plasma, this equation reduces to Eq. (3.5-7), which was used to derive the Boltzmann relationship for plasma electrons.

In Eq. (3.6-20), the total resistivity of a partially ionized plasma is given by (3.6-2 1)

where the total collision time for electrons, accounting for both electron-ion and electron-neutral collisions, is given by

1

Vei + ''en

2, = (3.6-22)

By neglecting the electron-neutral collision terms in Eq. (3.6-19), the well- known expression for the resistivity of a fully ionized plasma [1,9] is recovered:

(3.6-23) In ion and Hall thrusters, the ion current in the plasma is typically much smaller than the electron current due to the large mass ratio, so the ion current term in Ohm's law, Eq. (3.6-20), is sometimes neglected.