Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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Atoms in Dense Plasmas 86.4 Simulations 1313

86.4 Simulations

Most of the challenges pertaining to atomic phenomena

in dense plasmas arise from the many-body nature of the

atom-plasma interaction. Simple models are therefore

subject to inaccuracies that may arise from inconsisten-

cies or severe approximations which, in turn, degrade

our understanding of experimental data. Because of

this complexity, simulations are playing an increasingly

important role in this ﬁeld.

Historically, simulations of dense plasmas have

used simpliﬁed plasma models to address issues rang-

ing from equations of state [86.88, 89]toplasma

microﬁelds [86.27, 28, 90]. Atomic physics is typic-

ally ignored within the simulation by assuming that

an average charge





, somehow known, can be as-

signed to each ion. For these simulation methods,

there are many excellent textbooks that introduce the

basic ideas [86.91–93]. Here, we focus instead on

simulations that attempt to describe both the dense

plasma and the atomic physics self-consistently. These

simulation methods are categorized in terms of the un-

derlying algorithm used and fall either into Monte Carlo

or molecular dynamics methods; we discuss each in

turn.

86.4.1 Monte Carlo

Monte Carlo is a method to evaluate average quantities

in thermodynamic equilibrium using random numbers.

For example, to obtain a property of a classical system

we might write







r d



, p



exp

(

−βH

)



r d

p exp

(

−βH

)

(86.19)

and, in principle, we could sample the multidimensional

integral randomly to obtain a good estimate of the av-

erage value of O given a many-body Hamiltonian H .

In practice, however, there are large portions of phase

space that give very little contribution to the average and

some method of “importance sampling” must be carried

out. This problem was originally solved by Metropolis

and coworkers [86.94] who introduced the Metropo-

lis method, which uses a Markov chain of states in

phase space that preferentially migrates towards states

of higher probability [86.92].

The computation of properties of atomic systems in

dense plasmas requires a quantum Monte Carlo (QMC)

method because atomic systems are inherently quantum

systems and the plasma itself can be degenerate. Al-

though there are a variety of QMC methods [86.93,95],

the most important for our purposes is the path-integral

Monte Carlo (PIMC) method, which is formulated at ﬁ-

nite temperatures and, in fact, exploits this condition by

constructing an equivalent system of many more par-

ticles at a higher effective temperature. This is achieved

by writing spatial matrix elements of the quantal version

of the Boltzmann factor of (86.19)as



exp



−β















exp



−β

H /2















exp



−β

H /2









(86.20)

This expression can be used recursively to obtain matrix

elements evaluated at higher and higher temperatures;

this allows a high-temperature approximation to be

made, albeit at the expense of having many more ma-

trix elements to evaluate. It can be shown that in the

simplest approximation each quantum particle can be

replaced by a polymer of M classical particles linked by

springs; this picture is referred to as the “classical iso-

morphism” [86.96]. All electrons in the system (bound

and free) are treated on an equal footing.

Although the PIMC method is, in principle, simple to

implement and can be quite accurate, there are several

issues that arise when it is applied to dense plasmas.

There are difﬁculties with the deep attractive Coulomb

well that is crucial for describing atomic physics; this

leads to the need for enormous numbers of ﬁctitious

classical particles. A partially analytical or numerical

solution can greatly mitigate this problem [86.93, 97].

PIMC also suffers from difﬁculty when treating fermion

systems because of the so-called fermion sign problem

in which many terms of opposite sign arise from the

antisymmetric form of the N-electron wave function.

Progress has been made in this direction as well [86.95].

Finally, the long-ranged nature of the Coulomb potential

causes additional difﬁculties for describing bulk systems

with periodic boundary conditions [86.98]. When these

additional considerations are taken into account, good

agreement with other methods is found, and results have

led to important conclusions about experiments [86.99],

which are detailed below.

86.4.2 Molecular Dynamics

Molecular dynamics is a simulation method based on

the time evolution of a many-body system [86.91–93].

Part G 86.4

1314 Part G Applications

The vast majority of MD simulations are based on the

solution of Newton’s equations for N classical particles

in a main cell with periodic boundary conditions. Sim-

ulations can be carried out in various ensembles and

one assumes that long simulations in the canonical en-

semble will correctly sample canonical averages in the

same manner as (86.19). Atomic properties in dense

plasmas are usually obtained in the Born–Oppenheimer

approximation, which freezes the ionic dynamics be-

tween time-steps and performs an electronic structure

calculation to obtain the electronic density for that par-

ticular ionic conﬁguration. The ions are then advanced

using the forces from the resulting electronic density, and

the procedure is repeated. Many methods are available,

including basic TF [86.100], Hartree–Fock [86.101], and

tight-binding [86.102] to compute the electronic struc-

ture. A commonly-used DFT code for warm matter is the

Vienna Ab-Initio Simulation Package (VA S P ), which is

widely distributed [86.103]. In contrast to most PIMC

implementations, the deep Coulomb potential is often

treated in VA S P by softening the electron-ion interaction

with some form of pseudopotential [86.103,104]. These

methods trade between accuracy and range of validity;

for example, TF theory is less accurate than DFT-MD,

butisusefulathighertemperatures(Θ>50 eV) and for

very dense plasmas.

There are three advantages to the MD method. First,

the MD approach greatly extends the AA model by

including many nuclear centers in the main simula-

tion cell, and these centers may arise from different

elements. (This can also be done with PIMC.) Futher-

more, DFT-MD, which can be very accurate for cold

systems ([86.105], p. 117), has the additional advan-

tage that single-particle (Kohn–Sham) orbitals can be

used in formulae for linear, frequency-dependent re-

sponse properties, such as the electrical conductivity.

(It may seem paradoxical that electron dynamical in-

formation can be obtained from a static calculation.

Strictly speaking this is not possible, although reason-

able results can be obtained for some quantities [86.105,

p. 49]). And ﬁnally, MD has the additional advan-

tage that dynamical ionic properties can be obtained

from the time evolution that is simulated. For exam-

ple, DFT-MD simulations for the computation of the

self-diffusion coefﬁcient of dense hydrogen [86.106]

have recently been performed; in principle, a wide

range of other dynamical ionic properties are available,

such as viscosity, thermal conductivity, and collective

behavior.

Quantum simulations beyond the Born–Oppen-

heimer approximation that treat electrons and ions on

an equal, dynamical footing are much more difﬁcult

since they involve a numerical solution of the N-body

Schrödinger equation. Such simulations are necessary,

however, for obtaining dynamical electron properties,

especially under nonequilibrium conditions – those with

time-dependent temperatures or nonthermal momen-

tum distributions. Furthermore, these simulations can

describe electronic properties, such as atomic physics,

strong scattering, and degeneracy.

Very simple models have been developed by Deutsch

and coworkers, who constructed effective interactions

between the electrons that yield some known property,

such as the high-temperature pair correlation func-

tion [86.107]. These interactions can then be used

directly in a simulation, but with modiﬁed equations

of motion – an approach pioneered for dense plas-

mas by Hansen and coworkers [86.108]. Diffractive

and symmetry effects can be accounted for in the

high-temperature limit. For example, one model of

the diffractive potential for the electron-ion interaction

(r) =−



1 −exp

(

−r/λ

)



(86.21)

where λ is on the order of the electron deBroglie wave-

length. This potential is ﬁnite at the origin, which

prevents the classical collapse of a neutral system dur-

ing the simulation. Unfortunately, such a method suffers

from several weaknesses, including being limited to

high temperatures, only describing Pauli exclusion by

pair-interactions, and having incorrect atomic binding

energies.

There have been several attempts to improve upon

simple potentials of the form (86.21). Since one of the

main features of quantum mechanics is that conjugate

space- and momentum-dependent quantities do not com-

mute, it is natural to construct potentials v(r, p) that

depend on both r and p. Such momentum-dependent

potentials have been formulated in the context of nu-

clear physics and have recently been applied to dense

plasmas by Ebeling and coworkers [86.109]. Although

quite good atomic properties can be obtained, these po-

tentials suffer from the fact that they are ad hoc, and one

does not know how to choose adjustable parameters for

unexplored conditions.

A direct approach is to solve the time-dependent,

many-particle Schrödinger equation, albeit approxi-

mately. This can be done by reducing the (inﬁnite)

degrees of freedom to a smaller, more manageable

set. For example, Heller [86.110] ﬁrst suggested using

a Gaussian wavepacket to describe electron semiclas-

Part G 86.4

Atoms in Dense Plasmas 86.4 Simulations 1315

sical dynamics. This has been applied to ionization

in a dense plasma by Ebeling [86.111]. In general,

a many-body, antisymmetric wave function can be

parametrized in terms of a few parameters for each

particle and a time-dependent variational principle can

be used to obtain the equations of motion of the

parameters. For example, each particle in a Slater de-

terminant can be chosen to be a Gaussian with a width

w(t) with its conjugate momentum p

(t). This method

is referred to as “Fermion molecular dynamics” or

“wavepacket molecular dynamics” (WPMD). A review

of this approach has recently been given by Feld-

meir and Schnank [86.112]. Wavepacket shapes other

than Gaussians, which can better reproduce atomic

properties, have been proposed by Murillo and Timmer-

mans [86.113].

86.4.3 The Deuterium EOS Problem

Experiments can be notoriously difﬁcult to perform in

the dense plasma regime because of the enormous pres-

sures produced. Recently, experiments on compressed

deuterium have been performed that pass from the mo-

lecular ﬂuid phase into the dense plasma phase and

therefore probe the physics of atomic and molecular

states in a dense environment. The ﬁrst of these was

conducted at Livermore [86.114]usingtheNovalaser

to shock compress liquid deuterium to 2 Mbar. The ex-

periments indicated a higher compressibility compared

with commonly used equation of state properties. These

interesting results led to new experiments, again at Liver-

more [86.115], with pressures exceeding 3.0Mbar,and

at Sandia using a magnetically driven ﬂyer plate on the

Z machine [86.116] to achieve 0.7 Mbar. The Sandia

results did not show the unexpected higher compress-

ibility. Later, additional laser-based experiments were

carried out at the Naval Research Laboratory [86.117]

to 6 Mbar, which agreed with the orignial laser-based ex-

periments but had signiﬁcant error bars. The Livermore

and Sandia results are shown in Fig. 86.4. (Error bars are

not shown.) Interpreting these experiments has, in turn,

led to increased activity in the use and development of

various simulation methods.

Also shown in Fig. 86.4 are results from PIMC and

WPMD simulations. Early PIMC results, which treated

the Fermion sign problem using properties of free par-

ticles, did not agree well with either experiment. A later

calculation [86.99] improved upon that treatment and

0.1

0.5 0.6 0.7 0.8 0.9 1.0 1.1

Pressure (Mbar)

Density (g/cm

)

PIMC

WPMD

LLNL/Nova

SNL/Z

Fig. 86.4 The equation of state of shocked deuterium; here

pressure (in Mbars) versus density (in g/cm

) is plotted

along the Hugoniot. Results from two experiments are

shown: Sandia (triangles) and Livermore (circles). Also

shown are the results from two simulations: path inte-

gral monte carlo (diamonds) and wavepacket molecular

dynamics (squares)

showed that details of antisymmetric wave functions be-

come important below 2 Mbar. These results are shown

in the ﬁgure, and better agreement with the Sandia re-

sult is found. Also shown in the ﬁgure are results from

a WPMD calculation [86.118], which tend to agree with

the Livermore data. The WPMD calculations, however,

did not include full antisymmetrization of the electron-

electron interaction. The temperatures predicted by the

simulations tend to be in the fraction of an eV range at

the lower part of the ﬁgure and up to tens of eV toward

the top of the ﬁgure; thus the experiments and the sim-

ulations are probing the very interesting WDM regime

in which molecular and atomic species are heated into

a cool plasma state.

Together with experiments on resistivity and





a more complete picture of the physics of atoms in dense

plasmas is emerging. But, more theoretical and exper-

imental developments are needed before we can tackle

with conﬁdence the wide range of dense plasmas that

occur in the HEDP regime.

Part G 86.4

1316 Part G Applications

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Lett. A 218, 297 (1996)

86.112 H. Feldmeier, J. Schnack: Rev. Mod. Phys. 72, 655

(2000)

Part G 86

1318 Part G Applications

86.113 M. S. Murillo, E. Timmermans: Contr. Plasma Phys.

43, 333 (2003)

86.114 L. B. Da Silva, P. Celliers, G. W. Collins, K. S. Bundil,

N. C. Holmes, T. W. Barbee Jr., B. A. Hammel,

J. D. Kilkenny, R. J. Wallace, M. Ross, R. Cauble,

A. Ng, G. Chiu: Phys. Rev. Lett. 78,483

(1997)

86.115 G. W. Collins, L. B. Da Silva, P. Celliers, D. M. Gold,

M. E. Foord, R. J. Wallace, A. Ng, S. V. Weber,

K. S. Bundil, R. Cauble: Science 281, 1178 (1998)

86.116 M. D. Knudsen, D. L. Hansen, J. E. Bailey, C. A. Hall,

J. R. Asay, W. W. Anderson: Phys. Rev. Lett. 87,

225501 (2001)

86.117 A. N. Mostovych, Y. Chan, T. Lehecha, A. Schmitt,

J. D. Sethian: Phys. Rev. Lett. 85, 3870 (2000)

86.118 M. Knaup, P.-G. Reinhard, C. Toepffer, G. Zwick-

nagel: J. Phys. A 36, 6165 (2003)

Part G 86

1319

Conduction of

87. Conduction of Electricity in Gases

The conduction of electricity through gases has

played ubiquitous roles in science and technology.

It was responsible for many of the fundamental

discoveries in atomic and molecular physics;

gas discharge lighting is essential to every

night operations; gas discharge lasers are still

important in research and manufacturing; and

all of advanced microelectronics depends on

plasma enhanced processing. To a large extent,

the efﬁciencies of the above cited applications of

gaseous electronics depend on the maintenance

of the distinct non-equilibrium between the

electrons and the gas or vapor. This non-

equilibrium can be achieved by operating at low

pressures or under pulsed excitation, where the

duration of the energy input is less than the

energy equilibration time between the electrons

and the heavy particles. The term gas discharge

originally described a transient or spark condition,

but has been extended to mean the continuous

conduction of electricity through gases.

Section 87.1 treats the electron-velocity dis-

tribution and its effect on various measurements

involving a swarm or distribution of electron

velocities. In this section, low fractional ioniza-

tion (< 10

−6

) is assumed; electron–ion collisions

are negligible relative to electron–atom (and

electron–molecule) collisions in so far as they

affect electron mobility, diffusion or energy loss.

Section 87.2 introduces the glow discharge

and considers the cold cathode and hot cathode

discharge phenomena. Section 87.3 discusses

87.1 Electron Scattering and Transport Phe-

nomena..............................................1320

87.1.1 Electron Scattering Experiments ..1320

87.1.2 Electron Transport Phenomena ...1321

87.1.3 The Boltzmann Equation ............ 1321

87.1.4 Electron-Atom Elastic Collisions ..1322

87.1.5 The Electron Drift Current ...........1322

87.1.6 Cross Sections

Derived from Swarm Data...........1326

87.2 Glow Discharge Phenomena ................. 1327

87.2.1 Cold Cathode Discharges ............ 1327

87.2.2 Hot Cathode Discharges..............1327

87.3 Atomic and Molecular Processes ............1328

87.3.1 Ionization ................................1328

87.3.2 Electron Attachment ..................1329

87.3.3 Recombination .........................1330

87.4 Electrical Discharge in Gases: Applica-

tions ..................................................1330

87.4.1 High Frequency Breakdown ........ 1331

87.4.2 Parallel Plate Reactors

and RF Discharges .....................1331

87.5 Conclusions ......................................... 1333

References .................................................. 1333

ionization by electron collision, electron attach-

ment, ion mobility, ion–ion and electron–ion

recombination, and other important processes

that affect the conduction of electricity in gases.

Section 87.4 illustrates the importance of gaseous

electronics with several important phenomena

and technical applications.

The physics and chemistry of the conduction of elec-

tricity in ionized gases involves the interactions of the

electrons and ions in the gas among themselves, with

ground-state and excited-state gas atoms or molecules,

with any surfaces that may be present, and with any

electric or magnetic ﬁelds that are externally applied or

generated by movement of the charged species. The elec-

tron collision-induced excitation and dissociation can

result in new compounds being formed at rates which are

orders of magnitude larger than those without a plasma

present. For a gas pressure of 1 torr, the electron mean

free path λ

mfp

between collisions is ∼1mm(∼10

colli-

sions/second). The assumption that λ

mfp

is small relative

to the plasma physical dimensions is usually valid, how-

ever it is often comparable to, or larger than the space

charge sheaths that form near electrodes and surfaces.

Since the electron–atom interactions, in most cases

of interest, have ranges shorter than the average gas-atom

Part G 87

1320 Part G Applications

separation, they may be treated as binary collisions and

the effects of neighboring gas atoms on a given electron–

atom encounter can be neglected. Quantities involving

such interactions in a partially ionized gas, where the

electrons are distributed both in velocity and position,

can be directly related to the corresponding quantities in

beam experiments, where the collisions are binary and

the electrons have a deﬁned velocity. The measurements

and calculations of the electron velocity distributions, or

of the electron energy distribution functions (EEDF), are

therefore of fundamental importance in the description

of partially ionized gases.

87.1 Electron Scattering and Transport Phenomena

87.1.1 Electron Scattering Experiments

In 1903, Lenard [87.1] determined the attenuation of

a beam of mono-energetic electrons by several gases.

He measured the fraction of an electron beam that was

transmitted without scattering through a ﬁeld-free region

containing the gas. Let an electron beam of density n

electrons/cm

, traveling with velocity v, pass a distance

dx through the gas. Let N be the number of atoms/cm

The loss of electrons by scattering per unit time due to

collisions with atoms is then given by

=−Nvσn

, (87.1)

where σ is the collision cross section for electrons of

velocity v. Writing vdt = dx, we obtain

= n

−Nσx

. (87.2)

In relation to the conduction of electricity in gases,

a more useful concept than σ is the momentum transfer

100

0.1

0.001 0.01 0.1 1 10

(10

–16

)

(eV)

Helium

Neon

Argon

Krypton

Xenon

Fig. 87.1 Momentum transfer cross section for noble gases

cross section σ

. In general, σ

is not equal to σ . Exper-

imental determinations of σ

, generally from electron

swarm experiments, are summarized in [87.2]. Results

for the noble gases are presented in Fig. 87.1 and for

representative diatomic gases in Fig. 87.2. Above en-

ergies of about 10 eV, σ

decreases as the electron

energy increases. Towards higher electron energies in

the monatomic gases, σ

is inversely proportional to

the ionization potential and directly proportional to the

gas polarizability. The fact that σ

has extremely low

values in certain gases at low energies is known as the

Ramsauer–Townsend (RT) effect. This can be explained

by partial wave scattering theory as the point where

the radial wave function has exactly one more oscilla-

tion inside the interaction potential well than does the

corresponding free function. Outside the well the two

functions are indistinguishable and there is no  = 0

(s -wave) scattering [87.3] (Sect. 45.2.7). For the scatter-

ing potentials of the heavier noble gases, this occurs at

0.1

0.01 0.1 1 10

(10

–16

)

(eV)

Fig. 87.2 Momentum transfer cross section for some di-

atomic gases

Part G 87.1

Conduction of Electricity in Gases 87.1 Electron Scattering and Transport Phenomena 1321

low energies where the partial cross sections for the

higher angular momenta ( = 1, 2,...) are still neg-

ligible. Methane and silane (Fig. 87.3, [87.4, 5]) and

possibly other nonpolar polyatomic gases, also display

Ramsauer–Townsend (RT) minima. There is great sim-

ilarity between the σ

curves for atoms and molecules

with similar external electron arrangements. The σ

curves for H

and CO (Fig. 87.2) show that

these cross sections are large over the range 0.1 to 10 eV.

Around 1.5eV, N

and CO display resonances which

are associated with the formation of temporary negative

ions followed by efﬁcient vibrational excitation of these

molecules [87.6].

87.1.2 Electron Transport Phenomena

Electron transport phenomena in an ionized gas, such

as diffusion under the inﬂuence of a density gradient

and drift under the inﬂuence of an electric ﬁeld, are

directly related to the electron current density Γ

which

is caused by these inﬂuences. In order to calculate Γ

it is necessary to know the electron spatial and velocity

density distribution f(v, r, t). The electron density n

and current density Γ

are given by

(r, t) =



f(v, r, t) d

v, (87.3)

(r, t) =



v f(v, r, t) d

v. (87.4)

100

0.1

0.01 0.1 1 10

(10

–16

)

(eV)

SiH

Fig. 87.3 Momentum transfer cross section for CH

and

SiH

The electron drift velocity w

is related to Γ

and n

= Γ

. (87.5)

The electron drift velocity is usually small compared

with the electron random velocity. The other limiting

case is important in connection with ionic mobilities.

It can also occur in electron transport in Ramsauer

gases or gas mixtures that are used in fast particle coun-

ters [87.7]. The theoretical analysis of electron motion

in gases uses primarily the Boltzmann transport equa-

tion approach [87.8]. Mean free path methods [87.9]and

Monte-Carlo methods [87.10] have also been applied.

87.1.3 The Boltzmann Equation

The electron density distribution f(v, r, t) of a given

kind of particle in phase space (conﬁguration and veloc-

ity space) is determined by the combined effect of all the

interactions to which the particle is subjected. If there

are no sources or sinks, the number of particles in a vol-

ume element of the six-dimensional space which moves

with the particles does not change with time. In con-

ﬁguration space, the continuity equation for f(v, r, t)

∂ f

∂t

+∇

· v f = 0 ; (87.6)

in phase space, it has the form

∂ f

∂t

+∇

· v f +∇

· a f = 0 , (87.7)

where the subscripts on the divergence operators denote

the independent variable, v is the particle ﬂow velocity

in conﬁguration space, and a is the particle acceleration

(i. e., the ﬂow velocity velocity space). This may be

written as q(E+v × B)/m for a particle of charge q and

mass m in the presence of electric and magnetic ﬁelds,

E and B respectively. If there are sources or sinks of

particles, then terms representing these appear on the

right-hand side of (87.7).

In principle, the effect of elastic and inelastic colli-

sions may be included in the expressions for Eand B,but

they are usually treated as source terms which transport

the colliding particles instantaneously from one volume

element of velocity space to another. This is a good

approximation, for example, in the case of electron–

atom collisions, but it is not valid for the longer range

electron–ion and electron–electron interactions in the

presence of high-frequency electric ﬁelds. Other colli-

sions, such as ionizing collisions, act to change the total

as well as transporting electrons from one element of

velocity space to another.

Part G 87.1

1322 Part G Applications

The electron–electron collisions will produce

a Maxwellian velocity distribution when the fractional

ionization is sufﬁciently large. For molecular gases, this

generally requires n

> 10

−5

. Since the electron–

electron Coulomb collision frequency varies as v

−3

these collisions are especially effective in the low energy

elastic regime where there are no inelastic energy loss

processes, and they may have an inﬂuence for n

low as 10

−7

for the noble gases [87.11]. Spitzer [87.12],

Dreicer [87.13], and Butler and Buckingham [87.14]

treat cases where the electron–electron term is included

explicitly. A full treatment of this subject is given by

Shkarofsky [87.15].

Using ∇

· a =0, and including collisions as a source

term, (87.7) becomes

∂ f

∂t

+v · ∇

f +a· ∇

f =



∂ f

∂t



collisions

, (87.8)

where the subscript ‘collisions’ identiﬁes the source term

due to all types of collisions.

87.1.4 Electron-Atom Elastic Collisions

For electrons in a weakly ionized gas, where only

electron–atom elastic collisions need be explicitly con-

sidered, the collision term is given by the Boltzmann

collision integral [87.16]. This integral describes the rate

at which electrons are brought into and removed from

an element of volume in velocity space in terms of the

differential elastic scattering cross section .

The Boltzmann equation may be solved by a per-

turbation method [87.8]inwhichE, B, and the density

gradients are the perturbations, starting from

f(v, r, t) ≈ f

(v, r, t) +

· f

(v, r, t), (87.9)

where f

is the unperturbed isotropic term and f

is the

anisotropy due to E and B. Substituting into (87.8)and

equating terms of the same angular type then yields

∂ f

∂t

∇

· f

−

3mv

∂

∂v



E· f



∂

∂v





∂ f

∂v



(87.10)

and

∂ f

∂t

+v∇

−

∂ f

∂v

+ω

× f

=−ν

(87.11)

where ν

= eE/(w

m) is the momentum transfer colli-

sion frequency from the Boltzmann collisions integral,

=eB/m is a vector whose magnitude is the cyclotron

frequency, and g = 2m/(M +m) controls the partition

of kinetic energy in an elastic collision between particles

with masses M and m.

The right-hand term of (87.10) is the ﬁrst-order cor-

rection due to elastic kinetic energy exchange between

electrons and atoms. The E· f

term arises from work

done by Eon the electrons, and the ∇

· f

term accounts

for particle loss from the volume element in phase space

due to electric ﬁeld drift and density gradients. Simi-

larly, in (87.11), the E term accounts for loss due to

electric ﬁeld acceleration, and the v∇

term accounts

for changes due to electron drift across a density gradi-

ent. The −ν

term results from randomization of the

electron velocity direction in electron–atom collisions.

The neglect of higher order terms in (87.9) is justiﬁed

if δT

 1, where T

is the gain in electron kinetic

energy between collisions, and δ f

/ f

 1, where δ f

is the change in f

over λ

mfp

. For the case of static

E and B with E ⊥ B, the orbital time 1/ω

replaces

the time between collisions if it is shorter, making the

above conditions less stringent. However, for an ac ﬁeld

with the resonant frequency ω

, the original conditions

apply. The off-resonant case in between. See [87.8, 17]

for a further discussion of validity.

87.1.5 The Electron Drift Current

Transport properties of a partially ionized gas are

calculated from the moments of f(v, r, t). To obtain

,(87.11) must be solved for f

. Assume that ω

= 0

and that E is constant in time. Then, since f

usu-

ally changes slowly compared with ν

, ∂ f

/∂t may be

neglected to a ﬁrst approximation, and

−



v∇





−

∂ f

∂v



(87.12)

where n

(r, t) f

(v, r, t) = f

(v, r, t).Useof(87.3)

gives

=−∇

) −n

E , (87.13)

which deﬁnes the electron free diffusion coefﬁcient



3ν

4πv

dv =





3ν

, (87.14)

and the static electron mobility

=−



3ν

∂ f

∂v

4πv

dv. (87.15)

In (87.13), ∇

should be interpreted as acting only on that

part of D

whose spatial dependence enters through an

Part G 87.1