by B.M. SMIRNOV
cr BO.
SN s
spe
OVEN:
6b. M. CuuprHos
Beenenne
B GMSHHY nNNnasmMpl
VisgatenbctBo «HayHa»
MockBa
Introduction
to Plasma Physics
by B. M. Smirnov
Translated from the Russian
by Oleg Glebov
Mir Publishers
Moscow
lirst published 1977
Revised from the 1975 Russian edition
Ha aneauticrom aaoixe
© Faaninan popacina usnko-MaTeMaTwyeckKkow JIMTepaTypbl
naylaTepeTBa «TlayKa», 1975 r.
© English translation, Mir Publishers, 1977
Preface
This book is based on a series of lectures delivered by the
author over a ten year period at the Moscow Power-Engineer-
ing Institute to technology and chemistry undergraduates
not specializing in physics. The book aims at providing a
concise yet general description of the physics of weakly
ionized plasma so that a budding engineer or chemist can
obtain a general understanding of the phenomena occurring
in the plasma of a laboratory setting. This understanding is
necessary since low-temperature plasma is increasingly
used in technology.
The character of the book’s intended readership demands
that the mathematics of the book be relatively unsophisticat-
ed. The author believes that the purpose of this book cannot
be achieved merely by including descriptive material and
formulas without derivation. This approach can hardly
contribute to understanding the subject since the student
cannot see all the conditions providing for the validity of
the result. We use another approach. The book extensively
employs various evaluative techniques, which show the
dependence of the result on the parameters of the problem
and give its value within order of magnitude. Moreover, for
some functions only their limiting values are determined or
simple assumptions are made to find these functions. These
methods yield a correct qualitative picture of the subject
and considerably simplify the discussion. However, the
simplicity of discussion is essentially thought-provoking
and creates a profound understanding of the subject.
The amount of the material used in the book and the form
of its presentation were chosen to provide engineering stu-
dents with the general knowledge of the fundamentals of the
plasma physics, which they will need when working with
plasma-containing systems,
6 Preface
The reader is assumed to know the material given in a
basic course of general college physics. The choice of prob-
lems and their treatment in the book were to some extent
prompted by the author’s experience in the applied plasma
research.
A list of literature for further reading is given at the end
of the book.
B. M. Smirnov
Contents
1 Plasma in Nature and in Laboratory Systems
What is plasma? Laboratory equipment for maintaining
plasma. Space plasma.
2 Statistics of Weakly Ionized Gas
Distribution of particles by state. The Boltzmann distri-
bution. The statistical weight of a state and the distribu-
tion of particles in a gas. The Maxwell distribution. The
Saha distribution. Dissociation equilibrium in the mole-
cular gas. Planck’s distribution. The laws of black body
radiation.
3 The Ideal Plasma
The ideality of a plasma. Charged particlesin a gas. Screen-
ing of charge and field in a plasma. Oscillations of plasma
electrons. The skin effect.
4 Elementary Processes in Plasma
Act of collision of particles in a plasma. Elastic collision
of particles. The total cross section of scattering and the
cross section of capture. The condition of gaseousness and
the ideality of plasma. The types of elementary processes.
Inelastic collisions of atomic particles. Charge exchange
and similar processes.
Oo Formation and Decomposition of Charged
Particles in Weakly Ionized Gas
Ionization of an atom in a single collision with an electron.
Recombination of pairs of positive and negative ions. Triple
processes. Thomson’s theory for the constant of the triple
process. Triple recombination of electrons and ions. Triple
recombination of the positive and negative ions. Processes
involving formation of a long-lived complex. Dissociative
recombination of an electron and a molecular ion. Ioniza-
tion processes in collisions between an atom in an excited
state and an atom in the ground state. Stepwise ionization
of atoms. Thermodynamic equilibrium conditions for ex- ‘
cited atoms.
6 Physical Kinetics of Gas and Plasma
The kinetic equation. Macroscopic equations for a gas.
The equation of state for a gas.
7 Transport Phenomena in Weakly Ionized Gas
Transport phenomena in gas and plasma. Transport of par-
ticles in a gas. Energy and momentum transport in a gas.
Thermal conductivity due to the internal degrees of freedom.
The mobility of particles. The Einstein relation. The Navier-
Stokes equation. The equation of heat transport. The diffu-
we)
17
29
30
o4
72
78
Contents
sion motion of particles. Convective instability of a gas.
Convective motion of a gas. Convective heat transport. The
instability of convective motion.
Transport of Charged Particles in Weakly
Ionized Gas
The mobility of charged particles. The conductivity of a
weakly ionized gas. Ambipolar diffusion. The mobility
of ions in a foreign gas. The mobility of ions in the parent
gas. Recombination of ions in a dense gas. The recombina-
tion coefficient of ions as a function of gas density.
9 Plasma in External Fields
The electron motion in a gas in an external field. The con-
ductivity of a weakly ionized gas. The Hall effect. The
cyclotron resonance. The mean electron energy. The magne-
tohydrodynamic equations.
10 Waves in a Plasma
11
Acoustic oscillations. Plasma oscillations. Ion sound. Mag-
netohydrodynamic waves. Propagation of electromagnetic
waves in a plasma. Damping of plasma oscillations in a
weakly ionized plasma. The interaction between plasma
waves and electrons. The attenuation factor for waves in
plasma. The beam-plasma instability. The Buneman
instability. Hydrodynamic instabilities.
Radiation in Gas
Interaction between radiation and gas. Spontaneous and
stimulated emission. Broadening of spectral lines. The
Doppler broadening. Broadening due to finite lifetimes
of states. Impact broadening of spectral lines. Statistical
broadening theory. The cross sections of emission and
absorption of photons. The absorption coefficient. The
conditions of laser operation. Propagation of the resonance
radiation.
12 Plasma of the Upper Atmosphere
The balance equations for the parameters of weakly ionized
gas. The distribution of particles and temperature in the
atmosphere with height. The heat balance of the earth. The
elemental oxygen in the atmosphere. Charged particles
in the upper atmosphere.
A ppendices
Bibliography
Index
100
108
119
138
157
168
170
171
1 Plasma in Nature and in Laboratory Systems
What is plasma? The term “plasma” first appeared in phys-
ics in the 1920s. Plasma is a conducting gas, that is, a gas
which contains a noticeable proportion of charged particles
(electrons and ions). To understand the conditions of plasma
formation let us compare a plasma and a mixture of chemi-
cally active gases. For instance, the following chemical
reaction can occur in the air, which is basically a mixture
of nitrogen and oxygen:
N, +0, <2 2NO— 41.5 kcal-mol-! (1.4)
Hence, a small amount of nitric oxide NO is present in the
air at the equilibrium between nitrogen and oxygen. Accord-
ing to the Le Chatelier principle, increasing the air tempera-
ture results in a larger equilibrium amount of nitric oxide.
The equilibrium between the neutral and charged particles
is similar to the above case. An atom or molecule consists of
bound positively charged nuclei and negatively charged
electrons. At high temperatures the bonds can be broken
giving rise to electrons and positively charged ions. For in-
stance, the respective chemical reaction for the nitrogen
molecule is
N, z= Ni + e—360 kcal-mol"! (1.2)
The bonding energy of an outer electron in an atom or mole-
cule is roughly ten times the chemical bond energy. There-
fore, production of charged particles in this reaction becomes
noticeable at temperatures of the order of tens of thousands
of degrees. For instance, the sun’s photosphere, which emits
the main part of the solar radiation and where the tempera-
ture is about 6000 K and the hydrogen atom density is of the
order of 10!” cm~*, has been found to have the ratio between
10 Introduction to Plasma Physics
the densities of the charged particles and the neutral particles
(the degree of ionization) of about 5 x 10-3.
The equilibrium density of the charged particles at room
temperature is practically zero owing to the high bonding
energies of the electrons in atoms and molecules. However,
by placing the gas into an electric field, the gas can be made
electrically conductive even at room temperature. The
electrons become strongly heated when moving in the elec-
tric field and receiving energy from it. The electric field does
not affect the gas molecules and even if the degree of ioniza-
tion of the gas is low, the temperature of the molecules re-
mains at the room level. Such a conducting gas in an external
electric field is called the gas discharge.
Plasma can be created in different ways. We shall discuss
below in brief the principal types of plasma encountered
in practice or research. The diagram in Fig. 1 illustrates
the parameters of the plasmas found in various systems.
Laboratory equipment for maintaining plasma. The gas
discharge is the most popular technique for producing plasma
under laboratory conditions. The gas discharge is used for
exciting most gas lasers; gas discharge as well as radiation
sources and lamps which generate radiation in a wide wave-
length range are the basis of plasmatrons. There are various
useful applications of the gas discharge.
The gas discharge is a gas space across which a voltage is
applied. Charged particles are produced in this space and
they move in the electric field and take energy from it. If
the charged particles are produced owing to the effect of an
external agent, the resulting gas discharge is called nonself-
maintained discharge, in contrast to the self-maintained dis-
charge. The principal useful types of the self-maintained dis-
charge are the glow discharge and the arc discharge; the
essential difference between them consists in the process of
electron production in the cathode’s vicinity. The electron
density is 107-10! cm- for the glow discharge and higher
for the arc discharge.
The density of the charged particles (electrons and ions)
in the gas discharge is much lower than the density of the
neutral particles (atoms, and molecules). This plasma is
called weakly ionized or low-temperature plasma since the
mean energy of electrons or ions in it is much lower than the
Plasma in Nature and Laborafory Systems 11
ionization potential for the gas particles. Another extreme is
the hot plasma where the mean energy of ionsis much higher
than the ionization potential of the gas particles. Such plas-
ma contains ions and electrons and practically lacks neutral
particles.
An example of hot plasma is the thermonuclear plasma,
that is, the plasma which must be created for the course of
a controlled thermonuclear reaction. The most practicable
thermonuclear fusion reaction involves deuterium nuclei
or nuclei of deuterium and tritium (the hydrogen isotopes).
To make possible this reaction the deuterium or tritium ions
must be able to enter the reaction during the time of plasma
confinement, that is, when the ions are in the reaction volume.
In the existing laboratory installations, this condition
is satisfied at the ion temperature over hundreds of millions
degrees and when the product of the density N of charged
particles by the plasma confinement time t exceeds
10** s-cm~*. For the plasma containing deuterium and tritium
nuclei, Nt must be more than 1014 s-cm=? (the Lawson crite-
rion). If this condition is met, the self-maintaining thermo-
nuclear reaction can occur when the heat released by the
reaction maintains the temperature of the particles needed
for continuation of the reaction. The existing laboratory
installations do not achieve these values.* However, the
thermonuclear plasma is carefully studied and some advance
is forseeable in this field.
Note that the energy of radiation from the sun and stars
also is produced in thermonuclear fusion reaction involving
the hydrogen nuclei, or protons. This reaction is less effi-
cient than the reaction with the deuterium or tritium ions,
but since the reaction volume in stars is very large, the tem-
perature proves to be of the order of ten million degrees,
rhat is, considerably lower than the temperature of the labo-
ratory thermonuclear plasma for a more favourable reaction.
* The Tokamak-type installations, which are considered at pres-
ent to be the most promising ones for controlled thermonuclear fusion,
make it possible to reach values of Nt less than 10!° s-cm~-? and ion
temperature less than 107 K [see M. S. Rabinovich, Fizika plazmy 1,
335 (1975): “Results of the V International Conference on plasma
physics and controlled thermonuclear fusion, Tokyo, 10-15 Novem-
ber 1974”; English translation in Soviet Journal of Plasma Physics}.
12
FIG. 1.
The plasma
parameters:
Te is the electron
temperature,
and an asterisk
denotes plasmas
for which the ion
temperature 7;
is given;
Ne is the electron
density.
Introduction to Plasma Physics
log 1 K
11
10
Proton belt of the earths
@ ee radiation belt of the earth
CQ) inner radiation belt of the earth
Solar wind*
Solar corona
4
Exosphere
of the earth
(1000 km
3 and beyond)
BF,
)
1D) Ee: ae
lonosphere of the earth
(80 -250 km)
Interstellar
Plasma in Nature and Laboratory Systems 13
H-bomb -
The boundary of controlled
“ OD reactions in Tokamaks
7
tnd-stopper devices
CC} — Tokamak
Solar core —C)
Argon laser
CO, laser
He-Ne laser a Cathode spot
/
© MHD generator Spark, lightning
Cc» Mercury -vapour lamp
Chromosphere
of the sun ore
Thermoemission converter
Photosphere of the sun — :
Plasma of metals
10 12 14 16 18 20 22 24 26
log .cm~*
14 Introduction to Plasma Physics
The low-temperature plasma is used in the laboratory in-
stallations of various types, apart from the gas discharge
which can produce plasmas with different parameters. Let
us discuss some of these installations. If a magnetic field is
applied perpendicular to the flow of weakly ionized gas, an
electric current passes perpendicular to the flow and to the
magnetic field. If an electric field is applied opposite to this
current, we obtain an electric power generator which trans-
forms the kinetic energy of flow into electric energy. Such
installations are called the magnetohydrodynamic (MHD)
generators.
The greater the magnetic field, the density of the charged
particles, and the gas flow velocity, the greater the energy
that is produced by the unit volume of the MHD generator.
The magnetohydrodynamic transformation of energy is a
very promising method since it provides for high power pro-
duction per unit volume of the installation and high efficien-
cy. There are two types of the MHD generator configuration:
open-cycle and closed-cycle. In the MHD generators of the
open-cycle type, the working gas passes the conversion vol-
ume only once and then is discharged. Application of such
MHD generators already has been started at heat power
plants where they contribute to increasing the total plant’s
efficiency. In the elosed-cycle MHD generator the working
gas repeatedly passes the conversion volume. Practicable
closed-cycle MHD generators are still being developed.
If we connect two parallel metal plates with different
work functions*, this will give rise to a potential difference
across the vacuum gap between the plates. If we then heat
one of the plates to a high temperature, there will be some
electron emission from it and part of the electrons will reach
the cold plate. We shall heat the plate with the higher work
function and interconnect the plates via a load. Since the
electrons spend energy when passing from one plate to an-
other through the vacuum gap, the electric energy will be
liberated at the load. Hence, this system, which is called the
thermoemission converter, converts thermal energy into
electric energy.
* The work function is the energy needed by an electron to leave
a metal’s surface.
Piasma in Nature and Laboratory Systems 45
The efficiency of the thermoemission converter is low (less
than 20%) and for a high temperature of the plates its main
advantage is compactness, that is, it produces a high electric
power per unit area of the plates. The uncompensated charge
of electrons in the gap between the plates gives rise to an
electric field HE given by the Poisson equation:
dE
< =4ne (Ni—N,) (1.3)
where JV, is the electron density, N; is the ion density which
is zero in this case, and z is the distance to one of the plates.
Hence, the electrons give rise to the following potential
difference (EF = —dg/dz):
g = 2neN ,d* = 2njd?/ve (1.4)
The output voltage is decreased by this value; the output
voltage of the thermoemission converters amounts, typically,
to about one volt. In (4.4) d is the distance between the
plates, j = eN,v, is the electric current density, and v, is
the electron current velocity.
From the above equations it readily may be estimated that
the effect is absent for the practicable energy flux of about
1 W-cm~? if the width of the gap between the plates is much
less than 10 um. This must be done for the plate temperature
of about 2000 K, when there is intense evaporation of the
material from the plate surface. Hence, the above condition
is technologically unfeasible. However, this problem can be
solved by filling the gap between the plates with plasma,
which essentially will determine the parameters of the
thermoemission converter. .
The electrogasodynamic (EGD) generator is a less well-
known device than the MHD generator. In the EGD genera-
cor, the gas flow containing ions of the same polarity (only
negative or. only positive) is directed through an electric
field so that the ions are carried by the gas flow opposite to
the field. Hence, the ions “produce” electric power by con-
verting the energy of the gas flow. The output voltage of the
EGD generator can be rather high, but its power and specific
power are not high since the ion densities in the gas flow are
small.
16 Introduction to Plasma Physics
Interestingly, the concepts of the MHD and EGD genera-
tors and the thermoemission converter were suggested as
early as the end of the last century. But the high-temperature
materials necessary for constructing practicable systems
have been developed only recently.
The same is true for the plasmatrons, the gas-discharge
devices in which the electric energy is used for carrying out
chemical reactions. Plasmatrons first were developed at the
beginning of this century. However, because of the high
cost of electric power at that time, it was too expensive to
convert into chemical energy. Now plasmatrons are increas-
ingly used in industrial applications, which make it pos-
sible to decrease considerably production areas, to obtain
higher-quality products and to carry out processes in one
stage, thus getting rid of the useless intermediate products.
The above examples illustrate the fact that technological
innovations are not necessarily due to the advances in pure
science but can be initiated by developments in the technolo-
gy itself.
Space plasma. Apart from the laboratory plasma, the
attention of the scientists is increasingly drawn to the plas-
mas in the atmospheres of the earth and the planets, in the
stars, including the sun, and in outer space.* Each of the
above plasma types exists under rather special conditions.
For instance, the plasma of the earth’s atmosphere (hundreds
of kilometers above the earth’s surface) is created by the
ultraviolet solar radiation. This plasma’s parameters sharply
vary according to certain processes occurring on the solar
surface and to the parameters of the atmosphere itself. A few
successful experiments have been carried out with temporary
variation of the atmospheric plasma parameters in limited
volumes of space.
The plasmas of stars differ greatly in their parameters.
For instancein the inner part of the sun where the thermo-
nuclear fusion reaction occurs, the temperature is as high as
16 million degrees. The surface region of the sun about
1000 km thick which radiates most of the solar energy is
referred to as the photosphere; the temperature of the photo-
* Over 90% of the matter in the universe consists of charged par-
ticles, that is, it is in the plasma state.
Statistics of Weakly lonized Gas 17
sphere is about 6000 K and its distance from the sun’s centre
is 700 000 km. The region which is closer to the sun’s centre
is called the convective region since the energy is transported
there with convective movement of the solar plasma in
strong magnetic fields. Such movement of the solar plasma
gives rise to the granular structure of the photosphere, devel-
opment of the sun spots and other nonstationary phenomena
on the sun’s surface. However, the total solar power radiated
in the optical range is fairly stable despite the nonstationary
effects.
Over the sun’s surface there is a low-density high-tempera-
ture region (the temperature of about 10° K) called the solar
corona; it is arather powerful source of ultraviolet radiation.
The sun emits plasma from its surface. The stationary proton
flux emitted by the solar corona is referred to as the solar
wind. The plasma flow from the sun’s surface gives rise to
the interplanetary plasma. The electrons in this plasma are
captured by the magnetic field of the earth and give rise to
the radiation belts around the earth (at a distance of a few
thousand kilometers). The high-energy electrons and protons
produce various effects in the earth’s atmosphere, in particu-
lar the auroras.
The interstellar plasma has a very low density and a tem-
perature of about 3 K. The energy exchange between the par-
ticles of this plasma proceeds in a peculiar way via the in-
teraction with the electromagnetic radiation field. The in-
terstellar plasma is a source of information on the develop-
ment of the universe.
2 Statistics of Weakly lonized Gas
Distribution of particles by state. Let us assume that
we consider an ensemble of a large number of particles and that
each of the particles can be in one of the various states des-
cribed by a set of quantum numbers i. We have to find how
many particles of this system are in a given state. For in-
stance, we consider a molecular gas and have to find the
number of molecules in a given vibration-rotational state.
Discussed below are problems of this type. ©
2—01607
18 Introduction to Plasma Physics
Let us consider a system of particles containing a definite
number N of particles which does not vary with time. Let
us denote the number of particles in the ith state by n;,;
then the following relation must hold:
N= 2d nj (2.1)
Furthermore, our system of particles is closed, that is, it
does not exchange energy with the outside world. Hence,
if the total energy of the system is € and the energy of the
particle in state i is €;, then the following relation is satis-
fied owing to conservation of the total energy of the system:
é = > iN; (2.2)
Our closed system is in the state which is termed thermo-
dynamic equilibrium.
When the particles collide, they change their states, so
that the number of particles in a given state is continuously
changed. Hence, the probability that a given number of par-
ticles are in a given state is proportional to the number of
possible realizations of this distribution.
Let W (n,, Mo,...;;,.. .) denote the probability that n,
of the particles are in the first state, n, of the particles are
in the second state, and so on, and let us calculate the number
of possible realizations of this distribution. First, select
from the total number JN of the particles n, particles which
are in the first state; there are Cr, = ve ways to
do that. Next, select n, particles which are in the second state
from the remaining N — n, particles; this can be done in
Cx, ™ ways. Continuation of the procedure yields the follow-
ing expression for the probability:
W (14, Me, 20-5 My oe) = ia (2.3)
where A is the normalization constant.
The Boltzmann distribution. Let us find the most probable
number of particles, n;, in a given state i. It should be taken
into account here that n; > 1, and for n; = n; the probabili-
Statistics of Weakly lonized Gas 49
ty W of distribution of particles by state and the logarithm
of this probability have maximums. Let us denote dn; =
= n; — n; where n; > dn; > 1. Assuming that n; > 1, we
expand In W at the maximum. Using the relation
Tm Tr n;
Innj!=In [[ m= >) nme \ In x dx
m=1 m=1 0
we find d In n,!/dn; = 1n n;. From this relation and Eq. (2.3)
we obtain
In W (ny, Mg, ..-, Mi, -e-
)
= InW (n, ne, Ses Wig aca)
2
a 4 n:
The maximum condition for this quantity gives
> Inn; dn; = 0 (2.5)
Making use of Eqs. (2.1) and (2.2), we find the following
relations for dn;:
>) dn; =0 (2.6)
and
> 6:dn; =0 (2.7)
The mean number of particles in a given state, n;, can be
found from Eqs. (2.5)-(2.7). Multiply Eq. (2.6) by —In C and
Eq. (2.7) by 1/T where C and T are characteristic parameters
of our system. Adding the resulting relations, we find that
D (Inn; —InC+@;,/T) dn; =0
Since this equation holds for any dn,;, the term in the paren-
theses is equal to zero. This equation yields the following
expression for the most probable number of particles in a
given state:
n; == C exp (—6;,/T) (2.8)
This distribution is termed the Boltzmann distribution.
O*
20 Introduction to Plasma Physics
Let us determine the physical meaning of the parameters C
and 7 in Eq. (2.8). These parameters describe the particle
system being considered and their values can be found
from the additional conditions (2.1) and (2.2) which this
system should meet. For instance, condition (2.1) yields
C >) exp (—6,;/T) = N. This shows that C is a normaliza-
4 \
tion constant proportional to the total number of particles.
The energy parameter 7’ is termed the temperature of the
system; according to Eq. (2.2) 7 can be related to the mean
energy per particle.*
Before considering specific cases, we must make sure that
for large n; the probability that the number of particles in
this state noticeably differs from n,; is low. Transform
Eq. (2.4) taking into account Eq. (2.5):
W (nm, No, eee, Ny eid)
i)”
= W (nm, No, saan Tes ...)exp[ — 5) Siam)
This shows that the probability is noticeably decreased if
the difference between the number of particles and the mean
value is An; ~ ni? If the number of particles in the state
is high, the relative variation An,/n,;~ nj 1/? is small. Hence,
the observed number of particles in this state practically
coincides with the most probable number.
The statistical weight of a state and the distribution of
particles in a gas. In the above discussion, the subscript i
denoted one state of a particle. Now, let us take into account
the fact that this state can be a degenerate one. Let us
introduce the quantity g; referred to as the statistical weight,
which is equal to the number of states with the same quan-
tum number. For instance, a rotational state of a molecule
with the rotational quantum number J has the statistical
weight of 2J +41, that is, it equals the number of possible
angular momentum projections on a given axis. Taking the
sum over the degenerate states in Eq. (2.8), we can trans-
* We express here the temperature in energy units and, hence,
do not write the Boltzmann constant k = 1.38 X 10-16 erg-K- as
is sometimes done.
Statistics of Weakly lonized Gas 21
form it into =
nj = Cg, exp (—6;/T)
where the subscript 7 designates now a group of states. This
equation yields a relation for — densities:
N;= No— exp (—7) | (2.9)
Here NV, and WN, are the densities of particles in the jth and
ground ‘states, €; is the excitation energy for the jth state,
and g; and g, are the statistical weights of the jth and ground
states.
Let us find the statistical weight of the continuous spec-
trum states. The wave function of a free particle with mo-
mentum p, moving along the axis z is given, up to an arbi-
trary factor, by exp (ip,x/h) if the particle moves in the posi-
tive direction and by exp (—ip,.2/h) if the particle moves in
the negative direction (A is Planck’s constant h divided by
2). Let us put the particle into a potential well with infi-
nitely high walls so that the particle can move freely only
inthe region0 < 2x < L. Let us construct the wave function
of the particle in the potential well as a combination of the
above functions. The wave function of the particle must be
zero at the walls of the well; the boundary condition for
x = 0 shows that the wave function of the particle is pro-
portional to sin (p,a/h), and the boundary condition for
x = Lyields the possible values of the particle’s momentum:
p,L/h = nn where n is an integer.
Hence, a particle with a momentum in the range from p,.
to Put Gx can be in dn = L dp,/(2nh) states if we take
into account the sign of the momentum; if the particle is in
the interval dz, the number of states for a free particle is |
dp az
dn = oe (2.10a)
The formula for the three-dimensional case is
dp,dx dpydy dp,dz _ dpdr
Onh Onh Qnh (2nh)3 (2. 10b)
The number of states given by Eq. (2.10) is the statistical
weight for the continuous spectrum states since it deter-
mines the number of states corresponding to a given range
of continuously varying parameters. The quantity dp dr is
termed an element of phase space.
22 Introduction to Plasma Physics
Now let us consider some particular Boltzmann distribu-
tions. First, let us study the distribution of diatomic mole-
cules among the vibration-rotational states. For not too large
vibrational quantum numbers v, the excitation energy of the
vth vibrational level of the molecule is hwv where hw is the
gap between the neighbouring vibrational levels in the ener-
gy space. Hence, according to Eq. (2.9) we find that
N,= No exp (— hov/T) (2.11)
Since the total density of the molecules is V = > Ny
= N, >) exp (—hov/T) = N, [1 — exp (xs ho/T)|- iy the
density N, of the molecules at the vth vibrational level is
N,=N exp (— “a~) [4—exp (— ey)y" (2.42)
For the rotational state with the angular momentum J,
the excitation energy is BJ (J +1) where B is the rotational
constant of the molecule. Since the statistical weight of a
rotational state is 2/7 +4, Eq. (2.9) yields the following ex-
pression for the density of molecules at a given vibration-
rotational state:
B BJ (J+1
Nyy =Ny— (2F + 4) exp[ AY (2.13)
Here we made use of the normalization condition >; N, =
J
= N, and assumed that B < T, which is typically the case.
Let us now consider the spatial distribution of particles
in a uniform field. The particles are in a half-space; the force
F acts upon each particle so that the potential energy U
of each particle is U = Fx. Equation (2.9) yields the follow-
ing distribution of the particles in space:
N (x) = N (0) exp (—Fz/T)
where N (Q) is the particle density at the origin, and N (z)
is the particle density at the point zx. A particular case of
this formula is the distribution of the molecules in the
earth’s atmosphere by height under the effect of the gravi-
tational field:
N = N (0) exp (—Mgh/T) (2.14)
Statistics of Weakly lonized Gas 23
Here M is the molecule’s mass, g is the acceleration of gravi-
ty, and h is the height above the earth’s surface. For nitrogen
Meg/T is 0.11 km= at room temperature, and so the atmos-
pheric pressure varies noticeably when going up a few kilo-
meters. Equation (2.14) is called the barometric height for-
mula.
The Maxwell distribution. Let us consider oné more dis-
tribution of particles by state, namely, the distribution over
the velocities of gas particles. First, we shall discuss the
one-dimensional problem. The number of the particles with
the velocities in the range from v,, to v,-+dv,, is designated
as n (v,) dv,. The energy of these particles is Muz/2 (M is
the particle mass) and the statistical weight is proportional
to the number of states corresponding to the velocity range.
The number of states is dz dp,/(2nh) where p, = Mv, is
the particle’s momentum, and dz is the coordinate range of
the particle. The statistical weight in this case is seen to be
proportional to the given velocity range and Eq. (2.8) yields
n (v,,) dv, = C exp ( — Mes dv,,
where C is the normalization factor. The normalization con-
+ oo
dition \ n (v,) dv, = N (Nis the total number of particles)
MM
2nT
a3 1/2
yields C = N { “> Let us introduce a new function
+00
@ (v,) = n(v,)/N normalized to unity: { (v,) dv, = 1;
hence, the probability that a particle has the velocity v, is
(r= (he) exp( aE) (2.45)
Equation (2.15) is termed the Maxwell distribution.
Write down the above result for the three-dimensional
case. The number of particles having velocities in the range
from v tov + dv is n(v) dv where
ni(v) = NQ (Vx) 9 (Vy) F (Vz)
= N( ul )"" exp(— 42) (2.15a)
anT 2T
24 Introduction to Plasma Physics
where v = (vx +- vy + v2)? is the speed of a particle.
Using Eq. (2.15a) we can determine the mean kinetic energy
of a particle:
r Mv2 Mv?
ee! 2
\ exp ( OT )v dv
0
opr inal®? =57 = (2,16)
where a does not depend on the temperature. Hence, the
mean kinetic energy of a gas particle is 37/2 and the mean
kinetic energy per one degree of freedom is 7/2. Equation
(2.16) may be used for the definition of temperature.
The Saha distribution. Another case of interest we shall
consider here is the equilibrium between continuous-spectrum
and discrete-spectrum states. Let us find the relationship be-
tween the densities of electrons, ions, and atoms involved in
the following processes:
A* teva
where A* is the ion, e the electron, and A the atom. Let us
assume the plasma to be quasineutral, that is, the ion density
equals the electron density.
Equation (2.9) yields the following expression for the
ratio between the mean number of the electrons, n, = 7n;,
and the mean number of the atoms, n,, in the ground state:
rn; geei ( apa T+ p2/2m
Pi oft | rE ex ( — or
Na Ea
Here g, is the statistical weight of electrons, g; and g, are
the statistical weights of the ion and the atom corresponding
to their electron states, J is the ionization potential of the
atom, p is the free electron momentum so that J -+ p?/2m
is the energy needed for removing the electron from the atom
Statistics of Weakly lonized Gas 25
and transferring to it the kinetic energy p?/2m, and
dp dr/(2xh)> is the number of states in an element of phase
space, that is, the states in the given range of coordinates
and momentum of the particle.
Integration of this expression over the electron momentum
yields
a= EE (say) exp (—p) | at
Na
Let the total volume of the system be V. When integrating
this equation over volume, we should take into account that
the state of the electron system is not changed if the coordi-
nates of two electrons are interchanged. Therefore, to calcu-
late the number of states per one electron, we must take
into account only the volume per one electron. Hence we find
\ dr = V/n,. Using as notation for the electron density
N.= n /V, the ion density V; =n,/V, and the atom density
N, = 7n,/V, we can find the following relationship between
these quantities:
pa ttt (Bie )em(—t) 0
This equation is called the Saha distribution.
Equation. (2.17) can be written in the form of the Boltz-
mann: distribution (2.9):
Ne __ &cont. I
ye = Sent exp (—F] Faas)
; 3/2
where cont. = (so) is the effective statistical
weight of the continuous spectrum. It can be readily seen
that this weight is rather high for the ideal plasma. Owing
to the high statistical weight of the continuous spectrum, the
degree of plasma ionization is about unity for the tempera-
tures T< I. These temperatures are low compared to the
excitation energy of the atom. Hence, the relative number of
excited atoms is small; at the temperature comparable to the
excitation energy this is because almost all the atoms dis-
Sociate into ions and electrons,
26 Introduction to Plasma Physics
Dissociation equilibrium in the molecular gas. Let us
consider the equilibrium between atoms and molecules in
the molecular gas where the following reaction occurs:
a ae aoa 34
The relationship between the densities of the atoms Ny and
Ny and the molecules NV xy which are in the ground vibra-
tion-rotational state is given by the Saha distribution (2.17):
NxNy _ exgy ( pr \3/2 _D
Nxy v=0, J=0) gxy (oe) exp ( r) (2.19)
Here pu is the reduced mass of the atoms X and Y, and D is
the dissociation energy of the molecule. In contrast to the
above case of ionization equilibrium where all the atoms
were in the ground state, here most molecules are in excited
states.
Making use of Eqs. (2.12) and (2.13), we can find the rela-
tionship between the total density of molecules N yy and the
density of molecules in the ground state Nxy (v = 0,
J = 0):
h B
Nxy(v=0, J=0) = [ 1 —exp ( ——) | FN xy
Substituting this relation into Eq. (2.19), we obtain finally
NxNy __ exgy (42) B
Nyy a &xy 2nh2 T
x [1—exp (—) Jexp (—=+) (2.20)
Planck’s distribution. Let us assume that radiation is
in thermodynamic equilibrium with the walls of the vessel
it fills and with the gas in the vessel. This radiation can be
described by the temperature 7 equal to the temperature of
the gas and the walls, and it is called black body radiation.
Let us find the mean number of photons in one state. The
energy of a photon in a given state is hw. Since photons obey
Bose-Einstein statistics, any number of photons can be in
a given state. From the Boltzmann formula (2.11) we find
that the relative probability of m photons being in a given
state is exp (—fwn/T). The mean number of photons in
Statistics of Weakly lonized Gas 27
the same state with a given energy is
Bivon(—“F) 7 | ; ”
Selsey eee
n
Ny =
Equation (2.21) is referred to as Planck’s distribution.
The laws of black body radiation. The energy of the elec-
tromagnetic radiation field per unit volume and unit fre-
quency range is termed the spectral radiation density U,.
Hence, the energy of the electromagnetic radiation field in
the frequency range from » to w + dw filling volume V is
given by VU,, dw. On the other hand, this energy can be writ-
ten as 2hwn,V dk/(2n)* where factor 2 accounts for the two
types of polarization of the transverse electromagnetic wave,
V dk/(2n)° is the number of states in the given volume of the
phase space, 7, is the number of photons in one state, and hw
is the energy corresponding to this state. When we equate
the above two expressions for the energy and make use of the
dispersion relation wo = kc between the frequency o and the
wave vector k of the electromagnetic wave (c is the velocity
of light), we find that the spectral radiation density is
ha?
Us=5 Mo — (2.22)
Replacing in Eq. (2.22) n, by Planck’s distribution (2.21),
we obtain
| ho
0. = ——_—__——_—- 220
m2¢3 (exp = 1) :
T
Equation (2.23) is called Planck’s radiation formula. For
the extreme case when fiw/T < 1, it yields the Rayleigh-
Jeans formula
w?T
hw
Vo=s ai (2.24)
For the other extreme case, iw/T > 1, it yields the Wien
formula
hw? h h
Ue =~35 exp ( —+), = > 1 (2.25)
28 Introduction to Plasma Physics
Let us calculate the flux of radiation emitted by the sur-
face of a black body, that is, the energy radiated from the
unit surface area per unit time. Alternatively, this quantity
may be interpreted as the radiation flux coming from a hole
in a cavity with opaque walls filled with black body radia-
tion. The black body surface radiates an isotropic flux
oo
c \ U.,, dw so that the energy flux =c | U., dw is emitted
0 0
in the elementary solid angle dQ. Let us take the projections
of the elementary radiation fluxes on the resultant flux
vector which is normal to the emitting surface and take into
consideration only the part of the flux which leaves the
emitting body. Then we derive a formula for the resultant
radiation flux:
t/2 oo
J = \ a \ U.,. dw 2n cos 8d cos 0
0 0
IU
oo
=<£ \ U, dw =oT* (2.26)
0
where 8 is the angle between the normal to the surface and
the direction of the emitted photon, and the constant o =
co
dae \ Fe oe = 567 x 10-8) W-em 2K
— Gries | ex—i - = Boca — °° ,
0
is termed the Stefan-Boltzmann constant. Equation (2.26) -
represents the Stefan-Boltzmann law.
The radiation flux as a function of the parameters used in
Eq. (2.26) can be determined by considering the dimensions
involved. We deal with the following parameters: the char-
acteristic energy of the photon 7, Planck’s constant # and
the velocity of light c. The only combination of these param-
eters which has the dimension of the energy flux erg-cm~*s~'
is T4/(h8c?), so that J ~ T4/(h3c?) as in Eq. (2.26).
The Ideal Plasma 29
3 The Ideal Plasma
The ideality of a plasma. We shall consider a plasma whose
properties are similar to those of gas. When we observe a
particle of such a plasma, the particle travels most of the
time in a straight line at a constant velocity. To make this
possible the energy of interaction between the particle and
the surrounding particles at the mean distance between them
must be considerably less than the mean kinetic energy of
the particle. This condition is termed the condition of gase-
ousness for a system of particles, and the ideal plasma is the
plasma which complies with this condition with respect to
the interaction between charged particles.
Let us formulate the condition of ideality for a plasma.
The interaction between two charged particles is described
by the Coulomb potential whose absolute value is| U (R) | =
= e*/R where e is the charge of electron or singly charged ion,
and AR is the distance between the particles. The mean dis-
tance between the charged particles is of the order of NVz1/8
where JV, is the density of electrons equal to the density of
ions. Ilence, at the mean distance between the particles the
energy of interaction between them is | U| = e?NV1/3. The
mean kinetic energy of the charged particle is of the order of
T where 7 is the temperature of the plasma expressed in
energy units. Hence, we see that the parameters of an ideal
plasma must satisfy the following condition:
Ne <1 (3.4)
In the discussion below we shall deal only with the ideal
plasma. The nonideal plasma is not found in nature and so
far it is impossible to create it under laboratory conditions.
Charged particles in a gas. We consider here the weakly
ionized gas, that is, the gas in which the density of charged
particles is considerably lower than the density of atoms or
molecules. Nevertheless, many properties of the weakly
ionized gas, in particular the electric ones, are due to the
charged particles in it. For instance, the degree of ionization
in the powerful discharge-driven molecular gas lasers is
10-7-10-°. The electric energy in these lasers is transferred
from an external source to the electrons and then converted
30 Introduction to Plasma Physics
into the laser radiation energy. The presence of the electrons
in the gas, though their concentration is low, determines
the operation of the laser driven by electric discharge.
The interaction and collisions between charged and neutral
particles determine many properties of weakly ionized gas.
Some properties of weakly ionized gas are due only to the
interaction between the charged particles. Though the
concentration of charged particles in this case is low, the
long-range Coulomb interaction between them can prove in
some cases to be more significant than the short-range interac-
tion between the charged and neutral particles. Below we
shall discuss such properties of the weakly ionized gas which
are due to the long-range interaction between the charged
particles and which are not affected by the short-range in-
teraction involving the neutral particles.
Screening of charge and field in a plasma. Let us con-
sider penetration of the electric field produced by external
charges into plasma. Since this field affects the distribution
of charged particles in plasma, redistribution of charged
particles affects in its turn the field. The result is that at a
certain distance the external field is completely screened by
the plasma.
Let us find the variation of potential of the external elec-
tric field in a plasma. Then we write down Poisson’s equa-
tion:
div E = — V’o = 4ne (NV; — N,) (3.2)
Here E = —grad 9 is the electric field strength, @ is the
potential of the field, V, is the density of the ions which are
assumed to be singly charged, and J, is the density of the
electrons. Let us determine redistribution of the charged
particles in the external field. The densities of the ions and
electrons can be found from the Boltzmann distribution (2.9):
N; = Ny exp (—eqg/T), N,. = No exp (eq/T) (3.3) °
Here N, is the mean density of the charged particles in the
quasineutral plasma (that is, the plasma with the equal
average densities of electrons and ions), and 7 is the tem-
perature of the plasma.
Substitution of Eq. (3.3) into Poisson’s equation (3.2)
yields
V7 = 8nNee sinh (e~/T) (3.4)
The Ideal Plasma 31
Assuming eg/T < 1, we transform Eq. (3.4) into
V9 = girs, (3.9)
where
T \1/2
rp= ( (3.6)
BN ge
is the so-called Debye-Hiickel radius.
Equation (3.5) has a solution which exponentially de-
creases far from the boundary. For instance, if a constant
electric field penetrates a plasma through a plane boundary,
Eq. (3.5) is transformed into d*q/dx? = q/rp where the x
axis is normal to the boundary plane. This equation yields
the following expression for the electric field strength:
E = E> exp (—2/rp) where Eg is the electric field strength
at the gas boundary, and z is the distance between a given
point in the plasma and the gas boundary.
For the field of a test charge in a plasma, Eq. (3.5) has
the following form:
1 ad Ty
2 = — —o— —_ —
V oa r dr2 (rq) = r2,
where r is the distance from the test charge. If this test charge
g is in the vacuum, the right-hand side of the equation,
which is proportional to the density of charged particles in
a plasma, vanishes and we obtain g = q/r. The equation
has the same solution for r — 0, where there are no charged
particles and no plasma effects. Hence, we derive the follow-
ing expression for the potential of the test charge:
p= exp(—=} (3.7)
Thus, the Debye-Hiickel radius is the characteristic distance
at which plasma screens the external field. The effect of the
field of a charged particle on the surrounding particles is
compensated at the same distance. Let us verify the validity
of the condition eg/T < 1 for the interaction of particles in
a plasma. This condition was used to simplify Eqs. (3.4)-
(3.5). Since the interaction between the charged plasma par-
ticles and the field of the test charge is manifested at dis-
tances of the order of the Debye-Hiickel radius, we have to
32 Introduction to Plasma Physics
introduce into the above condition the potential at this
distance. This, up to a numerical factor, yields the follow-
ing form for the condition:
2 (sie)? < 4
T3
FIG. 2.
The potential drop
in the gas gap as
a function of coordinate z.
1—the density of charged
particles is zero.
2—the gas contains
charged particles and
the Debye-Hiickel radius
is smaller than the gap
width.
which proves to be the same as the condition for an ideal
plasma (3.1).
Let us find the number of charged particles involved in
screening the field of the test charge by a plasma. This num-
ber can be estimated as the number of charged particles in
the sphere with the radius of the order of the Debye-Hiickel
radius; up to a numerical factor, it is given by
T3 \1/2 —
rbNo~ (yz) >!
that is, this number is high for the ideal plasma.
ke Consider a gas gap in the external electric field. If the
gas does not contain charged particles, the external field
is uniform. If there are charged particles, the potential
drops mainly at the edges at a distance of about the Debye-
Hiickel radius (Fig. 2). Let us evaiuate the density of
charged particles for uniform potential distribution in the
gap 10 cm wide. For the characteristic electron temperature
T of about 1 eV we find V, <3 x 104 cm. This is a very
low density of electrons. For instance, in the glow gas dis-
charge N, ~ 107-10" cm-3.
The Ideal Plasma 33
Oscillations of plasma electrons. The characteristic size
for plasma, as shown above, is the Debye-Hiickel radius.
Let us determine the characteristic time of plasma response
to external fields. To do this, find what happens if all the
electrons of plasma are removed a certain distance z, to the
right starting from the plane z = 0 (Fig. 3). This will give
FIG. 3.
Distribution
of electrons for plasma
oscillations.
rise to an electric field whose strength is given by Poisson’s
equation (3.2):
dE
sy = ane (N, — Ne)
If we assume that the electric field strength is zero forz < 0,
for x> 2, Poisson’s equation yields EK = — 4neN ox
where N, is the mean density of charged particles in the
plasma. The movements of all the electrons in this field
produce a change in the position of the gas boundary. The
equation of motion for each of the electrons can be written as
a*(z-+ 2, 7
met 20) as ) ef
where m is the electron mass, and zx is the distance of the
electron to the boundary z,; the distance is not related to the
phenomena being considered and does not depend on time.
Thus, the equation of motion for the electron may be writ-
ten as
Tr =__ (W2Zp (3.8)
where
Wp == (Anker eo (3.9)
is the Langmuir, or plasma, frequency.
3—01607
34 Introduction to Plasma Physics
The solution of Eq. (3.8) shows that in this case the motion
of the electrons has an oscillatory character with a frequency
W ). Hence, 1/@, is the characteristic time of plasma re-
sponse to an external effect (for instance, if the external field
is applied instantly, the distribution of the field in the
plasma shown in Fig. 3 is established in time of about 1/a,).
Note that rp@) = (27/m)'/* is the thermal velocity of
electrons. Hence, the characteristic time of plasma response
to external effects is the time during which the electrons
cover the distance of about the Debye-Hiickel radius.
The skin effect. Let us consider penetration of slow-
varying fields into the plasma. The characteristic frequency
w of variation of these fields is small compared to the plasma
frequency; hence, we can apply Ohm’s law for the plasma,
j = oE where j is the current density in plasma, E is the
electric field strength, and o is the plasma conductivity
corresponding to the constant electric field. To describe the
variation of the fields, we must add to Ohm’s law the Max-
well equations
4n . 1 dE
crlE-—+ 28 giv H=0
c dat
where H is the magnetic field strength.
Let us assume that the characteristic frequency o of
variation of the external fields is small compared to the
plasma conductivity o so that, taking into account Ohm’s
law, the first Maxwell equation can be transformed into curl
H = 4noE/c. Substituting the resulting expression for the
electric field strength into the second Maxwell equation
(here we make use of the relation curl curl a = grad diva —
— V’a) and taking into account the third Maxwell equation,
we derive an equation for the magnetic field:
oH c2
Tt = 7 V’H (3.11)
A similar equation can be derived for the electric field.
Using Eq. (3.11) and dimensional analysis, we find that
the characteristic size corresponding to the distribution of
Elementary Processes in Plasma 35
fields is
1 ~(75)"" (3.42)
4nwo
If this size is small compared to the size of the plasma, the
external fields and the currents in plasma are concentrated
only at the surface of the plasma and penetrate into it to a
depth of about 2. This phenomenon is referred to as the skin
effect, and the layer at the plasma surface where the external
fields penetrate and where the plasma currents flow is termed
the skin layer. Equation (3.12) shows that the smaller the
skin-layer thickness is, the higher is the plasma conductivity
and the frequency of variation of the fields.
Let us make some numerical estimates. The conductivity
of the plasma of the upper atmosphere at the height of about
100 km is about 10° s-! and the plasma frequency is about
3 x 10’ s-!. For the frequencies of the order of the plasma
frequency the electromagnetic waves penetrate into. plasma
for fractions of a meter, that is, the depth of penetration
is considerably less than the depth of the atmosphere. It
can be seen that the electromagnetic signals whose frequen-
cies are lower than the plasma frequency cannot pass through
the upper atmosphere.
4 Elementary Processes in Plasma
Act of collision of particles in a plasma. Let us analyze
the collision of two particles in plasma. We have to find a
parameter for describing this act of collision. Denote the
first particle by A and the second particle by B. Let us
consider the collision which alters the inner state of the
particle A; describe the state of the particle A by the sub-
script i before the collision and by the subscript f after the
collision. Assume that each collision of the test particle A
with particle B can result only in transition of particle A
into state f. The probability W(t) that particle A has not
altered its state by time ¢ is given by the following equation:
dw
Te = —vVisW (4.1)
where v;; is the frequency of transition of particle A from
state i to state f following the collision with particle B.
3*
36 Introduction to Plasma Physics
Let us introduce a frame of reference where tle test par-
ticle A is at rest. The higher is the frequency of variation
of the particle’s state, the larger the incident flux j of parti-
cles B. The quantity v;;/j does not depend on the density of
the B particles. Hence, this ratio characterizes the act of
collision between two particles. It is termed the cross section
of inelastic collision of particles. This cross section can be
defined as the ratio between the probability of transition
from one state of a particle to another state per unit time
and the incident flux of particles.
If all the B particles travel at the same velocity, the flux
of the B particles in the frame of reference linked to particle
A is|v, — vg | [B]. Herev, and vz, are the velocities of the
particles A and B, and [B] is the density of the B particles.
Hence, we can write the following relationship between the
frequency v;; of transitions between the states and the cross
section 0;; of transition:
Vis =[B] | Va — Va| Sis (4.2)
where the cross section 0;; of transition depends only on the
relative velocity of collision of the particles.
When particles A and B have different velocities described
by a certain distribution, the frequency of transitions of
particle A following collisions with particles B is
Vit =1[B] (| Va— Va | O19) = [B] (his) (4.3)
The angle brackets here denote averaging over the relative
velocities of the particles, and k;; =|v, —vg|ojiy; is
the rate constant of the process. This rate constant also
characterizes. the act of collision between the particles. The
rate constant of the transition process is useful when meas-
uring or analyzing the frequency of transitions between the
states in a gas or a plasma, that is, when we are interested
in the frequency of transitions averaged over the velocities
of the particles.
Using the rate constants for the process of transition, let
us derive the balance equation for the A particles which are
in a given state i. The balance equation describes variation
of the particle due to appearance of new particles in the
given state and transitions of the particles from this state
into other states. The form of the balance equation depends
Elementary Processes in Plasma 37
on the processes resulting in the transitions involving the
given state. Denote the density of A particles in the state f
by N; and assume that all the transitions between the states
of particle A are due to collisions with B particles. Using the
definitions of the rate constant for the transition process, we
can write the balance equation for particles A in the follow-
ing form:
OE = [B\ SS hyNj—(BIN: Sy (4A)
f f
Here NV; is the density of A particles in state i and k;; is
the rate constant for the process of transition of particle A
from state i to state f. The balance equation (4.4) can be
readily extended to cover other processes.
Elastic collision of particles. Let us consider elastic colli-
sion of two particles, that is, the collision which does not
alter the internal states of the particle and changes only
their directions of motion. The motion of these particles is
described by the following equations (Newton’s equations):
ee 0U ee oU
MR, =-s> MR: = — oR,
oR, °
Here R, and R, are the radius vectors of the respective par-
ticles, M, and M, are their masses, U is the potential of the
interaction between the particles, which depends on the
distance between them: U = U (R, — R,), 0U/0R; is the
force with which the other particle acts on a given particle,
and 0U/dR, = —dU/OR,.
Let us introduce new radius vectors: the vector of the
centre of mass of,the particles, Re=(,R,+ M,R,)/(M,+M,),
and the separation between them, R = R, — R,. The New-
ton equations in these variables have the following form:
(M,+M,)BR.=0, pR= ———
Here »=M,M,/(M,+M,) is the reduced mass of the
particles. It is seen that the centre of mass travels at a con-
stant velocity, so that to find the cross section of elastic
collision we have to analyze the motion of one particle with
i. mass tt in acentral force field. Though the above discussion
was in terms of classical mechanics, in quantum mechanics
38 Introduction to Plasma Physics
the situation is the same. Indeed, in quantum mechanics
there is also the free motion of the centre of mass in the
absence of external forces, and collision depends only on the
distance between the particles R.
Figure 4 shows the path of a particle with mass pw in the
centre-of-mass frame of reference when the central potential
of interaction between the colliding particles depends only
FIG. 4.
The trajectory of
a particle in a central
field in the
centre-of-mass frame
of reference: 0 is
the impact parameter,
R is the radius vector
of the particle
with the reduced mass,
6 is the scattering angle
in the centre-of-mass
frame of reference,
and 1ro is the distance
of closest approach,
that is, the minimum
distance between
the particles for
a given impact parameter.
on the distance | R, — R, | between the particles. The pa-
rameters describing the collision also are shown in Fig. 4.
Let us find the relationship between the impact parameter po
and the distance of closest approach ry. To do this, make
use of the conservation of the momentum, which is pvo for
large distances between the particles and wv,r, at the distance
of closest approach; here v = |v, —v,.| is the relative
velocity of the particles, and v, is the tangential component
of the velocity at the distance of closest approach where the
normal component of the velocity is zero. Conservation of
energy gives pvi/2 = wv?/2 — U (ry) and we find the follow-
ing relationship between the impact parameter and the
distance of closest approach of the particles:
02 U (r9)
a no (4.5)
Elementary Processes in Plasma $0
where ¢ = pv*/2 is the energy of the particles in the centre-
of-mass frame of reference.
To determine the cross section of scattering of the parti-
cles, we use the centre-of-mass frame of reference where
scattering can be considered as motion of one particle in a
central force field. Let us define the differential cross section
of elastic scattering as the ratio of the number of collisions
per unit time that scatter the particles into the elementary
solid angle dQ at a definite angle to the flux of incident par-
ticles. Consider a beam of particles of density N and velocity
v falling on the scattering centre so that the flux of particles
is Nv. In a central force field the particles scattered into the
elementary solid angle dQ = 2nd (cos 9) at the angle @
have an impact parameter from p to 9+ dp since the scat-
tering angle 6 depends on the impact parameter. The number
of particles scattered per unit time into the given elemen-
tary solid angle is 2xp dp Nv, so that by definition the
differential cross section in the central force field is
do = 2np dp (4.6)
Elastic scattering of particles gives rise to many macro-
scopic parameters of gas and plasma: these™ parameters
depend on the cross sections of elastic scattering of the gas
and plasma particles averaged over the scattering angles.
Of importance here is the average cross section, the main
contribution to which is given by scattering at large angles.
An estimate of this cross section can'be made from Eq. (4.6).
Let us estimate the cross section of scattering at large
angles. For large-angle scattering the interaction potential
at the distance of closest approach is comparable to the
kinetic energy of the particles; hence, the cross section for
large-angle scattering is given by
o= mp2 (4.7)
while
U (Po) |
&
The most often used averaged cross section of elastic
scattering is the so-called diffusion, or transport, cross sec-
40 Introduction to Plasma Physies
tion, which is defined as
o* = \ (1 — cos 0) do (4.8)
where 6 is the scattering angle. The small scattering angles
do not contribute to the diffusion cross section since they
appear in the integrand with a weight factor 62/2. All the
macroscopic parameters related to the elastic scattering of
electrons by atoms can be expressed through the diffusion
cross section. Some transport parameters due to collisions
of atoms and molecules can be expressed through another
averaged cross section, o® = \ (4 — cos? 6) do. It may be
seen that this averaged cross section also depends on the
large-angle scattering.
Another widely used parameter is the so-called gas-kinetic
cross section og, which is defined as the averaged cross sec-
tion of elastic scattering of gas atoms or molecules at large
angles for thermal energies. This rough parameter amounts
to about 10-!© cm? (see Appendix 1).
Since v ~ Nvo is the frequency of collisions between the
test particle and the gas particles, the quantity t ~ 1/v is
the characteristic time between two successive collisions
for the given test particle, and the quantity ’ = vt ~
~ (No)-! is the distance covered by the test particle be-
tween two successive collisions referred to as the mean free
path. As follows from the definition of the mean free path,
its value can be only estimated similar to the characteristic
cross section for particle collision.
The total cross section of scattering and the cross section
of capture. Let us consider the total cross section of the
elastic scattering of particles, o, = ( do. In classical terms
the total cross section must be infinite. Indeed, the classical
particles are scattered at any distance from each other and
at any energy of interaction, and the total cross section must
take into account all these scatterings. Hence, the total
cross section of scattering is essentially a quantum quantity
and depends on A.
Let us estimate’ the total cross section of particles gov-
erned by laws of classical mechanics. The variation of the
particle’s momentum following a collision with another
Elementary Processes in Plasma 41
particle is given by
+00
Ap = ) F dt
where F = —0U/OR is the force with which one particle
acts upon the other particle, and U is the potential of the
interaction between the particles. Hence, we find that Ap ~
~ U (p)/v where p is the impact parameter. According to the
Heisenberg uncertainty principle, the value of Ap can be
determined up to an accuracy of h/p. Hence, the main
contribution to the total cross section of scattering is given
by the impact parameters that satisfy the relation Ap (p) ~
~ h/p, and the total cross section estimate is given by
0, ~ 07 (4.9)
while 0,U-(p;)/(Av) ~ 1.
In particular, if U (R) = CR", the total cross section
is given by
0, ~ (Cihivy!"- (4.40)
Since the value of the total cross section is determined by a
quantum effect, it tends to infinity for the classical limit.
This can be demonstrated with the above equations by tend-
ing Planck’s constant to zero.
Let us consider the scattering of particles in the case of
attractive interaction with the potential increasing at small
distances faster than 1/R?. Assuming that the variation of
the interaction potential is monotonic, we can use Eq: (4.9)
to derive the function p(r,) shown in Fig. 5. We see that no
distance of closest approach exists for impact parameters p
less than Pcapture- Such collisions result in particles approach-
ing each other to! infinitesimal] distances, or one particle
is said to capture the other one. For the inverse-power poten-
tial U (R) = —CR- the capture cross section is
nn [ C(n—2) J2/n
Ocapture = "Peapture = Rod ce) (4.11)
Here the relationship between the cross section and the
parameters is the same as in Eq. (4.7). For instance, for the
42 Introduction to Plasma Physics
polarization interaction between an ion and an atom U (R) =
= —fe?/(2R*) where 6 is the polarizability of the atom and e
is the charge of the electron, the cross section for the po-
larization capture of the ion by the atom is
2\14
Ocapture = 20 ( Be . (4.12)
pv
FIG. 5.
The impact parameter 9
as a function
of the distance of
closest approach ro for
an attractive potential
which increases faster
than 1/R2 at
small distances.
The condition of gaseousness and the ideality of plasma.
The condition of gaseousness for a system of particles can be
formulated in terms of the cross section of particle collision.
We shall define a gas as a system of weakly interacting par-
ticles in which every particle most of the time behaves like
a free particle, that is, travels in a straight line at a constant
velocity and only for a relatively short time interacts strong-
ly with other particles, thus changing its velocity, direction
of motion and, possibly, the internal state. The strong in-
teraction between the particles will be characterized by the
cross section of elastic scattering at large angles. Then a
test particle interacts strongly with another gas particle if
the second particle is in the sphere of the volume of about
o°/? around the test particle. The gas volume per particle
is 1/N where N is the density of particles. This leads to the
following form of the condition of gaseousness:
Nol <4 (4.43)
Elementary Processes in Plasma 43
Under this condition the average potential of interaction
between the test particle and the surrounding particles is
much lower than the particle’s mean energy.
Let us apply the above condition to a system of charged
particles. Since the interaction between charged particles is
governed by the Coulomb law, | U (R)| = e?/R,. Eq. (4.7)
yields the following expression for the cross section of elastic
collision between charged particles:
o ~ et/T? (4.14)
where 7 is the mean energy of the charged particles, that is,
their temperature expressed in energy units. Note that
Eq. (4.13), which is a condition of the ideality of plasma,
coincides with Eq. (3.4):
N.e/T? <1 (4.15)
The types of elementary processes. Given below is a gen-
eral description of the elementary processes occurring in a
weakly ionized plasma, of the mechanisms of these proces-
ses, and of the magnitudes of their cross sections. The facts
are presented in Tables 1-3. More detailed information on
some of the processes will be presented later. Here we shall
discuss some common features of the mechanisms of certain
processes.
Some processes of collision between electrons and mole-
cules occur via the stage of formation of intermediate bound
states of the electron and the molecule. For instance, Fig. 6
shows the terms of the diatomic molecule and the negative
molecular ion. The molecular terms are the energy levels of
the molecule depending on the distance between the nuclei.
The negative molecular ion is in the auto-ionization state,
that is, it has a finite lifetime and can decay into a molecule
and a free electron. Therefore, the energy level of a negative
ion cannot be determined precisely and possesses a certain
width.
Taking into account the auto-ionization state of the nega-
tive molecular ion, let us describe the process of collision
between the electron and molecule which occurs via formation
of this state. Assume that at the moment of collision be-
tween an electron and a molecule, the distance between the
nuclei is R,. Since the characteristic time of the electronic
UWOTJOES SSOID JIUIOZB 0}
JO Jopio eq} JO JO sso] SI UOTJOes ssO1D UINUITXeU
ey} pue ‘aouepuedep ASioue soUeUOSe & SBY UOT}
-008 SSOID OY, ‘[OAOT WOTPEZTUOI-OyNe oy} 728 ABIOUe
WOATS ey} JO WOIJOOTa 94} Butanydeo Jo AyI[Iqeqoid
eq} uo spuedep Ayt{Iqeqoad oy} ‘sseooid Asieue
UCAIZ B JOY ‘Gp E[NIe[OW Jo -gp 27e4s HOT{eZIUO!
-0jNB OY} JO UOlJyeuI0J oY} BIA pesooid sassod0Ig
UOISI{[09 Jo
AB1eue oy} pue setoeds e_NoeTOUl 9Y} 0} SUIpI0IN8
f—f VUssHIp oY} SeUlWJejJep o[NI UOTJDeTeS Y
‘g[Noe[OU B PUL TO1}9E[9 MOTS B UedMJoq UOIST[[O9
8 JO WOI}005 SssoID OY} UeY} opnyuseu jo siops0
OM4 10 ouo Aq ssaT ‘ATeOIdAy ‘st WOTJOes ssoID OY (,/f) gqv+3<(r)av+a
G '038G 88S
(o8}s U01}00[9 PoyTOXe Ue sejOUp ysl10}se oq)
1013908 SSOID O1JOUTH-ses
6y} JO Jopio oY} Jo St UOT}IeS ssoID OG} ‘ATTBOIGAT,
a9 +V +49 +2
q+-V< avr?
(tequinu winjuenb
euoljeqqia ay} st 4)
UOI}VUIqUIODEI OATBLOOSSI
gjnoefoul eB 04 U0d}99[9
ue JO JUeMITORIIe VATJELOOSSIG
@[Noe[OUl B JO STeAaT [eUOl}
(agy +a<(2) gy +a -BIqlA oY} UseMjoq SUOl}IsuBlL
(e[NIe[OUL
eq} jo wnjuemom
qejnsue oy} st s) e[NIe[OU B JO s[aAdT
+aa<p+a
aV +9<pt+a
[euoTejoI Usemjoq SUOT}IsSUeI,
qoed mI UoI.eTe Aq e[No
-9[0Ul 10 W0}e Ue JO UOI}BZIUO]
e[noejoul
Io woye ue jo seje}s U01}00Te
usemjog suolyisuel} d1}se[oU]
e[noe
-[oW eB 10 W0je UB pue WOIyO0Ta
ue UseMjeq UOISI[[Oo d1ISse[q
.
VtIsyra
———————
UIsTUB yOu
sso00id 943 ‘sso00id JO 8}UB{SUOD JO 014908 SSO)
eUlayos ssIv0Id ssoo0id Ale Use [A
re a
SO[NDaIOW JO SWO}Y PUL SUOI0]q USSMJOG UOIS]]]OD jo Sassaz01g AsejuOWwe]z O4L
b 21qe)
45
G °00S 99S
UOIJIOTS We YALA
WOTSI{[Oo Aq sfaAe] Je[NeTOU, Jo WORKS ot}
JOJ se owes 94} SI doUspuedep UOT}9eS SsOLD OTL,
‘Wd9} WOTs{Nder aq} OJUO sTNoajoUl B JO UOIVeIOXY
Ite HIF +2
q+-¥<q+yvt+e
q+y+e<av+4
SUOISI[[09
ejo1jied-0014} Ul wor ue pue
UoIjIe7Toe Ue JO OTBUIqUIOIOY
suois!|
-[09 a[9Tj1ed-se1q} UI e[noejoul
BO} UOIJIOTO Ue Jo JUOMIYIBIV
U01j99[9 UB YIM UOISI[[Oo
Aq o[Noejou & Jo UOTzBLOOSSIC
46
[jews
SI JI sa}ejJs JoqJO IO} ‘aUuO OIJOUTY-ses e Si UOT}IeS
SssOId 04} ‘sozejs poejyloxe Jo Jeqminu [[eWs e& JOY
Ajttiqeqoid MO] B sey Woye Ue YIIM UOISI[[OI
Aq UWolyeyOxe-aq ‘“UWOT}OES ssOID OIJOUIY-Sed eB 0}
S]JUNOWP UOISI[[O9 JO UOTJOAS SSOID *97e}S [BUOIPVIGIA
peqloxe Ue Ul SI e[Noajoul ay} jouUeYo JsIy oY} U]
WO1}OeS SSOID DIUI0Je Jo JapIO oY} JO SI UOIS
-I[]09 JO UOT}J09S ssoID 904} ‘soIsIoUs [eUIIOY} 104
SoTNve[our ey} Jo
einjeu oy} uO spuedep A,suoI3s pue opnyluseu jo
Slopdio [e1oAes Aq WOI}OeS SSOIO O1JOULY-ses e Ue}
daT[eWs SI U0T}9ES ssoId OY} ‘sols1aue [etUIey} 104
97e1S U0I}08[9 peyloxe Ue sejJOUep ysis}
-se 904} ‘opnyluseul Jo siopi1o Mey e Aq sUOrzIeS
SsO1D O1M0}Je UeY} Jo[[eUIs ale SUOI}oeS SSOID @AT}
-dedsal oy} pue AD Moy & JO solsIEUe JOJ AyI[Iqeqg
-o1d [jews B@ aAey sassa00id osayy ‘A[[BoTVeqeIpy
epnyiuseu jo Jepio0 ue
Aq solieA pue (zU19 oy-(O}) ON[BA OTJOUTY-ses oy} sey
U01499S SSOID BY} ‘UWOISI[[OO JO Sold1eUs [eUINy} JOg
a+4gV —xI@ t+V
ITETvVN
Id +«V
gat+p/ .
(NIg+V<(ragty
(2) 99 +¥ <(2)9a +P
at+.gGt+tV<-adtV
w7@ +¥ -—@t+V
q+v 2qa+yV
WOl}eZIUOL OAIJBIOOSSY
a[noe[ow
YUM UOISI][OO UI o3e}s OI}
-00[9 pojloxe JO WO1}BIIDK9-9q
S9[NIe[OUL JO S[aAd] [eUOT}
-8101 UWaeAjeq suUOlzIsueLy,
SO[NIB[OUL JO S[eAeT [eUOT}
-BIQIA UWaanjoq suUorpisuely,
sul0}e
JO SUOISI[[OO G}IM UOTzezZIUO]
Sm10}8 JO SUOISI[[OO UI
S]9A9] 01}99T9 JO U01}8z1NXY
se[hoejou pue
sul0}e JO UOISI[]OO 14Sse[q
wistue youl
ssa0id ay} ‘ssadoid JO sjuezJSUOd JO UOI}99S SSOID
gUIsYOS ssa001g
ssad0id Alg,UusULa A
Soj|NIIjOW Pue SWOJY 10} Sasse20/g UOISIjOoD AsejuoWe]y SUL
t 819e1
47
(G '09G 99S) ;-SglD ze_OF
jo Jepio oy} Jo ‘A[peordéy ‘st ssoooid opotyaed
~g01Y} @ JO JURJSUOD oYeI ayy ‘seldI0Ue [eUIEq} 104
(su01}09[9 94} Jo UOT}e4
-Ual10 UWIN}UsMOU Je[NZue [e}iqio ojJ0Uep SsMoOIIe)
souo O1JOULy-sed 04} UeYy
JeZ1e] Su01}I0sS SsOID OAR UeD sassav0id sdmeUOSeL
gsoy} ‘ASI0Ua UOlzIsUeI} JO MOLJeIIeA [[eWS e JOY
{aioje JO WNnjUeMOU Je[nsue [e}0} 9Y} sojJOUep f)
(suo}0a[9 ay} JO UOl}eJUSIIO UIds oJOuEp SMOJIB)
QUO 91}JeUTy-Sses e JO JepIO ay} Jo st ssaooid
ay} JO WOIJOeS sSsO1D a} ‘esBIOAe 94} UO “Gg Woe
jo jetjuajod uoljeziuol 94} Uy) JeYysIG st ABIau
UOTJVJIOXS S}I ‘a}e}S a[qeysejoul eB UI SI yp WO}Y
|
g+av< It+atyV
9+4aV¥ <Ia+V
(thaty<eWatV
(Qatyv<-(Qatv
(nNatWr<
<—(Wat(tHy
a2 +V Ita
a+sGq tv <a tay
SUOISI[[09 o[01,1ed-901T4
GI so[nosjoul, Jo woryonpoig
SUOI}ORveI [BOIMEY)
w0oye ue Jo UOTWezZIIe[Odeq
g0j4e84s oInjondjs
-aug) «= WeeMJOq «= SUT} ISURLY,
§9} 81S
einjonijs-ouyaedAY JO worze!
-IBA 9} pue oesueyoxe ulds
UoT}e}LOXe JO JojsueI],
qooge Suruueg
G *985 9805
(y-SgWL9 »z-O} 03 dN) Jes1e] A[qeiepisuoo oq ued 1Ue1s
-009 9}e1 94} sSo[hoejoul xXe,dwo0d Joy pue ‘sule}sAs
@[AUIIS IO} ;-SgWD yeOf JO JOpIO ay} Jo st ssevoid
oy} JO JUBISUOD 0981 OY} ‘seIsIaUe [eUI0q} 10g
SU01}0801 [BOIWOYD 0} Ie[IWIs ere sassov0dd esou],
UOT OAI}BZ0M B UI OI
-99[9 84} jo AZ1oue SZutpulq oy} Uey} Jesse, Alqe
-IapIsuod oq UBD P[OYseIy} sIyYL ‘sUlIe, Jo IMoAeYy
-oq ey? Aq peutulJeJep ployseryy B sey sso00id 904}
‘(}X0} 04} 90S) Wstueqoour ssa00id 9y} uo Sulpusdeq
(G ‘98S 99S) UWOI}OeS ssOID DIMIO}JB OY} UeY) JadIe]
‘Aqpeoidy ‘st sseoord oy} JO UWOl}0es ssold OU],
adueyoxe os1eyo ooueu
-OSAI ay} JO 4B} SB JepIO aus 04} JO SI UWOT}00S
Sso19 04} ‘peysizes you si (97°%) MOIjIpuoo ueyM
(} xtpueddy oes) Wol}0es ssoId O1J0UTH-ses
eq} Uey} Jedie] st ssad0id 04} JO U01}008 ssOID OY],
It+dtV¥-I+-I+4V
It+sdvV<—-IOt+ateyV
+O+avn
ga
a+qvy<-qt+yV
Ate <—-I+4V
44 TV a tay
sVt+VY <—pt+ay
SUOISI[[0D o[9T}1ed-s014}
Ul UWOlBUIQMIODEL UO1-U0]
suols
-Ij[09 aporyred-ce1q} =U
SUOL JB[Noe[OW OUT suo
D1WI0}B JO NOIJVULIOJSUBI],
WOTJOVaI WOT Je[NoI]oO}{
T1078 UB YIM UOISI[[OO UI
UoI oAIyeZou B Jo ABdeq
suo!
JO UOI}EZI[BIJNOU [BN4INW!W
esueyo
-xo adIeyo eouBuoseIUON
esueyo
-xo of18q9 oouBuOsoY
UlslueyIeUl
ssa00id ay} ‘ssed0id JO s}Ue4ySUOD IO SU01}0eS ssO1D
QUIEYOS ssev01g
ssoo0id Are}USUla[y
suo] BujA[OAU] Sesse201g UO[s|IJOD AsqjudWwe]y OUL
€ e1qe1
Elementary Processes in Plasma 49
processes is considerably less than the characteristic time
of the nuclear processes, the distance between the nuclei in
the molecule will-not be changed during this process of
electronic transition (the Franck-Condon principle). There-
fore, the molecule can capture the electron if the energy of
the incident electron is close to the energy difference be-
tween the negative ion and the molecule for a given distance
Fig. 6.
The terms of a molecule
(solid line)
and a negative ion
(dashed line).
& is the energy
of the captured
electron when
the distance between
the nuclei is Ry.
between the nuclei (see Fig. 6). Since this energy difference
varies with the distance between the nuclei, the range of the
electron energies at which it can be captured by the molecule
producing a negative ion is extended to a width of a fraction
of an electron volt. The cross section of the electron capture
by the molecule exhibits a resonance with a width of a frac-
tion of an electron volt.
A variety of processes of electron scattering on atoms or
molecules depend on the formation of the bound state of the
electron attached to these particles. For elastic and inelastic
collisions between the electron and the atom this gives rise
to resonances on the cross section of scattering versus the
collision energy curve. Some processes proceed only via the
auto-ionization state of the electron and the atom or mole-
cule. For such processes the cross section versus the electron
energy curve exhibits one or a group of resonances whose
position and character depend on the properties of the auto-
ionization state. Among such processes are the excitation of
the vibrational levels of the molecule, dissociative attach-
ment of the electron to the molecule, and dissociative recom-
4—014607
50 Introduction to Plasma Physics
bination of the electron and the molecular ion. An illustra-
tion of the processes of this type is presented in Fig. 7,
which shows the cross section of the dissociative attachment
of the electron to the carbon dioxide molecule as a function
of the electron energy.
Inelastic collisions of,atomic particles. The term atomic
particle refers here to an atom, ion, or molecule. Let us
FIG. 7.
The cross section
of dissociative
attachment of
an electron
to a carbon dioxide
molecule
(e+ CO, —> O- + CO)
as a function of
the energy of incident
>
Ww
—_—,
Cross section. 107'9 cm2
NO
electron.
The maxima 0
at the electron 4 6 8 10
energies 4.4 and 8.2 eV Electron energy, eV
correspond to
the positions
of the auto-ionization
levels of the negative
ion COz.
consider inelastic collision of atomic particles which results
in a change in the internal state of one of them, for instance,
in its electronic state. If the characteristic times of the
particle approach are high compared to the characteristic
electronic times, then the electrons have time “to follow” the
motion of the nuclei and for each nuclear configuration to
establish the same distribution as for the infinitely slow
motion of the nuclei. It is clear that for infinitely slow col-
lisions the system finally will return back to the initial
state and no transitions will occur. Therefore, if the time of
approach of the nuclei is small compared to the characteris-
tic electronic times, the probability of inelastic collision
with electronic transition is small.
The characteristic time of approach (or collision) of the
nuclei is,of the order of a/v where v is the relative velocity of
the particles, and a is of the order of atomic size. The charac-
teristic electronic time for the given transition is of the order
Elementary Processes in Plasma 54
of h/Ae where Ag is the energy )difference of the levels be-
tween which the transition occurs. Hence, the suggested
criterion can be written as follows:
Aga
—>1 (4.46)
The criterion (4.16) is called the adiabatic Massey principle;
if this inequality is satisfied, the transition proves to be
adiabatically unlikely.
There is also low probability, adiabatically, of such events
as the transitions between the electronic states of atoms
following collisions with atoms, ionization in the atom-atom
or atom-molecule collisions at the energies of the order of
a few electron volts, and the transitions between the vibra-
tional molecular levels at thermal energies. The cross sec-
tions of these processes are smaller than the gas-kinetic cross
sections by a few orders of magnitude. For other processes,
the adiabatic criterion (4.16) is satisfied only at large dis-
tances between the nuclei. The interaction between the atom-
ic particles draws together the electronic, levels between
which the transition occurs. Hence, the adiabatic criterion
(4.16) is not satisfied starting from a certain distance be-
tween the particles and the process proceeds with a high
probability.
This can be illustrated by the process of decomposition of
a negative; ion following, the collision with the atom:
A-+ B>~AB+e—>A+B-+e. At a certain; distance
between the nuclei the term of the A~B system intersects
the boundary of the continuous spectrum, that is, the AB
term. Therefore, there is a higher probability for the elec-
tron to be detached from the negative ion.
Another process of this type is the associative ionization
when an atom and an excited atom collide: 4 + B* >
—+ AB* + e. Figure 8 presents the terms for the initial
and final channels of the reaction. Similar to the above
processes, the mechanism of this reaction is determined by
the initial-state term moving into the continuous-spectrum
region. The reaction results in decomposition of the auto-
ionization state AB* giving rise to a free electron.
Some processes occur with asmall variation in the electron-
ic energy; they are termed the resonance processes. These
4*
52 Introduction to Plasma Physies
processes satisfy a condition opposite to criterion (4.16)
even for large distances between the nuclei. Hence, the’
resonance processes are effective when the colliding particles
pass at large distances from each other, the distances con-
siderably greater than the sizes of the particles. Hence, the
resonance processes can have cross sections which are con-
siderably larger than the areas of the particles cross sections.
FIG. 8.
Atomic terms
in associative
ionization and
dissociative
recombination:
A+ B¥ =z ABt-+e.
ABt
The resonance processes include the resonance charge ex-
change, the processes resulting in depolarization of the atom
or in variation of its fine or hyperfine electronic structure,
mutual neutralization from collision between ions, transfer
of excitation, etc.
Charge exchange and similar processes. Let us consider
the processes in which the electron is transferred from the
field of one atomic core to the field of another atomic core.
Among these processes are the resonance charge exchange,
mutual neutralization of negative and positive ions, spin
exchange, and some types of excitation transfer accompanied
by exchange of one or two valence electrons. For instance,
Fig. 9 represents the potential of the interaction between
the electron and the atomic cores in the case of the resonance
charge exchange for the cores of the same type.
Let us estimate the cross section of the resonance charge
exchange as a function of collision velocity. If the distance
between the nuclei of the atomic cores remains constant
(Fig. 9), the transfer of the electron from one atomic core
Elementary Processes in Plasma 03
to another will occur with a certain frequency w. The main
contribution to the cross section of the charge exchange is
given by those collisions for which the characteristic time
of particles collision a/v is comparable to the time 1/ of
the electron transfer from one atomic core to another. Here v
is the relative velocity of the nuclei and a is a characteristic
size which can depend on the distance between the nuclei.
Fig. 9.
Cross section of the
equipotential surface
for an electron in the
field of two identical
atomic cores.
1 and 2 are
the positions of the
nuclei of the cores,
R is the distance
between them, and
I is the ionization
potential, that is,
the binding energy
of the electron
to one of the cores.
The frequency of transfer w(R) is proportional to the proba-
bility of the potential barrier penetration by the electron;
quantum mechanics gives w(R) ~ exp (—yf) where y =
= (2mI/h?)1/2, J is the binding energy of the electron in the
atomic core (the ionization potential of the atom), and m
is the electron mass.
Hence, the main contribution to the cross section of the
charge exchange is given by the electron transitions at cer-
tain distances R,, which are determined by the collision
velocity: exp (—yRy) ~ v. This yields the cross section of
the resonance charge exchange (Gres, ~ RG) as a function of
the collision velocity:
Ores. = Oo In? = (4.17)
where dy and v, are certain parameters of the system. A simi-
lar velocity dependence is found for other resonance proces-
ses which do not involve elastic scattering and are accompa-
nied by the exchange of one or two electrons. The cross
section of the resonance charge exchange is seen to be only
D4 Introduction to Plasma Physics
slightly dependent on the collision velocity. The same con-
clusion follows from Appendix 2 where the cross sections of
the resonance charge exchange are given for specific atomic
pairs. These data show that the cross section of the resonance
charge exchange is larger than the gas-kinetic cross section.
5 Formation and Decomposition of Charged
Particles in Weakly lonized Gas
Since the electric characteristics of the weakly ionized gas
are due to the charged particles, the processes of their for-
mation and decomposition are of special importance.
The charged particles in the weakly ionized gas are pro-
duced in the collisions between the electrons and the atoms
or molecules. Ionization by electron impact may occur in
two ways according to the state of the gas. The ionization
of the atom or molecule following one collision with the elec-
tron is termed the single ionization. When the collisions
with the electrons make the atom or molecule pass a number
of excited states and ionization occurs from an excited state,
this ionization is called the multistage ionization. Ionization
of this type occurs when the electron density is sufficiently
high and the excited atom or molecule does not have time
to decay between two successive collisions with the electrons.
The charged gas particles decompose in various ways. For
instance, neutralization of the charged particles occurs at
the walls of the vessel filled with the weakly ionized gas.
Mutual neutralization of the charged particles (the collisions
between electrons and positive ions or positive and negative
ions) is termed the recombination of charged particles. This
process is characterized by the recombination coefficient, a,
which appears in the balance equation for the density of
charged particles:
—- = —aN,N;, (5.4)
where J, is the electron density and N; is the ion density.
Ionization of an atom in a single collision with an electron.
Let us consider the process of detachment of the atomic elec-
Charged Particles 58
tron in collision between the atom and another electron:
e+ A-—>2e + At (5.2)
The incident electron interacts with the valence electron of
the atom; between them there occurs an exchange of energy
exceeding the ionization potential of the atom, that is, the
binding energy for the valence electron in the atom. .Let
us analyze this process in terms of the simplest model devel-
oped first by J.J. Thomson. Assume that the valence elec-
tron is at rest, the motion of electrons is governed by the
classical laws, and the electrons do not interact with the
atomic core during the collision. Far from providing a quan-
titative description of the process, this model gives a correct
qualitative explanation of it.
In the framework of this model, ionization occurs each
time the valence electron receives energy exceeding the
ionization potential J of the atom. Hence, we have to find
the cross section for such collision in which the electrons
exchange a given energy As. Consider the process in which
the energy transferred, Ae, is much lower than the energy of
the incident electron, € = mv?/2. Then the incident elec-
tron travels in a straight line, the valence electron may be
assumed at rest during the collision, and the variation of the
electron momentum is perpendicular to the path of the
incident electron in the plane of. motion.
Newton's equation for variation of the momentum of the
incident electron is F = dAp/dt where F — e?R/R’ is the force
with which the valence electron acts on the incident elec-
tron, and R is the distance between the electrons. Hence, the
component of the momentum variation in the direction per-
pendicular to the path satisfies the equation dAp,/dt =
= e*o/R® (p is the impact parameter of the collision; see
Fig. 10) and this variation is
+0co
7e20 2e2
4p4= j Rr = pv
Ilere we made use of the following relation for the free motion
of the electron: R? = p? + v*t?. Now we can find the energy
lost by the incident electron and received by the valence
56 Introduction to Plasma Physics
electron:
where @ is the energy of the incident electron. Hence, we
find the following expression for the cross section of colli-
sions accompanied by the exchange of the energy Ae:
do —2no dp = meee (5.3)
Owing to the method of its derivation, Eq. (5.3) is valid
for Ae < €.* Now we shall extrapolate it to cover the
range Ae ~ &. Ionization in the Thomson model occurs when
the energy transferred, Ae, exceeds the ionization potential
of the atom, J; hence, we obtain the following expression for
the cross section of atom ionization:
Sin= § do = 5S (+--+) (5.4)
Ae>I
This expression for the ionization cross section is called the
Thomson formula. The formula was derived for the atom
with one valence electron. For an atom with several valence
electrons the cross section (5.4) must be multiplied by their
number.
Note that we can make directly some conclusions about
the ionization cross section using classical concepts. Indeed,
we can make use of the following classical parameters: the
electron mass m, the interaction parameter e?, the energy of
the incident electron €, and the ionization potential of the
atom J. The most general form of the cross section expressed
in terms of these parameters is
Sn= a t(+) 5.5)
where f (€/Z) is a universal function (that is, identicai for
all atoms). In particular, for the Thomson model, , where ad-
ditional assumptions are made, this — is ’
ie
f (z)= —
* Actually, Eq. (5.3) is valid over! the whole range Ae°<
Charged Particles 97
The Thomson formula gives a correct qualitative expres-
sion for the ionization cross sections. In the vicinity of the
threshold, the ionization cross section is proportional to the
energy; it exhibits a maximum when the energy of the inci-
dent electron is comparable to the ionization potential, and
this maximum is of the same order of magnitude as the cross-
sectional area of the atom. For high energies of the incident
FIG. 40. The path of the incident electron
Collision between
an electron and a resting
electron for a large
impact parameter.
The resting electron
electron, the ionization cross section decreases with the
reciprocal of the energy.
Recombination of pairs of positive and negative ions.
This process.can be represented by the following scheme:
A*4B->A4B (5.6)
The recombination consists in the valence electron going
over from the field of the atom B to the field of the ion A®*.
This tunnelling of the electron is effective when the distance
R, between the ions is comparable to or somewhat larger
than the size of the negative ion. Let us simplify the problem
and assume that the electron is transferred when the distance
of closest approach, ry, is less or equal to Ry and is not
transferred for larger impact parameters. Using Eq. (4.5),
we obtain the following expression for the cross section of
the ion pair recombination:
Oreo, = 12 = WRo [1 + e4/(eRo)] (5.7)
Here p, is the impact parameter for the collision in which
the distance of closest approach is Ro. For low collision
58 Introduction fo Plasma Physics
energies e < e?/Ry, Eq. (5.7) yields
Sree, = Ro (5.8)
Hence, we find the constant of the ion pair recombination
& = VOzec. = NV 2e?R,/(eu)'/?. Averaging this quantity over
the Maxwell distribution (2.15a) for the relative velocity of
the ions yields
& = (VOze9.) = 2V 20 Roe2/V wT (5.9)
where the angle brackets denote averaging over the ion
velocities.
Triple processes. The processes represented by the scheme
A+B+tC+AB+4C (5.10)
are referred to as the three-particle or triple processes.
Among them area number of processes described in Tables 1-3.
The triple process results in the binding of two colliding
particles while the third particle carries away the energy
released in the process. _
The balance equation for the densities of the particles
produced in the process (5.10) has the following form:
<1" =O {A[BIIC] (5.41)
where [X] is the density of the respective particles, and & is
the constant of the triple process, which does not depend on
the particle densities and has the dimensions cm®s~!. The
triple process can involve a large number of pair collisions.
Typically, the process (5.10) first gives rise to an excited
state of the bound system AB, which later goes over to its
ground state following multiple pair collisions.
Thomson’s theory for the constant of the triple process.
Let us follow Thomson’s argument in evaluating the constant
of the process (5.10) under the following conditions*: the
energy required to break the A-B bond is considerably higher
than the thermal energy of the particles, the motion’ of the
particles is governed by classical laws, and formation of
* Thomson developed his theory for the process A~ + B+ + C >
+A+tB4.+4C, but it can be readily extended to the general case.
Charged Particles 59
the bound state of particles is due to their interaction. These
conditions make it possible to estimate & for some triple
processes and to determine the relationship between & and
the parameters of the problem.
Formation of the bound state of the particles A and B
occurs in the following way. When particles A and B ap-
proach each other, their kinetic energy increases since the
potential energy of attraction between these particles converts
into kinetic energy. Assume that when these particles are
close by, a third particle C hits one of the two particles and
takes from it some of its energy. If the energy transferred to
the particle C is greater than the initial kinetic energy of
particles A and B, the bound state of these particles is
produced.
Our calculations will be based on the above considera-
tions. Let us assume that the characteristic energies of the
particles are of the order 7 and the mass of the particleC
is comparable to the mass of one of the first two particles
so that the characteristic energy exchange is of the order of
the thermal energy of the particles, 7. Since this energy
exchange must exceed the initial kinetic energy of the parti-
cles A and B, the potential of the interaction between these
particles during collision with the particle C must also be
of the order of 7. Thus, let us define the so-called critical
radius b by the following relation;
U(b)~T (5.42)
where U is the potential of the interaction between the
particles A and B.
Now let us estimate #@, the constant of the triple process
(5.10). The frequency of conversion of particle B into particle
AB is equal, by order of magnitude, to the product of the
probability for the particle B to be in the critical region
[B] b® and the frequency [C] vo of collisions with particle C.
Here v is the characteristic relative velocity of collision
and o is the cross section of the collisions between particlec
and particles A or B, which result in the energy exchange of
the order 7. If the masses of the colliding particles are com-
parable, this cross section equals the cross section of the
elastic collision of the particles,
60 Introduction to Plasma Physics
This estimate yields the following expression for the rate
of production of particles AB:
S1APL ~ [A] [B] 88 [C] vo
A comparison of this expression with the definition of the
constant of the triple process (5.11) yields the following
estimate for &%:
KH ~ bvo (5.13)
The process (5.10) is a triple process if the density [C] of
particles C is small and hence there is a low probability
that the colliding particles A and B will interact with par-
ticle C in the critical region. This condition is satisfied if
the critical radius b is small compared to the mean free path
1 ~ ([C] o)-! of particle A or B:
[Clob <1. (5.44)
If this condition is not satisfied, the process (5.10) is not a
triple process and Eq. (5.13) is inapplicable.
Triple recombination of electrons and ions. The process
of triple recombination of electrons and positive ions is im-
portant forghigh-density plasma; it can be represented by the
following scheme:
2e + A*—>A +e (5.15)
In the framework of the Thomson theory, the triple collision
produces initially an excited atom whose ionization poten-
tial is of the order of the thermal energy 7, and later this
atom goes over to the ground state following multiple colli-
sions with electrons. The Thomson theory can be applied to
this process since the classical laws are valid for the highly
excited state of the atom (7 < J, where J is the ionization
potential of the atom).
Let us evaluate the constant of the process (5.15) proceeding
from Eq. (5.13). According to Eq. (4.14), the cross section
of the elastic collision of electrons is o ~ e*/T?, and accord-
ing to Eq. (5.12), the critical radius for the Coulomb inter-
action between an electron and an ion is b ~ e?/T; hence,
the constant of the process (5.15) is
a @10
=N, ~ ape (40)
Charged Particles 61
where @ is the recombination coefficient defined by Eq. (5.4).
Equation (5.16) is valid if the condition (5.14) is satisfied;
in this case this condition coincides with the condition of
plasma ideality (3.1) and has the following form:
Ne/T® <1
Equation (5.16) presents a correct relationship between
the recombination coefficient and the parameters of the
problem. This relationship can be derived in a simpler way
using dimensional analysis. Indeed, for the suggested mecha-
nism of the process (5.15), its constant depends only on the
parameter of the interaction e?, the electron mass m, and
the thermal energy 7 (the temperature of the electrons).
There is only one combination of these three parameters which
has the dimensions cm®s-! of the constant of the triple proc-
ess; it can be readily seen that this combination coincides
with Eq. (5.16).
A similar procedure can be used to determine the constant
of the process of formation of the molecular ions from the
atomic ions by the following scheme:
At+2A->A$+A (5.47)
This process is due to the polarization interaction between
the ion and the atom; the potential of this interaction is
U (R) = —fe?/2R* where R is the distance between the ion
and the atom, and f is the polarizability of the atom. Only
the combination of the interaction parameter Be?, the mass
of the nucleus M, and the thermal energy of the particles 7
has the dimension of &; this combination is
(Be2)5/4
hh ~ mi/2p3/4
(5.418)
The same expression for the constant for the process (5.17)
can be derived directly from the Thomson formula (5.13).
Triple recombination of the positive and negative ions.
The process of triple recombination of ions in triple colli-
sions may be described by the following scheme:
A*4+B-4+C>+AGHB4AC (9.19)
The process gives rise initially to a bound state of the positive
and negative ion A*-B-; after a while a valence electron is
62 Introduction to Plasma Physics
transferred from the field of the atom B to the field of the
ion A* and the bound state decays into two atoms A and B.
The second stage of this process occurs spontaneously
following the capture of the negative ion by the positive ion
so that the triple recombination is determined by the for-
mation of the bound state of these ions. Hence, let us estimate
the constant for the process (5.19) assuming the Coulomb
interaction between the ions and the polarization interaction
between the ion and the atom. If the atom mass M is com-
parable to the mass of one of the ions, Eq. (4.12) for the cross
section of the atom-ion collision and Thomson’s formula
(5.13) yield
a e6 / Be2 \ 1/2
«= 5, ~ as (ar) (9.20)
Equation (5.20) is valid if the condition (5.14) is satisfied,
which in our case means that
2 1/2
(q\ — <1 ~~ (5.24)
Inserting into Eq. (5.21) the polarizability of the atom, which
is about several atomic units, we find that at room tempera-
ture [C] is much less than 107° cm-3, that is, the process of
triple recombination of positive and negative ions is dis-
rupted in this case at the gas pressure of about one at-
mosphere.
Processes involving formation of a long-lived complex.
If collision of particles produces unstable bound states with
large lifetimes, the constant of the triple process can prove
to be larger than the value found from Thomson’s theory.
Such continuous spectrum bound states are termed auto-
ionization states. An auto-ionization state can appear when
the excess kinetic energy of the particle is transferred to the
internal degrees of freedom. For instance, in the interaction
between the ion and the electron the bound state of the
particles can be produced by exciting the inner or valence
electrons of the ion and forming a bound state of the incident
electron and the ion. If the energy of the incident electron
is less than the excitation energy for the inner electrons of
the ion, such a system can decompose into an ion and a free
Charged Particles 63
electron owing to transfer of the excitation energy to an
inner electron of the ion.
In a similar way, we can produce the auto-ionization state
of a molecule with an ion or atom by transferring the excess
energy to an excited vibration-rotational state of the mole-
cule. There are other types of the auto-ionization states, and
all such states are the bound states of particles whose dis-
crete energy levels lie in the continuous spectrum. The life-
times of the auto-ionization states are great compared to the
characteristic times for such systems, that is, the times in
which the particles pass the interaction region. This is due
to the fact that decay of the auto-ionization state involves
transition between various degrees of freedom. It is the large
lifetimes of the auto-ionization states that determines their
significance for production of the bound states of colliding
particles.
Formation of the bound states of particles via the auto-
ionization states can be described by the following scheme:
k
A+ BSP AB**, — (5.22a)
AB**—> A+B, (5.22b)
Rk
AB** + CWP AB+C, — (5.22c)
k
ABY*4+ COMP. AL BIC (5.224)
Here AB** is the auto-ionization state of the particles; the
quantities written over the arrows are the constants for the
respective processes and the lifetime of the auto-ionization
state AB**, Hence, the balance equation for the density
of the auto-ionization states has the following form:
d [AB** eae
dt J0= [A] [B} keapture———
aad [AB**] [C] (Aguench. F Kaecomp.)
The equation yields
xe) [A] [B] keapture
[AB = t1+ [C] (decomp. + Aquench.)
64 Introduction to Plasma Physics
The equation for the process (5.22c) is
d[AB
che) = Kquench, [AB**] (C] = [A] [B] [Cl]
Keapture*quench. Tt
1+1[C] (Aquench. -+ kdecomp.) t
A comparison of this balance equation and the definition ot
the constant of the triple process (5.13) shows that this
process is a triple process if the condition
[C] (Aquench. + Kaecomp,) t € 4 (5.23a)
is satisfied, and the constant of the triple process in this
case is given by
x
#H = keapturequench, T (5.23b)
For another limiting case
[C] (Aquench. + Mdecomp.) t > 1 (5.24a)
the density of the particles C is so high that the rate of
production of the bound state AB does not depend on [C]
and we have
yore kquench.
cia a Feapture kquench. + “decomp. ere)
The particles C affect the factor Aguench./(Aquench. + Kaecomp.)
in Eq. (5.24b); this factor is the probability that the colli-
sion between particle C and particle AB in the auto-ioniza-
tion state will force AB to transfer to the stable bound state.
The lifetime of the auto-ionization state increases sharply
with an increase in the number of the internal degrees of
freedom since the excess energy can be distributed over a
larger number of the degrees of freedom. The auto-ionization
state produced in collision of systems with large numbers of
degrees of freedom is called a long-lived complex. It is the
formation of a long-lived complex that makes it possible to
satisfy the condition (5.24a). For instance, the lifetime of the
long-lived complex made up of large molecules and elec-
trons is of the order of 10-°-10-* s, that is, the condition
(5.24a) is satisfied for [C] > 10'°-10!" cm-*. Typically, the
lifetime of the long-lived complex is considerably smaller
Charged Particles 65
and the process has a triple character. Note that we can
obtain Thomson's formula (5.13) by replacing t in Eq. (5.23b)
with the transit time of the interacting particles.
Dissociative recombination of an electron and a molecular
ion. This process can be described by the following scheme:
e+ ABt+>A* iB (5.25)
FIG. 11.
The terms
of a molecular ion
and an excited
molecule involved
in dissociative
recombination.
The high importance of the process is due to the fact that
the coefficient of dissociative recombination does not depend
on the densities of the particles and the rates of recombination
are relatively high.
Figure 11 illustrates the mechanism of dissociative recom-
bination. It shows the term of the ground state of the mole-
cular ion and also one of the terms of the molecule consisting
of atom B and atom A* in an excited state. These terms
intersect for a certain distance Ry between the nuclei so
that for smaller distances the state A*B is an auto-ionization
state. It is this auto-ionization state that makes possible
dissociative recombination.
In the course of dissociative recombination, the electron
colliding with the molecular ion is captured to the auto-ioni-
zation level. Since the interaction between the atoms in the
auto-ionization state is repulsive, they move apart. If the
5—01607
66 Introduction to Plasma Physics
auto-ionization state does not have time to decay while the
atoms move apart to the distance Roy, the result is a stable
state of the particles, that is, there occurs recombination of
the electron and the molecular ion giving rise to the excited
atom and the atom in the ground state.
Dissociative recombination is a fairly complicated process
since the number of the auto-ionization states of the system
A*-B is large (sometimes, infinite) and the lifetime of these
states (with respect to decomposition into an electron and a
molecular ion) depends on the distance between the nuclei.
Moreover, the recombinating molecular ion can be in excited
vibrational states and this fact also influences the magnitude
of the recombination coefficient. Owing to the above diffi-
culties the general expressions for the recombination coeffi-
cient can hardly be derived, while the expressions for special
cases which are highly involved carry insufficient informa-
tion and we shall not discuss them here. Note that when the
conditions for dissociative recombination are satisfied, the
recombination coefficient has the same or higher order of
magnitude as the respective quantity for atoms.
Let us consider a simple model of dissociative recombina-
tion of an electron and a complex molecular ion. The latter
has a large number of internal degrees of freedom which can
absorb the excess energy. Therefore, any capture of an elec-
tron on the auto-ionization level can be assumed to result
automatically in recombination, and the electron is captured
if it gets into the region of strong interaction with the mole-
cular ion, that is, into the region where this ion is located.
This model can be used for estimating the coefficient of
dissociative recombination of an electron and a complex
molecular ion. Let us assume that the behaviour of the
electron is governed by classical laws and that if it gets into
the region of the size Ry, the recombination will occur. This
model was used above in the analysis of the ion pair recom-
bination and it yielded Eq. (5.9), which in this case gives
a ~ 5Rye2/V mT (5.26)
where Rj is of the order of the size of the molecular ion, and
m is the electron mass.
Equation (5.26) can be used for estimating the coefficient
of dissociative recombination. At room temperature the coef-
Charged Particles 67
ficient of dissociative recombination of an electron and a
complex molecular ion is of the order of 10-® cm®’s"!, which
agrees with experimental results.
Ionization processes in collisions between an atom in an
excited state and an atom in the ground state. Associative
ionization is the process opposite to dissociative recombina-
tion and can be described by the following scheme:
A* + B+ AB* +e (5.27)
The colliding atoms occur on the term which intersects the
molecular ion term (see Fig. 11), and the respective state
is the auto-ionization state for the distances between the
nuclei less than Ry. It is this state that decomposes into a
molecular ion and an electron. The cross section of associa-
tive ionization depends on the lifetime of the auto-ionization
state. Hence, at low energies only a limited number of the
excited states have significant cross sections of- associative
ionization. This process is important for production of
charged particles in plasma.
Another practically important ionization process in col-
lisions of atoms is the Penning effect, which can be described
by the following scheme:
A* +B>A44+Bt +e (5.28)
where A* is a metastable atom whose excitation energy is
higher than the ionization potential of atom B. The meta-
stable atomic state is an excited state which does not decay
or which decays slowly due to radiation transition. Such
metastable states are found, for instance, in atoms of inert
gases or atoms of alkali-earth metals but not in atoms of
alkali metals. Since the metastable atoms have large life-
times, their concentration in the weakly ionized gas can be
high and exceed the concentration of the charged particles.
Ilence, the rate of production of charged particles is changed,
owing to the Penning effect, by addition of an easily ionized
impurity to the weakly ionized gas even in such small quan-
tities as a few per cent of the density of the gas.
The Penning effect is essentially the decomposition of the
auto-ionization state of system A*B. The initial state of the
system is the auto-ionization state for any distance between
the nuclei. However, for large distances the lifetime of
5*
68 Introduction to Plasma Physics
this state is large because of the weak interaction in the
system. The lifetime of the auto-ionization state decreases
with decreasing distance between the nuclei, and the Penning
effect occurs if this state is able to decay during the collision.
Stepwise ionization of atoms. In the gas-discharge plas-
ma, the mean electron energy is usually considerably lower
than the ionization potential of the atoms. Under these
circumstances single ionization of atoms can occur only
in collisions with the high-energy electrons from the tail
of the distribution function. Ionization can occur also when
the atom passes a number of excited states in collisions
with electrons and is ionized from an excited state; this
process is termed stepwise ionization. Let us find the con-
stant of stepwise ionization assuming that the electrons
have the Maxwell velocity distribution and the electron
temperature is considerably lower than the ionization po-
tential of -the atom:
Tom d (5.29)
Assume that the electron density in the system is sufficiently
high and all the transitions between the excited states of
the atoms are due to collisions with the electrons. Under
these assumptions stepwise ionization proves to be the detai-
led reverse of the process of triple recombination of an elec-
tron and ion (5.15). In these processes the atoms undergo
the same transformations but in the opposite directions.
Assume that the electrons in the system are in thermo-
dynamic equilibrium with the atoms. Under these condi-
tions the charged particles are produced as a result of step-
wise ionization while their decay is due to triple recombina-
tion; hence, the balance equation for the electron density is
oNe =0=NNokstep. —4NeN ;
Here N, is the atom density and kstep, is the constant of
stepwise ionization of the atoms. Since the electrons and
atoms are in thermodynamic equilibrium and the Saha dis-
tribution (2.17) gives the relationship between their den-
Sities, we obtain the following relationship between the
constants of the above processes:
k a Efi | mT ia exp —z) (5.30)
step. = Ne gq \ Oni
Charged Particles 69
where g,, g; and g, are the statistical weights of the electron,
ion and atom, respectively, and 7, is the electron tempera-
ture.
It should be noted that relationship (5.30) is valid for
any ratio between the atom and electroni densities since the
constants kstep. and a/N, do not depend on the densities
- of the particles. The assumption of the thermodynamic
equilibrium in the system was made only to help us find
the relationship between the constants.
Inserting Eq. (5.16) for the triple recombination coeffi-
cient into Eq. (5.30), we obtain the following expression
for the constant of stepwise ionization:
- melo
Kstep. — AB h3T3 exp ( =4 7)
(5.31)
where A is a numerical factor identical for all the atoms.
Let us compare the constant of stepwise ionization with
the constant of single ionization, which is given in this
case by
Cost? 2 B® y 7 28172
ion = qiepare °*P ( ~z) (=} Sion (6) dé
where € is the energy of the incident electron. If the con-
dition (5.29) is satisfied, this integral convergesin the vicin-
ify of the ionization potential where the cross section is
given by Ojon = 0, (6/7. — 1); here o, is of the order of
the atomic cross section. Taking into account the condition
(5.29) and using this cross section, we find
hicn= ( ore a Oy EXP (—+) 5.32)
A comparison of Eqs. (5.31) and (5.32) yields (o) ~ a?
where a, is the Bohr radius)
kion ( RT, _
Kstep. meA
This ratio is much less than unity owing to the condition
(9.29) and to the fact that the ionization potential of the
atom, J, is of the order of me*/h?. Hence, if the conditions
are suitable for stepwise ionization, this process at low
70 Introduction to Plasma Physics
electron temperatures proves to be more efficient than the
single ionization of an atom.
Thermodynamic equilibrium conditions for excited atoms.
When deriving Eq. (5.31), we made an implicit assumption
that the excited atoms involved in ionization are in thermo-
dynamic equilibrium with the atoms in the ground state.
Let us find out the conditions under which this assumption
is valid.
Let us consider two states of the atom, the ground state
which we denote by the subscript “gr.”, and the excited
state denoted by “exc.”. A transition between these two
states can be due to collisions with electrons; the excitation
constant averaged over theelectron velocities will be denot-
ed by Kerexc., the constant for the opposite transition
by kexc.cr.. Apart from that, there are other mechanisms
of decay of excited states (radiation, decay at the walls,
etc.) for which the lifetime is +. Let us determine the condi-
tion of thermodynamic equilibrium under these circum-
stances.
The balance equation for the density of excited atoms in
our case is
aNexc. _
dt
O = N gr. Nekgr.exc. — Nexc.Nekexc. gr.
hee
—Aexe: (5,33)
First, we shall consider the plasma in which the transitions
between the states occur in collisions with the electrons,
that is, t—> oo. Then the density of the excited atoms is
given by the Boltzmann distribution (2.9):
AS
Nexe. = Ngr, #22 exp (—F
Ser. e
Hence, we obtain the following relationship between the
constants:
; AS
Ker -exc. = Kexc.gr. a exp ( —7) (5.34)
where A€ is the energy difference between the states, and
f, is the electron temperature. Since fgy.exc, and kexe.gr.
do not depend on the density of the atoms, relationship
Charged Particles 74
(5.34) is valid when the thermodynamic equilibrium between
the atom states is disrupted. This relationship follows di-
rectly from the principle of detailed balancing.
The balance equation (5.33) and Eq. (5.34) can be used
to determine the density of the excited atoms:
Nexe. = Nee. sense: ( 1 eee eee -
Kexe.er. Nekexc.gr.t
AS 1 -{
—N Eexc. py (—+)(1 wa)
ah 8er. P Te as Nekexc.gr.t
It can be readily seen that our system is in thermodynamic
equilibrium when
N ekexc.gr.t > 4 (5.35)
that is, when the time of decay of a given excited state
via the collision with an electron is considerably less than
the time of its decay via other decay channels. This condi-
tion is satisfied at high electron densities. The lifetime of
excited atoms with respect to radiative transition is larger
for higher excited levels. Therefore, condition (5.35) is not
satisfied, typically, for the lower excited states of atoms
while the higher excited states are in thermodynamic equi-
librium.*
Now let us discuss once again Eq. (5.31) and determine
how it can be applied taking into account the radiative
transitions in the atoms and also the decay of the excited
atoms at the walls of the vessel filled with plasma. Equa-
tion (5.31) is valid if radiative transition does not affect
the process, that is, if condition (5.35) is satisfied for the
excited atoms (here t is the lifetime of the excited states
with respect to radiative transitions). Actually, this con-
dition should be satisfied only for the lower excited states.
Moreover, the excited atom must have time to undergo all
the transformations leading to ionization long before it
* This means that the Boltzmann distribution (2.9) gives the rela-
tionship between the densities of the atomic excited states i and k,
the transitions between which occur owing to collisions with the elec-
trons. These excited atoms may be not in thermodynamic equilibrium
vither with the free electrons or the atoms in the ground state if there
is no general thermodynamic equilibrium.
72 Introduction to Plasma Physics
can drift to the wall of the vessel. Hence,
N Kstep.Taritt > f
where tTgrirt is the characteristic time for the atomic drift
to the wall.
6 Physical Kinetics of Gas and Plasma
The kinetic equation. We have to give the mathematical
description of the system of particles comprising the gas.
Though collisions in the gas are rare and each particle
spends only a small fraction of its time, No*/*, strongly
interacting with other particles (see Sec. 4), the significance
of these collisions is that they result in energy exchange
between the particles. Hence, a description of the system
behaviour must take into account collisions of the particles
though we can confine ourselves only to the pair collisions,
since the probability of the triple collisions is by a factor
of No®/? less than the probability of the pair collisions
while the probability of collisions between a larger number
of particles is even lower.
Now we have to choose a parameter for describing the
system. The state of an individual particle is described by
the velocity v and the inner quantum numbers J. We have
to find at each moment of time how many particles in each
state are at each point in space. Hence, we describe the
system by a distribution function f(v, J, r, t) so that
f(v, J, r, t) dv. is the number of particles in unit volume
at point r at moment ¢ which have the inner quantum num-
bers J and the velocities in the range from v to v + dv.
Hence, the density of particles at point r at moment ¢ is,
by definition,
Nir, dj=> \ f(v, J, r, t) dv (6.4)
J
The evolution of the system can be described by the
equation which is satisfied by the distribution function
f(v, J, r, t) and which is called the kinetic equation. We
assume in this equation that the variation of the number of
particles in a given state is due to the pair collisions be-
Physical Kinetics of Gas and Plasma 73
tween the particles. The equation can be written as
d
H=To. (6.2)
where I¢9;}, is the so-called collision integral, which takes
into account the variation of the number of particles in
a given state owing to collisions with other particles.
Let us write down the left-hand side of the kinetic equa-
tion which describes the motion of particles in external
fields in the absence of collisions
of _ _f(v+day, r-+dr, t+dt)—f(v, r, t)
dt dt
In the absence of collisions we have dv/dt = F/M where F
is the force of the external field acting on the particle, M
is the mass of the particle, and dr/dt = v. Hence,
Fy Hy OF
dt — ot M ov
so that the kinetic — is transformed into
ad F of
re M “OV. =— = I o01. (f) (6.3)
Equation (6.3) describes the development of the system in
time and is called the Boltzmann kinetic equation.
Let us consider the collision integral. It characterizes the
variation of the distribution function owing to collisions
between the particles. The collision integral’s dimensions
are f/t where f is the distribution function, and t is time.
The.characteristic time t for variation of the distribution
function owing to collisions is of the order of the time of
collisions between the particles, (Nvo)-!, where N is the
density of the particles, v is the characteristic velocity of
collision, and o is the cross section of the large-angle col-
lisions. The collision integral is zero if the velocity distri-
bution function is the Maxwell function fo.
Proceeding from the above properties of the collision
integral, we can derive the following simple approximation:
Teo. ee Phe (6.4)
where f is the distribution function, and the characteristic
time t is of the order of the time between collisions and
TA Introduction to Plasma Physics
doponds on the velocity. Approximation (6.4) is called the
“tau” approximation; it yields approximate solutions of
(he kinetic equation which give a correct form of the
distribution function.
Lot us illustrate our discussion by applying the tau ap-
proximation to study the evolution of the system returning
lo the equilibrium state following a disturbance. There are
no external fields and the system is uniform, so we obtain
for the kinetic equation (6.3) in the tau approximation
(he following form:
Of __ _ f=fo
at tT
The solution of this equation is
f=fo +f (v, 0) — fol exp (—#/t) (6.9)
where ¢ is time, f (v, 0) is the velocity distribution func”
tion at zero time, and f, is the Maxwell distribution func-
tion. Relaxation of the system to its equilibrium occurs
during a period of time which is of the order of the time
between two consecutive collisions of the particles.
Macroscopic equations for a gas. The distribution func-
tion yields detailed information about a system of particles
and can be used to express all the macroscopic parameters
of the system. Hence, the kinetic equation may be used to
derive all the equations for these macroscopic parameters.
Below we shall derive the simplest macroscopic equations
from the kinetic equation.
Let us integrate the kinetic equation (6.3) over . the velo-
cities of the particles. The right-hand side of the equation is
the total variation of the density of particles per unit time
due to collisions. Assuming that in the volume considered
there is no production or decomposition of particles, we
find that the right-hand side of the resulting equation
equals zero:
of of F Of a
) 3 V+ \ Vv 3, V+ \ sb av=0
Let us change the order of differentiation and integration
in the first and second terms and make use of the fact that
\ fav = N and | v/ dv — Nw where N is the density of
the particles, and w is their mean velocity referred to as
Physical Kinetics of Gas and Plasma 75
the drift velocity. The third term is zero since the distri-
bution function for the infinite velocity is zero. Hence, we
obtain
a tdiv(Nw)=0° (6.6)
This equation is termed the continuity equation.
To derive another equation, we shall multiply the kinet-
ic equation by Mv; and integrate over the velocities of the
particles (M is the particle mass and v, is the respective
component of the particle’s velocity, i = z, y, z). The right-
hand side of the resulting equation is the variation of the
total momentum of the particles due to collisions per unit
time. Let us consider a system of identical particles. In
this system, collisions do not change the total momentum of
the particles so that the right-hand side of the equation is
zero and the macroscopic equation has the following form:
0} 0
| Mv, oe dv+ | Muy, ¢¢ dv+ Fr \ v1 sL dv=0
Here the subscripts i and k denote the components of the
appropriate vector (i, k = x, y, 2) with summation over
k, and zx, is a coordinate. Let us change the order of inte-
gration and summation in the first two terms and integrate
the third term by parts:
| dvv, se sul — | dvfbin= —N6in
where 6;, = 1 wheni = k and 6;, = Owheni +k. Finally,
we obtain
< (MNw,) +; se (N (Mv; Vz)) —NF;=0
where the aii brackets denote averaging over the distri-
bution of particles.
Let us define the pressure tensor as
P i, = (M (vj — w;) (vp — Wp) (6.7)
= this tensor into the above equation, we find that
= (MNw w)+% ih po (MNw,w,) —F; =0
76 tntroduction to Plasma Physics
Subtract se this equation the continuity equation
Mw [> -—— = (Nw,) | = = (0. Finally, we obtain
4 OP iz Ow; _
M Fi Gz, a MWe a ,=0 (6.8)
The form of this equation for the mean momentum depends
on the,representation of the pressure tensor, which is deter-
mined by the properties of the system. We shall discuss
below a special form of representation for this tensor. The
macroscopic equation for the mean energy has an even more
complicated form.
The macroscopic equations (6.6) and (6.8) have been de-
rived for a one-component gas. Let us transform these equa-
tions to apply to a multicomponent system. The right-hand
side of the continuity equation is the variation of the num-
ber of particles of a given species in unit volume per unit
time due to production or loss of the particles of this spe-
cies. In the absence of production or loss of this species,
the continuity equation for this species has the form (6.6)
irrespective of the presence or absence of other species in
the system.
The right-hand side of Eq. (6.8) for the multicomponent
system should contain the variation of the momentum of
a given species per unit timejowing to collisions with other
species. If the mean velocities of two species are different,
there occurs the momentum transfer betweenthem. Hence,
the momentum transfer between two species is proportional
to the difference between their mean velocities. Therefore,
we can write Eq. (6.8) in the following form:
( ) (9g
aw’? 1 IPG mo au FY”
at M,Nq 92h - “Ot, Mg
(s) \9)
qs
The subscripts s and q denote here the particle species, and
Tgs is the characteristic time of the momentum transfer from
the species g to the species s. Since this transfer does not
change the total momentum of the system, the characteris-
tic time of the momentum transfer satisfies the following
Physical Kinetics of Gas and Plasma 77
equation:
q‘Yq
PEE 1640)
os t
The equation of state for a gas. The relationship be-
tween the macroscopic parameters of the gas (the pressure p,
the temperature 7, and the particle density NV) ts given by
the equation of state. Let us derive this equation for a homo-
geneous gas. Let us express the gas pressure in terms of the
distribution function. Let us introduce a new coordinate
system in which the gas (or a given gas volume) is resting.
In this coordinate system, the pressure is the force acting
on a unit area of an imaginary surface in the system.
The flux of particles with the velocities in the range from
v, to v, +- dv, through the given unit area is dj =v,f dv,
with the x axis being perpendicular to the area. Reflection
of each particle from the area is elastic, that is, the normal
component of its velocity is reversed: v,,— — v,. Hence,
the particle of the mass M transfersto the areaa momentum
of 2Mv,.. The force acting on the area is the variation of the
momentum per unit time. Hence, the gas pressure is giv-
en by
p= \ 2Mv, Xv,f dv,=M \ v2f dv, = MN (v2)
V_>9
The integration in the first term is over the velocities v,,
that exceed zero; it should be taken into account that the
gas pressures on both sides of the area are the same.
In the above formula, v, is the particle velocity com-
ponent in the frame of reference where the gas is resting
as a whole. The gas pressure in the general case is given by
p = MN ((v, — wz)”) (6.11)
where w,, is the component of the mean velocity. In the coor-
dinate system in which the gas is resting the distribution
function for an isotropic gas f (v,.) does not depend on direc-
tion. Under these circumstances the gas pressure is the
same in all directions:
p = MN (vy — w;)") = MN (vy — wy)’
= MN ((v, = Wz)") (6.12)
78 Introduction to Plasma Physics
Here we have a simple relationship between the pressure
tensor (6.7) and the gas pressure:
Fin=p0in (6.18)
The definition of the gas temperature (2.16) relates the
temperature to the mean kinetic energy of particles in the
frame of reference in which the mean velocity is zero; hence,
we obtain
37/2 = M (wv — w)* 9/2
Using Eq. (6.12), we find the following relationship between
the pressure and the temperature:
p=NT (6.14)
Equation (6.14) is known as the equation of state for a gas.
Let us write the macroscopic equation (6.8) for the mean
momentum in the case of the isotropic gas when the pressure
tensor has the form (6.13).* Inserting Eq. (6.13) into Eq.
(6.8), we obtain
Ow grad ‘p F
+ (wy)w+EP_—=0 (6.45)
Here w is the mean velocity of the gas, 9p = MN is the
mass density, and F is the force acting on one particle.
In the absence of external fields (F = 0), this equation
is called the Euler equation. Equation (6.9) for a multi-
component system is transformed in this case into
grad Pg Fa _y¥ Ws— Wq (6.16)
Pg M, +
OWg
Gr (WaV) Wa +
where the subscripts g and s denote the gas particle spe-
cies,
7 Transport Phenomena in Weakly lonized Gas
Transport phenomena in gas and plasma. Transport of parti-
cles in a gas. The transport phenomena in gas and plasma
primarily are due to thegradients ofthe macroscopic param-
* Since Eq. (6.13) is valid also for the isotropic liquid, the equa.
tion derived below describes not only a gas but a liquid as well.
Transport Phenomena 79
eters. These gradients give rise to fluxes that finally equal-
ize the macroscopic parameters over the gas or plasma vol-
ume.
Let us consider the simplest transport phenomenon due
to the transport of particles. If the density of a given particle
species is not spatially uniform and this gas state is hydro-
dynamically stable, there is a directional flow of particles
tending to equalize in space the density of the given particle
species. If the density of the given species varies slightly
over the mean free path, that is, the density gradient is
small, the diffusion flux density j is proportional to that
gradient:
j= — Degrad N (7.1)
The factor M in Eq. (7.1) is called the diffusion coefficient.
Let us estimate the magnitude of the diffusion coeffi-
cient. The net diffusion flux density equals the difference
of the flux densities in opposite directions; each is, by order
of magnitude, equal to Nv where JN is the density of a given
species, and v is the characteristic velocity. Prior to colli-
sion each particle travels the distance of the order of the
mean free path A ~ (N,zo)7! (here o is the characteristic
cross section of collisions between the given species and
other gas particles, and N, is the total gas density). Hence,
the net diffusion flux density of the particles is ANv where
AN is the difference in the densities of the given species at
two points separated by the distance of the order of the
mean free path. And so we obtain AN ~ A grad N and
the diffusion flux density j ~ Av grad N. A comparison
with Eq. (7.1)* yields the following estimate for the diffu-
sion coefficient**:
eis 7.2
D~v ~ MPN 6 ( .2)
* We neglect here the sign of the flux density. Naturally, the
direction is determined by the fact that it tends to equalize the par-
ticle density, so that the sign in Eq. (7.1) reflects this fact. The same
can be said about the signs for other transport phenomena.
** This formula can be derived using dimensional analysis. The
parameters 7, M, and A, which describe the motion of an individual
particle in gas, make up only one combination with the dimensions
of the diffusion coefficient M, which is A71/2/ 41/2,
80 Introduction to Plasma Physics
Here 7 is the temperature of the gas, M is the mass of the
particles, and it is assumed that the mass of the diffusing
species are of the same order of magnitude as the mass of
the other particles. Let us analyze the restoration of the
equilibrium gas densily after a density gradient has been
established. Inserting (7.1) into the continuity equation
(6.6), we obtain the equation for the particle density, which
is known as the equation of diffusion:
oN — DVN (7.3)
Let us denote by Z the characteristic distance at which there
occurs a noticeable variation of the gas density. Then
Eq. (7.3) yields the characteristic time of the density var-
iation, t, ~ L/D ~ L?/(va).
Macroscopic analysis of the transport of particles in
gas is valid if the characteristic variations of the density
occur in time intervals which are considerably larger than
the characteristic time between two consecutive collisions
of the particle t ~ A/v. Hence, we obtain t/t, ~ (A/L)? < 1,
that is, the macroscopic analysis of transport phenomena
is valid if the gas density varies markedly at a distance con-
siderably larger than the mean free path of the particles.
Energy and momentum transport in a gas. We have
discussed the transport of {particles in a gas. Transport of
energy and momentum occurs in a similar way when the
appropriate gradients exist in the gas. For instance, the
temperature gradient gives rise to the heat flux vector
defined as
2
a= | ve f dv (7.4)
This heat flux vector tends to equalize the temperature of
the gas in different regions; the flux vector is zero when
the temperature is constant over the gas volume. Hence, the
heat flux vector is proportional to the temperature gra-
dient:
= — x grad Tf (7.5)
The factor x is the thermal conductivity of the gas. In
terms of our analysis, Eq. (7.5) is valid if the temperature
Transport Phenomena : 81
of the gas varies only slightly over a distance of the order
of the mean free path of the gas particles.
Let us estimate the magnitude of the thermal conductiv-
ity and its dependence on the system’s parameters as it
has been done for the diffusion coefficient (7.2). Imagine
a plane perpendicular to the vector grad 7 and calculate
the heat fluxes passing through it in both directions. Accord-
ing to Eq. (7.4), each of the fluxes is
q~ Nv X Mv? ~ NvT
where JV is the density of particles, and v is their charac-
teristic velocity.
These heat fluxes are due to the motion of the particles,
which travel prior to collision a distance about the mean
free path 2. Since the heat fluxes in both directions through
the imaginary plane are due to the particles moving on
different sides of the plane where the temperatures are
different, the net flux is
gq ~ NvAT
Here AZ is the temperature difference over the distance of
the order of the mean free path (the characteristic size of
the region from which the particles get to the plane without
collisions with other gas particles, that is, without energy
exchange). Hence, we obtain AT ~ A grad T and q ~
~ Nvi grad 7. A comparison with Eq. (7.5) yields the
following estimate for the thermal conductivity:
1/2
v T
mw NUN mo ~ eth
(7.6)
where o is the characteristic cross section for the colliding
particles.
The thermal conductivity is seen to be independent of the
density of the particles. Indeed, an increase in the particle
density gives rise to a proportional increase in the number
of particles transporting heat and to a proportional decrease
in the mean free path of the particles, that is, the charac-
teristic distance of heat transport. These two effects are
mutually cancelled.
In the case of a directional motion of gas with the average
velocity not being uniform over the cross section, the mean
6—01607
82 Introduction to Plasma Physics
momentum is transported between the regions with differ-
ent mean velocities of the gas. This gives rise to friction
forces, which tend to slow the motion of gas with higher
mean velocity and to equalize the mean velocity. The force
F that acts (per unit area) opposite to the direction of the
gas motion is
Ow
Here w,, is the mean velocity of the gas (see Fig. 12), and
the factor y is called the viscosity coefficient.
Let us estimate the magnitude of the viscosity coefficient
and its dependence on the parameters of the system as it
has been done for the diffusion coefficient and the thermal
conductivity. The force (7.7) acting on the unit area is due
to the exchange of particles with different mean velocities.
Let us calculate this force.
Variation of the mean momentum of the particles due
to their exchange of position is MAw, where M is the par-
ticle mass, and Aw,, is the difference between the mean veloc-
ities of directional motion. The flux density of particles
is, by order of magnitude, equal to Nv where JN is the den-
sity of particles, and v istheir characteristic velocity. There-
fore, the force acting on the unit area and equalling the
variation of the mean momentum of particles per unit time
is F ~ NuMAw,,.. Since the exchange of particles occurs
between the regions separated by a distance of the order of
the mean free path, we find that Aw, ~ A’ Ow,,/0z and
F ~ NvMi dw,,/dz. A comparison with Eq. (7.7) yields the
following estimate for the viscosity coefficient:
1/2741/2
1 ~ NvMi ~ 4 (7.8)
According to Eq. (7.8), the viscosity coefficient is inde-
pendent of the density of particles. Similar to the thermal
conductivity, this independence is due to compensation of
two opposite effects occurring with momentum transport.
Indeed, the number of momentum carriers is proportional
to the density while the characteristic distance of transport,
the mean free path of the particles, is inversely proportion-
al to the gas density.
Transport Phenomena 83
We have discussed the simplest transport phenomena
caused by gradients of the macroscopic properties: the
transport of particles caused by the density gradient, the
heat transport caused by the temperature gradient, and
the momentum transport caused by the gradient of the
directional velocity. Therecanalsooccur transport phenom-
ena of mixed origin. For instance, the density gradient can
FIG. 12. : me es
The mean velocity aa
of the gas particles =
in the presence
of viscosity. ———
eo
give rise to the heat transport, the temperature gradient
can give rise to the transport of particles, and so on. These
effects are due to specific relationships between the macro-
scopic parameters NV, 7, w, etc., which characterize the
system. The picture can be even more complicated in the
multicomponent system where a pressure gradient can be
maintained for a long time for a given species. To analyze
all these effects, we can use the general approach applied
for deriving Eqs. (7.2), (7.6), and (7.8).
Thermal conductivity due to the internal degrees of free-
dom. The characteristic cross section for the elastic col-
lisions of particles is considerably larger than the cross
section for inelastic collisions. Hence, the cross sections
in Eqs. (7.2), (7.6), and (7.8) are those for the elastic col-
lisions of particles while the inelastic collisions do not
affect transport phenomena. One exception is the thermal
conductivity of gas where the internal degrees of freedom
affect the heat transport in a peculiar way. Let us assume
that the gas at each point is in thermodynamic equilibrium
with respect to the internal degrees of freedom. Then trans-
port of particles between regions with different temperatures
results in the transport of the internal energy of the particles
6*
84 Introduction to Plasma Physics
though the internal energy difference is converted into
the energy of translational motion (or vice versa) in the
course of numerous elastic collisions. This phenomenon
results in an increase in the thermal conductivity.
Let us represent the heat flux vector as a sum of two.
terms:
q = — transi. grad L’ — Xjntern. grad T (7.9)
Here the first term is due to the transport of translational
energy of particles, similar to Eq. (7.0), while the second
term is due to the internal degrees of freedom. Hence, the
thermal conductivity is a sum of two terms:
% = Xtransl. 1 “intern. (7.10)
and we have to find the second term.
Let us describe the internal state of the particle by the
label i. Since the spatial density of particles with a given
internal state i is not uniform because of the temperature
gradient, there is a diffusion flux density of these particles
i:= —D; grad N; = —D, grad T
where Q&; is the diffusion coefficient of the particles with
a given i. Denote by €; the excitation energy of the ith
internal state of the particle. Then the heat flux vector due
to the internal degrees of freedom is
: ON;
Gintern. = > éiii = — > Ei Di OT grad T
i i
A comparison with Eq. (7.9) yields the following expres-
sion for the thermal conductivity for the heat transport due
to internal degrees of freedom:
ON;
*intern. = by E:D; ar (7.41)
a
Equation (7.11) is simplified if the diffusion coefficient
does not depend on the excitation of the internal degrees of
freedom of the particle. This is the case, for instance, when
the respective internal degrees of freedom correspond to
the rotational or vibrational excitation of molecules. In
89
Transport Phenomena
this case we obtain
ON 0
*intern. = D >i Eis = Dor > EiN;
i
i
=D (€N)=DNey — (7.12)
Here N = > N, is the total density of the gas particles,
7
é= N71) €:N; is the mean excitation energy of the
t
particles, and cy is the specific heat per molecule correspond-
ing to excitation of the internal degrees of freedom. For
instance, if excitation of the internal degrees of freedom is
sufficiently effective, in other words, if cy ~.1, then Eqs.
(7.12) and (7.2) yield intern. ~ T1/2/(M4/?0), that is,
inten. lias the same order of magnitude as the thermal
conductivity (7.6) due to the translational motion of par-
ticles.
Let us consider another special case when the thermal
conductivity due to internal degrees of freedom is
nificant, namely, the thermal conductivity of a monoatgmic
gas with a slight admixture of molecules. The density of
0.20): N,, is related to the density of atoms JV, th ‘ough
2.20): i
Na/Nm= f (T) exp (— D/T)
Here f (7) is a power function of temperature, and D is
the dissociation energy of the molecule. Since D > T, we
obtain
0Nm _ D
or = px Nm
Inserting this relation into Eq. (7.41), we find that
D\2
Mintern. =D (a) Nm (7.13)
where & is the diffusion coefficient for the molecules in the
monoatomic gas. Compare this parameter with the thermal
conductivity % transi, (7.6) due to the heat transport in
the translational motion of molecules. Equations (7.2),
86 Introduction to Plasma Physics
(7.6), and (7.13) yield
intern. D \2 Ny
=e aad Me a
In the case being considered, we have NV, < N,. However,
since the complete dissociation of the molecules occur at
the temperatures which are low compared to the molecule
dissociation energy (7 < D), the ratio (7.14) can be about
unity for comparatively low concentrations of the mole-
cules in the gas. |
The mobility of particles. The Einstein relation. When
a particle travels in vacuum in a field of external forces,
it is uniformly accelerated. If this particle travels in gas,
the collisions with the particles of the gas give rise to a fric-
tion force whose action determines the mean velocity of
directional motion of the particle. When the external force
is small, that is, the mean velocity w of directional motion
of the particle is small compared to the thermal velocity
of the gas particles, the velocity w is proportional to the
acting force F:
w = OF (7.15)
The proportionality factor b is known as the mobility of
the particle. In the small-force approximation, the mobil-
ity does not depend on the magnitude of the external
force F.
Let us consider the motion of the test particles in a gas
under the action of an external field, the particles being in
thermodynamic equilibrium with the gas. According to
the Boltzmann formula (2.9), we have the following spatial
distribution of the density of the test particles:
N = N, exp (—UIT) (7.16)
where U is the potential of the external field, 7 is the gas
temperature, and N, is the density of the test particles at
the point where the potential U is assumed to be zero.
Since the density of the test particles is not uniform over
the cross section of the gas, the gradient of density gives
rise to a diffusion flux density
i= — JD grad N= Zandt N= — FN
Transport Phenomena 87
Here we used the Boltzmann formula (2.9) for the density of
the test particles and the relation F = — grad U for the
external force. Under thermodynamic equilibrium, when
there is no net directional motion of the test particles, the
diffusion flow of the test particles is compensated by the
flow of the test particles under the action of the external
force, the flow being wN = bFN according to Eq. (7.15).
Since the net flow is zero, we obtain the following relation-
ship between the diffusion coefficient and the mobility of the
test particles:
b = DIT (7.17)
Equation (7.17) is known as the Einstein relation, and it is
valid for the small fields which do not disturb the thermo-
dynamic equilibrium between the test particles and the gas
particles.
Equation (7.2) yields the following estimate for the
mobility:
b~ (MT)~*? (No) (7.18)
The Navier-Stokes equation. Let us derive the equations
for transport of momentum and energy in a viscous gas.
The equation for momentum transport can be derived from
Eq. (6.15) by taking into account the viscosity of the gas.
Confining ourselves to the one-component gas, we can use
Eq. (6.8) as the basis for the equation of momentum trans-
port where we should add to the pressure tensor a term ac-
counting for the viscosity. Using the frame of reference of
Fig. 12 and Eq. (7.7), we find that this term, proportional
to the viscosity coefficient, is given by
’ Ow
Pig 2 aN) an
Under our conditions the only nonzero component of the
velocity is along the z axis and the acceleration is along the
Zz axis..
Assuming that under the above conditions the pressure
tensor is determined only by viscosity and noting that this
tensor is symmetrical, we can write it in the following gen-
eral form:
, had Ow
Fi, = — 7 (+s 3a, 7 Sip ain 5 ) (7.19)
88 Introduction fo Plasma Physics
Here summation is assumed over the subscripts appearing
two times. The factor a can be found in the following way.
The forces of viscous friction in gas are due to the fact that
adjacent gas layers move with different velocities. If the
gas is decelerated as a whole, that is, if the derivative
dw;/dx; is made nonzero, this mechanism of friction dis-
appears and the force due to the gas viscosity must vanish.
Hence, the trace of the pressure tensor due to the viscosity
must be zero, that is, » Fi; = 0. Thus, we find a = — 2/3,
1
and the viscosity term of the pressure tensor can be written
as
— aN tad ac
Dip Ty (=! Ox; 3 Sin =) (7.20)
Let us add to the viscosity term of the pressure tensor
(7.20) the term (6.13) due to the gas pressure. Inserting
this pressure tensor into Eq. (6.8), we obtain
0 d
ar t (wv) w= — SP 4 A vew
++ gh grad div w+ a (7.24)
Equation (7.21) describes momentum transport and is
called the Navier-Stokes equation.
The equation of heat transport. Let us derive the heat
transport equation for the moving gas in which there occurs
heat transport due to heat conduction. Let us write the
balance equation for the variation of the energy of the gas
in an elementary volume V. Denote by e« the mean energy
of a gas particle and by N the density of the gas (to simplify
the treatment we consider a one-component gas). The varia-
tion of energy in the volume V is given by
3 | Nar= —ads
14 S
Here q is the heat flux vector, S is the surface enclosing
the volume V, and ds is the element of the surface. The
minus accounts for the fact that the heat outflow through
the surface around the volume decreases tle energy con-
tained in the volume.
Transport Phenomena 89
Making use of the Gauss theorem, we can transform the
above equation into
) [+ (eN) + diva] dr =0
yl
Since this integral is zero for an arbitrary volume V, the
integrand must be zero:
<-(@N)+divq=0 = (7.22)
This equation of heat transport has the same form as the
continuity equation (6.6) for transport of particles.
The heat flux vector in Eq. (7.22) can be written as a
sum of two terms. The first term is the heat flux eNw due
to the gas motion. The second term is the heat flux due to
thermal conduction, which is —x grad T according to Eq.
(7.5) where x is the thermal conductivity coefficient. Insert-
ing these heat fluxes into Eq. (7.22), we obtain the following
equation for heat transport:
<_ (eN) + div (gNw) — “vf =0
Subtraction from the above equation of the continuity
equation (6.6) yields
N ad + Nw grade = xV2T (7.23)
Assume that the mean energy of the molecule, which is
the sum of the internal energy and the translational energy,
is proportional to the temperature. This is the case if the
gas temperature is the same as the temperature of the in-
ternal degrees of freedom. Hence, we obtain e@=¢yT
where cy is the specific heat per molecule*. Thus, the equa-
tion of heat transport (7.23) can be transformed into
or +werad T= oy VT (7.24)
* For a monoatomic gas cy = 3/2; if the gas temperature of a
binary molecular gas is identical to the rotational temperature and
if the majority of molecules are in the ground vibrational state and
in the rotational states with high quantum numbers, then for this
gas cy = 5/2.
90 Introduchon to Plasma Physies
The diffusion motion of particles. Let us study the diffu-
sion motion of particles in gas. Denote by W (r, ¢) the pro-
bability that a test particle is at point r at time ¢. Assume
that at the zero time the test particle is at the origin so
that the function W (r, #) is spherically symmetric. The
function W (r, t) is normalized thus:
| Wor, t)4ur%dr=1 (7.25)
"0
This probability function satisfies the diffusion equation
(7.3) for the test particle travelling in the gas:
= DPW (7.26)
Since the function W is spherically symmetrical, Eq.
(7.26) can be written in the — form:
4 92
= DF aa (rW)
We are interested in average characteristics. Let us multi-
ply the above equation by 4zr‘dr and integrate the result
over r. The left-hand side of the equation yields
d dr2.
& ee 2 2dr _
| dar dr =z \e W x Sur dr = —-
0 0
where r? is the mean square of the distance. Integrating
twice by parts and using Eq. (7.25), we transform the right-
hand side of the equation into
D \ A4nr‘ dr — 2 <3 = (rW) = =—3D j 4nr? dr 2? om
0
‘or2
0
[oo
= 65 \ Aart drW = 6D
0
The resulting equation is
dr2
= = 6D
Transport Phenomena 91
Since at zero time the particle was at the origin, the solution
of the equation is given by
In the case of spherical symmetry, we have z? = y? = 2? =
= r*/3 so that
PSP] —29i (7.28)
Equations (7.27) and (7.28) describe the diffusion motion
of the particle characterized by numerous collisions with the
gas particles so that each collision changes the direction
of the particle’s motion. This motion is sometimes called
Brownian motion. For Brownian motion the distance trav-
elled varies as the square root of time.
Convective instability of a gas. If the temperature gradient
is large in a gas which is in a field of external forces, there
may appear a more effective mechanism of heat transport
than thermal conduction, which is called convection. This
process consists in the movement of the warmer gas into
the colder regions and the colder gas into the warmer re-
gions.
Let us analyze the stability of a gas at rest with respect
to development of convection. Consider a gas at rest in
which a temperature gradient is maintained in the field of
external forces. The parameters of the gas are subjected to
a small perturbation, which is due to a slow motion of the
gas and corresponds to convection. If this perturbation
proves to be possible, convective instability can develop in
such a gas. We have to find what conditions are necessary
for the development of the convective instability, that is,
for convective heat transport.
Let us formulate the simplest problem, namely, the so-
called Rayleigh problem. The gas fills a gap between two
infinite parallel walls. The temperature of the lower wall
is T,, the temperature of the upper wall is 7,, and 7, is
higher than 7. The force of the external field is directed
downwards perpendicular to the walls. The distance be-
tween the walls is L. Find the conditions for the develop-
ment of convective instability.
92 Introduction to Plasma Physics
Let us represent the parameters of the gas as sums of
two terms: the first term is the parameter for the resting
gas and the second term is a small perturbation of the para-
meter due to the convective motion of the gas. Thus, the
gas density is N + N’, the gas pressure is pp) + p’, the
gas temperature is J + 7’, and the gas velocity is w (it
is zero in the absence of convection). Insert these parame-
ters into the stationary equations of continuity (6.6), of
momentum transport (7.21), and of heat transport (7.24).
The zero-order approximation is
grad pp = — FN, y°T =0
In the first small-parameter approximation these equations
yield
divw=—0O,
grad (po+ p’) nV2w -
~ N+N T WENT tees 0 (7.29)
(Tg—T) ’
Wy L = iN VT
The parameters of the above problem are used in the last
equation. Here the z axis is perpendicular to the walls.
Transform the first term in the second equation (7.29).
Up to the first order of approximation, this term is
grad (po+p’) _ grat Po a grat p’ grad po N’
N-+N’
(1M) 4
According to the equation of state (6.14), fora gas N = p/T,
and so we find that NV’ = (ON/0T),T’ = — NT'/T. Insert-
ing this relation into the second equation, we can write the
system (7.29) in the following form:
divw= 0,
grad p’ T’ n -
yo Gy (7.30)
“*L v27"
V2 = ey (T2—T1)
Let us reduce the system of equations (7.30), which con-
nect the parameters of the gas, to an equation for one para-
Transport Phenomena 93
meter. First, we apply to the second equation of (7.30) the
operator div and take into account the first equation of
(7.30). We find that
V2p' F oT’
No & ae (7.31)
Here we assume that (7, — T,)/T, < 1. Therefore, the
unperturbed parameters of the gas do not vary much inside
the volume being considered; we shall neglect their varia-
tion and assume that the unperturbed gas parameters are
spatially constant.
Inserting w, from the third equation of (7.30) into the
2th component of the second equation and applying the
operator V* to the result, we obtain the following equation:
t Oo 2,/ Vet" Hx 2\3 JD! __
i ae ar eal cyN* (T2—T) Oe
Using the relation (7.31) between V?p' and 7’, we obtain
finally
r R 02 \ m
(v2)3 T’ = —= (V3) T (7.32)
where the dimensionless combination of parameters
(2 —T2) ey PN2L8
i= Sage (7.33)
is called the Rayleigh number.
Equation (7.32) shows that the Rayleigh number deter-
mines the possibility of the development of convection. For
instance, in the Rayleigh problem the boundary conditions
at the walls are 7’ = 0, w, = O. Also, the tangential forces
y(Ow,,/0z) and y(d0w,/dz) are zero at the walls. Differentiat-
ing the equation div w = 0 with respect to z and using the
conditions for the tangential forces, we find that at the
walls (62w,/0z”) = 0. Hence, we have the following boundary
conditions:
’ 0*wz
T'=0, w,=0, aod = 0
Denote by z = 0 the coordinate of the lower wall and by
z = L the coordinate of the upper wall. The general solu-
tion of Eq. (7.32) with the boundary condition 7’ = 0 at
94 Introduction to Plasma Physics
z = 0 can be expressed as
T’ =C exp [i (k,2 + ky,y)] sin kz (7.34)
The boundary condition 7’ = 0 at z = L yields k,L = an
where n is an integer. Inserting the solution (7.34) into
Eq. (7.32), we obtain
(k2L2-+ n2n2)3
R= ae (7.35)
where k? = kj +kj. The solution (7.34) satisfies all bound-
ary conditions.
Equation (7.35) shows that convection can occur for
Rayleigh numbers not less than a minimum number Rypi,
corresponding to n = 1 and kyin = n/(LY 2):
274
Ryin = oY ea
The magnitude of Ryin varies according to the geometry of
the problem, but in all problems the Rayleigh number char-
acterizes the possibility of convection.
Convective motion of a gas. Let us analyze the simple
case of motion of gas in the plane zz. Inserting the solution
(7.34) into the equation
? ae
divw=-—— ae Os = =;
we find the following expressions for the components of the
gas velocity:
IUNZ
W, = WwW) cos kz sin ae
Tinz
Wp Sin kx cos ae
(7.36)
Ws= — FL
where nm is an integer, and the characteristic velocity of
gaS W» is assumed to be small compared to the respective
parameters in the gas at rest; for instance, wy is small com-
pared to the thermal velocity of the particles.
Let us trace the motion of a gas element. The equations
of motion for this element are dz/dt = w,, and dz/dt = w,;
using Eqs. (7.36) for the components of the gas velocity,
Transport Phenomena 95
we obtain
dz
dz
wn TNZ
= — 7, tan kx cot ——
This equation describes the path of the gas element. The
equation yields
sin kx sin a =C° (7.37)
where C is a constant determined by the initial conditions.
FIG. 13.
The paths of the gas
elements in the Rayleigh
problem for n=1, k=n/L.
The constant C variesfrom —1 to +1; its value depends on
the initial positions of the gas elements being considered.
Of special significance are the lines at whichC = 0. These
lines are given by the following equations:
ate
k
where m, and m, are integers, and m, >0O and m, > 0.
The straight lines (7.38) divide the gas into cells. The gas
Me (7.38)
pee t=
n 49
96 Introduction to Plasma Physics
particles which are inside a cell can travel only inside this
cell and cannot leave it. Indeed, Eqs. (7.36) show that the
component of the gas velocity normal to the cell boundary
is zero, that is, the gas cannot cross the boundary between
the cells. These cells are known as the Benard cells.
Equation (7.37) shows that inside each cell the gas ele-
ments travel along closed paths around the centre where
the gas is resting. Figure 13 snows the paths of the gas ele-
ments in the Rayleigh problem for n = 1 and k = a/L,
that is, for the Rayleigh number R = 8x’. In the Rayleigh
problem, the Benard cells are pyramids with regular poly-
gon bases; in the general case, these cells have a more com-
plicated shape.
Convective heat transport. Compared to the thermal con-
duction, convection is a more effective mechanism of heat
transport. Let us analyze heat transport in the Rayleigh
problem. The resting boundary layers of thickness 6 are
formed at the walls whose temperatures are constant. The
thickness of the boundary layer is determined by the vis-
cosity of the gas, and the heat transport on the boundary
layer is due to thermal conduction. Then this heat is trans-
ported by gas flows so that the heat flux transported in the
Rayleigh problem equals the heat flux due to thermal con-
duction in the boundary layer:
qg=%|grad T | ~ este)
Applying the Navier-Stokes equation (7.21) to the bound-
ary region, we can estimate the thickness of the boundary
layer. This equation describes continuous transition between
the wall and the moving gas flows. Now to the second equa-
tion of (7.30) we have to add the term M(wV)w, which
cannot be neglected here. A comparison of separate terms of
the zth component of the resulting equation yields, by
order of magnitude,
2
w F(T,—T w
MY ~ SP) Me (7.39)
Hence, we obtain
ee Lert, | (7.40)
Transport Phenomena 97
Let us compare the heat flux transported by gas in the
above problem and the heat flux transported due to heat
conduction in the resting gas. The heat flux due to heat
conduction is qcona. ~ * (1, — T,)/L where L is the
separation of the walls; thus, we have
q L bees (T,—Ts) 11/3
-Geond. 5 "2
~Gi? (7,44)
Here G is the dimensionless combination of the parameters,
_ N2FM (T,—T,) 13
G eT
(7.42)
known as the Grashof number. A comparison of the Rayleigh
number (7.33) and the Grashof number (7.42) yields the
ratio between them
R_ cym
G Mx
Note that the continuity equation (6.6), the equation of
momentum transport (7.21), and the equation of heat trans-
port (7.24) used above are valid not only for a gas but also
for a liquid. Therefore, the results obtained are applicable
to liquids as well. However, gas has some peculiarities in
this respect. For instance, Eqs. (7.6) and (7.8) yield that
for gas the ratio y/(Mx) is about 1. Furthermore, the specific
heat cy of one molecule is also about 1. Hence, the Rayleigh
number has the same order of magnitude as the Grashof
number for gas. Since convection develops at high Rayleigh
numbers, we find that for convection G >> 1. Hence, accord-
ing to Eq. (7.41), we find that heat transport via convection
is considerably more effective than heat transport in a
resting gas via heat conduction.
The ratio (7.41) between the heat fluxes due to convection
and thermal conduction has been derived for an external
force directed perpendicular to the boundary layer. Let
us derive the respective relation for the external force di-
rected parallel to the boundary layer. Let the z axis be
normal to the boundary layer and the external force be di-
rected along the x axis. Then to the second equation of the
system (7.30) we add the term M (wV)w and compare the
terms of the zth component of this equation. The result is,
7—01607
98 Introduction to Plasma Physies
instead of Eq. (7.39), equal to
wy (T1—T2) nn w
M—- ~ Ft 2 Mia ae (7.43)
Equation (7.43) yields an estimate of the thickness of the
boundary layer:
5 ~ n°TL 1/4
N2FM al
Hence, we find the ratio between the heat fluxes of con-
vection and thermal conduction:
Gg, L ~ (4/4
Ycond. 5 2 (7.44)
The ratio between the heat fluxes in this case is seen to
be different from that for the external force perpendicular
to the boundary layer [see Eq. (7.41)]. However, the con-
vective heat flux in this case is still considerably larger than
the heat flux due to the thermal conduction in the resting
gas.
The instability of convective motion. New types of
convective motion develop when the Rayleigh and Grashof
numbers increase. This disturbs the ordered convective
flow and finally disrupts the stability of the convective
motion of gas giving rise to disordered or turbulent flow
of gas even if it is contained in a resting closed system.
Let us analyze the development of turbulent gas flow.
We consider once again the Rayleigh problem: a gas at
rest is in an external field between two infinite and parallel
planes maintained at different constant temperatures. We
analyze the convective motion of gas (7.36) corresponding
to sufficiently high Rayleigh numbers with n >2. In
this case there can simultaneously develop at least two differ-
ent types of convective motion. Figure 14 shows two types
of convective motion for the Rayleigh number R = 1084
corresponding to the wave number k, = 9.4/Z for n = 14,
and k, = 4.7/L for n = 2.
Let us analyze the example of convection illustrated in
Fig. 14. Using the above parameters, we combine the solu-
tions so that the gas flows corresponding to n = 1 and
n = 2 travel in the same direction in some regions of the
Transport Phenomena 99
gas volume. Then in other regions these flows will necessarily
travel in different directions. Two types of motion can
develop for any Rayleigh numbers.
The fact that the problem has two solutions correspond-
ing to opposite gas flows in some regions does not necessar-
ily mean that the ordered flow of gas is disturbed. Indeed,
we can always make a combination of two solutions describ-
FIG. 14.
The types of
convective motion
in the Rayleigh
problem for R=108n!.
The mixing of the
gas flows travelling
in opposite
directions finally
results in a random
gas motion,
or turbulence.
ing a mixture of the two types of motion. For instance, if
two gas flows travel in the opposite directions with the
same velocity and amplitude, such a combination of solu-
tions corresponds to a gas that is stationary in the given
volume.
Nevertheless, the fact that increasing the Rayleigh num-
ber gives rise to new types of solutions means that the
convective flow can become turbulent. We assume that in
the system there is ordered convective flow corresponding to
one of the solutions. Then a small perturbation in one of
the regions of the gas volume gives rise to another type of
flow. At the boun lary of this region two opposite gas flows
meet so that the kinetic energy of motion of the gas flows
transforms into the thermal energy of gas. This results in a
disordered motion of the gas. The development of turbulence
changes the character of heat transport.
100 Introduction to Plasma Physie¢s
8 Transport of Charged Particles
in Weakly lonized Gas
The mobility of charged particles. The conductivity of a
weakly ionized gas. Let us analyze the motion of charged
particles in a gas in the presence of an external electric
field. The mobility of charged particles K is defined as the
proportionality factor between the mean velocity of charged
particles w and the electric field strength E:
w= KE (8.1)
The force of the electric field acting on the electron or singly
charged ion is F = eE, and a comparison of Eqs. (8.1)
and (7.15) yields
b = Kile (8.2)
Hence, for charged particles the Einstein relation (7.17)
is transformed into
K = eD/T (8.3)
where & is the diffusion coefficient of the charged particle
in the gas. Equations (8.3), (7.2), and (7.18) yield the fol-
lowing estimate forthe mobility of a charged particle (an
electron or a singly charged ion) in gas:
K ~ e(MT)"*/2 (No)! (8.4)
where JN is the gasdensity, and o is the characteristic cross
section of collision between the charged particle and the
gas particle.
Let us find the conductivity % of a weakly ionized gas,
that is, of a gas which has a small concentration of charged
particles, that is, a small ratio between the current of the
charged particles and the electric field strength. The electric
current due to the electrons is
ewN, = eK NE
where JV, is the density of electrons, K, is the mobility of
electrons, and w is the mean velocity of electrons in the
electric field FH. A similar expression can be written for
the electric current due to the ions. Equation (8.4) shows
that the mobility is inversely proportional to the square
root of the particle’s mass. Since the densities of electrons
Transpori of Charged Particles 101
and ions in the quasineutral plasma, which is the one we
usually consider, are the same, the electric current in the
weakly ionized gas is due mostly to electrons, and the
gas conductivity is
1=eK,N, (8.5)
Using Eqs. (8.4) and (8.5), we can estimate the conduc-
tivity of a weakly ionized gas:
=~ Ne? (mT)-42 (Noeq)™! (8.6)
Here m is the electron mass, and o,, is the characteristic cross
section for collisions between electrons and the neutral
gas particles whose density is NV. |
Ambipolar diffusion. Assume that a plasma has been
formed in a small volume of gas and it tends to spréad over
the whole gas volume. Let us analyze the decay of this
plasma. In our case, the charged particles are spreading in
the gas volume via the diffusion mechanism. Equation
(7.2) shows that the electrons have a considerably larger
diffusion coefficient than the ions; therefore, the electrons
are spreading over the gas volume considerably faster than
the ions. This results in a disturbance of quasineutrality of
the plasma and in the emergence of electric fields in the
plasma created by the charged particles.
The electric field E created due to the spatial distribu-
tion of the charged particles satisfies Poisson’s equation:
div E = 4ne (N; — N,) (8.7)
where NV, and WN; are the densities of the electrons and
ions*. The flux densities of the electrons and ions in the
system are the sums of the diffusion flux density and the
flux density due to the electric field:
ji. = — D, grad N, — KNEE; (8.8a)
1 a D; grad N; = K;NiE (8.8b)
In the above equations HD, and JF; are the diffusion coeffie
cients of the electrons and ions in the gas, and K, and K,
are the mobilities of the electrons and ions.
* From now on the ions are assumed to be singly charged,
102 Introduction to Plasma Physics
Let us consider such a mode of development that the
plasma remains quasineutral in the process of motion.
This is the case for relatively high densities of charged par-
ticles; the separation of charges gives rise to large fields
which prevent further separation and preserve the quasi-
neutrality of the plasma. This phenomenon is termed the
ambipolar diffusion. In this case VN, = N; = N so that
AN =|N.—N;|<N, and the fluxes of the electrons
and the ions are the same.
Let us analyze Eqs. (8.8) for the electron and ion flux den-
sities in the case of the ambipolar diffusion. Equation (7.2)
for the diffusion coefficient and Eq. (8.4) for the mobility
show that each of the terms in Eq. (8.8a) for the electron
flux is much larger than the respective term in Eq. (8.8b)
for the ion flux. For the electron flux to be equal to the ion
flux, the first term in Eq. (8.8a) must almost cancel the
second term. This means that the electric field strength
must be E = — (9,/K,) X (grad N)/N = — (T,/e) X
X (grad N)/N. Inserting this into Eq. (8.8b), we find the
flux density of the charged particles:
j= — XH, grad N (8.9)
where
D,=5;1 +77; (8.10)
is known as the ambipolar diffusion coefficient. In the
above analysis, we made use of the Kinstein relation (7.17)
and assumed that the electron temperature 7, differs from
the ion temperature 7;. Thus, the ambipolar diffusion of
the charged particles is a diffusion-like motion with the
time parameter corresponding to the ions.
Let us determine the conditions under which the ambi-
polar diffusion occurs. Assume that the characteristic size
of the plasma at which the plasma parameters vary notice-
ably is L. The above expression for the electric field strength
yields the following estimate (grad N ~ N/ZL):
E w~ T JeL
Poisson’s equation yields
= | div E | T
AN =|Ne-Nil—"Ga ™ dea
Transport of Charged Particles 103
Using the Debye-Hiickel radius (3.6), we find that
ies ( 2)
N L
With the above condition for ambipolar diffusion, AN < N,
we obtain the following condition for the realization of
ambipolar diffusion:
rm<bL (8.11)
that is, the Debye-Hiickel radius of the plasma must be con-
siderably smaller than the characteristic size.
The mobility of ions in a foreign gas. Equation (8.4)
gives an estimate of the mobility of ions for small electric
fields. Now we shall analyze ion motion in gas under rea]
conditions for arbitrary electric fields. First, consider the
motion of ions in a foreign gas where the scattering of the
ion is due to elastic collisions between the ion and the
gas particles. For low collision energies, the cross section
of the collisions is determined by the polarization interac-
tion between the ion and the gas particle: UV = — Be?/ (2R*)
(here B is the polarizability of the gas particles, e is the ion
or electron charge, and A is the distance between the ion
and the particle). Equation (4.11) then gives the cross
section for the capture of an ion by an atom:
2 \1/2
Ocapture = an (=) (8.12)
Here ¢ = pv?/2 is the energy of the relative motion of the
particles, up is the reduced mass, and vis the relative velocity
of the collision. The frequency of collisions between the
ions and the gas particles (atoms) does not depend on the
velocity of collision.
Taking into account that the mean energy of ions, s, is
of the order of the thermal energy, we find from Eqs. (8.4)
and (8.12) the mobility of the ions as a function of the
parameters of the problem:
1
K~ yg (8-43)
The mobility of ions in the parent gas. When ions travel
in the parent gas, their mobility is determined by the reso-
nance charge exchange between ions and gas particles,
104 Introduction to Plasma Physics
This is the case when an ion can be produced from the gas
particle (atom or molecule) by removing one electron. The
resonance charge exchange can be described by the following
scheme:
At++A—>A +A? (8.44)
Here the tilde designates one of the nuclei. The process
FIG. 15. ~
The relay process lon
of resonance charge
exchange between lon scattering
an ion and an atom. angle
Before collision
After collision
consists in the electron transferring from one atomic core
to another.
For thermal collision energies, the cross section of the
resonance charge exchange is considerably larger than the
cross section of the elastic collision between the ion and
the atom. Therefore, the main contribution to the cross
Section of the resonance charge exchange is made by the
collisions in which the atom and the ion travel along straight
lines. Under these circumstances the ion travels in the gas
via a relay mechanism (see Fig. 15). The resonance charge
exchange changes the path of the ion since it is associated
now with another nucleus. Therefore, into Eq. (8.4) for
the ion mobility for low electric fields we must insert the
cross section of the resonance charge exchange instead of
the cross section of the elastic collision between the ion
and the gas particle.
Let us analyze the ion mobility in the parent gas for
high electric fields when the mean ion energy is consider-
ably higher than the thermal energy of the gas particles.
Under such conditions the gas particles colliding with the
ion can be assumed to be at rest. Hence, we can visualize
the motion of the ion in the following way. The resonance
Transport of Charged Particles 105
charge exchange between the ion and the gas particle gives
rise to an ion with a zero velocity, the ion then being accel-
erated by the electric field until the next charge exchange
between it and a gas particle takes place. Therefore, the
only nonzero velocity component of the ion, v, = (eE/M;)t,
is along the electric field (MV; is the ion mass, and ¢ is the
time elapsed since the last charge exchange).
Denote by W (¢) the probability that the ion will not
make a charge exchange with a gas particle during the time
t after the last charge exchange. This probability function
satisfies the equation of radioactive decay:
dWw
ap — vW
here v = Nv,Ores, iS the frequency of the resonance charge
exchange, N is the density of the gas particles, and Gres.
is the cross section of the resonance charge exchange. The
solution of this equation is
t
W (1) =exp[ — | vr) ae’]
0
The velocity distribution of ions is described by the function
f (v,,.). Since the ion velocity v, along the field is determined
by the time ¢ elapsed since the last charge exchange, the
distribution function f (v,.) is proportional to the probability
that the ion has not exchanged charge up to the moment f.
Using the normalization condition, we find the distribution
function :
2M; 1/2 M; x
f (Vx) =N; ( nme AX ( = a ) ? Vv, >0 (8.15)
Here N; is the ion density, A = 1/(No;es.) is the mean
free path of the ions, and the cross section of the resonance
charge exchange is assumed independent of the velocity,
as is the case in reality. Hence, we obtain the following
expression for the mean velocity and the mean energy of the
ion:
1/2
v= W;)= ( a (8.16)
(zMwty=ZeEh (8.17)
106 Introduction to Plasma Physics
It may be seen that for high electric fields the ion mobility
K varies with the field strength as E-'/?. The above equations
hold when the characteristic ion energy is considerably
higher than the thermal energy:
eEXS>T (8.48)
Equation (8.18) is the criterion for a high field in terms of
this problem. When the condition (8.18) is not satisfied, the
drift velocity of the ion is small compared to the thermal
velocity, the velocity distribution function for the ions is
close to the Maxwell distribution, and the ion mobility does
not depend on the electric field strength for any types of
collisions between the ion and the gas particle. Under these
circumstances the electric field is assumed to be small. When
the condition (8.18) is satisfied, these relations do not
hold.
Recombination of ions in a dense gas. Let us analyze
recombination of the positive and negative ions in high-
density gas for which the conditions opposite to conditions
(5.14) and (5.21) are satisfied. Under such circumstances,
frequent collisions of the ions with the gas particles prevent
them from approaching each other and thus the time of
recombination is, primarily, the time required for approach-
ing. If the separation of the ions is R, each ion is under the
action of the field produced by another ion whose strength
is e/R*. This field makes the ions move towards each other
with a velocity w = e (K; + K_)/R? where K, and K_ are
the mobilities of the positive and negative ions in the gas.
The expression for the drift velocity w is valid for R >A
where A is the mean free path of the ions in the gas.
Now let us determine the frequency of decay of the posi-
tive ions due to recombination. Imagine a sphere of radius R
around the positive ion and compute the number of negative
ions entering this sphere per unit time. The number is the
sphere surface area times the flux density of the negative
ions N_w = N_e (Ky + K_)/R*. Proceeding from this,
we write the balance equation for the density N, of the
positive ions as
aN,
dt
= — N,N_Ane (K++ K-)
Transport of Charged Particles 107
A comparison with Eq. (5.1) yields the recombination coef-
ficient
‘q@ = 4me (Ky + K_) (8.19)
Equation (8.19) is known as the Langevin formula.
Let us consider this type of recombination in more detail.
The ions approach under the action of the Coulomb force
FIG. 16.
The recombination
coefficient of
positive and negative
ions as a function
of gas density.
and collide with the gas particles in the course of this ap-
proach. When the distance between the ions is R, we have
e"/R? < T; this relation is the opposite of the condition
(8.18). Hence, the mean kinetic energy of the ions is the
same as for an infinite distance between the nuclei. This
is the case also for R ~ b [see Eq. (5.12)], when the poten-
tial of the interaction between the ions is of the order or more
than their thermal energy, that is, the ions are in a bound
state. Thus, the main difficulty encountered in this type
of ion recombination is not the production of the bound state
as is the case for the triple recombination but the approach
of the ions up to the distances at which charge exchange can
occur. The higher the gas density the more difficult is the
approach, and Eqs. (8.4) and (8.19) indicate that the recom-
bination coefficient for ions in a dense gas decreases with
increasing gas density.
The recombination coefficient of ions as a function of
gas density. We have determined the recombination coef-
ficient of the positive and negative ions in a gas under various
conditions. Thus we have the general picture of the ion-
ion recombination. Figure 16 presents the qualitative rela-
tionship between the recombination coefficient and the gas
density. Let us analyze this relationship.
For low gas densities (region 7) recombination is due to.
the pair collisions of ions, the recombination coefficient
108 Introduction to Plasma Physics
is given by Eq. (5.9), and the order-of-magnitude estimate
of the coefficient is a, > h?/(m*uT)'/? since in Eq. (5.9)
we have Ry > ay. Here p is the reduced mass of the ions,
m is the electron mass, and a) = h?/(me?) is the Bohr radius.
The recombination coefficient in region 2 is given by Eq.
(5.20); we have a, ~ IC] (e§/T%) (Be?/M)1/2 where B£ is the
polarizability of the particle species C, [C] is its density
and M is the mass of particle C.
The gas densities of the order of N, ~ a, (T/e)*/*B-'/? cor-
respond to the transition from region / to region 2.
Region 3 is described by the Langevin theory and Eqs.
(8.13) and (8.19) give the following order-of-magnitude
estimate for the recombination coefficient in this field:
a; ~ e([C]/(Bu)/?. The transition between regions 2 and
3 corresponds to the density VN, ~ (T/e*)?/*B-1/2, This density
corresponds to the greatest recombination coefficient @max ~
~ ey-1/?7-3: it is found for the pair process assuming the
Coulomb interaction cross section. Thus, the greatest recom-
bination coefficient has the same order of magnitude as the
rate constant for the elastic collision of ions.
Given below are the numerical values of the qualities
indicated in Fig. 16 for the ions produced in air at room
temperature: NN, ~ 107 cm-’, N, ~ 10 cm, a ~
~ 10-® cm®s-!, and Gmax ~ 107® cm?s7!.
9 Plasma in External Fields
The electron motion in a gas in an external field. Let us
consider the motion of electron gas to which electric and
magnetic fields are applied. To make a correct qualitative
description of the relevant phenomena let us make use of
the simplest form of the collision integral, that is, the tau
approximation (6.4). The kinetic cquation for the electron
distribution function can be written as
of , F Of f—fo 7
ot ' mov ))6ht (9.4)
where F is the force of the external field acting on the elec-
tron, m istheelectron mass, f, is the equilibrium distribu-
tion function realized in the absence of external fields,
1/,; = N vO, is the frequency of the elastic collisions of elec-
Plasma in External Fields 109
trons and the gas particles, NV is the gas density, v is the
electron velocity, and o,, is the cross section of elastic col-
lisions of electron and the gas particles. To simplify the
analysis we shall assume that t is independent of the electron
velocity.
Let us study the force F of the external magnetic and
electric fields acting on the electron. We assumé that the
electric field varies harmonically and the magnetic field is
constant and normal to the electric field. From this gene-
ral case we can derive all the special cases of interest for us.
The force which acts on the electron is
F = cE exp (iot) +— vXH (9.2)
where E and H are the electric and magnetic fields, w is
the frequency of variation of the electric field, and v is the
electron velocity. Let the vector H be parallel to the z axis
and the vector E lie in the zy plane, that is, H = kf and
E = iE, + j-,. Here i, j, k are the unit vectors along the
axes zr, y, Z, respectively.
The kinetic equation (9.1) yields the equation for the
‘mean velocity of the directional motion of an electron;
w = Nz} \ vf dy. We multiply Eq. (9.1) by the electron
velocity, divide it by the electron density, and integrate
over the electron velocities. The first term gives
1 of > alt _ dw
ne \ Var 8 = ar (az | viev) =F
Using expression (9.2) for the force in the second term in the
modified Eq. (9.1) and integrating by parts, we get for this
term the following formula:
e ; Of eE :
aN, °%P (iat) | v (E=-) dv = — man, exP (iwt) \ fdv
eK :
= aa exp (iwt)
The third term transforms into
e
war) v((v XH) st) dv
140 Introduction to Plasma Physics
The right-hand side of the equation can be transformed as
follows, taking into account that f, is a spherically symmetri-
cal function and t is independent of the velocity:
4 (f — fo) _ Ww
oye \ Sree Ne ne are
Finally, we obtain the following equation for the mean ve-
locity of the electron:
was. 03)
at ct 386m
ee ed
Equation (9.3) is the equation of motion for the electron
travelling in the field of an external force F. The collisions
between the electron and the gas particles are accounted for
by the frictional force mw/t.
The conductivity of a weakly ionized gas. Let us write
the components of Eq. (9.3). We denote by w) = eH/(mc) the
so-called cyclotron frequency and introduce the parameters
a, = eE,/m and a, = eE,/m. Hence, we obtain the following
system of equations:
dw w ;
—* +-—* = a, exp (iwt) + wy,
dt tT
ad ke
at t = Oo“ x)
dwz Wy
7 ae = a, exp (iat)
We wish to find the steady-state solution of this system of
equations, which does not depend on the initial conditions.
It has the following form:
io+ 11 :
Wy = 7-2-1 (@2— o2) + 2i0t-! a, exp (iw?)
° A)
Wy = oF a) poor! (oF a) Toit tx EXP (iwt) (9.4)
citing ae
wW,= joe! az exp (io )
The solution (9.4) may be used for analyzing the behav-
iour of a weakly ionized plasma in the external electric and
magnetic fields. The electric properties of a weakly ionized
plasma are due to the motion of electrons whose mean veloc-
Plasma in External Fields 111
ity is given by Eq. (9.4). For instance, for a constant elec-
tric field (@ = @) = 0), Eqs. (9.4) yield
eE
w=—tT (9.5)
m
and the electron mobility is
Ke=—1 (9.6)
which is in agreement with Eq. (8.4): t7! ~ No (T/m)¥?.
The conductivity of a weakly ionized plasma in this case
is given by Eq. (8.5):
dip = N,€?t/m (9.7)
which agrees with Eq. (8.6).
The Hall effect. When a magnetic field is applied to
a weakly ionized plasma, its conductivity ceases to be a sca-
lar. The motion of electrons under the action of the magnetic
field gives rise to electric current perpendicular to the
electric field, apart from the current parallel to the electric
field. The electric current in the weakly ionized plasma in
the direction i is >} E,;, where E, is the electric field
P
strength in the direction of k, and 2;,is the conductivity
tensor. According to Eqs. (9.4), this tensor has the form
1+ iot WoT
1+ (@3 —@2) t2-+- 2iwt §=1-++ (02 — 2) T2-++- 2iwt
Wot 4+ iot
2p 1-+ (2 — w2) 12-4 2iwot = 1+ (w27— 2) 12+ 2iwt 0 (9.8)
1
0 0
1-++ iwt
where 2 is the conductivity in a constant electric field
lsee Eq. (9.7)]. The imaginary terms in the conductivity
tensor describe the phase shift of the electric current in the
plasma with respect to the electric field. We can now use
(9.8) to analyze various limiting cases.
(i) A weakly ionized gas in a variable electric field (wy = Q).
The conductivity in this case is 2 = 2,/(1 + iwt). If the
variation frequency of the electric field is small (wt < 1),
an electron is slowed down because of the collisions with
the gas particles, and the conductivity of the plasma is deter-
{42 Introduction to Plasma Physics
mined by the conductivity 2» for a constant electric field.
For high electric field frequencies the electron changes its
direction of motion because of the variation of the polarity
of the electric field. Since collisions with the gas particles
in this case are not significant, t does not enter the expres-
sion for the conductivity:
a2“ (9.9)
uma)
Because the conductivity is an imaginary quantity, the
phase lag between the electric current and the field is 1/2.
(ii) A weakly ionized gas in skewed constant electric and
magnetic fields. In this case the current has two components:
one along the electric field and the other perpendicular to
the electric and magnetic fields; the respective components of
the conductivity tensor are
z = Doron (9.10)
1
N= oT pore?
For high magnetic fields (w)t > 1)* the resulting current
is perpendicular to the electric and magnetic fields. The
conductivity in this limiting case does not depend on the
collision time t and is given by
2
a i (9.14)
Wot mMWo
If the transverse electric current does not pass the bounda-
ry of plasma, it results in a separation of charges, thus giving
rise to an electric field that stops the electrons. This gives
rise to a potential difference in the direction perpendicular
to the electric and magnetic fields. The phenomenon is
known as the Hall effect.
The cyclotron resonance. When wt > 1 and a) t > 1, the
conductivities in the directions parallel or perpendicular
* We assume that o,t < M/m where M is the ion mass, and m
is the electron mass. Otherwise, the ions will drift in the same direc-
tion and with the same velocity as the electrons do, and the conduc-
tivity of plasma will be small compared to the value given by Eq. (9.11).
Plasrna in External Fields 443
to the electric field are
2) =
ay
| Saeaeee (9.12)
s 9 © — i(w@§—o?) |
Wo 2W ot}
The conductivity is seen to have a resonance at © = @p,
and its value is 2) = 2, = 2/2. The resonance width is
Aw ~ 1/t. This conductivity resonance at an electric
field frequency w equal to the cyclotron frequency @, is
called the cyclotron resonance.
The cyclotron resonance has a simple physical meaning.
In a magnetic field, an electron travels along a circular orbit
with the cyclotron frequency @,. If an electric field is ap-
plied in the plane of circular orbit and if this field varies
so that its direction remains parallel to the electron velocity,
the electron continuously receives energy from the field.
Similar to electron motion in the constant electric field, the
electron is accelerated until it collides with an atom. Hence,
in both cases the conductivities are of the same order of
magnitude and are expressed in terms of the frequency 1/t
of collisions between the electron and the atom. If o ~@p,
the conductivity is considerably lower since the conditions of
interaction between the electron and the field are not opti-
mal.
The mean electron energy. Let us estimate the mean
energy of electrons travelling in a gas in a constant electric
field. Since the electrons take on energy from the electric
field, their mean energy can be much higher than the ther-
mal energy of the gas particles. When determining the mean
energy of electrons, we shall take into account only their
elastic collisions with the gas particles.
To do this we have to write the balance equation for the
electron energy since the electrons obtain energy from the
electric field and dissipate it in the collisions with the gas
particles. Let us determine the mean energy given up by
the electron and obtained by the gas particle in the elastic
8—01607
114 Introduction to Plasma Physies
collision between them. This energy is
__(P+AP)2 P2 es) 4 PP
SS REE ED
— —— =—
— Fon) Py er Pr
Here P is the momentum of the gas particle before the colli-
sion, AP is the variation of the momentum of the gas particle
due to the collision, Ap is the variation of the momentum of
the electron due to the collision (conservation of momentum
yields AP =— Ap), and M is the mass of the gas particle.
The mass of the gas particle is considerably larger than
the electron mass, so that in calculating the electron momen-
tum we can assume the gas particle to be at rest and to have
an infinite mass. Therefore, the momentum p of the electron
and its variation Ap in collision with the gas particle do not
depend on the momentum | of the gas particle. Averaging
of the expression for Ae over all the directions of the electron
momentum yields Ae =Ap?/(2M). Since Ap ~ p and p ~
~ (me)1/* where p is the electron momentum and ¢ is the
mean electron energy, the energy given by the electron to
the gas particle in one collision is Ae ~ meé/M, and the
energy given up by the electron per unit time is Ae/t ~
~ me/(Mt).
The energy obtained by the electron per unit time from
the electric field is e(Ew) where w is the velocity of the
directional motion of the electron given by Eq. (9.9): w =
= eEt/m. Equating the energies obtained and dissipated
by the electron per unit time, we find the following estimate
for the mean electron energy:
e~ M (== )" (9.13)
This estimate holds if the mean electron energy is high
compared to the thermal energy of the gas particles, so that
the electron only gives up the energy to the gas, and alsoif
the mean energy is small compared to the characteristic
electron energies at which there is a significant contribution
from inelastic processes.
Plasma in Externa/ Fields 445
Using a similar approach, let us find the mean energy of
the electron travelling in the gas in scewed constant electric
and magnetic fields. Equation (9.4) indicates that the
electron takes on from the field the energy ekw=t(eE)* X
x (4 + w2t?)-1/m so the order-of-magnitude estimate of
the mean electron energy is
e~M(2=)"(S+03)" (9.44)
Let us, finally, consider the motion of an electron in the
gas in the variable electric field. If the electric field is
Ecos wt, we obtain the following expression for the drift ve-
locity of the electron along the field by taking the real
part of Eq. (9.4):
We = Zz (o? +=) (— cos wt +wsin at)
Mm
Hence, we find the energy obtained by the electron per unit
time (averaged over time):
ee (eE)2 ‘ oS -1
CHW, = 2mt (o pg 2
Thus, we find the mean energy of an electron travelling in
a gas in a variable electric field:
= eH \27/ 1 =—4
The magnetohydrodynamic equations. If the plasma den-
sity is sufficiently high, we must take into account the fields
produced by plasma motion. These fields, which are due to
distribution and motion of plasma, affect the motion of
plasma, that is, the plasma parameters and the fields pro-
duced by plasma are interrelated. The motion of plasma and
variation of its parameters can be described by the conti-
nuity equation for the density of electrons and ions (6.6),
the equation for the mean momentum of the electrons and
ions (6.8), Poisson’s equation (3.2), and the Maxwell equa-
tions. The resulting system is called the system of equations
3*
116 Introduction to Plasma Physics
of magnetohydrodynamics:
oN + div (Nw) =0, |
0 d F
B+ ww) wt Sige — Fp =
div K=4ne(N;—N,),
4n. 1 0B (9.16)
curlH =——j—— a
1 oH
curl BS 5
div H=0O
The first two equations can be written both for the electrons
and for the ions; therefore, w and N are the velocity and
‘the density of the electrons or the ions, respectively. To (9.16)
we must add the equation of state of the type (6.14) and the
thermodynamic equation for the process (for instance, the
adiabatic equation of the process if the variation of the
parameters of the system is an adiabatic process). These
equations interrelate the density, temperature, and pressure
of the plasma particles. We must add to these equations also
Ohm’s law relating the plasma current and the electric field
strength. The system of magnetohydrodynamic equations
with the addition of the above-mentioned equations and
the initial conditions will giveadefinite description of plas-
ma development.
Let us analyze the plasma motion in which the velocity
of electrons is considerably greater than the velocity of ions.
Then the electric current is due to the electrons and is
given by
j= — eN.w
where w is the directional velocity of the electrons, and NV,
is their density. If the motion occurs in a magnetic field,
an additional electric field is produced in the laboratory
frame of reference, the field given by
pan 1,
BE’ = —wXH = — 3 iXH (9.17)
This field acts on the electrons giving rise to an additional
force acting on the entire plasma. The force acting on the
Plasma in External Fields 417
unit volume of plasma is
ecE'N.=—+iXH (9.18)
If the plasma conductivity is sufficiently high, its response
to the electric field (9.17) will result in the movement of
electrons. This movement will continue until separation of
the electrons and ions gives rise to an internal electric
field in plasma
E=—+wXH = (9.19)
which will compensate the field (9.17). We insert Eq. (9.19)
into the Maxwell equation —(0H/dt) = c curl E and obtain
the following equation for the variation of the magnetic
held:
&* -- curl (wXH) (9.20)
Let us analyze the variation of the magnetic field and
the plasma motion when the electric current is due to the
electrons and the plasma conductivity is high. Transform
Kq. (9.20) by writing curl (w & H) = wdivH + (HV)w—
—(wy) H — Hdiv w, noting that according to the Maxwell
equation div H = 0, and using the expression for div w
from the continuity equation for the electrons, N, xX
<div w = —(0N,/dt) — (wy)N.-. We obtain
oH sO ON, H 7
We divide this equation by NV, and find that
£(H)=(Ey)~ oa
where
d H ) H H
a (a) =a (w,) +O) x,
is the derivative at the point which travels together with
the plasma.
Let us analyze the motion of the elementary plasma vol-
ume with the length dl and the cross section ds which
118 Introduction to Plasma Physics
contains V.dsdl electrons. At first, assume that the vector
dl is parallel to the magnetic field H so that the magnetic
flux through this elementary plasma volumeis Hds. Let us
determine the time dependence of the magnetic flux. If the
plasma velocity at one end of the segment dl is w, then at the
other end the velocity is w + (dl V)w so that the variation
of the segment length during a small time interval 6¢ is
5¢ (dlV)w. Hence, the length of the segment satisfies the
equation
d
—— (dl) = (div) w
which is identical to Eq. (9.21). This fact indicates that
(1) at all times the segment dl has the same direction as the
magnetic field and (2) the length of the plasma element re-
mains proportional to the quantity H/N,, that is, the
magnetic flux through this plasma element does not vary with
time during plasma motion. Thus, the lines of magnetic force
are “frozen” into the plasma, that is, their direction is
such that the plasma electrons travel along these lines.
Remember that this is the case when the plasma conductiv-
ity is high.
Let us consider the steady-state motion ofa high-conductiv-
ity plasma. According to Eq. (9.19), each plasma electron
is under the action of a force F = —eK = (e/c) w X H =
= —(cN,)“! j & H. Inserting into the expression for the
force the current density j = c (4n)7! curl H, we obtain
1
. 4
= — Ne XH = 7, HxXcurl H
= os E grad H?—(HV)H] — (9.22)
Let us substitute Eq. (9.22) in the second equation of the
system (9.16).We assume that the velocity of the direction-
al motion of the electrons is considerably greater than
their thermal velocity. Hence, we can neglect the term
(wy)w compared to the term (WN ,)~' grad p and obtain the
following equation:
grad ( p+ =) ~7-(Hv)H=0 (9.23)
Waves in a Plasma 119
The quantity H?/(8x) is termed the magnetic field pressure;
it is the pressure with which the magnetic field acts on
the plasma.
Let us analyze the properties of a cylindrical plasma col-
umn maintained by a direct current. Here the lines of mag-
netic force are straight lines and Eq. (9.23) in the direction
perpendicular to the field has the following form:
grad, [p + H?/(8x)] = 0. This equation shows that the
total pressure p + H?/(8x), which is the sum of the gas-
kinetic pressure and the magnetic field pressure, is independ-
ent of the transverse coordinate. Let the radius of the plasma
column be a and the current in it be J so that the magnetic
field at the surface of the column is H = 2I/(ca). The total
pressure outside the column near its surface equals the mag-
netic field pressure [?/(2c?a*), and the total pressure inside
the plasma column equals the gas-kinetic pressure p. Equa-
ting these two pressures, we find the radius of the plasma
column |
I
ee Tire Onp)ia (9.24)
An increase in the current in the plasma column accompanied
by the respective increase in the magnetic field is seen to
give rise to a contraction of the plasma column. This phe-
nomenon is called the pinch effect, and the state of the
plasma column itself is known as the z pinch.
10 Waves in a Plasma
Although plasma, as a consequence of the interaction be-
tween particles, is a gas, oscillations and noises play a much
greater role than in ordinary gas. First, in a plasma that is
located in external fields and is not homogeneous, a wide
variety of oscillations can occur because of the long-range
interaction between particles. Secondly, these oscillations
vary frequently and become amplified to a relatively high
energy. In this case, the plasma oscillations determine its
parameters and development. We shall analyze the simplest
types of oscillations in gas and plasma.
120 Introduction to Plasma Physics
Acoustic oscillations. The natural vibrations of gas are
the acoustic vibrations, that is, the waves of compression
and rarefaction which propagate in gas. We have to find the
relationship between the frequency @ of the vibrations and
their wavelength 4, which is related to the wave vector k
as follows: k = 2n/i.
In our analysis, we shall assume that the amplitude of
the system’s oscillations is small. Thus, any macroscopic
parameter of the system can be expressed as
A=A,+ >) Ao exp [i(kx—ot)] (10.1)
where Ay is the macroscopic parameter in the absence of vi-
brations, Aj is the amplitude of vibrations, w is the fre-
quency of vibrations, and k is the respective wave number
(the wave propagates along the z axis). Since the vibration
amplitude is small, the vibration with a given amplitude
does not depend on the amplitudes for other frequencies.
That is to say, there is no interaction between waves of differ-
ent frequencies owing to the smallness of their amplitudes.
Therefore, we can retain only one termin the sum in Eq.(10.1)
and express the macroscopic parameter A in the following
form:
A = Ay +A’ exp li (kx — ot)] (10.2)
In our analysis of the acoustic oscillations in gas let us
write the gas density NV, the gas pressure p, and the mean
gas velocity w in the form (10.2); we shall assume further
that the gas is at rest, that is, wy) = 0. Using the continuity
equation (6.6) and neglecting the terms with squared oscil-
lation amplitudes, we obtain
oN’ = kNww’ (10.3)
Note that the velocity w’ of gas propagation with the acous-
tic wave is directed along the wave vector k.
Similarly, the Euler equation (6.15) yields, up to quad-
ratic terms, the following equation:
ow’ = a (10.4)
Assume that the regions of compression and rarefaction of
gas do not have time to exchange energy during propagation
Waves in a Plasma 121
of the acoustic wave, that is, the process is adiabatic. The
larger the wavelength, the better satisfied are the adiabati-
cily conditions. The parameters of the acoustic wave satisfy
the adiabatic equation
pN-Y = const (10.5)
where yp =c,/cy is the adiabatic exponent, cp is the specific
heat al constant pressure, and cy is the specific heat at con-
stant volume. Using the expansion (10.2), we find the fol-
lowing relationship between the wave parameters:
p’ N’
ee (10.6)
Eliminate the amplitudes from Eqs. (10.3), (10.4), and
(10.6) and make use of the equation of state (6.14), pp) = NT.
The result is the relationship between the acoustic fre-
quency w and the wave number k:
T \1/2
Q) = (y ar) k (10.7)
Hquation of the type (10.7), which relates the wave fre-
quency lo the wave number, is called the dispersion relation.
The sound propagation velocity (@w/0k) = (yT/M) '7, which
is called the group velocity, is seen to be of the order of the
thermal velocity of the gas particles.
Plasma oscillations. Let us analyze the oscillations
which are due to the motion of charged particles in a plasma
or a weakly ionized gas. In the simplest case of a homo-
geneous plasma and in the absence of external fields, there
are two kinds of the natural plasma oscillations since plasma
has two species of charged particles. These kinds of oscilla-
tions differ considerably since the electrons and ions res-
ponsible for them differ greatly in mass.
Let us study the high-frequency oscillations of the homo-
geneous plasma. These oscillations are due to electron mo-
tion; they are referred to as plasma waves. Because of their
large mass, the ions are not involved in these oscillations,
and when analyzing plasma waves, we shall assume the
ions to be at rest and their charge uniformly distributed
over the gas volume.
Similar to acoustic oscillations in gas, we shall derive
the dispersion relation for the plasma waves from the con-
422 Introduction to Plasma Physics
tinuity equation (6.6), the Euler equation (6.15), and the
adiabatic equation (10.5) for the wave. Moreover, we must
take into account the electric field produced by the motion
of electrons owing to disturbance of the quasineutrality of
plasma. We shall introduce the electric field term into the
Euler equation (6.15) while the electric field strength will
be given by Poisson’s equation (3.2).
Similar to the derivation of the dispersion relation for
acoustic oscillations, let us assume further that the macro-
scopic parameters of the oscillating plasma can be written
in the form (10.2) and that in the absence of oscillations the
mean velocity w of electrons and the electric field strength EF
are zero. Hence, we obtain
—ioNe + ikNow' = 0, )
—iow’ +i HP -| = 0, «|
aa (10.8)
Pp No’ |
ikE’ = —4neN,
Here k and @ are the wave number and the frequency of the
plasma oscillations, N, is the mean density of charged
particles, py = N,mi(v;z) is the electron gas pressure in the
absence of oscillations, m is the electron mass, v, is the
electron velocity component in the direction of oscillations,
and the angular brackets denote averaging over the electron
velocities. The quantities Ni, w’, p’, and LE’ in Eq. (10.8)
are the oscillation amplitudes of the electron density, mean
velocity, pressure, and electric field strength, respectively.
Eliminate the oscillation amplitudes of the macroscopic
quantities from the system of equations (10.8). Hence, we
obtain the following dispersion relation for the plasma os-
cillations:
w? =: wf +p (Ux) Ke (10.9)
where w, = (4nN,e?/m)'/? is the plasma frequency [see
Eq. (3.9)].
Note that plasma oscillations are longitudinal in contrast
to electromagnetic oscillations. Hence, the electric field due
Waves in a Plasma 423
to plasma waves is directed along the wave vector. This fact
was used in deriving the system of equations (10.8).
Ion sound. We consider now the oscillations that are
due to the motion of the ions in a uniform isotropic plasma.
The peculiarity of these oscillations is determined by the
large mass of the ions. Owing to its small mass, an electron
can follow the motion of an ion so that the plasma remains
quasineutral in any large volume containing a large number
of charged particles:
N= WN;
Moreover, the electrons have time to redistribute due to the
electric field in the plasma; this distribution is given by the
Boltzmann formula
Ne = No exp (e/T) ~ Ny (1 + e@/T)
where @ is the electric potential due to the oscillation of
plasma. Hence, we obtain the amplitude of oscillation of
the ion density:
Ni=N, tr (10.10a)
where 7’, is the electron temperature.
Let us use the motion equations for ions. The continuity
equation
aN; , @
” Sr tae (Nii) = 0
yields
oN} =kNyw; (40.40b)
where w is the frequency, k is the wave number, and w, iS
the mean ion velocity due to the oscillations. Here wé as-
sume the usual harmonic dependence of the oscillation pa-
rameters on position and time [see Eq. (40.2)].
Furthermore, let us make use of the equation of motion
for ions due to the electric field of the wave M (dw;,/dt) =
= eE = — e grad @ where &/ is the ion mass. Taking into
account the harmonic dependence (10.2) on position and
time, we obtain |
Mow; = ekp (10.10c)
When we eliminate the oscillation amplitudes of Ni, p,and
w; in the system of equations (10.10), we obtain the following
124 Introduction to Plasma Physics
dispersion relation between the frequency and the wave
number:
Pa yt
=: (<5 ] k (40.44)
The oscillations caused by the ion motion are known as the
ion sound. Similar to the plasma oscillations, the ion sound
is a longitudinal wave, that is, the wave vector, k, is paral-
lel to the oscillating vector of the electric field, E. The
dispersion relation for the ion sound is similar to that for
ordinary sound. This is due to the fact that both types of
oscillations are characterized by a short-range interaction.
In the case of the ion sound, the interaction is short-ranged
because the electric field of the propagating wave is screened
by the plasma. This screening is effective if the wavelength
of the ion sound is considerably larger than the Debye-ITtickel
radius for the plasma where the sound propagation occurs:
Arp <1
The dispersion relation (10.11) for the ion sound is valid
if this condition is satisfied.
Magnetohydrodynamic waves. Let us study the waves
developing in a high-conductivity plasma which travels in
a magnetic field. The magnetic lines of force in this case are
“frozen” into the plasma so that if the plasma current is
“shifted” somewhat so as to displace the magnetic lines of
force, the magnetic field acts so as to shift the plasma current
back. This gives rise to waves known as magnetohydrody-
namic waves.
Consider the magnetohydrodynamic waves whose wave-
length is much less than the radius of curvature of the
magnetic field:
1 H
ES | grad H |
where k is the wave number, and #H is the magnetic field
strength. Hence, the magnetic force lines can be assumed
to be straight. The system of magnetohydrodynamic equa-
tions (9.16) can be used to describe the waves. As above, we
Waves in a Plasina 135
write the plasma parameters as sums of two terms:
‘H = Hy +h exp [i (kr — o/)],
N = Ny +N’ exp Li (kr — od)],
P = Po + P’ exp Li (kr — of)]
and also
w= w’ exp li (kr — of)]
The first term refers to the plasma in the absence of oscilla-
tions and also when the oscillation amplitude is small and does
not depend on position and time. Furthermore, we assume
that the plasma is at rest in the absence of oscillations and
its pressure is zero, that is, small compared to the magnetic
field pressure. Inserting the plasma parameters in the above
form into the system of magnetohydrodynamic equations
(9.16), we obtain the following system of equations for the
oscillation amplitudes of the plasma parameters:
kh--0, —oh=k X<(w>X H,), |
wN’ = kwNo, 10.12)
ow = aaa «(KX h)
Introduce a new coordinate system in which the wave
vector k is parallel to the zx axis and the magnetic field
strength H,is inthe zy plane.The first equation (10.12) indi-
cates that magnetic field amplitude h has two components:
h, and h,. Then we write the remaining equations (10.12) in
components:
—= wh, = kw,H ., )
-— why = k (wyHo, — WHoy),
wN’ = kw,No,
H
ow, = —k aha, (10.13)
H
WW, == —k Tany by
ow, =k tay
0
136 Introduction to Plasma Physies
The system (10.13) can be divided into two independent
systems. One of these systems corresponds to magnetic
field oscillations along the z axis. The oscillation frequency
is related to the wave number by the following dispersion
relation:
Hok
(4N o)1/2
Another system of equations describes the oscillations in
the zy plane; it yields the same dispersion relation:
___Hok
—~ “GaN 1/2
OoO=
(10.14)
0)
(10.45)
Thus, when the wave travels perpendicular to the magnetic
field, the dispersion relations for the waves of both types
are the same. The phase velocity of the wave coincides with
the group velocity and both are equal to
Ho
The velocity u is known as the Alfven velocity, and the mag-
netohydrodynamic waves are known as the Alfven waves.
Propagation of electromagnetic waves in a plasma. Let
us derive the dispersion relation for the electromagnetic
wave propagating in a plasma. The plasma exerts the fol-
lowing effect on the propagation of the electromagnetic
wave. The electric field of the wave gives rise to an electron
current which produces a magnetic field which, in its turn,
affects the field of the wave. Make use of the Maxwell equa-
tions for the electromagnetic wave:
1 OH
curl E= —— curl H = —-j——-—— ___ (10.17)
ot? c c Ot
Here E and H are the electric and magnetic fields in the
electromagnetic wave, j is the density of the electron cur-
rent produced by the effect of the wave’s field, and c is the.
velocity of light. |
Applying the operator —c~1(0/0t) to the first equation
of (10.17) and the operator curl to the second equation and
eliminating the magnetic field from the resulting equations,
Waves in a Plasma 49?
we obtain the following equation:
grad div E—vy2E 2% 21 1 @E _
We assume that the plasma is quasineutral and that div E=
= (0 according to Poisson’s equation (3.2). The electric
current j is due to the motion of the electrons: | = —eN yw
where JV, is the density of electrons, and w is the electron .
velocity due to the effect of the electromagnetic wave. The
equation of motion of the electron is m(dw/dt) = —eE so
that
Oj a dw __ e2No
“Ot No a= m BE
Hence, we obtain the following equation for th electric
field of the electromagnetic wave:
where w, is the plasma frequency.
Write the electric field in the form (10.2) and substitute
it into the above equation; the result is the dispersion rela-
tion for the frequency w and the wave number k of the elec-
tromagnetic wave propagating in the plasma:
@?—=wi+c?k* (10.18)
If the plasma density is low (Ny > 0, w) > 0), Eq. (40.18)
is the same as the dispersion relation for the electromagnetic
wave propagating in a vacuum: @ = kc. The dispersion rela-
tion (10.18) shows that the electromagnetic waves whose
frequency is lower than the plasma frequency cannot propa-
gate in a plasma. Such waves are damped in the plasma and
the dispersion relation (10.18) shows that the characteristic
distance of this damping is of the order c/(w?—?)!”.
Damping of plasma oscillations in a weakly ionized plasma.
When deriving the dispersion relation for the plasma oscil-
lations, we neglected the interaction between the electrons
and the neutral particles. This interaction results in the
fading of the plasma oscillations because the energy of the
electron oscillations is transferred to the neutral particles.
To account for this interaction in the dispersion relation for
128 Introduction to Plasma Physics
the plasma waves, let us use Eq. (6.16), rather than
Eq. (6.15), as the equation for the mean electron momentum.
Then the second equation in the system (10.8) is transformed
into
ee ikp’ eh’ ow’
m
and the remaining equations of this system are not changed.
Here t is the characteristic time of the elastic collision
between an electron and a neutral gas particle giving rise to
a noticeable change in the electron momentum. We assume
that the mean velocity of directional gas motion is zero.
We replace the second equation in (10.8) with Eq. (10.19)
and eliminate the amplitudes from the resulting system.
The result is the following dispersion relation instead of
Eq. (10.9):
wo = (@f + y (vk) )'27—— (40.20)
To derive this relation we assumed that
wt > 1 (10.24)
Substituting (10.20) into Eq. (10.2), we find that the
wave amplitude decreases with time as exp (—2/t) and this
decrease is due to the scattering of electrons by the gas
particles. The condition of the existence of the plasma
waves is that the characteristic time of the wave damping
must be considerably less than the oscillation period, namely
condition (10.21) must hold. The frequency of collisions be-
tween an electron and the gas particles is 1/t ~ Nuvo where
N is the gas density, v is the characteristic electron velocity,
and o is the cross section of the collision between the electron
and gas particles. Assuming that this cross section is of
the order of the gas-kinetic cross section (~ 107° cm’), the
characteristic electron energy is of the order of 1 eV, and the
frequency w is of the order of the plasma frequency (3.9),
we obtain the following estimate for condition (10.21):
Ne!7/N > 107** em?”
This indicates that in some gas-discharge plasmas the condi-
tion of existence of plasma oscillations will not be satis-
fied.
Waves in a Plasma 429
The interaction between plasma waves and electrons. The
above mechanism of damping of plasma waves is due to the
collisions between electrons and gas particles. The energy
exchange can occur also in the absence of collisions. Let
us analyze the interaction between electrons and plasma os-
cillations. We introduce the frame of reference in which the
wave is at rest. This wave captures the electrons whose veloc-
FIG. 17. 7)
Interaction of electrons
with plasma
oscillations.
Noncaptured electron
Captured
ity is close to the wave’s velocity (Fig. 17). There is a con-
tinuous energy exchange between the wave and the electron
captured in the potential well of the wave.
For instance, if in the frame of reference in which the
wave is at rest an electron travels along the wave with a ve-
locity w and reverses its direction of motion in a certain
time period, then it transfers to the wave during this time
period the following energy:
m (Upp. tu)? =m (vp. — 4)?
hae aie = 2MVpn.U
Here Un. = w/k is the phase velocity of the wave. The
characteristic velocity of the captured electrons in the frame
of reference connected with the wave, u, is of the order of
(ep/m)'/? where is the amplitude of the wave potential.
It may be seen that the energy exchanged between the wave
and the noncaptured electron is of the order of ep while for
the captured electron this energy is of the order of upp. (meq),
that is, for small wave amplitudes, considerably higher
than the energy exchanged by the noncaptured electron.
Hence, when analyzing the energy exchange between elec-
9—01607
130 Introduction to Plasma Physics
trons and plasma waves, we can consider only the electrons
captured by the wave.
Since the captured electron oscillates in the wave’s field,
the energy exchanged between the electron and the wave must
be zero when averaged over a large time interval. But this
is not the case when we take into account the collisions be-
tween the captured and noncaptured electrons. The energy
exchange between these electrons results ultimately in the
energy exchange between electrons and the wave. In other
words, the interaction between the electrons and the wave al-
ters the distribution function for the electrons with veloc-
ities close to the phase velocity vp. of the wave. Indeed,
the result of the interaction is that the number of electrons
with the velocities vp,, + u and v,;», — u become equal.
However, the collisions tend to restore the distribution func-
tion of electrons, and this process results in the energy ex-
change between the electrons and the wave.
Let us assume that the energy exchange between electrons
takes less time than the period of oscillation of the captured
electron in the potential well of the wave. The frequency of
these oscillations is of the order of k(eq/m)}/2 ~ (eE'k/m)/?~
~ (e?N2/m)/* where k is the wave number, and g, E’, Nz are
the amplitudes of the electric potential, of the electric field
strength of the wave, and of the electron density, respectively.
To derive this frequency, we made use of the last equation
of the system (10.8). The frequency of the energy exchange
between the electrons equals the frequency of collisions be-
tween the electrons, N,vo, of the order of N,(e4/T*) x
x (T/m)'/? [see Eq. (4.14)] where JV, is the electron density,
and 7 is the electron temperature. Hence, we find that the
interaction between the electrons and the wave does not
alter the distribution function of electrons when
Ne Nee®
We S73
Provided this condition is satisfied, let us determine the
direction of the energy exchange between the wave and the
electrons in a plasma. Now the electron distribution func-
tion is not altered owing to the interaction with the wave,
and we have to compare the number of electrons with veloc-
ity Upn.-+u that transfer the energy to the wave and the
(10.22)
Waves in a Plasma 431
number of electrons, with a velocity v,,.—wu that take the
energy from the wave. The number of captured electrons with
a given velocity is proportional to the electron distribution
function f(v). Hence, the wave gives its energy to electrons
and is damped if f(vp,,—w) is larger than f (vpn. +-w). Since
f(Upn. -E U).='f(Vpn.) ULOf(Vpn.)/Ov,,], the wave is damped
when
mil <0 (10.23)
Vx I0~=Vpp,
llere v, is the component of the electron velocity in the
direction of the wave propagation, and the derivative is
taken for the electron velocity being equal to the phase
velocity Upp, of the wave. When the condition opposite to
Eq. (10.23) is satisfied, the wave takes the energy from
electrons and its amplitude increases.
In deriving the condition (10.23) of the wave damping,
we have used the condition (10.22) which is satisfied when
the field of the wave does not affect the distribution func-
tion of electrons. However, (10.23) holds also when the
condition (10.22) is not satisfied and the electron distri-
bution function is altered by the effect of the wave. The
interaction with the wave tends to equalize the number of
the electrons with velocities vp,, -+-u and vp,.—u and, hence,
it decreases the derivative Of/dv,, but does not change its
sign. Thus, the condition (10.23) is valid even when the
interaction with the wave alters the electron distribution
function.
The condition (10.23) is satisfied if the electron distribu-
tion function has the Maxwellian form or is a monotonically
decreasing function, and the oscillations in such a plasma
decrease with time owing to the interaction with electrons.
However, when an electron beam is injected into the plasma
producing the electron distribution function shown in
Fig. 18, the plasma oscillations in this system will be ampli-
fied getting the energy from the electrons. The interaction
between the plasma oscillations and the electrons will reduce
the velocity of the electrons until the electron distribution
function becomes a monotonically decreasing one.
Amplification of the oscillations means that their ampli-
tude increases with time. While the oscillation amplitude is
g*
132 Introduction to Plasma Physics
small and the oscillations do not alter the electron distri-
bution function, that is, while (10.22) is satisfied, the oscil-
lation amplitude increases exponentially. When the plasma
particles transfer their energy to the plasma wave, which
is thus amplified, such a state of plasma is termed the
unstable state.
FIG. 18. f
The electron
distribution function
for the electron
beam injected into
a plasma.
The solid line
represents the
initial distribution
and the dashed line
is the distribution
established after
a period of time.
The attenuation factor for waves in plasma. Let us in-
troduce into Eq. (10.2) the attenuation factor y for the
waves so that the oscillation amplitude varies as exp (—y?).
Let us estimate the attenuation factor when the wave in plas-
ma does not affect the distribution of charged particles, that
is, a condition of type (10.22) is satisfied. Variation of the
energy W of the plasma wave per unit volume per unit time
may be estimated as follows:
+u0
dw
ey \ f (v) Ae du
—uo
Here v~ u,k is the oscillation frequency for the electron
captured in the potential well of the wave, uy = (2eq/m)'/?,
~ is the amplitude of the potential oscillations, and Ae =
= 2mVyp.Uo is the variation of the electron energy when the
direction of the electron motion is reversed (we take into
account only the interaction between the wave and thecap-
tured electrons). The right-hand side of the above relation
may be estimated as
af Of \ eo
Uk x (,-) Ug X MVph Uo X Uo ( OV. mk2
Waves in a Plasma 133
where we have used the relations upp, = w/k and W ~ E?~
~ g?’/k? (E" is the amplitude of the wave’s electric field).
From the definition of the attenuation factor for the plasma
wave, dW/dt = — yW, we obtain the following estimate:
eu | Of
aid ake Wa (10.24)
Attenuation occurs when (10.23) is satisfied. The attenua-
tion of type (10.24), which is due to the interaction between
charged particles and the wave, is known as the Landau
damping.
The condition of existence of the plasma waves and the ion
sound has the following form:
v< Oo (10.25)
Let us transform the condition (10.25) assuming the Maxwell
distribution function for the particles. For the plasma os-
cillations taking @ ~ @»), we obtain
krp << 1 (10.26)
where rp is the Debye-Hiickel radius (38.6) When this condi-
tion is satisfied, the phase velocity of the wave is considera-
bly higher than the thermal velocity, so that the electrons
captured by the wave are at the tail of the distribution
function.
When the ion sound propagates in a plasma in which the
temperatures of electrons and ions are the same, the phase
velocity of the sound is of the order of the thermal velocity
of ions and the attenuation factor is of the order of the
sound frequency. Therefore, the ion sound can propagate only
in the plasmas in which the electron temperature 7, is con-
siderably higher than the ion temperature 7;:
T.>T; (40.27)
The beam-plasma instability. Assume that an electron
beam penetrates a plasma and that the velocity of the elec-
trons in the beam is considerably higher than the thermal
velocity of the plasma electrons while the density Nz, of the
electrons in the beam is considerably lower than the density
N, of the plasma electrons. Deceleration of the electron beam
can occur owing to the scattering of the electrons of the
134 Introduction to Plasma Physics
beam on the electrons and ions of the plasma. There is,
however, another mechanism of deceleration of the electron
beam, which is known as the Langmuir paradox* or beam in-
stability. This kind of deceleration can be more effective
than deceleration due to scattering on the charged plasma
particles.
This mechanism acts as follows. The plasma oscillations
are generated in plasma. Interacting with the electrons of
the beam and taking energy from them, these oscillations are
amplified. Thus, the energy of the electron beam is trans-
formed into the energy of plasma oscillations and it remains
in the plasma. Further, this energy may transfer to other
degrees of freedom in the plasma.
Let us analyze the amplification of the plasma oscilla-
tions in the above case assuming that the amplitude of os-
cillations is small and that the temperatures of the elec-
trons in the plasma and in the beam are zero. Hence, the
pressure p, of electrons in the plasma and the beam is zero.
Applying to the plasma electrons the continuity equation
(6.6) and the Euler equation (6.15), we derive equations for
the amplitudes of the plasma, parameters following from the
first two equations of the system (10.8) at p’ = 0. Elimina-
tion of w’ from these equations yields
Ne=—ik“272 (10.28a)
We can obtain the expression for the amplitude of oscilla-
tions of the electron density Nz, in the beam in a sim-
ilar way writing the electron density in the beam as
Nz + Nz exp li(kx — wt)) and the velocity of the electrons
in the beam as wu + wy, exp [i(kx — wt)] where the z axis is
* Langmuir studied the equalizing of the temperature of the beam
of electrons ejected from a cathode surface and the temperature of
electrons of the gas-discharge plasma which this electron beam pene-
trates. He found that equalization occurred at much smaller distances
from the cathode than the calculations suggested assuming that decel-
eration of the electrons in the beam is due to their scattering on the
charged particles in the plasma. This phenomenon was called the
Langmuir ‘paradox.
Waves in a Plasma 135
parallel to the velocity of the beam:
(10.28b)
Similar to the last equation of the system (10.8), Poisson’s
equation (3.2) yields the following equation for the ampli-
tudes of the system’s parameters:
ikE’ = —4ne(Ne+N 4) (10.28c)
Eliminating from the system of equations (10.28) the ampli-
tudes Ne, Ng, and E’, we obtain the following dispersion
relation:
a a No
oe | 98 (o— ku? No (10.29)
Here wy = (4nNoe?/m)1/? is the frequency of plasma oscilla-
tions. When the density of the beam electrons is zero (Vp =
= 0), Eq. (10.29) reduces to Eq. (10.9) where the electron
temperature is taken to be zero.
If the phase velocity of the plasma waves w/k is equal to
the velocity u of the electron beam, the interaction between
them is the strongest. Let us analyze this case. Since the
density of the beam electrons Ny is small compared to the
density N, of the plasma electrons, the frequency of the
plasma oscillations is close to the plasma frequency w, of the
plasma. Hence, we shall consider the waves with a wave
number k = w/u, which have the most effective interaction
with the electron beam. We write the frequency of these
oscillations as m = @, + 46 and insert it into Eq. (10.29).
Expanding the result in a series of the parameter 5/9, we
obtain
Ny \1/3 Onin
6 = (3) exp ( 5
where nv is aninteger. We see that |5|/w. ~ (N,/N,)? < 1,
that is, the above expansion is valid.
If the imaginary component of the frequency, which is
equal to the imaginary component of 6, is negative, the
wave is attenuated; if it is positive, the wave is ampli-
fied. The maximum value of the amplification factor is giv-
136 Introduction to Plasma Physics
en by (n = 1)
1/3 Np \ 1/3
gsi (Fe) op =0.69 (28) a, (40.30)
The amplitude varies with time as exp (yé); this result
is valid if the plasma oscillations are small and do not affect
the properties of the plasma. This type of instability is
known as the beam-plasma instability.
The Buneman instability. Let us consider instability
of another type which develops if the mean velocity of elec-
trons differs from the mean velocity of ions. Let us formulate
the problem. All the plasma ions are at rest and all the
electrons travel with a velocity u with respect to the ions.
The plasma is quasineutral, that is, the densities of the
electrons and ions are equal. We have to determine the
maximum amplification factor of plasma oscillations. The
electron beam is decelerated owing to the transfer of energy
from the beam to the plasma oscillations.
With this formulation the problem is equivalent to the
preceding one. In both problems there is an electron beam
penetrating the plasma so that the dispersion relation can
be derived in a similar way. Denoting the ion mass as M
and taking into account that the ion density is equal to
the electron density, we obtain the dispersion relation
2
t= tele (10.34)
instead of (10.29). Tending the ratio m/M to zero, we obtain
the following dispersion relation: w = @,) + ku. Hence,
we can write the frequency of the plasma oscillations as
© = o,+ kut+ 6
Substituting this frequency into Eq. (10.31) and expand-
ing the result in a series of the small parameter 4/wo, we
obtain
26 om w?
@o M (mp+ku+6)2
The electron beam has the strongest interaction with the
wave whose wave number k = —q@,/u. For this wave
6 = (a) Wy EXP ( ey
Waves in a Plasma 137
where n is an integer. The highest amplification factor cor-
responds to n = 1 and is given by
1/3 1/3
—y=Ims-43 (4) Wp = 0.69 (<7 } Wo (10.32)
Note that the frequency of oscillation is of the order of the
attenuation factor. This type of instability of the electron
beam due to the interaction with the plasma electrons is
known as the Buneman instability.
Hydrodynamic instabilities. The types of instability
discussed above are the so-called kinetic instabilities for
which the amplification of oscillations is due to the differ-
ences in the character of motion of various groups of par-
ticles. The development of the oscillations ultimately results
in a change in the velocity distribution of the charged parti-
cles. Another type of instabilities is known as the hydrody-
namic instabilities. The development of hydrodynamic
instabilities involves a displacement of the plasma regions
and results, finally, in a variation of the spatial configura-
tion of the plasma. We shall analyze the simplest type of
the hydrodynamic instability, namely, the instability of
the pinch.
Let us consider the stability of the pinch with respect to
the so-called “sausage” instability. This instability changes
the radius of the pinch but leaves the axial symmetry of the
pinch conserved. We have to find under what conditions an
accidental distortion of the pinch will not develop further.
Let us assume that the distortion of the pinch results only
in a slight curving of the magnetic lines of force, that is,
the radius of curvature of the magnetic lines of force ‘is
considerably larger than the radius of the pinch. According
to Eq. (9.23), in the plasma region the following equation is
then satisfied:
H2
pt aa const
Let us analyze the variation of the parameters of the
pinch due to the variation of its radius. The total current
and magnetic flux through the cross section of the pinch
must be conserved in the process. The electric current is
I, = caH,,/2 where a is the pinch radius, and A, is the
138 Introduction to Plasma Physics
axial magnetic field strength. The condition 6/, = 0 yields
(Sa/a) + (5H,/H,) = 0 where Sa is the variation of the
pinch radius, and 6, is the variation of the axial magnetic
field at the pinch surface on the outside of the plasma. The
longitudinal magnetic field is frozen into the plasma so
that a displacement of the plasma elements does not change
the magnetic flux through them. The condition of conserva-
tion of the magnetic flux @, = na’H, yields (25a/a) +
+ (5H,/H,) =0 (here A, is the longitudinal magnetic
field inside the plasma). Hence, (6A ,/H,) = 2(6H,/H,).
The variation of the magnetic field pressure inside the plas-
ma 6(H2/8n) = H,6H,/(4m), and the variation of the mag-
netic pressure outside the plasma is #A,6H,/(4n) =
= A36H,/(8nH,). It may ke seen that if
H?>>H2/2 (40.33)
holds true, the additional internal magnetic field pressure
produced by the above distortion of the pinch is larger than
the additional outside magnetic field pressure. When the
condition (10.33) is satisfied, the pinch is stable with re-
spect to displacements of the sausage type.
11 Radiation in Gas
Interaction between radiation and gas. Let us discuss the
interaction between an electromagnetic field and an atomic
medium, which gives rise to transitions between atomic states.
Table 4 presents a summary of one-photon processes due
to interaction between the radiation field and atomic sys-
tems. This interaction is weak; the small parameter character-
izing the weakness of this interaction is the fine structure
constant e?/fic = 1/137 < 1, which is the ratio between the
characteristic velocity of the valence electrons and the
velocity of light. Another parameter which proves to be
small in reality is the ratio between the electromagnetic
radiation field and a certain characteristic quantity. Since
these parameters are small, the transitions in the atomic
systems involving absorption or emission of photons are slow
compared to the characteristic atomic times or to similar
atomic processes.
139
oy+s6ra+apta
oY+dV ~atV
oy+-p+p+a
oy+ty~+.V+a
a+Vv + qv+oy
a+V~+--p+oy
at+4V <—V+oy
V+OYt + ah + OY
OY -+V < sV
eV Voy
Wot JO T10}8 Ue YPM SUOISI[[OO UI SUOI}Oe[9 JO Zun[Ye1j}ssmolg
SUl0Je JO UOT}BUIqUIODeI10}0Tg
W078 UB 0} UOI}A[a Ue JO JUBMMYOeI}e OATVIPY
u0I}0e[9 pUe UOT Ue JO UWOT}eUIqMIOD0I0}0N
@[NdejouL B JO UWOI}eID0SSIPO}ON
UOT BAT}esau e Jo Aedapojog
@[Noo[OU IO WO} Ue JO UOIZeZIUOI0}0UY
uojoyd e& Jo UOIssIma poye[NWI3S
epotzaed ormioje poyloxe ue JO UO1}e{I0xe-ap snosuejyuods
uor}diosqe uojoqd 0} enp o[d1,1ed OIMo}e Ue Jo UOTB}IOXY
aulayos sse00ld
ssovo0id Ale }UIUII[Y
swojsks 2WOJy PUR UOHeIpey UseMjog UOHIeJBjU] JO Sassaz0Jq Alejuowsajy =o -O/qGeL
140 Introduction to Plasma Physics
For instance, the lifetime of an excited atomic particle, t,
which is determined by its de-excitation in the absence of
external fields, is considerably larger than the characteristic
atomic times. The reciprocal quantity 1/t (the frequency of
spontaneous de-excitation), apart from the atomic param-
eters, depends on the factor (e?/hc)? and, hence, is lower
by at least six orders of magnitude than the frequency of
FIG. 19. “Excited”
Radiative transition state
between two
levels.
“Ground”
state
the emitted radiation. Since the interaction between the
radiation field and the atomic system is weak, we can sim-
plify the description of the radiative transitions. In partic-
ular, we can represent the electromagnetic field as made up
of noninteracting photons and neglect all two-photon
transitions. The times of these transitions are large com-
pared to the times of one-photon transitions (that is, tran-
sitions resulting in emission or absorption of one photon).
Spontaneous and stimulated emission. Let us analyze
the transitions resulting in emission or absorption of a pho-
ton between two states of an atom or molecule, the states
denoted by the subscripts “gr.” (ground) and “exc.” (excited),
respectively (Fig. 19). (The names given to the states are
purely conventional.) Assume that the gas contains ny
photons in a definite state. The number of these photons
can increase due to emission caused by transition of the
atom from the “excited” state to the “ground” state or can
decrease due to absorption caused by the reverse transition.
Assuming the one-photon transition, we write the probability
of photon absorption by one atom per unit time:
W(gr., Ng > exc., No — 1) = Any (11.1)
Here we made use of the fact that no transitions occur in the
absence of photons (n, = 0) and only one-photon transi-
Radiation in Gas 141
tions take place (that is, the higher-power terms in n, can be
neglected). The quantity A is independent of the electro-
magnetic field and is determined only by the atomic pa-
rameters.
The probability of the atomic transition with emission
of a photon is given by
W (exc., Ny —egr., Ny +1) = 1/1 + Bn,y (11.2)
Here 1/t is the rate of spontaneous emission of the atom
which occurs in the absence of external fields, and the
coefficient B refers to the radiation stimulated by the exter-
nal electromagnetic field. Both parameters depend only on
the properties of the atom. The coefficients A and B are
known as the Einstein coefficients.
Let us find a relationship for the quantities A, B, and 4/t
in the case of thermodynamic equilibrium between the
radiation and the atomic system. The densities of the atoms
at the “excited” and “ground” states are related by the
Boltzmann distribution (2.9):
exc. h
Negee Hex Nar. €XP ( +} (41.3)
where fiw is the energy difference betwcen these states, and
Zexc. and g,e,, are the statistical weights of the “excited”
and “ground” states of the atom. The mean number of photons
in a given state can be found from Planck’s distribu-
tion (2.21):
No = [exp (hw/T) — 1} (11.4)
Under thermodynamic equilibrium the number of emission
transitions per unit time must be equal to the number of
absorption transitions per unit time. We write this equality
for a unit volume:
Ner.W (gr., No > eXC., Ng — 1)
= Nexc.W (exc., No > 2L., Ny +1)
Using Eqs. (41.1) and (11.2), we obtain
Nor. Ano = Nexe, (1/t + Bro) (11.5)
Equations (11.5), (11.3), and (41.4) yield A = gexe./(Zgr.T)
and B = 1/t. Now we can write Eqs. (41.1) and (41.2) in
142 Introduction to Plasma Physics
the following form:
W (gr., Ng exc., Ng— 1) = ae Nes (11.6)
gr.
W (exc., %o— QT., ny +1) == —2. (11.7)
The second term in Eq. (41.7) is the so-called stimulated
radiation, which is of fundamental importance.
Broadening of spectral lines. Let us determine the energy
distribution of the photons emitted in the transition between
the two atomic states. We introduce the frequency distribu-
tion function a, of photons: a,dw is the probability that
the frequency of the emitted electromagnetic radiation is in
the interval from o to w + dw. Conservation of energy
indicates that the energy of the emitted photon is ha, =
= Kexc. — Eg,.. Therefore, the width of the photon fre-
quency! distribution function is small compared to the fre-
quency of the emitted photon w,. We shall determine the
form of the frequency distribution function a, for photons
in various specific cases.
The Doppler broadening. Let us analyze the simplest
mechanism of spectral line broadening which is due to the
motion of the emitting atoms. The electromagnetic wave
emitted by a moving atom with a frequency w, is received by
a stationary detector as a wave with a frequency w; the
frequencies wm and wy, are related by the Doppler equation
@ = Wo( 1 + v,/c) (41.8)
where v,, is the velocity of the atom in the direction of pro-
pagation of the emitted wave, and c is the velocity of light.
If we assume the Maxwellian velocity distribution (2.15)
for the atoms and the Doppler relationship (11.8) between
the atom velocity and the frequency of the emitted photon,
we can transform the equation a,do = @ (v,) dv, into the
following expression for the profile of the spectral line:
1 ( Me? \4/2 Me? (w—«o)?
= tie ( xT exp [—- “a (141.9)
The broadening of the spectral line due to the motion of the
emitting particles is known as the Doppler broadening.
Broadening due to finite lifetimes of states. Let us analyze
the broadening of spectral lines due to the fact that the
@
Radiation in Gas 143
states between which the transition occurs have finite life-
times. The lifetimes of the states can be determined by the
transitions owing to collisions as well as by the radiative
transition being considered.
The amplitude of the electromagnetic field (that is, the
strength of the electric or magnetic field of the electromagne-
tic wave) is proportional to the product of the wave func-
tions axc, (¢) and z,, (t) of the “excited” and “ground” states
the transition between which produces the photon. The
steady-state time dependence of the wave function is given
by exp (—i€pr. exc.t/h) where €,,, and exc, are the energies
of the “ground” and “excited” states. Hence, we obtain
the following time dependence for the amplitude of the
electromagnetic field:
f (t) ~ exp (i,t) (14.10)
where Wo = (€exc. — &gr.)/h. It may be seen that the
frequency distribution of the emitted photons in this case
is given by Eq. (11.7) since the photon energy is determined
by the conservation of energy.
Now assume that the states of the atomic system between
which the transition occurs have finite lifetimes. Let t,
be the lifetime of the & state. Hence, the probability | ,|?
that the system is in state k satisfies the equation of radioac-
tive decay
d\n |? 4
ala coe i
and the probability of being in a given state decreases as
|»p, |? = exp (—2/t,) (at the beginning the system is in the
given state). Then the time dependence of the wave function
is given by , ~ exp [—i€,t/hk — t/(2t,)], and we obtain
the following expression for the amplitude of the electro-
magnetic field
f(t) ~ exp (i@pt — vit), 2v = Tet. + Texe, (11.41)
instead of Eq. (41.10). Here t,,. and tee, are the lifetimes
of the “ground” and “excited” states. Figure 20a gives the
field amplitude as a function of time when the field is
damped due to the finite lifetimes of the states.
144
Introduction to Plasma Physics
The Fourier transform of the amplitude f (¢) yields the
frequency distribution for the amplitude of the electro-
magnetic field:
+00
=k | fQexp(— ion dt ~ (v+i@—o)
— oO
FIG, 20.
The electromagnetic
radiation amplitude
as a function of time:
(a) for finite lifetimes
of the states and
(b) for a collision
between the emitting
atom and a gas particle;
At is the duration
of collision,
and y is the phase shift
due to collision.
(a)
Such parameters as the radiation intensity and the fre-
quency distribution function a, of photons are quadratic
functions of the field amplitude. We derive the function a,
by taking into account that a, ~|f,|? and using the normal-
Radiation in Gas 145
4-00
ization condition \ a,dwo = 1:
— oO
Vv 1
Oy =a (11.12)
v2 -+- (W — Wo)”
The frequency distribution function (11.12) of the emitted
photons is known as the Lorentz profile. If the quantity
t = (2v)-! in Eq. (11.12) is the lifetime of the “excited”
state depending on the radiative transition to the “ground”
state, then this type of broadening is called the radiative
broadening. |
Impact broadening of spectral lines. Let us analyze the
broadening of the spectral lines due to the interaction be-
tween the emitting atom and the surrounding gas particles.
First, let us consider the case when the duration of the
interaction between the emitting atom and the gas particles
is much smaller than the lifetime of the atom states. This
means that the emitting atom occasionally collides with
a gas particle and the time of collision is much shorter than
the time interval between two successive collisions. This
type of broadening of spectral lines is known as the impact
broadening.
The impact broadening has the following mechanism. The
electromagnetic radiation field is produced by the transition
of an atom from the “excited” state to the “ground” state and
this field is given, up to a phase factor, by Eq. (11.10). When
the emitting atom collides with a gas particle, the transi-
tion frequency is changed and the general form of the radia-
tion field in this case is shown in Fig. 200.
Assuming that the duration of the collision is small com-
pared to the time interval between two successive collisions,
we can write the radiation field as
f (t) ~ exp [iwot +i pa xn (t— te) ]
Here y (x) = 0 when z <0 and y (x) = 1 when x > 0, and
Xz 1S the variation of phase following the Ath collision. The
10—01607
146 Introduction to Plasma Physics
Fourier transform of f (z) is
-++ 00
1 _ ( F(t) exp (—iot) dt
ho = Byte Lan P
-— 0O
te > 1 — exp (iwpTp) exp i (S 4+)
W— Wo
k j<k
Here t, is the timeinterval between the kth and the
(4 + 1) st collisions.
To determine the frequency distribution function for pho-
tons, let us average it over the phase shifts X, occurring due
to collisions. Assume that the phase shifts y, are random
and not small so that we have exp [i (vy, — x;)] = Sj, (Oj, =
= 0 when j “~k and 6;, = 1 when j = k). Taking into
account that |1— exp(iz)|? = 2 (1—cos zx), we find the
following expression for the frequency distribution of the
emitted photons:
A
ice
° (@— Wo)?
(1 — cos (@ — @y) t) (11.13)
Here A is the normalization factor, ¢ is the time interval
between two successive collisions, and the angle brackets
denote averaging over time ¢. Let us express the result of
averaging in terms of the mean time t between two successive
collisions.
Denote by W (¢) the probability that a collision occurs
at the moment t; when t > 0, we have W (t) = vt. The fol-
lowing equation may be written for the probability W:
W(t +h) = W(t) +11 — W (d)] W (e)
Tending ¢, to zero, we obtain the equation dW/dt = v(1—W)
whose solution is W (¢) = 1 — exp (—vé). The probability
that a collision occurs in the time interval dt is (dW/dt) dt;
hence, the value of a function X (¢) averaged over the time
between two successive collisions is given by
C aw
(X)= | X That
0
Radiation in Gas 147
Therefore, the mean time between two successive collisions is
t= \ texp (—vt)vdt =~
0
Thus, we have
W (t) = 1 — exp (—t/t)
and
(1 — cos (W — Wp) =| grim ~ [1—cos (W—Wp) ft]
0
__ (@— po)? T4
~ 1-+(@ — @)? t2
Inserting this relation into Eq. (11.13) and taking into
account the normalization condition, we find the frequency
distribution function for photons
T 1
m [(@ — @o)2 2+ 4]
It may be seen that impact broadening also gives rise to
the Lorentz profile and Eq. (11.14) is identical to Eq. (14.12)
(ignoring the definition of parameter t). Hence, if the broad-
ening of the spectral line is caused by collisions with gas
particles, the shape of the line is given by identical expres-
sions of the type of Eqs. (11.12) or (41.14) irrespective of
whether the collisions are elastic or not. The parameter t
in Eqs. (441.12) and (41.14) may be estimated as follows:
1/t ~ Nvo (11.15)
where JN is the density of the gas particles, v is the charac-
teristic velocity of collision between the emitting atom and
the gas particle, and o is the cross section of such a collision.
Assume that the broadening of the spectral lines is due
to the interaction between the gas particles and the “excited”
state of the emitting atoms and that the contribution of the
“ground” state of the emitting atoms into this broadening is
small. Let us determine the cross section in Eq. (44.15)
for this case assuming the classical motion of the colliding
particles. The mechanism of impact broadening indicates
that the main contribution to the cross section o is given
by the impact parameter 0, for which the phase shift y is
10*
Ag =
(41.14)
148 Introduction to Plasma Physics
of the order of unity. Hence, we obtain
o~p, (11.16)
and for the impact parameter p, we find that \ LU at ~ 1
where U (R) is the potential of the interaction between the
gas particle and the emitting atom in the “excited” state.
Since the particles travel in straight lines with this impact
parameter, we find that
\ U(R)dt — eoV (Po)
h hv
Thus, this cross section is of the order of p* and p)U (0,)/(hv)~
~1. It may be seen that the cross section (11.16) is identical
to the total cross section of the collision between the gas
particle and the emitting atom in the “excited” state [see
Eq. (4.9)].
Let us find the criterion of applicability of the above theo-
ry of impact broadening. The time it takes the particle to
pass the region of size 0), where scattering occurs, must be
much smaller than the time interval between two successive
collisions:
£2. < (Nvo,)!
where o; is the total cross section of the collisions. Hence,
we obtain
No?’ <1. (14.17)
Equation (11.17) is the criterion of applicability of the
impact broadening theory in the main frequency range of the
photon emission.
Statistical broadening theory. Let us analyze another
mechanism of spectral line broadening due to the interaction
with the gas particles, which is the opposite of the impact
broadening. Assume that the gas particles are not able to
travel noticeable distances during the characteristic times
of broadening development. Hence, the gas particles can be
assumed to be stationary and the observed shift of the
Radiation in Gas 149
spectral line is given by
o—o=— S'V(R;) — (11.18)
where V is the difference between the potentials of inter-
action of the emitting atom in the “excited” and “ground”
states with the gas particle *, and R; is the radius vector of
the ith gas particle in a frame of reference with the origin
at the nucleus of the emitting atom.
Equation (11.18) expresses the spectral line shift for
a given configuration of the interacting gas particles. To
find the frequency distribution function for. the emitted pho-
tons, we must average Eq. (41.18) over all such configurations.
Let | V (R) | monotonically decrease with an increase in
the distance between the emitting atom and the interacting
gas particles. According to Eq. (11.18), each configuration
of the gas particles gives rise to a spectral line shift in the
same direction, so that the mean width Aw of the spectral
line is of the same order of magnitude as the mean shift.
Since the mean distance between the gas particle is of the
order of N~-'/3 (N is the density of particles), we obtain
the following order-of-magnitude estimate according to
Eq. (11.18):
Ao~ —V(N-'%) (44.49)
Let us determine the frequency distribution function of
photons at the wing of the line profile. The shift at the wing
of the line profile is larger than the mean shift. It is due to
interaction with the gas particles which are very close to
the emitting atom in the region where they have a low
probability of being. The probability for a gas particle
to be at a distance from R to R + dR from the emitting
atom is 4nR? dR WN; thus
a,dw = 4nR*NdR (411.20a)
Here the line shift @ — a, is
@—@=—V(R) (14.20b)
"* Tf the radiative transition removes the atom to the real ground
state, the quantity V is practically identical to the interaction poten-
tial of the atom in an excited state.
150 Introduction to Plasma Physics
In particular, when V (R) = CR, Eqs. (11.20) yield the
following frequency distribution function of photons at the
line wing:
Annee! do
Let us-derive a criterion of applicability, for the statisti-
cal theory of broadening. The theory holds if the gas parti-
cles are not able to change their positions in the time of the
order of (Aw)"! in which the broadening develops. Hence,
the criterion is
WN" <do~ ZV (N~*) (14.22)
where v is the characteristic velocity of collision, and N~'/3
is the mean distance between the gas particles or between
the emitting atom and the interacting gas particles. .
Assume that the broadening is caused only by the inter-
action with the excited atom state, that is, V (R) in Eqs.
(11.16) and (11.22) is identical to the potential U (R) of
interaction between the emitting atom in the excited state
and the gas particle. Writing the criterion (11.22) as
N71/8y N71/8
ae a
and comparing it with Eq. (11.18), we find that for a mono-
tonic potential 0, > N-/?. Hence, we can derive the follow-
ing criterion of applicability for the statistical theory of
broadening taking into account that o; ~ 0%:
No}? S14 (44.23)
Comparing Eqs. (41.28) and (41.17), we see that the impact
broadening theory and the statistical broadening theory de-
scribe two opposite extreme cases of interaction between the
emitting atom and the surrounding gas particles.
The cross sections of emission and absorption of photons.
The above mechanisms of spectral line broadening make it
possible to estimate the line width due to the motion of the
emitting atoms, the finite lifetimes of their excited states,
and their interaction with the surrounding gas particles.
The frequency distribution function a, cf the emitted
Radiation in Gas 151
photons, which accounts for the most effective broadening
mechanism, can be used for determining the cross sections of
absorption and stimulated emission of photons.
Indeed, by definition the cross section of a process is
the ratio between the probability of transition per unit time
and the incident particle flux. The probability of the stim-
ulated emission per unit time is Bn,= n,/t, according
to Eq. (41.2), where n, is the number of the photons of fre-
quency ow in the same state, and t is the lifetime of the atom
state with respect to the radiative transition. Hence, we
find that the probability of stimulated emission per unit
time due to photons in the frequency range from w to o +
+dw is (t)“!n,a,do.
The photon flux in this frequency range is c dN, where c is
the velocity of light, and dN, is the density of photons in
this frequency range (dN, = 2n,dk/(2n)? where the factor 2
accounts for two polarizations of photons, and k is the wave
vector of the photon which is related to the photon frequen-
cy as w = kc). Thus, the photon flux in this frequency range
is n, (w/nc)*dw. Hence, the cross section of the stimulated
photon emission, which equals the ratio between the fre-
quency of stimulated emission and the incident photon
flux, is given by
Sem. == 2% (14.24)
@2
The absorption cross section is given by Oaps. =
= (nc/w)*Aa, where A is the Einstein coefficient; according
to Eqs. (441.1) and (11.6), A = Sexye./(Ger.t). Here gexec.
and g,,, are the statistical weights of the “excited” and
“ground” atom states and t is the time of the respective
radiative transition. Then we can easily derive
m2c2 goxc, a _
Oabs. = ae 2ae = (11.25)
Let us find the maximum cross section of absorption. It
corresponds to the centre of the minimum-width transition
line. According to Eq. (10.12), the distribution function at
the centre of the line is a, = 2t/n and the maximum cross sec-
152 Introduction to Plasma Physics
tion of absorption is given by
Zexe. C2 Zexc, A?
Gans. == 20 en G2 Beer 2m.
where A is the wavelength of the transition. For instance,
when photons are emitted in the visible-light spectrum, this
cross section is of the order of 10-!9°-10-° cm?, that is, it
is larger than the characteristic gas-kinetic cross sections.
The absorption coefficient. Let us define the absorption
coefficient for photons, k,, by the following equation:
dly
<%= —hyly (41.26)
Here /,, is the intensity of radiation of the frequency pass-
ing through the gas, and z is the distance travelled by the
photons in the gas. Assume that absorption or increase of the
photon flux is due to transitions between two atomic states.
Hence, the absorption coefficient can be written as
ky = Ner.Sabs. nn Nexc.Sem.
Nexc. &gr. a
= Nee Gans. (1 —o) (11.27)
where Oajys. and Oey, are the cross sections of absorption
and emission of photons defined by Eqs. (41.25) and (11.24),
and N,,, and MNexc, are the densities of the atoms in the
“ground” and “excited” states.
Equation (11.27) shows that when
Sexe. > Fexc. (11.28)
‘Vor, &er.
the absorption coefficient k, is negative, that is, a photon
flux passing through this system is amplified. The situ-
ation when the “excited” level has a greater population than
the “ground” level is known as the inversion or inverted
population of the levels, and the medium where condition
(11.28) is satisfied is called the active medium. Active
mediums are used for constructing lasers, which are genera-
tors of monochromatic radiation.
The conditions of laser operation. A prerequisite of laser
operation is the inversion of levels, that is, condition (11.28)
must be satisfied.‘ There are various means for producing
Radiation in Gas 153
a greater population at an “excited” level of a certain system
than at a“ground” level (for instance, with a gas discharge,
optical pumping, or electron beam). This gives rise to an
inversion of the levels, but by itseff this is net-sefficient for
laser operation.
The essential condition of laser operation is that the
amplification coefficient must not be too small. Let the
size of the active medium be LZ. A mirror is placed at one
end of the active medium providing for total reflection of
the incident radiation, and means for partial reflection of
radiation are provided at the other end of the active medium’*.
The total probability of the laser radiation escaping or being
absorbed at the second end is a. A photon travelling in the
active medium is reflected at the ends, and the mean dis-
tance it travels until it escapes or is absorbed at the end is
2L/a. This distance must be larger than the distance —1/k,,
the photon must travel to give rise to another photon of the
same energy. I{ence, the condition of the laser operation is
given by
—k,, > a/2L (11.29)
Let us analyze the operation principles of a laser. Assume
that the condition (11.29) is satisfied for a certain frequen-
cy range at the initial moment when no photons exist in the
system. Then the photons of this frequency range accidentally
produced in the system and travelling along its axis will
stimulate emission of more photons and the photon flux will
be amplified. This will give rise to a radiation field in the
active medium between the mirrors while the inversion of the
levels and the amplification factor will decrease. The condi-
tion (11.29) will be satisfied for a narrower frequency range
and a further intensification of the radiation field will pro-
duce a situation during which only the photons corresponding
to the centre of the spectral line with the maximum ampli-
fication factor are emitted.
Therefore, the width of the laser line is small; it is deter-
mined by the properties of the resonator system, that is,
the mirrors. The number n, of the photons in the same state
* This is typically accomplished with a half-silvered mirror.
a mirror with a hole at the centre, or a prism.
154 Introduction to Plasma Physics
proves to be large, n, > 1, and the frequency of the stimu-
lated emission is considerably higher than the frequency
of spontaneous emission. Thus, all the emitting (excited)
atoms contribute to the useful radiation. Hence, the laser
emits a parallel beam of photons in a narrow frequency range.
Propagation of the resonance radiation. Let us discuss
the propagation of the resonance radiation, that is, the
radiation produced with de-excitation of one of the lowest
excited states of the atom. This atom state is known as the
resonance state; it is the lowest excited state for which the
effective emission transition to the ground state is possible.
The photons emitted in this transition are known as the
resonance photons.
The free path of the resonance photons in a gas is small
since, first, they are absorbed by the ground-state atoms and,
secondly, their absorption cross section for the line’s centre
is greater than the gas-kinetic cross section by a few orders
of magnitude. Therefore, propagation of resonance radiation
is determined, to a great extent, by re-emission of photons.
If the free path of the resonance photon at the line’s centre is
much smaller than the dimensions of the system, the propa-
gation of the photon cannot be described by the diffusion
equation as can be.done for the gas particles. A more fa-
vourable process than multiple re-emission of the line-centre
photon is the radiation of the line-wing photon whose free
path is comparable to the dimensions of the system. This
possible process is of considerable importance in propagation
of the resonance radiation.
Let us discuss the escape of resonance radiation outside
the system assuming that the atoms are excited to the res-
onance state by the collisions with electrons and the escape
of radiation does not affect the density Nex-, of the excited
atoms. Assume that the free path of the line-centre photons
is much smaller than the size of the system L, that is, kjL >
> 1 where ky = Nor.Sabs.(@o) — Nexc.Sem.(Wo) is the absorp-
tion coefficient for the line-centre photons.
A statistical equilibrium is established under the above
conditions between the atoms and the line-centre photons
whose free path is small compared to the size of the system.
Let i,, be the flux of photons of a frequency o inside the gas.
Then the number of photons absorbed in a unit volume per
Radiation in Gas 155
unit time in the frequency range from wo to w + do is
i,k,do where k,, is the absorption coefficient (11.27). This
number must be equal to the number of photons emitted per
unit time in a unit volume in the same frequency range,
which is (Noxc./t)a,dw. Using Eqs. (11.27), (11.24), and
(14.25), we obtain
7 AyNexe. w ( Nor. &exc. iy:
—— oe i ee
Ket m2c2 Nexc.&gr.
This photon flux is isotropic and can be detected at any
point in the medium which is separated from the boundary by
at least the photon free path. The photon flux outside the
system is given by
m/2 +2/2
To = ( \ iy cos 0d cos 6) ( \ dcos0) =i,
0
—1/2
Here @ is the angle between the normal to the gas surface
and the direction of photon propagation; in the above equa-
tion we have taken into account the fact that the total
photon flux outside the system is normal to the system's
surface. This leads to the following equation for the photon
flux outside the system with a frequency o:
wo? Ner.exc.
lo = Gat (Weng. 1) heh >t (11.30)
When the atoms in the excited and ground states are in
thermodynamic equilibrium (2.9), Eq. (11.30) is identical
to Eq. (2.26) for black body radiation.
Let us estimate the line width of resonance radiation
escaping the system.
For the frequency range in which k,0 <1, the mean
radiation flux escaping outside the gas volume is
Nexc. V
Iy= ay (14.34)
where V is the gas volume, and S is the surface area of the
gas. Equations (11.30) and (41.31) show that the line of
the total radiation escaping the system is wider than the
line radiated by an individual atom and that its width can
156 Introduction to Plasma Physics
be estimated from the following relation:
hol ~ 1 (11.32)
.For instance, for the Lorentz profile Eqs. (41.14), (41.25),
and (11.27) show that the absorption coefficient is given
by k, = kpAwj/(@ — wo)? where ky is the absorption coeffi-
cient at the linecentre, and Aw, is the line width. Proceeding
FIG. 21.
Self-reversal
of a spectrum line.
The solid line is
the line in the absence
of re-emission,
and the dashed line
is the line profile
with re-emission.
from Eq. (11.32), we find that the line width of the
radiation escaping outside the system is
Aw = Aw, (koL)'? (14.33)
if kjkL > 1. Similarly, using Eqs. (41.9), (11.25), (411.27),
and (11.32), we find that the line width of the radiation
escaping outside the system with Doppler broadening is
given by
Aw = Awp (In kyL)'/* (11.34)
where Awp =, [T/(Mc?)]'/? is the line width with Doppler
broadening in the absence of re-emission [see Eq. (11.9)].
When k,Z > 1, the radiation of the frequency o which
escapes outside the system is produced by the de-excitation
of the excited atoms which are at a distance of the order of
1/k,, from the system’s surface. Hence, into Eq. (11.34)
we should insert the temperature of the excited atoms which
are in this region of the system. If the electron temperature
and, hence, the density of the excited atoms is considerably
lower at the boundary than in the bulk of the radiating
system, ‘then the emitted line has a profile illustrated in
Plasma of the Upper Atmosphere 157
Fig. 21, that is, a dip appears at the centre of the line.
This effect is known as the self-reversal of spectrum lines
and is found typically in arc discharges.
12 Plasma of ihe Upper Atmosphere
The balance equations for the parameters of weakly ionized
gas. We have discussed a number of processes of production
and decay of charged particles in plasma which are only
a fraction of the possible processes involving these particles.
What we really have to find is the density of charged parti-
cles and the variation of this density in space and time.
For instance, the useful parameter in the gas laser is the
density of charged particles in the emitting states, in the
gas-discharge radiation sources it is the density of the
emitting atoms, in the shock tube it is the density of the
molecules in the excited vibrational states which determines
the vibrational temperature of the molecules, and in the
conducting gas it is the density of electrons. To find the
parameters needed in a particular problem we have to derive
the balance equation (or system of equations) for them and
solve it.
The balance equation for the density of particles (or for
any other parameter) in plasma is simple in form; it accounts
for all the processes producing variations of the density of
particles. For instance, Eq. (4.4) is the balance equation for
the density of particles in one state when the transitions to
other states are due to collisions with particles of another
species. Generally, the main problem in deriving the bal-
ance equation for a parameter of a plasma is to identify the
process which gives rise to a variation of this parameter. If we
can correctly identify the principal processes or mechanisms
which determine the effect studied or the properties of
a given plasma, our problem will be solved. Otherwise, no
sophisticated mathematics will help. This is why we have
paid attention in this book mostly to those features which
make up the physical picture of the process being analyzed
rather than to solution of the equations which describe
this picture.
158 Introduction to Plasma Physics
Let us now illustrate our analysis by discussing the plas-
ma of the upper atmosphere and showing how the study of
elementary processes can yield a qualitative description of
the system. We wish to find the distribution of charged and
neutral species with height, the composition of the upper
atmosphere, and its heat balance. We shall make use of the
descriptions of the elementary processes made above and
the numerical values of the parameters of this system.
The distribution of particles and temperature in the atmos-
phere with height. The atmosphere near the earth’s surface
at sea level consists primarily of molecular nitrogen (78%),
molecular oxygen (21%), and the total gas density is 2.7 x
<x 101° cm~? (the pressure is 1 atm). Let us find the varia-
tion of the gas density with height. We are interested in
heights which are small compared to the earth’s diameter
(12 800 km) so that the earth’s surface can be assumed flat.
The total flux density of the molecules to the earth’s sur-
face is
j= — D grad N +wN = 0 (12.1)
Here w = bMg is the drift velocity of molecules under the
action of earth’s gravity, N is the density of molecules, & is
the diffusion coefficient of molecules, b is the mobility of
molecules, M is the mass of molecules being considered,
and g is the acceleration of gravity. According to the EKins-
tein relation, = bT where T is the gas temperature.
Hence, we obtain the steady-state (j = 0) distribution of
the molecules of a given species:
N = Noexp (— Ee ) (12.2)
0
Here NV, and WN are the densities of molecules at sea level
and at height h. If the temperature is independent of height,
Eq. (12.2) is identical to the barometric formula (2.14).
Let us make an order-of-magnitude estimate for the drift
velocity of heavy species using Eqs. (7.15) and (7.17) and
the estimate (7.2) for the diffusion coefficient:
_. Me M\1/2_ g
ee’ D~ (+) NmOg
T
Plasma of the Upper Atmosphere 159
Here o, ~ 10-15 cm? is the gas-kinetic cross section and V,
is the total gas density. Hence, we obtain
wN »y, 3 x 10 cms (12.3)
When estimating the temperature gradient, we shall ignore
heat transport processes. Hence, the system is ‘adiabatic
and NT*’-! is constant at all heights so that we have dN/N=
= —(y—1)dT7/T where the adiabatic exponent y for air
is 1.4. Equation (12.2) yields dN/N = —(Msg/T) dz, and
we find that d7/dz = Mg (y — 1) x 14 K-km7!. Thus, the
atmospheric temperature decreases with height. This is the
case up to the atmospheric layers where the energy of solar
radiation is absorbed. The value of d7/dz is overestimated
since we ignored the heat transport processes, which are
determined, primarily, by evaporation and long-wave radia-
tion. It can be readily {shown that heat transport due to
thermal conduction is inessential. Indeed, the thermal
conductivity x» is of the order Ujnermai/Og Where Ujhermal
~ 3 x 104 cm-s7! is the thermal velocity of the gas mole-
cules, and o, ~ 10°71° cm? is the gas-kinetic cross section of
‘the collisions between molecules. Hence, the heat flux q~
~ x (dT/dz) ~ 10-§ W-cm~™ due to thermal conduction is
considerably smaller than the radiation flux at the earth’s
surface (~ 0.1 W-cm~?).
The heat balance of the earth. Let us analyze the trans-
formations of the solar radiation energy. Assume that the
sun is an absolutely black body with a surface temperature
of 5800 K. Then the radiation flux from the sun’s surface
iS Ign == 6.4 kW-cm~?, the maximun} intensity of the pho-
ton flux corresponds to the photon energy of the order of an
electron volt, and the main part of the flux is in the optical
range. The solar radiation flux at the earth’s orbit is
Tearth = 0.14 W-cm~*. The quantity Jea;t, is referred to as
the solar constant.
The solar radiation absorbed by the earth must be radiated
back to space since during its existence the earth received
such an amount of energy that would have been sufficient to
evaporate it. Let us consider the mechanisms for returning
the solar energy received by the Earth into space. There are
two possible mechanisms: the radiation of photons and the
160 Introduction to Plasma Physics
escape of high-energy particles. Let us first estimate the
energy flux due to the latter mechanism.
It is obvious that only those particles whose velocity
component perpendicular to the earth’s surface is higher
than the escape velocity Vege. = (2gReartn)/? ~ 1.1 x
x 10° cm-s7! can escape from the earth’s gravitational field
(Rearth = 6400 km is the earth’s radius). The atmospheric
particles whose density NV ,, satisfies the condition N,o,L <
< 1 can escape without collisions (here 0, ~10-) cm? is the
gas-kinetic collision cross section, and L = Mg/T ~ 10 km
is the distance at which a noticeable variation of the atmos-
pheric density occurs). Hence, we obtain N,, ~ 10° cm™3
and the energy flux due to the particles escaping from this
layer is I ~ Nyvase. X exp [—Mv3,../(2T)] where 7 is the
gas temperature. The energy flux due to this mechanism is
of the order of 0.01 W-cm~? x exp [—Mv¢,-./(2T)]. Since the
gas temperature 7 is much lower than Muv;,, /2 = 20 000 K,
this energy flux is considerably lower than the solar energy
flux. This means that the solar energy received by the earth
cannot be returned to space by escaping high-energy atoms
and molecules.
Hence, the solar energy absorbed by the earth is emitted
to space as long-wave radiation. The diagram in Fig. 22
illustrates the mechanisms of transformation of energy re-
ceived by the earth’s surface and the atmosphere. In partic-
ular, the thermal radiation by the earth’s surface amounts
to 2.06 x 10!4 kW. Assuming that the earth is radiating
likea black body, we can estimate from the Stefan-Boltzmann
law (2.26) that the mean temperature of the earth’s surface
is 291 K or 18 °C.
The elemental oxygen in the atmosphere. At high alti-
tudes there occurs effective photodissociation of the atmos-
pheric molecular oxygen:
O, + ho > OP) + OD) — (12.4)
This process gives rise to radiation absorption known as
the Schumann-Runge continuum, in the wavelength range
from 1325 to 1759 A (the photon energy from 6 to 10.3 eV)
with the cross sections of the order of 10-!9-10-!? cm?.
Photodissociation of molecular oxygen changes the composi-
164
aa
P 9
ow
Uo}
oS
= 7,2
a’; as
= s
nw o
ce
as 3
ag 3
et ny fe)
ban
Oo Mm 3
1 ae 5
ose Ss
- —- O a
=" OO
~} sa 3
~§ 3
€°9 ‘yee ay} jo
"1°S Burdeosy
—_—_—
Buidedsq
L'Q ‘aoeyns s,yyea ayy Aq payoajjas U's
S’°Z4L
X
\
“OORJINS S YMCA Used By} Jo ‘4"|
ayy Aq paquosqe
JaYydsowye joss
L°8¢
“asayudsowye jo 1°) ;eJOL
*ujsea ayy Aq
paquosqe ‘1 ‘Ss
zoe I
‘a0epns\ S,yled aul jo
uoljydsosge
}EYOL
9°07 ‘adepns
S,Uysea ay jo '4'| {
G‘6L ‘aszaydsowje
Aq paquosge adejins
S$, Yeo ay} WO ‘4 *|
#
Sos "UOI]}BJOGRBAA
‘aoepins
S,yea ay} Je juapioul ‘4°s
ayedissip ‘4's
O°€ ‘asaydsowje
O'b ‘aoeyns
S,yUea ayy ye JuapIOU
pue asaydsouge
Aq paquosge °4°s
*g0eJINS S,Yyea au} Je
quapiou Ay oesIp “2 °S
if
=e
——i——_—_—_-—
Ep ‘“spnojo Aq paldajjas *4°S Huldeds3
¢
4
8, ‘uns ayy
“res
woud}
*‘(eBues
peiesu! aut)
UOI}EIPes DAPM
-Buo] au} Ss! ‘4°
(seBue,
BurunoqyuBieu
pue jerydo eu)
UOH}EIPes BACM
“JOYS OU} SI ‘4's
‘MA ¢1 0b JO
sjiun ul siaMmod
POe}eAUOD UBSW 9U}
ese oinbiy au} ul
usaiB saenjea oyy
“esaudsouye
$ji pue adeyins
S,YYeS SU} JO
esuejeq jeoy oy]
‘tT “Old
162 Introduction to Plasma Physics
tion of the atmosphere at high altitudes. The density of
elemental oxygen becomes equal to the density of molecular
oxygen at the altitude of 100-120 km and to the density of
molecular nitrogen at the altitude of 150-200 km. Thus,
elemental oxygen is a major component of the upper atmos-
phere.
First, let us analyze absorption of the short-wave radia-
tion coming to the earth, which produces photodissociation
of an oxygen molecule. The balance equation for the inten-
sity J, of this radiation can be written as follows:
dl
eo — LS [Oz]
Here z is the altitude, [O,] is the density of molecular oxygen,
and o, is the cross section of photodissociation of an oxygen
molecule. Assume that radiation flux is normal to the
earth’s surface and the density of molecular oxygen varies
according to the barometric formula
[O,] = No exp (—z/L)
Here L = Mg/T ~ 10 km. The above equation yields
I.(2) = I,(00) exp {—expl|—(z—z,)/L]} (12.5)
where I,, (co) is the-photon flux at frequency @ outside the
atmosphere, and the point z) is found from the equation
[(O,],-2, = (o,L)"'. Equation (12.5) indicates that the
greater part of the photons of this frequency are absorbed at
the altitude of about z, in the atmospheric layer with a
thickness of about L.
The cross secion of photon absorption by an oxygen mol-
ecule varies from 10-!® to 107!’ cm? for these photon fre-
quencies so that these photons are absorbed mainly in those
atmospheric layers where the molecular oxygen density [O,]
is of the order of 101-10! cm-°.
Let us verify the validity of the barometric formula for
the molecular oxygen density at these altitudes. The charac-
teristic time of photodissociation of the oxygen molecule
at the altitudes where [O,] ~ 1044 cm~? and where the
1300-1600 A photons are mostly absorbed is given by tas. ~
~ (z=) code) ~ 2x 10° s (the factor 1/4 is due to the
Plasma of the Upper Atmosphere 163
averaging of the solar radiation flux over the earth’s sur-
face). The characteristic time in which a molecule passes
the above layer (drifts from it) is
Tanitt ~ L/w ~ 3 x 10°8N,, cms (12.6)
where N,, is the density of particles in the atmosphere. As-
suming [0,] ~ N,,/4, we find Tart a 104 s, that is,
Taritt < Tais.. Hence, photodissociation cannot violate the
barometric formula for the distribution of molecular oxy-
gen.
Let us estimate the density of elemental oxygen in the
almospheric layers where the photodissociation of molecular
oxygen occurs. The flow w [O] of oxygen atoms drifting from
(his layer under the action of the earth’s gravity is equal to
the flow of atoms produced by photodissociation, which
equals the flow of photons of the appropriate frequency at
this altitude. Using Eq. (12.3), we find the following value
of the elemental oxygen density [O] at this altitude (the
‘olar radiation flux giving rise to the photodissociation of
molecular oxygen is IT ~ 10% cm~°s-!):
[(O] ~ I/lw~0.4Nm (12.7)
Lience, [O] is of the order of 10!* cm@* at this altitude. The
barometric formula may be used for finding the elemental
oxygen density at higher altitudes.
The barometric formula can be applied to elemental oxy-
gen at lower altitudes so that its density must increase with
decreasing altitude. With increasing elemental oxygen den-
sity, there increases the probability of the three- ope
process of recombination of oxygen atoms:
20 +M—> 0, +M, 0+0,4+M>0,4+M (12.8)
llere M is the nitrogen or oxygen molecule. The elemental
oxygen density is the highest at the altitude where the rate
of recombination of oxygen atoms equals the rate of produc-
tion of oxygen atoms. Let us estimate the highest elemental
oxygen density from this condition:
Here [O]lmax is the highest density of elemental oxygen,
HK ~ 10-*3 cm*s“! is the rate constant of the three-particle
11*
164 Introduction to Plasma Physics
recombination of oxygen atoms, and w is the drift velocity
of the oxygen atom towards the earth’s surface due to gravity,
which is given by Eq. (12.3). Using Eqs. (12.7) and (42.9)
and the numerical values of the parameters, we find that
[O]max ~ 104% cm-3.
Charged particles in the upper atmosphere. Photoioniza-
tion of atoms and molecules in the upper atmosphere produ-
ces ions, for instance, Nj, Of, N*, and O*. The photon flux
Iion, With photon energies exceeding the ionization poten-
tials, for these molecules is of the order of 3 x 10!° cm~?s-!
in the upper atmosphere, and the respective photoionization
cross section iS Ojon ~ 10718-1071? cm?. Equation (12.5)
indicates that this photon flux is absorbed mainly in the
atmospheric layers where the particle density N,, is of the
order of (GjonL)-! ~ 1011-10! cm-*. The primary produc-
tion of charged particles occurs just in these layers of the
upper atmosphere.
Molecular ions decay due to dissociative recombination,
and their density can be estimated from the relationaN .N,~
~ TIion/L where a ~ 10-* cm*s~ is the rate constant of dis-
sociative recombination, and the electron density N, is as-
sumed to be of the order of the ion density N;. Substituting
the numerical values, we obtain (Iio. ~ 3 X 10'° cm~*s—*)
the following expression:
Ny~ (=n) ~5 x10 em (42.40)
Here the characteristic time T,»-, of recombination of the
molecular ion is of the order of (aNV,)"! ~ 20 s while the
characteristic time Tg;ir_ of the drift of the molecular ion
from this layer, is of the order of 10* s. Therefore, we
could ignore the escape of ions to other layers when writing
the balance equation for the density of the molecular ions.
Since the upper atmosphere contains electrons, it does not
transmit long-wave electromagnetic waves whose frequency
is higher than the plasma frequency w, = (4nN ,e2/m)!/? =
= 3 x 10’ s“! [see Eqs. (3.9) and (10.9)]. This frequency
corresponds to the wavelength 4 ~ 60 m and the electro-
magnetic waves of lower frequencies will be either damped
or reflected from this atmospheric layer known as the iono-
sphere.
Plasma of the Upper Atmosphere 165
The atomic ions produced by photoionization in the upper
atmosphere react with molecules:
Ot-+N,—NOt+N+1.05 eV, (12.44)
O*+0,> Of +044.3 eV, (12.14b)
N+ +0,>NO*+N+4+5.7 eV - (12.44)
The rate constant k of the first reaction is of the order of
10-11 cm’s-!, that of the second reaction is of the order of
10-12 cm’s-!, and that of the third reaction is of the order
of 10-2° cm’s-!. Hence, the characteristic times of transfor-
mation of atomic ions into molecular ions in this atmospheric
layer are of the order of (kN,,)~! ~ 0.01-10 s. It may be
seen that the transformation of atomic ions into molecular
ions occurs faster than the recombination of molecular ions.
Therefore, the density of atomic ions at these altitudes is
lower than the density of molecular ions. In the atmospheric
layer where the density of molecular ions is the highest (the
altitude about 100 km), the principal ion species are Oj,
N3, and NO*.
Let us analyze the distribution of the atomic ions of
oxygen, whichconstitute the main ion component of the atmo-
sphere at large altitudes, where it consists basically of
elemental oxygen and molecular nitrogen. Oxygen ions are
produced by photoionization of oxygen atoms, and they dis-
appear in the ion-molecular reaction with nitrogen mole-
cules or drift away due to diffusion. The highest density of
oxygen ions corresponds to the equilibrium between these
processes:
Here w is the drift ion velocity (12.3), k is the rate constant
of the ion-molecule reaction (12.11a), and [N,] is the density
of nitrogen molecules. Assuming that this atmospheric
layer consists of elemental oxygen, we obtain
[N.] [0] ~ w [O]/(Lk)
Assuming, in accordance with Eq. (12.3), w[O] ~ 3 x
<x 107% cm=*s-!, and L ~ 10 km and k ~ 10-8 cm3s-!, we
obtain [N,] [0] ~ 3 x 10!® cm~® for the altitude which cor-
responds to the highest density of atomic ions. This altitude
is about 200 km.
166 Introduction to Plasma Physics
Let us estimate the highest density (N;)max of the atomic
ions of oxygen from the balance equation for the density of
atomic ions \ Sion dion 1O] = k [N.]N;. In the case of
photoionization of atomic oxygen, we have |\ Ojon dlion ~
~ 2 xX 10-? s-! and the highest density of the atomic ions of
oxygen is of the order of 2 x 10° cm? x [OI/(N,] <
< 10° cm™.
At greater altitudes, the density of atomic ions is given
by the barometric formula, and it slowly decreases with in-
creasing altitude since the temperature of ions is consider-
ably higher than the gas temperature.
Note that the characteristic times of decay of ions do not
exceed 100 seconds. Therefore, the ion density at night is
considerably lower than the ion density at daylight. We
made all our estimates for daylight conditions. The density
of ions at night can be estimated from the relation
aN;t ~ 1 where t is the duration of night. Hence, we find
that at night N,; ~ 107-10? cm=? in the atmosphere layer
where the negative ions are not produced (t ~ 3 xX 10' s,
a~ 10-7 cm's-4).
As discussed above, molecular ions are produced mostly
in the atmospheric regions where the gas density \V,, is of the
order of 1011-10!" cm-*. The molecular ions drift to the
regions of higher density under the action of gravity. The
density of molecular ions in these regions at daylight can
be estimated from the balance equation
wN ;/L ~ aN?
Since the drift velocity w of ions under the action of gravity
is close to the drift velocity (42.3) of atoms and molecules,
we find the following estimate:
NiNm~ 104% cm (42.12)
At smaller altitudes the molecules capture electrons pro-
ducing negative ions. For instance,
e+20,>0O;+0, (12.43)
The rate constant & of the process (12.13) is of the order of
10-3! cm®s“!, and this process is of significance at altitudes
where wNV /L ~ ZN, 10,]?, that is, where the character-
Plasma of the Upper Atmosphere 167
istic time of drift of charged particles is equal to the time
of electron capture by molecule. Using the formula for
ambipolar diffusion and Eq. (12.3) for the drift velocity of
electrons, we find that negative ions are predominant among
the negative species in the regions where the density of oxy-
gen molecules is
[Oo] > ( etal) 10 om — (12.44)
In these regions recombination occurs according to the reac-
tion A~ + Bt— A + B, and the recombination coefficient
a is of the order of 10-° cm’s~!so that we obtain the follow-
ing estimate for the ion density instead of Eq. (12.12):
N,Nm~ 1017 em-® (42.15)
We have considered the plasma of the upper atmosphere to
illustrate the techniques used in describing a specific system.
The parameters of the plasma have been estimated using
the relevant numerical values (the parameters of the solar
radiation and the rate constant and the cross sections of
elementary processes). Our results do not provide an accu-
rate and detailed description of the system, but they do
provide a correct qualitative picture. An incomparably
greater amount of work is required to increase the accuracy
of description and to account for the details of the process,
but the results will be fundamentally the same.
168
gL 209
(¢4°0) oy
e°9 : OD
(7r'0) (6b°0) § (84°0)
6°S °F 6°7 79)
(2r°0) (sb'0) (2270) (84'0)
g°9 es Vy esc oN
(17°0) (87°0) (7G 0)
Gy 6° 7'¢ eq
((9¢°0) (e9°0) (69°0) = (89°0) (Z'1) (¢°1)
Gy 6°¢ L’g ge g°¢ are ory
(1°0) (1°0) (S°0) (8%0°0)
6°9 eZ O'S 0°6 aX
(ho) (€t'o) (sto) § §=(9t'0) ~— (a0) (9°09) (¥90°0) (180°0) }
%'G 6°S o°¢ Q'Y ey ZY a L°9 Iy
(>1'0) (91°0) (9t°0) (9b°0) (99°0) (69°0) (G60°0) § (zt°0) ~— (9T°0)
L°S e"c eG 7c (ans Be L°9 9° o°¢ V
(6° 0) (e"0) = (8"0) Ss (460) Ss (640) ~— (220) (820) ~— (#0)
Vy (ans 0S LG yy O'F ye ¥°C oN
(1¢°0) (29°00) (¥9'0) (9°0) (T°) (71) (g%0) (Sc°0) (79°00) = (e6'0) = (89)
9° oe 6°C VE Aare Ce Lg ee 6'S 0% OF oH
2090 (eye) 20 oN %q eH xX IY Vv aN oH lied
es "(900 o1N4 819d
-wi3} pue Wye J BInssaid) SUO}}IPUOD PJEpUL}S 0} SUIPUOdSIIIOD g—UID 6101 X689'Z JO ATISUID B JO] (8/79 UI) §
UdAIS 91B SJUSIOYJa09 UOISNAIp syL ‘“selotysed jo Ayisuop ay} SI AY pue ‘AZID0[0A uOIsTIIO® aA iuiel Peels au} Ay a eepivied
SUIPI[1[09 942 JO SseU paonpal oy} St W a1ayM (CG Nen)/L = Bo UUl,eTaL VY} Sulsn G syuslogjaod uolsnyIp ayy Woy paAlap
are oq} ‘ZW o{-0F JO] S}{UN Ul WAALS ale sSa[Ndd[OUI pue SWO}e UaeM Jaq SUOISIT[OI J0J uOT\oe8 ssoro %o DIJOULY-Ses OULL
SJUBIDIHJEOD UOISNYIG PUR SUOI}IaS sso‘D DYaUIy-seD 4 Xjpuoddy
saoipueddy
Appendices 169
Appendix 2‘ The Cross Section of Resonance Charge Exchange
Between the Positive ton A+ and the Respective Atom A
The cross section is given in units of 10-15 cm2, and E is the ion
energy for the resting atom; the jon and the atom are in their
ground states
H He Li Be B C N O
E=0.1 eV 6.2 3.5 26 13 96 5.3 5 4.8
4 eV 5.0 2.8 22 11 7.4 4.3 3.8 3.5
10 eV 3.8 2.4 18 19 5.8 3.2 3 2.8
F Ne Na Mg Al Si Pp S
E=0.1 eV 3.4 3.2 31 19 16.14 8.7 8.1 8.5
4 eV 2.5 2.5 26 16 12.9 6.5 6.5 6.8
10 eV 4.9 1.9 22 13 10.0 4.9 5.0 5.3
Cl A K Ca Ti Vv Cr Mn Fe
E=0.1 eV 4.9 5.5 At 26 22 23 21 2() 21
14 eV 3.9 4.5 9 3d 21 19 19 18 16 18
10 eV 3.0 3.6 29 18 15 46 14 13 15
Co Ni Cu Zn Ga Ge AS £e
E=0.1 eV 22 19 49 16 17 9.4 9.8 9.3
1 eV 18 16 16 13 14 7.9 8.0 7.3
10 eV 15 13 43 11 11 6.14 6.3 5.7
Br Kk} Rb sr Zr Nb Mo Ag Cd
E=0.1 eV 5.9 7.3 45 30 24 22 24 20 17
1 eV 4.6 5.9 39 29 20 19 17 #17 «=~ «14
10 eV 3.7 4.6 32 21 16 15 14 14 12
In Sn Sb Te I Xe Cs Ba
E=0.1 eV 19.5 10.7 141.4 10.6 7.0 9.1 5d OD
1 eV 16 8.7 9.1 8.6 5.6 7.5 45 30
10 eV 13 6.9 7.2 6.8 4.4 6.0 38 29
Ta Ww Re Pt Au He Tl Pb Bi
E=0.1 eV 19 18 24 17 15 15 18.6 11 15.4
4 eV 16 15 17 #16 £14 «12 =~ «15.4 9.2 12.7
10 eV 13 13 14 +13 11 #10 = =«12.1 to 4053
Bibliography
1.
2.
3.
10.
11.
Artsimovich, L. A. Controlled Thermonuclear Reactions, Gordon
and Breach, Science Publishers, New York, 1960.
Brown, S. C. Basic Data of Plasma Physics, Wiley, New York,
1959.
Frank-Kamenetskii, D. A. Lektsii po fizike plazmy (Lecture Notes
of Plasma Physics), Atomizdat, Moscow, 1964.
Ginzburg, V. L. The Propagation of Electromagnetic Waves
in Plasmas, 2nd ed., Pergamon Press, Oxford, 1971.
Hirschfelder, J. O., Curtiss, GC. F., and Bird, R. B. Molecular
Theory of Gases and Liquids, 2nd printing with notes added,
Wiley, New York, 1964.
McDaniel, E. W. Collision Phenomena in Ionized Gases, Wiley,
New York, 1964.
Penning, F. M. £lectrical Discharges in Gases, Cleaver-Hume
Press, London, 1957.
Silin, V. P. Vvedenie v kineticheskuyu teoriyu gazov (Introduction
to the Kinetic Theory of Gases), Nauka, Moscow, 1971.
Smirnov, B.M. Fizika slaboionizirovannogo gaza (Physics of
Weakly Ionized Gas), Nauka, Moscow, 1972.
Spitzer, L., Jr. Physics of Fully Ionized Gases, 2nd ed., Inter-
science, New York, 1962.
Zel'dovich, Ya. B., and Raizer, Yu. P. Physics of Shock Waves
and High Temperature Hydrodynamic Phenomena, 2nd ed.,
2 vols, Academic Press, New York, 1966-7.
170
Index
Acoustic waves in plasma 120
dispersion relation for 124
propagation velocity of 121
Alfven velocity 126
Ambipolar diffusion 104
coefficient of 4102
condition for 103
Associative lonization 46, 51
Aurora 17
Auto-ionization state 43
lifetime of 64
Barometric formula 23, 158, 162,
166
Benard cells 96
Black-body radiation 27
Boltzmann distribution 18, 19
Boltzmann kinetic equation 73
Broadening of spectral lines
Doppler 142
due to finite lifetimes 142
due to interaction with the
gas particles 148
impact 145
Brownian motion 91
Buneman instability 136
Charge exchange 52
Coefficient of photon absorp-
tion 152
Conductivity of weakly ionized
gas 110
174
Continuity equation 75
Controlled thermonuclear
sion 14
Convective motion of gas 94
instability of 98
Cosmic plasma (see Space plasma)
Critica! radius 59
Cross section
differential 39
diffusion 39, 40
of capture 40
of emission and absorption
of photons 150
gas-kinetic 40
of inelastic collision 36
of ion pair recombination 57
of resonance charge exchange
105
Cyclotron frequency 113
Cyclotron resonance 112
fu-
Damping of plasma
tions 127
Debye-Hiickel radius 314
Diffusion 79
coefficient of 79
Dissociative recombination 44, 65
coefficient of 66
mechanism of 65
oscilla-
Einstein coefficients 1414
Einstein relation 86
derivation of 86, 87
172
Elastic collision 37
cross section of 39
Electrogasodynamic generator 15
Electromagnetic waves in plas-
ma 126
dispersion relation for 4127
Electron motion in external field
108
equation for 110
frictional force in 110
Excited atoms
decay of 71
density of 71
thermodynamic equilibrium
of 70
lifetime of 71
radiative transformations of
71
Franck-Condon principle 49
Gas discharge 10
arc 410
glow 10
nonself-maintained 10
self-maintained 10
Gaseousness criterion 29, 42
Grashof number 97
Hall effect 1411, 112
Heat transport 80
convective 96
equation of 89
Heisenberg uncertainty princi-
_ ple 41
Hydrodynamic instability 137
condition of 138
for pinch 137
Impact broadening 149
Impact parameter in collisions 38
Index
Inelastic collisions 50
cross section of 36
Interplanetary plasma 17
Interstellar plasma 17
Ionization of atoms 54
associative 51
Thomson model of 55
in collisions 51, 52
stepwise 68
Ton sound 123
condition of existence 133
dispersion relation for 124
phase velocity of 133
Kinetic equation 72
in external field 73
integration of 74
Langevin formula 107
Langmuir frequency 33
Langmuir paradox 134
Laser operation 152
condition of 153
principles of 153, 154
Lorentz profile of spectral line
due to impact broadening 147
due to radiative broadening
145
Lawson criterion 114
Magnetohydrodynamic (Alfven)
waves 124
damping of 127
dispersion relation for 126
velocity of 126
Magnetohydrodynamic equations
115
Magnetohydrodynamic generator
14
closed-cycle 14
open-cycle 14
Index
Massey principle 54
Maxwell distribution 23
Maxwell equations 34
Mean free path of gas particles 40
definition of 40
Mobility of charged particles 100
definition of 100
estimate of 100
of electrons 100
of ions in foreign gas 103
of ions in parent gas 104
Mobility of gas particles 86
estimate of 87
relation to diffusion coeffi-
cient 87
Molecular gas 26
dissociation equilibrium of
26
specific heat of 85, 89
Navier-Stokes equation 88
Oscillations of plasma electrons
33
frequency of 33
Penning effect 47, 63
Pinch effect 119
Planck’s distribution 26, 27
Plasma frequency 33
_ Plasma oscillations 124
amplification of 132
attenuation of 132
condition of existence 133
damping of 127
dispersion relation for 122
energy exchange with elec-
trons 130 ,
interaction with electrons
129
173
Landau damping of 133
propagation of 126
Pressure in gas 77
relationship to temperature
78
Pressure of magnetic field 119
Pressure tensor 795
relationship
sure 78
to gas pres-
Rayleigh-Jeans formula 27
Rayleigh number 93
minimum for convection 94
relationship to Grashof num-
ber 97
Rayleigh problem 93
Recombination coefficient for
ions 107
dependence on gas density
108
estimate of 108
Recombination of ions 57
constant of 58
cross section of 57
Resonance processes 51
definition of 54
condition of 52
properties of 48
Resonance radiation 154, 155
emission of 154
line width of 155
propagation of 154
Saha distribution 24, 25
Screening of charges and fields 30
Single ionization 54
Skin effect 34
Skin layer 35
Solar corona 17
Solar photosphere 9, 16, 17
Space plasma 16
174 Index
Specific heat 85 Triple processes 58
for binary gas 89 Thomson’s theory for 58
for monoatomic gas 89 Triple recombination 60
Spontaneous emission 140 of electrons and ions 60
Stefan-Boltzmann law 28 of ions 61
Stepwise ionization 68 Tokamak 14
constant of 68 Turbulent gas flow 98
Stimulated radiation 142 development of 99
emission of 141
Viscosity 82
coefficient of 82
estimate 6f 82
“Tau” approximation 74
Temperature 20
definition of 24
Thermal conductivity 80
due to internal degrees of
freedom 83 Wave damping in plasma 129
estimate of 81 conditions for 134
Thomson formula for ionization Wien formula 27
cross section 56 Work function 14
TO THE READER
Mir Publishers welcome your comments
on the content, translation, and design of
this book.
We would also be pleased to receive any
proposals you care to make about our fu-
ture publications.
Our address is:
USSR, 129820, Moscow 1-110, GSP
Pervy Rizhsky Pereulok, 2
Mir Publishers
Printed in the Union of Soviet Socialist Republies
Other Books for Your Library
ELEMENTS
OF APPLIED- MATHEMATICS
by Ya. B. ZELDOVICH and A. D. MYSKIS
This book describes useful methods of calculation
and gives the fundamentals of complex variables, linear
differential equations, vectors and vector fields,
and the calculus of variations. Formal proofs are largely
replaced by leading questions and pointers, thereby
achieving simplicity and clarity of exposition. Certain
physical problems, in particular those relating to optics,
mechanics, and the theory of probability, are analyzed
in detail. This text will be of interest to university
students as a supplement to their standard textbooks,
and to engineers, physicists, and anyone else wishing
to brush up on the elements of modern applied
mathematics.
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11