by B.M. SMIRNOV

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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

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© 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