INTRODUCTION TO

PLASMA PHYSICSAND CONTROLLED

FUSIONSECOND EDITION

Volume 1: Plasma Physics

Francis F. ChenElectrical Engineering Department

School of Engineering and Applied ScienceUniversity of California, Los Angeles

Los Angeles, California

PLENUM PRESSNEW YORK AND LONDON

Chapter One

INTRODUCTION

OCCURRENCE OF PLASMAS IN NATURE 1.1

It has often been said that 99% of the matter in the universe is in theplasma state; that is, in the form of an electrified gas with the atomsdissociated into positive ions and negative electrons. This estimate maynot be very accurate, but it is certainly a reasonable one in view of thefact that stellar interiors and atmospheres, gaseous nebulae, and muchof the interstellar hydrogen are plasmas. In our own neighborhood, assoon as one leaves the earth’s atmosphere, one encounters the plasmacomprising the Van Allen radiation belts and the solar wind. On theother hand, in our everyday lives encounters with plasmas are limitedto a few examples: the flash of a lightning bolt, the soft glow of theAurora Borealis, the conducting gas inside a fluorescent tube or neonsign, and the slight amount of ionization in a rocket exhaust. It wouldseem that we live in the 1% of the universe in which plasmas do notoccur naturally.

The reason for this can be seen from the Saha equation, which tellsus the amount of ionization to be expected in a gas in thermal equilibrium:

Here tzi and n, are, respectively, the density (number per m3) of ionizedatoms and of neutral atoms, T is the gas temperature in °K, K isBoltzmann’s constant, and U; is the ionization energy of the gas-that

2ChapterOne

is, the number of ergs required to remove the outermost electron froman atom. (The mks or International System of units will be used in thisbook.) For ordinary air at room temperature, we may take n,, =3 x 1O25 mm3 (see Problem 1-1), T 25: 300°K, and Vi = 14.5 eV (fornitrogen), where 1 eV = 1.6 x 10-l’ J. The fractional ionization ni/(n, +ni) z ni /n, predicted by Eq. [1-1] is ridiculously low:

ni- z 10-122n,

As the temperature is raised, the degree of ionization remains lowuntil Ui is only a few times kT. Then ni/n, rises abruptly, and the gasis in a plasma state. Further increase in temperature makes n, less thanni, and the plasma eventually becomes fully ionized. This is the reasonplasmas exist in astronomical bodies with temperatures of millions ofdegrees, but not on the earth. Life could not easily coexist with aplasma-at least, plasma of the type we are talking about. The naturaloccurrence of plasmas at high temperatures is the reason for the designa-tion “the fourth state of matter.”

Although we do not intend to emphasize the Saha equation, weshould point out its physical meaning. Atoms in a gas have a spread ofthermal energies, and an atom is ionized when, by chance, it suffers a

- - - -- - - - ---, A-a

-m-- c-- - -- - - - - e-

- - - e--+-- a.

Ar

FIGURE 1-1 Illustrating the long range of electrostatic forces in a plasma.

/

collision of high enough energy to knock out an electron. In a cold gas,such energetic collisions occur infrequently, since an atom must beaccelerated to much higher than the average energy by a series of“favorable” collisions. The exponential factor in Eq. [1-1] expresses thefact that the number of fast atoms falls exponentially with UJKT. Oncean atom is ionized, it remains charged until it meets an electron; it thenvery likely recombines with the electron to become neutral again. Therecombination rate clearly depends on the density of electrons, whichwe can take as equal to n;. The equilibrium ion density, therefore, shoulddecrease with n;; and this is the reason for the factor ni’ on theright-hand side of Eq. [1-1]. The plasma in the interstellar medium owesits existence to the low value of n; (about 1 per cm3), and hence the lowrecombination rate.

3Introduction

DEFINITION OF PLASMA 1.2

Any ionized gas cannot be called a plasma, of course; there is alwayssome small degree of ionization in any gas. A useful definition is asfollows:

A plasma is a quasineutral gas of charged and neutral particles whichexhibits collective behavior.

We must now define “quasineutral” and “collective behavior.” Themeaning of quasineutrality will be made clear in Section 1.4. What ismeant by “collective behavior” is as follows.

Consider the forces acting on a molecule of, say, ordinary air. Sincethe molecule is neutral, there is no net electromagnetic force on it, andthe force of gravity is negligible. The molecule moves undisturbed untilit makes a collision with another molecule, and these collisions controlthe particle’s motion. A macroscopic force applied to a neutral gas, suchas from a loudspeaker generating sound waves, is transmitted to the.individual atoms by collisions. The situation is totally different in aplasma, which has charged particles. As these charges move around, theycan generate local concentrations of positive or negative charge, whichgive rise to electric fields. Motion of charges also generates currents, andhence magnetic fields. These fields affect the motion of other chargedparticles far away.

Let us consider the effect on each other of two slightly chargedregions of plasma separated by a distance r (Fig. 1-1). The Coulombforce between A and B diminishes as l/r*. However, for a given solidangle (that is, ∆r/r = constant), the volume of plasma in B that can affect

4ChapterOne

A increases as r3. Therefore, elements of plasma exert a force on oneanother even at large distances. It is this long-ranged Coulomb forcethat gives the plasma a large repertoire of possible motions and enrichesthe field of study known as plasma physics. In fact, the most interestingresults concern so-called “collisionless” plasmas, in which the long-rangeelectromagnetic forces are so much larger than the forces due to ordinarylocal collisions that the latter can be neglected altogether. By “collectivebehavior” we mean motions that depend not only on local conditionsbut on the state of the plasma in remote regions as well.

The word “plasma” seems to be a misnomer. It comes from theGreek ~A&r~cy, -cum~, ~6, which means something molded or fabricated.Because of collective behavior, a plasma does not tend to conform toexternal influences; rather, it often behaves as if it had a mind of its own.

1.3 CONCEPT OF TEMPERATURE

Before proceeding further, it is well to review and extend our physicalnotions of “temperature.” A gas in thermal equilibrium has particles ofall velocities, and the most probable distribution of these velocities isknown as the Maxwellian distribution. For simplicity, consider a gas inwhich the particles can move only in one dimension. (This is not entirelyfrivolous; a strong magnetic field, for instance, can constrain electronsto move only along the field lines.) The one-dimensional Maxwelliandistribution is given by

f(u) = A exp (-imu2/KT) [1-2]

where f du is the number of particles per m3 with velocity between uand u + du, imu is the kinetic energy, and K is Boltzmann’s constant,

’K = 1.38 x 1O-23 J/°K

The density n, or number of particles per m3, is given by (see Fig. l-2)

The constant A is related to the density n by (see Problem l-2)

[1-4]

The width of the distribution is characterized by the constant T,which we call the temperature. To see the exact meaning of T, we can

5Introduction

A Maxwellian velocity distribution. FIGURE 1-2

compute the average kinetic energy of particles in this distribution:

I

00$nu ‘f (u) du

E,,= -*m

I-mfb4 du

Defining

vt/& = (2K7-/m)1’2 and y = UIVth [1-6]

we can write Eq. [1-2] as

and Eq. [1-5] asf(u) = A exp (-u2/vt2h)

$nAv f,,

Eav =I

cokxp C--Y 2)1~2 dy--co

I

coA&h exp (-y ‘1 dr-co

The integral in the numerator is integrable by parts :

I

coy - bp (--r2)lr dy = E-$Cexp (-Y~)IYI~~ - -i exp (-y2) dy

--co4,

1=5

Iew k-y ‘14

-a3

Cancelling the integrals, we have

E,, =$L‘iV $j,

A&t,= $&, = $KT

Thus the average kinetic energy is $KT.

[1-5]

[1-7]

6ChapterOne

It is easy to extend this result to three dimensions. Maxwell’s distribu-tion is then

where

f(u, v, w) = As exp [-$z(u* + tr* + w*)/KT) [1-8]

[1-9]

The average kinetic energy is

E,, =Asim(u2 + v* + w*) exp [-im(u2 + v* + w*)/KT] du dv dw

Asexp[-$m(u2+v2+w2)/KT]dudvdw

We note that this expression is symmetric in u, v, and w, since a Maxwelliandistribution is isotropic. Consequently, each of the three terms in thenumerator is the same as the others. We need only to evaluate the firstterm and multiply by three:

Eav =3As j $mu* exp (-kmu*/KT) du jj exp [-im(v* + w*)/KT] dv dw

As j exp (-$mu*/KT) du ljexp [-im(v* + w*)/KT] dv dw

Using our previous result, we have

E,, = $KT [1-10]

The general result is that E,, equals $KT per degree of freedom.Since T and E,, are so closely related, it is customary in plasma

physics to give temperatures in units of energy. To avoid confusion onthe number of dimensions involved, it is not E,, but the energy corres-ponding to KT that is used to denote the temperature. For KT = 1 eV =1.6 x lo-“J, we have

T =1.6 x 10-l’

1.38 X 1o-23= 11,600

Thus the conversion factor is

1 eV = 11,600°K [1-11]

By a 2-eV plasma we mean that KT = 2 eV, or E,, = 3 eV in threedimensions.

It is interesting that a plasma can have several temperatures at thesame time. It often happens that the ions and the electrons have separate

Maxwellian distributions with different temperatures Ti and T,. Thiscan come about because the collision rate among ions or among electronsthemselves is larger than the rate of collisions between an ion and anelectron. Then each species can be in its own thermal equilibrium, butthe plasma may not last long enough for the two temperatures to equalize.When there is a magnetic field B, even a single species, say ions, canhave two temperatures. This is because the forces acting on an ion alongB are different from those acting perpendicular to B (due to the Lorentzforce). The components of velocity perpendicular to B and parallel toB may then belong to different Maxwellian distributions with tem-peratures TL and Tll.

7Introduction

Before leaving our review of the notion of temperature, we shoulddispel the popular misconception that high temperature necessarilymeans a lot of heat. People are usually amazed to learn that the electrontemperature inside a fluorescent light bulb is about 20,000°K. “My, itdoesn’t feel that hot!” Of course, the heat capacity must also be takeninto account. The density of electrons inside a fluorescent tube is muchless than that of a gas at atmospheric pressure, and the total amount ofheat transferred to the wall by electrons striking it at their thermalvelocities is not that great. Everyone has had the experience of a cigaretteash dropped innocuously on his hand. Although the temperature is highenough to cause a burn, the total amount of heat involved is not. Manylaboratory plasmas have temperatures of the order of 1,000,000°K (100 eV), but at densities of 10’s-lO’g per m3, the heating of the walls isnot a serious consideration.

1-1. Compute the density (in units of m-“) of an ideal gas under the followingconditions:

(a) At 0°C and 760 Torr pressure (1 Torr = 1 mm Hg). This is called theLoschmidt number.

(b) In a vacuum of lo-” Torr at room temperature (20°C). This number is auseful one for the experimentalist to know by heart ( low3 Torr = 1 micron).

1-2. Derive the constant A for a normalized one-dimensional Maxwellian distri-bution

f(u) = A exp (-mu2/2KT)

PROBLEMS

such that

I

a,f(u) du = 1

-co

8ChapterOne

:II:

PLASMA- - - -- -

-- - +++

FIGURE 1-3 Debye shielding.

1.4 DEBYE SHIELDING

A fundamental characteristic of the behavior of a plasma is its ability toshield out electric potentials that are applied to it. Suppose we tried toput an electric field inside a plasma by inserting two charged ballsconnected to a battery (Fig. l-3). The balls would attract particles of theopposite charge, and almost immediately a cloud of ions would surroundthe negative ball and a cloud of electrons would surround the positiveball. (We assume that a layer of dielectric keeps the plasma from actuallyrecombining on the surface, or that the battery is large enough tomaintain the potential in spite of this.) If the plasma were cold and therewere no thermal motions, there would be just as many charges in thecloud as in the ball; the shielding would be perfect, and no electric fieldwould be present in the body of the plasma outside of the clouds. Onthe other hand, if the temperature is finite, those particles that are atthe edge of the cloud, where the electric field is weak, have enoughthermal energy to escape from the electrostatic potential well. The “edge”of the cloud then occurs at the radius where the potential energy isapproximately equal to the thermal energy KT of the particles, and theshielding is not complete. Potentials of the order of KT/e can leak intothe plasma and cause finite electric fields to exist there.

Let us compute the approximate thickness of such a charge cloud.Imagine that the potential 4 on the plane x = 0 is held at a value ~$0 bya perfectly transparent grid (Fig. 1-4). We wish to compute 4(x). Forsimplicity, we assume that the ion-electron mass ratio M/m is infinite,so that the ions do not move but form a uniform background of positivecharge. To be more precise, we can say that M/m is large enough that

9Introduction

0 X

Potential distribution near a grid in a plasma. FIGURE 1-4

the inertia of the ions prevents them from moving significantly on thetime scale of the experiment. Poisson’s equation in one dimension is

d24EoV2cp = t?o- =dx2

-e(ni - ne) (Z= 1) [1-12]

If the density far away is noo, we have

In the presence of a potential energy 44, the electron distribution func-tion is

f(u) = A exp [-&mu2 + W/K77It would not be worthwhile to prove this here.What this equation saysis intuitively obvious: There are fewer particles at places where thepotential energy is large, since not all particles have enough energy toget there. Integrating f (u) over u, setting q = -e, and noting that n,(q5 +0) = noo, we find

n, = na exp (t$IKT,)

This equation will be derived with more physical insight in Section 3.5.Substituting for ni and n, in Eq. [1-12], we have

eO$=enm[[exp(~)] - I}

In the region where 1 e4/KT,I << 1, we can expand the exponential in aTaylor series:

[1-13]

No simplification is possible for the region near the grid, where 1 e4/JXc 1may be large. Fortunately, this region does not contribute much to thethickness of the cloud (called a sheath), because the potential falls veryrapidly there. Keeping only the linear terms in Eq. [1-13], we have

d24 n,e2‘“&xl = KTe 4

Defining

[1-14]

[1-15]

where n stands for noo, we can write the solution of Eq. [1-14] as

4 =40exp(-bIlW [1-16]

The quantity λ n, called the Debye length, is a measure of the shieldingdistance or thickness of the sheath.

Note that as the density is increased, λD decreases, as one wouldexpect, since each layer of plasma contains more electrons. Furthermore,λD increases with increasing KT,. Without thermal agitation, the chargecloud would collapse to an infinitely thin layer. Finally, it is the electrontemperature which is used in the definition of λD because the electrons,being more mobile than the ions, generally do the shielding by movingso as to create a surplus or deficit of negative charge. Only in specialsituations is this not true (see Problem 1-5).

The following are useful forms of Eq. [1-15]:

AD = 69(T/n)1’2 m, T in °K

AD = 7430(KT/~)“~ m, KT in eV [1-17]

We are now in a position to define “quasineutrality.” If thedimensions L of a system are much larger than An, then whenever localconcentrations of charge arise or external potentials are introduced intothe system, these are shielded out in a distance short compared with L,leaving the bulk of the plasma free of large electric potentials or fields.Outside of the sheath on the wall or on an obstacle, V2qS is very small,and ni is equal to ne, typically, to better than one part in 106. It takesonly a small charge imbalance to give rise to potentials of the order ofKT/e. The plasma is “quasineutral”; that is, neutral enough so that onecan take ni = n, = n, where n is a common density called the plasma

11Introduction

density, but not so neutral that all the interesting electromagnetic forcesvanish.

A criterion for an ionized gas to be a plasma is that it be denseenough that AD is much smaller than L.

The phenomenon of Debye shielding also occurs-in modifiedform-in single-species systems, such as the electron streams in klystronsand magnetrons or the proton beam in a cyclotron. In such cases, anylocal bunching of particles causes a large unshielded electric field unlessthe density is extremely low (which it often is). An externally imposedpotential-from a wire probe, for instance-would be shielded out byan adjustment of the density near the electrode. Single-species systems,or unneutralized plasmas, are not strictly plasmas; but the mathematicaltools of plasma physics can be used to study such systems.

THE PLASMA PARAMETER

The picture of Debye shielding that we have given above is valid onlyif there are enough particles in the charge cloud. Clearly, if there areonly one or two particles in the sheath region, Debye shielding wouldnot be a statistically valid concept. Using Eq. [1-17], we can compute thenumber ND of particles in a “Debye sphere”:

ND = n$rAff, = 1.38 X 1 06P2/n “* (T in °K)

In addition to An << L, “collective behavior” requires

ND >>> 1

[1-18]

[1-19]

1.5

CRITERIA FOR PLASMAS 1.6

We have given two conditions that an ionized gas must satisfy to be calleda plasma. A third condition has to do with collisions. The weakly ionizedgas in a jet exhaust, for example, does not qualify as a plasma becausethe charged particles collide so frequently with neutral atoms that theirmotion is controlled by ordinary hydrodynamic forces rather than byelectromagnetic forces. If o is the frequency of typical plasma oscillationsand 7 is the mean time between collisions with neutral atoms, we require07 > 1 for the gas to behave like a plasma rather than a neutral gas.

12ChapterOne

The three conditions a plasma must satisfy are therefore:

1. An<< L.2. ND>.> 1.3. 07 > 1.

PROBLEMS 1-3. On a log-log plot of n, vs. KT, with n, from lo6 to 10z5 m-‘, and KT, from0.01 to 105eV, draw lines of constant An and ND. On this graph, place thefollowing points (n in rne3, KT in eV):

1. Typical fusion reactor: n = 102’, KT = 10,000.2. Typical fusion experiments: n = lOI’, KT = 100 (torus); n = 1023, KT =

1000 (pinch).3. Typical ionosphere: n = lo”, KT = 0.05.4. Typical glow discharge: n = 1015, KT = 2.5. Typical flame: n = 1014, KT = 0.1.6. Typical Cs plasma; n = lo”, KT = 0.2.7. Interplanetary space: n = 106, KT = 0.01.

Convince yourself that these are plasmas.

1-4. Compute the pressure, in atmospheres and in tons/ft2, exerted by a ther-monuclear plasma on its container. Assume KT, = KTi = 20 keV, n = 102i m-‘,and p = nKT, where T = Ti + T..

1-5. In a strictly steady state situation, both the ions and the electrons will followthe Boltzmann relation

nj = no exp (-qj4/KTj)

For the case of an infinite, transparent grid charged to a potential 4, show thatthe shielding distance is then given approximately by

Show that An is determined by the temperature of the colder species.

1-6. An alternative derivation of hr, will give further insight to its meaning.Consider two infinite, parallel plates at x = Itd, set at potential 4 = 0. The spacebetween them is uniformly filled by a gas of density n of particles of charge q.

(a) Using Poisson’s equation, show that the potential distribution between theplates is

t#J = $d’ - x2)

(b) Show that for d > A b, the energy needed to transport a particle from aplate to the midplane is greater than the average kinetic energy of the particles.

l-7. Compute AD and ND for the following cases:

(a) A glow discharge, with n = 10.16 m-‘, KT, = 2 eV.

(b) The earth’s ionosphere, with n = 10” m-‘, KT, = 0.1 eV.

(c) A &pinch, with n = 1023 m-‘, KT, = 800 eV.

APPLICATIONS OF PLASMA PHYSICS

Plasmas can be characterized by the two parameters n and KT,. Plasmaapplications cover an extremely wide range of n and KT, : n varies over28 orders of magnitude from lo6 to lo’* mA3, and KT can vary overseven orders from 0.1 to lo6 eV. Some of these applications are discussedvery briefly below. The tremendous range of density can be appreciatedwhen one realizes that air and water differ in density by only 103, whilewater and white dwarf stars are separated by only a factor of 105. Evenneutron stars are only 1015 times denser than water. Yet gaseous plasmasin the entire density range of 10z8 can be described by the same set ofequations, since only the classical (non-quantum mechanical) laws ofphysics are needed.

Gas Discharges (Gaseous Electronics)

The earliest work with plasmas was that of Langmuir, Tonks, and theircollaborators in the 1920’s. This research was inspired by the need todevelop vacuum tubes that could carry large currents, and thereforehad to be filled with ionized gases. The research was done with weaklyionized glow discharges and positive columns typically with KT, = 2 eVand lo’* <n < 10” rnm3. It was here that the shielding phenomenonwas discovered; the sheath surrounding an electrode could be seenvisually as a dark layer. Gas discharges are encountered nowadays inmercury rectifiers, hydrogen thyratrons, ignitrons, spark gaps, weldingarcs, neon and fluorescent lights, and lightning discharges.

Controlled Thermonuclear Fusion

Modern plasma physics had it beginnings around 1952, when it wasproposed that the hydrogen bomb fusion reaction be controlled to makea reactor. The principal reactions, which involve deuterium (D) and

13Introduction

1.7

1.7.1

1.7.2

1 4 tritium (T) atoms, are as follows:ChapterOne D+D+sHe+n+3.2MeV

D+D+T+p++.OMeV

D+T+4He+n+17.6MeV

The cross sections for these fusion reactions are appreciable only forincident energies above 5 keV. Accelerated beams of deuterons bom-barding a target will not work, because most of the deuterons will losetheir energy by scattering before undergoing a fusion reaction. It isnecessary to create a plasma in which the thermal energies are in the10-keV range. The problem of heating and containing such a plasma isresponsible for the rapid growth of the science of plasma physics since1952. The problem is still unsolved, and most of the active research inplasma physics is directed toward the solution of this problem.

1.7.3 Space Physics

Another important application of plasma physics is in the study of theearth’s environment in space. A continuous stream of charged particles,called the solar wind, impinges on the earth’s magnetosphere, whichshields us from this radiation and is distorted by it in the process. Typicalparameters in the solar wind are n = 5 X lo6 m-3, K7’i = 10 eV, KT, =50 eV, B = 5 x lo-‘T, and drift velocity 300 km/sec. The ionosphere,extending from an altitude of 50 km to 10 earth radii, is populated bya weakly ionized plasma with density varying with altitude up to n =lOi* m+. The temperature is only 10-l eV. The Van Allen belts arecomposed of charged particles trapped by the earth’s magnetic field.Here we have n I 10’ mW3, KT, I 1 keV, KTi = 1 eV, and B =500 x lo-‘T. In addition, there is a hot component with n = 103 mm3and KT, = 40 keV.

1.7.4 Modern Astrophysics

Stellar interiors and atmospheres are hot enough to be in the plasmastate. The temperature at the core of the sun, for instance, is estimatedto be 2 keV; thermonuclear reactions occurring at this temperature areresponsible for the sun’s radiation. The solar corona is a tenuous plasmawith temperatures up to 200 eV. The interstellar medium contains ion-ized hydrogen with n = lo6 mm3. Various plasma theories have been usedto explain the acceleration of cosmic rays. Although the stars in a galaxy

are not charged, they behave like particles in a plasma; and plasmakinetic theory has been used to predict the development of galaxies.Radio astronomy has uncovered numerous sources of radiation thatmost likely originate from plasmas. The Crab nebula is a rich source ofplasma phenomena because it is known to contain a magnetic field. Italso contains a visual pulsar. Current theories of pulsars picture themas rapidly rotating neutron stars with plasmas emitting synchrotronradiation from the surface.

MHD Energy Conversion and Ion Propulsion

Getting back down to earth, we come to two practical applications ofplasma physics. Magnetohydrodynamic (MHD) energy conversion util-izes a dense plasma jet propelled across a magnetic field to generateelectricity (Fig. l-5). The Lorentz force qv x B, where v is the jet velocity,causes the ions to drift upward and the electrons downward, chargingthe two electrodes to different potentials. Electrical current can then bedrawn from the electrodes without the inefficiency of a heat cycle.

The same principle in reverse has been used to develop engines forinterplanetary missions. In Fig, 1-6, a current is driven through a plasmaby applying a voltage to the two electrodes. The j x B force shoots theplasma out of the rocket, and the ensuing reaction force accelerates therocket. The plasma ejected must always be neutral; otherwise, the spaceship will charge to a high potential.

Solid State Plasmas

The free electrons and holes in semiconductors constitute a plasmaexhibiting the same sort of oscillations and instabilities as a gaseousplasma. Plasmas injected into InSb have been particularly useful in

I@| + ^ evxB

-v- evxB

ICI3

Principle of the MHD generator.

15Introduction

1.7.5

1.7.6

FIGURE 1-5

16Chapter

FIGURE 1-6 Principle of plasma-jet engine for spacecraft propulsion.

studies of these phenomena. Because of the lattice effects, the effectivecollision frequency is much less than one would expect in a solid withn z 10” mmSs Furthermore, the holes in a semiconductor can have a very low effective mass-as little as O.Olm,-and therefore have highcyclotron frequencies even in moderate magnetic fields. If one were tocalculate iVn for a solid state plasma, it would be less than unity becauseof the low temperature and high density. Quantum mechanical effects(uncertainty principle), however, give the plasma an effective tem-perature high enough to make Nn respectably large. Certain liquids,such as solutions of sodium in ammonia, have been found to behave likeplasmas also.

1.7.7 Gas Lasers

The most common method to “pump” a gas laser-that is, to invert thepopulation in the states that give rise to light amplification-is to use agas discharge. This can be a low-pressure glow discharge for a dc laseror a high-pressure avalanche discharge in a pulsed laser. The He-Nelasers commonly used for alignment and surveying and the Ar and Krlasers used in light shows are examples of dc gas lasers. The powerfulCO* laser is finding commercial application as a cutting tool. Molecularlasers make possible studies of the hitherto inaccessible far infraredregion of the electromagnetic spectrum. These can be directly excitedby an electrical discharge, as in the hydrogen cyanide (HCN) laser, orcan be optically pumped by a CO2 laser, as with the methyl fluoride(CHsF) or methyl alcohol (CHsOH) lasers. Even solid state lasers, suchas Nd-glass, depend on a plasma for their operation, since the flashtubes used for pumping contain gas discharges.

FIGURE P1-11

1-8. In laser fusion, the core of a small pellet of DT is compressed to a density PROBLEMSof 1033 m-” at a temperature of 50,000,000°K. Estimate the number of particlesin a Debye sphere in this plasma.

1-9. A distant galaxy contains a cloud of protons and antiprotons, each withdensity n = lo6 m-” and temperature 100°K. What is the Debye length?

1-10. A spherical conductor of radius a is immersed in a plasma and chargedto a potential &. The electrons remain Maxwellian and move to form a Debyeshield, but the ions are stationary during the time frame of the experiment.Assuming & << KT,/e, derive an expression for the potential as a function of r in terms of a, do, and An. (Hint: Assume a solution of the form emkr/r.)

1-11. A field-effect transistor (FET) is basically an electron valve that operateson a finite-Debye-length effect. Conduction electrons flow from the source S tothe drain D through a semiconducting material when a potential is appliedbetween them. When a negative potential is applied to the insulated gate G, nocurrent can flow through G, but the applied potential leaks into the semiconductorand repels electrons. The channel width is narrowed and the electron flowimpeded in proportion to the gate potential. If the thickness of the device is toolarge, Debye shielding prevents the gate voltage from penetrating far enough.Estimate the maximum thickness of the conduction layer of an n-channel FETif it has doping level (plasma density) of lo** m-“, is at room temperature, andis to be no more than 10 Debye lengths thick. (See Fig. P1-11.)

17Introduction