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Introduction
1.1 Occurrence of Plasmas in Nature
It is now believed that the universe is made of 69 % dark energy, 27 % dark matter, and 1 % normal matter. All that we can see in the sky is the part of normal matter that is in the plasma state, emitting radiation. Plasma in physics, not to be confused with blood plasma, is an “ionized” gas in which at least one of the electrons in an atom has been stripped free, leaving a positively charged nucleus, called an ion. Sometimes plasma is called the “fourth state of matter.” When a solid is heated, it becomes a liquid. Heating a liquid turns it into a gas. Upon further heating, the gas is ionized into a plasma. Since a plasma is made of ions and electrons, which are charged, electric fields are rampant everywhere, and particles “collide” not just when they bump into one another, but even at a distance where they can feel their electric fields. Hydrodynamics, which describes the flow of water through pipes, say, or the flow around boats in yacht races, or the behavior of airplane wings, is already a complicated subject. Adding the electric fields of a plasma greatly expands the range of possible motions, especially in the presence of magnetic fields.
Plasma usually exists only in a vacuum. Otherwise, air will cool the plasma so that the ions and electrons will recombine into normal neutral atoms. In the laboratory, we need to pump the air out of a vacuum chamber. In the vacuum of space, however, much of the gas is in the plasma state, and we can see it. Stellar interiors and atmospheres, gaseous nebulas, and entire galaxies can be seen because they are in the plasma state. On earth, however, our atmosphere limits our experi- ence with plasmas to a few examples: the flash of a lightning bolt, the soft glow of the Aurora Borealis, the light of a fluorescent tube, or the pixels of a plasma TV. We live in a small part of the universe where plasmas do not occur naturally; otherwise, we would not be alive.
© Springer International Publishing Switzerland 2016 1 F.F. Chen, Introduction to Plasma Physics and Controlled Fusion,
DOI 10.1007/978-3-319-22309-4_1

2 1 Introduction
The reason for this can be seen from the Saha equation, which tells us the amount of ionization to be expected in a gas in thermal equilibrium:
ni nn

2:410
21 T3=2 U =KT
Here ni and nn are, respectively, the density (number per m3) of ionized atoms and of neutral atoms, T is the gas temperature in K, K is Boltzmann’s constant, and Ui is the ionization energy of the gas—that is, the number of joules required to remove the outermost electron from an atom. (The mks or International System of units will be used in this book.) For ordinary air at room temperature, we may take nn
3  1025 m3 (see Problem 1.1), T
300 K, and Ui 1⁄4 14.5 eV (for nitrogen), where 1 eV 1⁄4 1.6  1019 J. The fractional ionization ni/(nn + ni)
ni/nn predicted by Eq. (1.1) is ridiculously low:
ni
10122 nn
As the temperature is raised, the degree of ionization remains low until Ui is only a few times KT. Then ni/nn rises abruptly, and the gas is in a plasma state. Further increase in temperature makes nn less than ni, and the plasma eventually becomes fully ionized. This is the reason plasmas exist in astronomical bodies with temperatures of millions of degrees, but not on the earth. Life could not easily coexist with a plasma— at least, plasma of the type we are talking about. The natural occurrence of plasmas at high temperatures is the reason for the designation “the fourth state of matter.”
Although we do not intend to emphasize the Saha equation, we should point out its physical meaning. Atoms in a gas have a spread of thermal energies, and an atom is ionized when, by chance, it suffers a collision of high enough energy to knock out an electron. In a cold gas, such energetic collisions occur infrequently, since an atom must be accelerated to much higher than the average energy by a series of “favor- able” collisions. The exponential factor in Eq. (1.1) expresses the fact that the number of fast atoms falls exponentially with Ui /KT. Once an atom is ionized, it remains charged until it meets an electron; it then very likely recombines with the electron to become neutral again. The recombination rate clearly depends on the density of electrons, which we can take as equal to ni. The equilibrium ion fraction, therefore, should decrease with ni; and this is the reason for the factor ni1 on the right-hand side of Eq. (1.1). The plasma in the interstellar medium owes its existence to the low value of ni (about 1 per cm3), and hence the low recombination rate.
1.2 Definition of Plasma
Any ionized gas cannot be called a plasma, of course; there is always some small degree of ionization in any gas. A useful definition is as follows:
A plasma is a quasineutral gas of charged and neutral particles which exhibits collective behavior.
ni
e i ð1:1Þ

1.2 Definition of Plasma 3
Fig. 1.1 Illustrating the long range of electrostatic forces in a plasma
We must now define “quasineutral” and “collective behavior.” The meaning of quasineutrality will be made clear in Sect. 1.4. What is meant by “collective behavior” is as follows.
Consider the forces acting on a molecule of, say, ordinary air. Since the molecule is neutral, there is no net electromagnetic force on it, and the force of gravity is negligible. The molecule moves undisturbed until it makes a collision with another molecule, and these collisions control the particle’s motion. A macroscopic force applied to a neutral gas, such as from a loudspeaker generating sound waves, is transmitted to the individual atoms by collisions. The situation is totally different in a plasma, which has charged particles. As these charges move around, they can generate local concentrations of positive or negative charge, which give rise to electric fields. Motion of charges also generates currents, and hence magnetic fields. These fields affect the motion of other charged particles far away.
Let us consider the effect on each other of two slightly charged regions of plasma separated by a distance r (Fig. 1.1). The Coulomb force between A and B diminishes as 1/r2. However, for a given solid angle (that is, Δr/r 1⁄4 constant), the volume of plasma in B that can affect A increases as r3. Therefore, elements of plasma exert a force on one another even at large distances. It is this long-ranged Coulomb force that gives the plasma a large repertoire of possible motions and enriches the field of study known as plasma physics. In fact, the most interesting results concern so-called “collisionless” plasmas, in which the long-range electro- magnetic forces are so much larger than the forces due to ordinary local collisions that the latter can be neglected altogether. By “collective behavior” we mean motions that depend not only on local conditions but on the state of the plasma in remote regions as well.
The word “plasma” seems to be a misnomer. It comes from the Greek πλάσμα, ατoς, τo
, which means something molded or fabricated. Because of collective behavior, a plasma does not tend to conform to external influences; rather, it often behaves as if it had a mind of its own.

4 1 Introduction
1.3 Concept of Temperature
Before proceeding further, it is well to review and extend our physical notions of “temperature.” A gas in thermal equilibrium has particles of all velocities, and the most probable distribution of these velocities is known as the Maxwellian distribu- tion. For simplicity, consider a gas in which the particles can move only in one dimension. (This is not entirely frivolous; a strong magnetic field, for instance, can constrain electrons to move only along the field lines.) The one-dimensional Maxwellian distribution is given by
f ðuÞ 1⁄4 A exp
12 mu2=KT ð1:2Þ where f du is the number of particles per m3 with velocity between u and u + du,
12 mu2 is the kinetic energy, and K is Boltzmann’s constant, K 1⁄4 1:38  1023J=K
Note that a capital K is used here, since lower-case k is reserved for the propagation constant of waves. The density n, or number of particles per m3, is given by (see Fig. 1.2)
ð1
n1⁄4 fðuÞdu
1
The constant A is related to the density n by (see Problem 1.2)  m 1=2
A1⁄4n 2πKT
ð1:3Þ
ð1:4Þ
Fig. 1.2 A Maxwellian velocity distribution

1.3 Concept of Temperature 5
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 compute the average kinetic energy of particles in this distribution:
ð1
12 mu2 f ðuÞdu Eav1⁄4 ð1
f ðuÞdu 1
1
ð1:5Þ
ð1:6Þ
Defining
vth 1⁄4 ð2KT=mÞ1=2 we can write Eq. (1.2) as
and y 1⁄4 u=vth
fðuÞ1⁄4Aexp
u2=v2  th
ð1 

exp y2 y2dy
and Eq. (1.5) as
The integral in the numerator is integrable by parts:
ð1
 ð1

12mAv3th
Eav1⁄4 ð1
2
where
y  exp y2 ydy 1⁄4 121⁄2expðy2Þy 11 
12exp y2 dy
ð1
 1 expy2 dy
1
1 mAv3 1
Eav 1⁄42 th2 1⁄414mv2th 1⁄412KT
Avth Thus the average kinetic energy is 12 KT.
1
1⁄412 Canceling the integrals, we have
Avth
1 1
exp y dy
 m 3=2
A3 1⁄4 n 2πKT ð1:9Þ
ð1:7Þ
It is easy to extend this result to three dimensions. Maxwell’s distribution is then f ðu; v; wÞ 1⁄4 A3exp12 mðu2 þ v2 þ w2Þ=KT ð1:8Þ

6 1 Introduction The average kinetic energy is
ððð1  
A312 mðu2 þ v2 þ w2Þexp 12 mðu2 þ v2 þ w2Þ=KT dudvdw
Eav1⁄41ððð1 1222
A3exp2mðu þv þwÞ=KTdudvdw 1
We note that this expression is symmetric in u, v, and w, since a Maxwellian distribution is isotropic. Consequently, each of the three terms in the numerator is the same as the others. We need only to evaluate the first term and multiply by three:
3A Ð 1 mu2exp
1 mu2=KTduÐÐexp1 mðv2 þ w2Þ=KTdvdw 3222
Eav 1⁄4 A Ð exp
1 mu2=KTduÐÐexp1 mðv2 þ w2Þ=KTdvdw 322
Using our previous result, we have
T1⁄41:381023 1⁄411,600 Thus the conversion factor is
Eav 1⁄4 32 KT
ð1:10Þ
The general result is that Eay equals 12KT per degree of freedom.
Since T and Eav are so closely related, it is customary in plasma physics to give temperatures in units of energy. To avoid confusion on the number of dimensions involved, it is not Eav but the energy corresponding to KT that is used to denote the
temperature. For KT 1⁄4 1 eV 1⁄4 1.6  1019 J, we have 1:6  1019
1eV 1⁄4 11,600 K ð1:11Þ
By a 2-eV plasma we mean that KT 1⁄4 2 eV, or Eav 1⁄4 3 eV in three dimensions.
It is interesting that a plasma can have several temperatures at the same time. It often happens that the ions and the electrons have separate Maxwellian distribu- tions with different temperatures Ti and Te. This can come about because the collision rate among ions or among electrons themselves is larger than the rate of collisions between an ion and an electron. Then each species can be in its own thermal equilibrium, but the plasma may not last long enough for the two temper- atures to equalize. When there is a magnetic field B, even a single species, say ions, can have two temperatures. This is because the forces acting on an ion along B are different from those acting perpendicular to B (due to the Lorentz force). The components of velocity perpendicular to B and parallel to B may then belong to
different Maxwellian distributions with temperatures T⊥ and T||.

1.4 Debye Shielding 7
Before leaving our review of the notion of temperature, we should dispel the popular misconception that high temperature necessarily means a lot of heat. People are usually amazed to learn that the electron temperature inside a fluorescent light bulb is about 20,000 K. “My, it doesn’t feel that hot!” Of course, the heat capacity must also be taken into account. The density of electrons inside a fluores- cent tube is much less than that of a gas at atmospheric pressure, and the total amount of heat transferred to the wall by electrons striking it at their thermal velocities is not that great. Everyone has had the experience of a cigarette ash dropped innocuously on his hand. Although the temperature is high enough to cause a burn, the total amount of heat involved is not. Many laboratory plasmas have temperatures of the order of 1,000,000 K (100 eV), but at densities of only 1018–1019 per m3, the heating of the walls is not a serious consideration.
Problems
1.1. Compute the density (in units of m3) of an ideal gas under the following conditions:
(a) At 0 C and 760 Torr pressure (1 Torr1⁄41 mmHg). This is called the Loschmidt number.
(b) In a vacuum of 103 Torr at room temperature (20 C). This number is a useful one for the experimentalist to know by heart (103 Torr 1⁄4 1 μ).
1.2. Derive the constant A for a normalized one-dimensional Maxwellian distribution
such that
^f ðuÞ 1⁄4 Aexp
mu2=2KT
ð1
^f ð u Þ d u 1⁄4 1
1
Hint: To save writing, replace (2KT/m)1/2 by vth (Eq. 1.6).
1.2a. (Advanced problem). Find A for a two-dimensional distribution which inte-
grates to unity. Extra credit for a solution in cylindrical coordinates. ^f ðu; vÞ 1⁄4 Aexpm
u2 þ v2=2KT
1.4 Debye Shielding
A fundamental characteristic of the behavior of plasma is its ability to shield out electric potentials that are applied to it. Suppose we tried to put an electric field inside a plasma by inserting two charged balls connected to a battery (Fig. 1.3). The balls would attract particles of the opposite charge, and almost immediately a cloud of ions would surround the negative ball and a cloud of electrons would surround

8
1 Introduction
Debye shielding
Potential distribution near a grid
in a plasma
the positive ball. (We assume that a layer of dielectric keeps the plasma from actually recombining on the surface, or that the battery is large enough to maintain the potential in spite of this.) If the plasma were cold and there were no thermal motions, there would be just as many charges in the cloud as in the ball, the shielding would be perfect, and no electric field would be present in the body of the plasma outside of the clouds. On the other hand, if the temperature is finite, those particles that are at the edge of the cloud, where the electric field is weak, have enough thermal energy to escape from the electrostatic potential well. The “edge” of the cloud then occurs at the radius where the potential energy is approximately equal to the thermal energy KT of the particles, and the shielding is not complete. Potentials of the order of KT/e can leak into the plasma and cause finite electric fields to exist there.
Let us compute the approximate thickness of such a charge cloud. Imagine that the potential φ on the plane x 1⁄4 0 is held at a value φ0 by a perfectly transparent grid (Fig. 1.4). We wish to compute φ(x). For simplicity, 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 positive charge. To be more precise, we can say that M/m is large
Fig. 1.3
Fig. 1.4

1.4 Debye Shielding 9
enough that the inertia of the ions prevents them from moving significantly on the time scale of the experiment. Poisson’s equation in one dimension is
ε∇2φ1⁄4εd2φ1⁄4eðnnÞ ðZ1⁄41Þ ð1:12Þ 0 0dx2 ie
If the density far away is n1, we have
ni 1⁄4 n1
In the presence of a potential energy qφ, the electron distribution function is
f ðuÞ 1⁄4 A exp
1 mu2 þ qφ=KT  ð1:13Þ
Defining
d2φ n1e2 ε0 dx2 1⁄4 KT φ
e
εKT 1=2 λ 0 e
D ne2
ð1:15Þ
ð1:16Þ
2e
It would not be worthwhile to prove this here. What this equation says is intuitively obvious: There are fewer particles at places where the potential energy is large, since not all particles have enough energy to get there. Integrating f (u) over u, setting q 1⁄4 e, and noting that ne(φ ! 0) 1⁄4 n1, we find
ne 1⁄4 n1expðeφ=KTeÞ
This equation will be derived with more physical insight in Sect. 3.5. Substituting
for ni and ne in Eq. (1.12), we have
d2φ  
ε0dx21⁄4en1 eeφ=KTe1
In the region where jeφ/KTej  1, we can expand the exponential in a Taylor series:
d2φ eφ2
ε0 1⁄4 en1 þ 1 eφ þ    ð1:14Þ
No simplification is possible for the region near the grid, where jeφ/KTej may be large. Fortunately, this region does not contribute much to the thickness of the cloud (called a sheath), because the potential falls very rapidly there. Keeping only the linear terms in Eq. (1.13), we have
dx2 KTe 2KTe
where n stands for n1, and KTe is in joules. KTe is often given in eV, in which case, we will write it also as TeV.

10 1 Introduction
We can write the solution of Eq. (1.14) as
φ 1⁄4 φ0expðjxj=λDÞ ð1:17Þ
The quantity λD, called the Debye length, is a measure of the shielding distance or thickness of the sheath.
Note that as the density is increased, λD decreases, as one would expect, since each layer of plasma contains more electrons. Furthermore, λD increases with increasing KTe. Without thermal agitation, the charge cloud would collapse to an infinitely thin layer. Finally, it is the electron temperature which is used in the definition of λD because the electrons, being more mobile than the ions, generally do the shielding by moving so as to create a surplus or deficit of negative charge. Only in special situations is this not true (see Problem 1.5).
The following are useful forms of Eq. (1.16):
λD 1⁄4 69ðTe=nÞ1=2m, Te in K
λD 1⁄4 7430ðKTe=nÞ1=2m, KTe in eV
ð1:18Þ
We are now in a position to define “quasineutrality.” If the dimensions L of a system are much larger than λD, then whenever local concentrations of charge arise or external potentials are introduced into the 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, ∇2φ is very small, and ni is equal to ne, typically to better than one part in 106. It takes only a small charge imbalance to give rise to potentials of the order of KT/e. The plasma is “quasineutral”; that is, neutral enough so that one can take ni ’ ne ’ n, where n is a common density called the plasma density, but not so neutral that all the interesting electromagnetic forces vanish.
A criterion for an ionized gas to be a plasma is that it be dense enough that λD is much smaller than L.
The phenomenon of Debye shielding also occurs—in modified form—in single- species systems, such as the electron streams in klystrons and magnetrons or the proton beam in a cyclotron. In such cases, any local bunching of particles causes a large unshielded electric field unless the density is extremely low (which it often is). An externally imposed potential—from a wire probe, for instance—would be shielded out by an adjustment of the density near the electrode. Single-species systems, or unneutralized plasmas, are not strictly plasmas; but the mathematical tools of plasma physics can be used to study such systems.
Debye shielding can be foiled if electrons are so fast that they do not collide with one another enough to maintain a thermal distribution. We shall see later that electron collisions are infrequent if the electrons are very hot. In that case, some electrons, attracted by the positive charge of the ion, come in at an angle so fast that they orbit the ion like a satellite around a planet. How this works will be clear in the discussion of Langmuir probes in a later chapter. Some like to call this effect anti- shielding.

1.6 Criteria for Plasmas 11
1.5 The Plasma Parameter
The picture of Debye shielding that we have given above is valid only if there are enough particles in the charge cloud. Clearly, if there are only one or two particles in the sheath region, Debye shielding would not be a statistically valid concept. Using Eq. (1.17), we can compute the number ND of particles in a “Debye sphere”:
ND 1⁄4 n 43 πλ3D 1⁄4 1:38  106T3=2=n1=2 ðT in KÞ ð1:19Þ In addition to λD  L, “collective behavior” requires
ND⋙ 1 ð1:20Þ
1.6 Criteria for Plasmas
We have given two conditions that an ionized gas must satisfy to be called a plasma. A third condition has to do with collisions. The weakly ionized gas in an airplane’s jet exhaust, for example, does not qualify as a plasma because the charged particles collide so frequently with neutral atoms that their motion is controlled by ordinary hydrodynamic forces rather than by electromagnetic forces. If ω is the frequency of typical plasma oscillations and τ is the mean time between collisions with neutral atoms, we require ωτ>1 for the gas to behave like a plasma rather than a neutral gas.
The three conditions a plasma must satisfy are therefore:

1. λD L:


2. ND⋙1:


3. ωτ>1:
   Problems
   1.3. Calculate n vs. KTe curves for five values of λD from 108 to 1, and three values of ND from 103 to 109. On a log-log plot of ne vs. KTe with ne from 106 to 1028 m3 and KTe from 102 to 105 eV, draw lines of constant λD (solid) and ND (dashed). On this graph, place the following points (n in m3, KT in eV):


4. Typical fusion reactor: n 1⁄4 1020, KT 1⁄4 30,000.


5. Typical fusion experiments: n 1⁄4 1019, KT 1⁄4 100 (torus); n 1⁄4 1023,
   KT 1⁄4 1000 (pinch).


6. Typical ionosphere: n 1⁄4 1011, KT 1⁄4 0.05.


7. Typical radiofrequency plasma: n 1⁄4 1017, KT 1⁄4 1.5


8. Typical flame: n 1⁄4 1014, KT 1⁄4 0.1.


9. Typical laser plasma; n 1⁄4 1025, KT 1⁄4 100.


10. Interplanetary space: n 1⁄4 106, KT 1⁄4 0.01.

12
1 Introduction
1.4.
1.5.
Convince yourself that these are plasmas.
Compute the pressure, in atmospheres and in tons/ft2, exerted by a thermonu- clear plasma on its container. Assume KTe 1⁄4 KTi 1⁄4 20 keV, n 1⁄4 1021 m3, and p1⁄4nKT, where T1⁄4Ti +Te.
In a strictly steady state situation, both the ions and the electrons will follow the Boltzmann relation


nj 1⁄4n0exp qiφ=KTj
For the case of an infinite, transparent grid charged to a potential φ, show that
the shielding distance is then given approximately by λ21⁄4ne2 1þ1
D 20 KTe KTi
Show that λD is determined by the temperature of the colder species.
An alternative derivation of λD will give further insight to its meaning. Consider two infinite parallel plates at x 1⁄4 d, set at potential φ 1⁄4 0. The space between 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 the plates is
φ 1⁄4 nq
d2  x2 2ε0
(b) Show that for d>λD, the energy needed to transport a particle from a plate to the midplane is greater than the average kinetic energy of the particles.
Compute λD and ND for the following cases:
(a) A glow discharge, with n 1⁄4 1016 m3, KTe 1⁄4 2 eV.
(b) The earth’s ionosphere, with n 1⁄4 1012 m3, KTe 1⁄4 0.1 eV.
(c) A θ-pinch, with n 1⁄4 1023 m3, KTe 1⁄4 800 eV.
Applications of Plasma Physics
1.6.
1.7.
1.7
Plasmas can be characterized by the two parameters n and KTe. Plasma applica- tions cover an extremely wide range of n and KTe: n varies over 28 orders of magnitude from 106 to 1034 m3, and KT can vary over seven orders from 0.1 to 106 eV. Some of these applications are discussed very briefly below. The tremen- dous range of density can be appreciated when one realizes that air and water

1.7 Applications of Plasma Physics 13
differ in density by only 103, while water and white dwarf stars are separated by only a factor of 105. Even neutron stars are only 1015 times denser than water. Yet gaseous plasmas in the entire density range of 1028 can be described by the same set of equations, since only the classical (non-quantum mechanical) laws of physics are needed.
1.7.1 Gas Discharges (Gaseous Electronics)
The earliest work with plasmas was that of Langmuir, Tonks, and their collabo- rators in the 1920s. This research was inspired by the need to develop vacuum tubes that could carry large currents, and therefore had to be filled with ionized gases. The research was done with weakly ionized glow discharges and positive columns typically with KTe ’ 2 eV and 1014 < n < 1018 m3. It was here that the shielding phenomenon was discovered; the sheath surrounding an electrode could be seen visually as a dark layer. Before semiconductors, gas discharges were encountered only in mercury rectifiers, hydrogen thyratrons, ignitrons, spark gaps, welding arcs, neon and fluorescent lights, and lightning discharges. The semiconductor industry’s rapid growth in the last two decades has brought gas discharges from a small academic discipline to an economic giant. Chips for computers and the ubiquitous handheld devices cannot be made without plasmas. Usually driven by radiofrequency power, partially ionized plasmas (gas discharges) are used for etching and deposition in the manufacture of semiconductors.
1.7.2 Controlled Thermonuclear Fusion
Modern plasma physics had its beginnings around 1952, when it was proposed that the hydrogen bomb fusion reaction be controlled to make a reactor. A seminal conference was held in Geneva in 1958 at which each nation revealed its classified controlled fusion program for the first time. Fusion power requires holding a 30-keV plasma with a magnetic field for as long as one second. Research was carried out by each individual country until 2007, when the ITER project was started. ITER stands for International Thermonuclear Experimental Reactor, a large experiment being built in France and funded by seven countries. Serendipitously, ITER is a Latin word meaning path or road. It is a road that mankind must take to solve the problems of global warming and oil shortage by 2050. Of all the “magnetic bottles” presented at Geneva, the USSR’s TOKAMAK has survived as the leading idea and is the configuration for ITER. First plasma in ITER is scheduled for about the year 2021.

14 1 Introduction
The fuel is heavy hydrogen (deuterium), which exists naturally as one part in 6000 of water. The principal reactions, which involve deuterium (D) and tritium
(T) atoms, are as follows:
D þ D !3He þ n þ 3:2MeV DþD!Tþ pþ4:0MeV D þ T !4He þ n þ 17:6MeV
These cross sections are appreciable only for incident energies above 5 keV. Accelerated beams of deuterons bombarding a target will not work, because most of the deuterons will lose their energy by scattering before undergoing a fusion reaction. It is necessary to create a plasma with temperatures above 10 keV so that there are enough ions in the 40-keV range where the reaction cross section maximizes. The problem of heating and containing such a plasma is responsible for the rapid growth of the science of plasma physics since 1952.
1.7.3 Space Physics
Another important application of plasma physics is in the study of the earth’s envi- ronment in space. A continuous stream of charged particles, called the solar wind, impinges on the earth’s magnetosphere, which shields us from this radiation and is distorted by it in the process. Typical parameters in the solar wind are n 1⁄4 9  106 m3, KTi1⁄410 eV, KTe1⁄412 eV, B1⁄47109 T, and drift velocity 450 km/s. The ionosphere, extending from an altitude of 50 km to 10 earth radii, is populated by a weakly ionized plasma with density varying with altitude up to n1⁄41012 m3. The temperature is only 101 eV. The solar wind blows the earth’s magnetic field into a long tail on the night side of the earth. The magnetic field lines there can reconnect and accelerate ions in the process. This will be discussed in a later chapter.
The Van Allen radiation belts are two rings of charged particles above the equator trapped by the earth’s magnetic field. Here we have n 109 m3, KTe 1 keV, KTi ’1 eV, and B’500109 T. In addition, there is a hot com- ponent with n 1⁄4 103 m3 and KTe 1⁄4 40 keV, and some ions have 100s of MeV.
Exploration of other planets have revealed the presence of plasmas. Though Mercury, Venus, and Mars have little plasma phenomena, the giant plants Jupiter and Saturn and their moons can have plasma created by lightning strikes. In 2013 The Voyager 1 satellite reached the boundary of the solar system. This was ascertained by detecting an increase in the plasma frequency there (!).
1.7.4 Modern Astrophysics
Stellar interiors and atmospheres are hot enough to be in the plasma state. The temperature at the core of the sun, for instance, is estimated to be 2 keV;

1.7 Applications of Plasma Physics 15
thermonuclear reactions occurring at this temperature are responsible for the sun’s radiation. The solar corona is a tenuous plasma with temperatures up to 200 eV. The interstellar medium contains ionized hydrogen with n ’ 106 m3 (1 per cc). Various plasma theories have been used to explain the acceleration of cosmic rays. Although the stars in a galaxy are not charged, they behave like particles in a plasma; and plasma kinetic theory has been used to predict the development of galaxies. Radio astronomy has uncovered numerous sources of radiation that most likely originate from plasmas. The Crab nebula is a rich source of plasma phenom- ena because it is known to contain a magnetic field. It also contains a visual pulsar. Current theories of pulsars picture them as rapidly rotating neutron stars with plasmas emitting synchrotron radiation from the surface. Active galactic nuclei and black holes have come to the forefront. Astrophysics now requires an under- standing of plasma physics.
1.7.5 MHD Energy Conversion and Ion Propulsion
Getting back down to earth, we come to two practical applications of plasma physics. Magnetohydrodynamic (MHD) energy conversion utilizes a dense plasma jet propelled across a magnetic field to generate electricity (Fig. 1.5). The Lorentz force qvB, where v is the jet velocity, causes the ions to drift upward and the electrons downward, charging the two electrodes to different potentials. Elec- trical current can then be drawn from the electrodes without the inefficiency of a heat cycle.
A more important application uses this principle in reverse to develop engines for interplanetary missions. In Fig. 1.6, a current is driven through a plasma by applying a voltage to the two electrodes. The j  B force shoots the plasma out of the rocket, and the ensuing reaction force accelerates the rocket. A more modern device called a Hall thruster uses a magnetic field to stop the electrons while a high voltage accelerates the ions out the back, giving their momenta to the spacecraft. A separate discharge ejects an equal number of warm electrons; otherwise, the space ship will charge to a high potential.
Fig. 1.5 Principle of the MHD generator

16 1 Introduction
Fig. 1.6 Principle of plasma-jet engine for spacecraft propulsion
1.7.6 Solid State Plasmas
The free electrons and holes in semiconductors constitute a plasma exhibiting the same sort of oscillations and instabilities as a gaseous plasma. Plasmas injected into InSb have been particularly useful in studies of these phenomena. Because of the lattice effects, the effective collision frequency is much less than one would expect in a solid with n ’ 1029 m3. Furthermore, the holes in a semiconductor can have a very low effective mass—as little as 0.01 me—and therefore have high cyclotron frequencies even in moderate magnetic fields. If one were to calculate ND for a solid state plasma, it would be less than unity because of the low temperature and high density. Quantum mechanical effects (uncertainty principle), however, give the plasma an effective temperature high enough to make ND respectably large. Certain liquids, such as solutions of sodium in ammonia, have been found to behave like plasmas also.
1.7.7 Gas Lasers
The most common method to “pump” a gas laser—that is, to invert the population in the states that give rise to light amplification—is to use a gas discharge. This can be a low-pressure glow discharge for a dc laser or a high-pressure avalanche discharge in a pulsed laser. The He–Ne lasers commonly used for alignment and surveying and the Ar and Kr lasers used in light shows are examples of dc gas lasers. The powerful CO2 laser has a commercial application as a cutting tool. Molecular lasers such as the hydrogen cyanide (HCN) laser make possible studies of the hitherto inaccessible far infrared region of the electromagnetic spectrum. The krypton fluoride (KrF) laser uses a large electron beam to excite the gas. It has the repetition rate but not the power for a laser-driven fusion reactor. In the semicon- ductor industry, short-wavelength ultraviolet lasers are used to etch ever smaller transistors on a chip. Excimer lasers such as the argon fluoride laser with 193 nm wavelength are used, with plans to go down to 5 nm.

1.7 Applications of Plasma Physics 17
1.7.8 Particle Accelerators
In high-energy particle research, linear accelerators are used to avoid synchrotron radiation on curves, especially for electrons. The 3-km long SLAC accelerator at Stanford produces 50-GeV electrons and positrons. Plasma accelerators generate plasma waves on which particles “surf” to gain energy. This technique has been applied at SLAC to double the energy from 42 to 84 GeV. A 31-km International Linear Collider is being built to generate 500 GeV colliding beams of electrons and positrons. In principle, plasma waves can add 1 GeV per cm.
1.7.9 Industrial Plasmas
Aside from their use in semiconductor production, partially ionized plasmas of high density have other industrial applications. Magnetrons are used in sputtering, a method of applying coatings of different materials. Eyeglasses can be coated with plasmas. Arcs, such as those in search lights, are plasmas. Instruments in medical research can be cleaned thoroughly with plasmas.
1.7.10 Atmospheric Plasmas
Most plasmas are created in vacuum systems, but it is also possible to produce plasmas at atmospheric pressure. For instance, a jet of argon and helium can be ionized with radiofrequency power. This makes possible small pencil-size devices for cauterizing skin. Industrial substrates can be processed by sweeping such a jet to cover a large area. Large atmospheric-pressure plasmas can also be produced for roll-to-roll processing.
Problems
1.8. In laser fusion, the core of a small pellet of DT is compressed to a density of 1033 m3 at a temperature of 50,000,000 K. Estimate the number of particles in a Debye sphere in this plasma.
1.9. A distant galaxy contains a cloud of protons and antiprotons, each with density n 1⁄4 106 m3 and temperature 100 K. What is the Debye length?
1.10. (Advanced problem) A spherical conductor of radius a is immersed in a uniform plasma and charged to a potential φ0. The electrons remain Max- wellian and move to form a Debye shield, but the ions are stationary during the time frame of the experiment.
(a) Assuming eφ/KTe  1, write Poisson’s equation for this problem in terms of λD.
(b) Show that the equation is satisfied by a function of the form ekr/r. Determine k and derive an expression for φ(r) in terms of a, φ0, and λD.

18 1.11.
1 Introduction
A field-effect transistor (FET) is basically an electron valve that operates on a finite-Debye-length effect. Conduction electrons flow from the source S to the drain D through a semiconducting material when a potential is applied between them. When a negative potential is applied to the insulated gate G, no current can flow through G, but the applied potential leaks into the semiconductor and repels electrons. The channel width is narrowed and the electron flow impeded in proportion to the gate potential. If the thickness of the device is too large, Debye shielding prevents the gate voltage from penetrating far enough. Estimate the maximum thickness of the conduction layer of an n-channel FET if it has doping level (plasma density) of 1022 m3, is at room temperature, and is to be no more than 10 Debye lengths thick. (See Fig. P1.11.)
Fig. P1.11
1.12. (Advanced problem) Ionization is caused by electrons in the tail of a Max- wellian distribution which have energies exceeding the ionization potential. For instance, this potential is Eioniz 1⁄4 15.8 eV in argon. Consider a one-dimensional plasma with electron velocities u in the x direction only. What fraction of the electrons can ionize for given KTe in argon? (Give an analytic answer in terms of error functions.)