1.1.3.1 Thermonuclear fusion

High-temperature thermonuclear plasma research was started in 1950 in both the United States and the Soviet Union. The main directions of research were concerned with heating and control of the plasma stability. Since then, for more than half a century, scientists from all over the globe are working on turning the promise of controlled fusion into a practical reality. In fusion reactions that are of interest for energy production, nuclei of light elements combine together to form heavier elements releasing enormous amount of energy. The reactions of greatest interest for fusion energy involve deuterium, a stable isotope of hydrogen whose nucleus contains one proton and one neutron. It should be pointed out that deuterium occurs naturally, making up about 0.015% of all hydrogen.

Steady-state reaction can be supported if power input is balanced by the losses. Power input is supplied externally or from the reaction itself, while power losses are typically due to particle convection to the walls and radiation. Lawson [2] demonstrated that a simple power balance leads to two separate requirements in terms of ion temperature that has to be about 5 keV and a product of the density and confinement time which has to be about 10²⁰ s/m³.

Such temperatures are necessary in order to overcome Coulomb repulsive forces and fuse the deuterium (D=2H) and tritium (T=3H) into helium:

Recall that the fusion reaction produces much more energy than the nuclear fission. For instance, U235 split releases about 0.8 MeV per nucleon, while D–T fusion into He produces about 3.5 MeV per nucleon.

Lawson’s criterion demonstrates the importance of the plasma confinement. Among various possible confinement approaches, magnetic confinement has been the mainstream approach from the beginning. Several practical magnetic confinement schemes were proposed such as the stellarator (Lyman Spitzer) and tokamak (that was proposed in the former Soviet Union). Tokamak configuration demonstrated the best confinement results. However, it became obvious that further progress will require resources of the international community. An example of this is the Joint European Torus (JET) in Culham, UK, which has been in operation since 1983. In 1991, the JET tokamak achieved the world’s first controlled release of fusion power. Scientists have now designed the next-step device—ITER—that will produce more power than it consumes: for 50 MW of input power, 500 MW of output power will be produced.

1.1.3.2 Vacuum arcs

The vacuum arc is an electrical discharge, which occurs in the vapor ejected from the electrodes. The vapor that forms the conductive path is supplied from small luminous areas on the cathode surface called cathode spots. The concept of the cathode spot unites two physically different regions: the metal surface, which can be heated by the plasma spot to extremely high temperatures, and the near-cathode plasma, which is generated as a result of cathode vaporization and atom ionization. The cathode surface phenomena (electron emission, heating) in the spot area are mutually depended on the phenomena occurring in the spot plasma. Consequently, the combination of these two interacting regions shall be determined by the cathode spot parameters. The near-cathode plasma in the spot is characterized by high particle density, current density, and temperature, which depend on the type of spot and the form of the discharge.

Various theoretical estimates and experiments have determined the following values of near-cathode plasma parameters: heavy particle density N~10²⁴−10²⁶ m⁻³, degree of ionization is about 0.1–100%, current density J~10⁹–10¹² A m⁻², electron temperature T_e~1–7 eV, ion temperature T_i~0.5–3 eV.

Research on vacuum arc was always closely related to applications. Arc discharges were considered primarily as a lighting source. Over time it was realized that vacuum arc in mercury behaves differently dependent on the electrode polarity. This led to idea that vacuum arc can be used as AC current rectifier. Significant development can be attributed to research at General Electric (Lafferty, Greenwood). Substantial contribution to the physics and application of the mercury vacuum arc was done by Weintraub, Child, Kesaev. Cathode spot is the most complicated phenomenon in the vacuum arc. Ecker, Kesaev, and most recently Beilis developed modern understanding of the complicated cathode spot mechanism.

One of first applications of the vacuum arc was a switching medium in power circuit breakers. The arc is an essential element in the current interruption process. When a pair of current-carrying contacts is separated, current is not interrupted immediately but continues to flow through an arc, which is at once established. One of the earliest definitive works on the subject was done by Sorenson and Mendenhall in 1926, but vacuum switches became available commercially for power system applications only in the beginning of 1960s.

The vacuum arc plasma jet is an excellent source of energetic ions for the deposition of high-quality films. The advantages of this technology are:

The applications include the deposition of hard coatings such as TiN and TiAlN, on cutting tools to improve their lifetime and performance, transparent conducting thin films, amorphous semiconducting silicon thin films, and diamond-like carbon film.

In addition, vacuum arcs are used in metal ion accelerators and recently for spacecraft micropropulsion. Microcathode thruster based on the vacuum arc was recently developed. Figure 1.4 illustrates the operation of this thruster in the laboratory conditions.

The discharge at the anode is distributed diffusely over the anode surface and this electrode does not suffer erosion. The material used to form the plasma stream is evaporated at microscopic spots on the cathode surface, i.e., on average, a cold, solid surface.

1.1.3.3 Cold plasma

Recent progress in the atmospheric plasmas led to production of the cold plasmas with ion temperatures close to room temperature. This makes these plasmas different from typical low-temperature plasma of a typical electrical discharge. Such plasmas represent very useful tool enabling interaction with biological tissue without thermal damage. Cold plasma technology development resulted in rapid formation of a new field of plasma medicine [3,4]. Typical cold plasma source is shown in Figure 1.5. The plasma source is equipped with pair of high-voltage (HV) electrodes—central electrode (which is isolated from the direct contact with plasma by ceramics) and outer ring electrode as shown schematically in Figure 1.5. Electrodes are connected to a secondary of HV resonant transformer (voltage U up to 10 kV, frequency ~30 kHz). A typical photograph of the plasma jet is shown in Figure 1.5 for U~5 kV and helium feeding corresponding to the flow rate of about υ_fl=5–10 l/min. The visible plasma jet had a length of approximately 5 cm and was well collimated along the entire length.

While effects of the electric field on cell was known for a long time [5], cold plasma offers much larger array of possible pathways due to the presence of various chemically active species and charged particles. Cold nonthermal atmospheric plasmas can have tremendous applications in biomedical technology. In particular, plasma treatment can potentially offer a minimum-invasive surgery that allows specific cell removal without influencing the whole tissue. Conventional laser surgery is based on thermal interaction and leads to accidental cell death, i.e., necrosis and may cause permanent tissue damage. In contrast, nonthermal plasma interaction with tissue may allow specific cell removal without necrosis. In particular, these interactions include cell detachment without affecting cell viability, controllable cell death, etc. It can be also used for cosmetic methods of regenerating the reticular architecture of the dermis. The aim of plasma interaction with tissue is not to denaturate the tissue but rather to operate under the threshold of thermal damage and to induce chemically specific response or modification. In particular, presence of the plasma can promote chemical reaction that would have desired effect. Chemical reaction can be promoted by tuning the pressure, gas composition, and energy. Thus the important issues are to find conditions that produce effect on tissue without thermal treatment. Overall plasma treatment offers the advantage that can never be thought of in most advanced laser surgery [6]. For instance, cold plasma demonstrated great promise in cancer therapy [7]. In recent years, cold plasma interaction with tissues becomes very active research topic due to aforementioned potential.

1.1.3.4 Plasma in nature

Atmospheric lightning is probably the most known phenomenon to humans since the early days of humankind. Lightning (shown in Figure 1.6) is an electric discharge, which typically occurs during the thunderstorms. Luminous flashes above thunderstorms have been reported by eyewitnesses for over a century although these flushes were registered only two decades ago [8]. Among them are “red sprites”—optical flashes predominantly in the red located at 50–90 km above ground and associated with giant thunderstorms. Optical observation made with photometers of high spatial resolution revealed that red sprites start as a luminous cloud which then propagates mostly downward developing a highly branched structure as shown schematically in Figure 1.7. Recently upward-propagating stratospheric flashes were identified, named blue jets (BJ) primarily due to blue color. At present, a number of BJ as well as gigantic blue jets (GBJ), propagating into the mesosphere/lower ionosphere, were captured. It appears that the GBJ short-circuit the thundercloud to the ionosphere and thus may have implications for affecting the global electric circuit [9].

1.2.1.1 Definitions

Plasma particles change their momentum, energy, and states of excitation or charge through particle collisions known as elementary processes. Two types of collisions in plasmas can be distinguished, namely, elastic and nonelastic. Elastic collisions can be broadly defined as events in which the total kinetic energy of particles is conserved and particles retain their original charges. Elastic collisions in which momentum is redistributed between particles involved are described by the following formula:

Nonelastic collisions are those in which kinetic energy is redistributed between particles and is transferred into some internal mode of one or more colliding particles or into creation of a new particle(s). Nonelastic collisions lead to direct and reverse processes, which include particle excitation and deexcitation, ionization and recombination, charge exchange, dissociation, and charge neutralization. These elementary processes are described by the following formulas:

1.2.1.2 Cross section: mean free path

The collision probability of each elementary process mentioned above is characterized by particle density and a parameter named cross section. Let us define this parameter in case of elastic collisions considering a single test particle moving through a cloud with a density n_a of particles at rest (as shown in Figure 1.8) and having only elastic encounters with target particles.

The number of target particles in the volume V is n_aV and the cross section σ of the target particle is defined as σ=(π/4)d²_p, where d_p is the diameter of particle. Let us introduce the mean free path for collisions which is the average distance between consecutive collision events as

(1.14)

The frequency of collisions with target particles can be defined as

(1.15)

where u is the test particle velocity.

Let us introduce the collisional mean free path in detail. To that end we consider a simple gas consisting hard-sphere particle. The distance along the line joining the center of any two particles is equal to d (Figure 1.9).

Consider a single test particle z moving in randomly distributed particles at rest as shown in Figure 1.10A. It is assumed an oversimplified situation in which the test particle z moves at a uniform speed equal to the mean molecular speed 〈C〉 and all the other molecules are standing still.

Along the molecule path (Figure 1.10B), the volume trace by the sphere of influence per unit time is πd²〈C〉. For a density of fixed particles, n_a, the number of particle centers lying within cylinder volume is πd²〈C〉n_a. Since each of these centers corresponds to a collision, this product must also represent the number Σ of collisions per unit time for test molecule z, i.e.,

(1.16)

In essence, parameter Σ characterizes the collision frequency. The distance travel in time between the collisions characterizes the mean free path λ of the molecule with average velocity 〈C〉 as

(1.17)

More complicated analysis based on the particle kinetics produces very similar results with just a factor in denominator (1.17):

(1.18)

The factor in the denominator arises from the fact that the correct speed to use for the evaluation of Σ in Eq. (1.17) is really the mean relative speed of the molecules, and this can be shown to be Equation (1.18) can also be written in terms of the mass density ρ=mn, where m is the mass of the assumedly identical molecules. This gives

(1.19)

Thus for a given value of ρ, the mean free path for the rigid-sphere model is independent of temperature and depends only on the gas density, mass, and diameter of the molecules.

In case of assembly of test particles, the flux density of the particles can be defined as Γ=nu, where n is the particle density (see Figure 1.8). Let us assume that collisions remove particles from the beam. The change of the particle density can be calculated as follows:

(1.20)

and the reduction of the particle flux can be calculated as

(1.21)

Integration of Eq. (1.21) will give the flux dependence on distance x as the following:

(1.22)

After a collision, the test particle scattered by an angle θ of scattered particle trajectory with respect to its incident direction as shown in Figure 1.11. This kind of collisions is characterized by another parameter defined as differential cross section σ(θ). Impact parameter (b, see Figs. 1.11, 1.12) is the perpendicular distance from the original center of a scattering particle to the original line of motion of a particle being scattered.

The probability of the particle emerging into the solid angle is dΩ=sin θ dθ dφ. Integrating over the full solid angle yields the so-named total cross section for a particular collision in the form

(1.23)

For many practical applications, it is important to consider interactions that lead to scatter by θ=90° or more as shown in Figure 1.12. The elastic collision with θ=90° occurs on χ=45°. Thus the cross section in this case is equal to cross section for scattering with angle larger than The collision cross in the hard-sphere model is One can see that 90° is smaller than that of the hard-sphere model by factor of 2.

1.2.1.3 Charge-exchange cross section

Both resonant and nonresonant charge-exchange collisions can take place. Resonant charge exchange occurs between the same atoms, while nonresonant occurs between different atoms. Consider reaction

Electron transfer from B to A⁺ occurs in two steps: first, release from B and then capture by A⁺. By calculating the potential energy of electron in the electrostatic field of A⁺ and B⁺ and considering kinetic energy of electron, one can calculate the distance between A and B leading to electron transfer. For the ground state, resonant charge exchange cross section (i.e., A=B) can be estimated as:

(1.24)

where e is the electron charge, ε₀ is the permittivity of vacuum, and E_i is the ionization potential. Detailed theoretical analysis and experiment show that cross section for charge exchange has weak dependence on the kinetic energy. While most of data on charge-exchange cross section are obtained experimentally, there are several simplified theories developed.

Sakabe and Izawa [14] calculated cross section for charge-exchange collisions by solving the time-dependent Shrodinger equation. The cross sections of resonant charge transfer between atoms and positive ions for all nontransition elements have been calculated by the impact parameter and close-coupling methods. The calculated results are in good agreement with the experimental results for the elements for which data are available. Figure 1.13 shows the calculation result and the experimental data on the cross section of helium atoms as a function of impact velocity. The calculated results are in good agreement with the experimental results, in particular in the low-velocity range less than 10⁸ cm/s.

1.2.1.4 Coulomb collision cross section

Let us consider collisions of two particles with charges q₁ and q₂. This type of interactions occurs between electron–electron, ion–electron, and ion–ion. Particles interact with one another via electrostatic potential:

(1.25)

Coulomb forces are long-range interaction as it depends inversely on distance r². However, potentials are screened in plasma at the scale of about Debye length L_D. Thus the maximum impact parameter can be taken equal to the Debye length since the Coulomb electric field force decays exponentially at the distances larger than L_D. In other words, the maximum distance for interactions is b_max~L_D. Yet, a collision cross section based on the Debye length, i.e., is still very large. Taking into account that the electric field is partially screened in the Debye sphere, a collision cross section based on the Debye length includes scattering over very small angles that does not affect significantly transport coefficients. Thus we will limit our consideration of the scattering process by only scattering over large angles, e.g., >90°.

Scattering to large angles can be due to a single collision scattering over a large angle or due to cumulative effect of many small-angle collisions. In plasmas, because of the long-range nature of the electrostatic interaction, much frequent small-angle collisions led to larger effect than the fewer large angle events. In order to evaluate this effect, one needs to consider the cumulative effect of many scattering of electron by many ions with different values of impact parameter b (Figure 1.12).

Consider electron having initial velocity v that undergoes many small-angle scattering collisions. Each collision event will lead to small increment in electron velocity Δv in all directions. Recall that electron energy is nearly conserved in collision since electron is a light particle that is scattered off a heavy ion, while electron losses its momentum. Thus there is a reduction of the electron velocity in the original direction. The average velocity reduction can be estimated as

(1.26)

where n is the ion density, Z is the ion charge, and ln Λ=ln(b_max/b_min) is called Coulomb logarithm. From this equation, one can define the electron–ion collisional frequency for loss of electron momentum as

(1.27)

To estimate the minimum impact parameter b_min, we need to point out that when the Coulomb potential energy becomes as large as the electron thermal energy, the scattering angle becomes the largest.

Thus b_min can be estimated from the condition that the Coulomb energy is equal to the thermal energy:

(1.28)

The Coulomb logarithm represents the sum of all scattering angles by Coulomb collisions within the Debye sphere. Detailed analysis leads to the following expression for the Coulomb logarithm (if T_i(m_e/m_i)<T_e<10Z² eV [15]):

(1.29)

where T_e is in eV and n_e is in cm³. In plasma of interest is 5–10 eV. ln Λ is about 17 in the case of fusion plasma and about 10 in the case of arc discharge plasmas.

The total cross section for electron scattering on ions can thus be calculated using relationship between the collision frequency and cross section:

(1.30)

1.2.1.5 Ionization cross section

In general, the quantum mechanical analysis should be employed to describe the cross section of atom ionization. However, some qualitative analysis with reasonable quantitative results was performed using classical approach in physics. In this case, the ionization process was considered as interaction of impact electron with a valence electron of a neutral atom. It was assumed that the atom was ionized if the transferred energy to the valence electron exceeds the ionization potential E_i. This is the basis of the model that was developed by Thomson back in 1912 [16]. According to the Thomson’s model, the ionization takes place if transferred energy is larger than the ionization potential, i.e., E>E_i. After integration over electron energies that is larger than E_i, the following expression for the ionization cross section was obtained [14]:

(1.31)

At very high energies, Thomson model predicts that ionization cross section fall as ~1/E. The cross section has its maximum at E=2E_i. Thus the maximal cross section is

(1.32)

The dependence of the ionization cross section on the electron energy for different atoms is shown in Figure 1.14.

It should be pointed out that the electron temperature is typically in the order of few electron volts which is smaller than the ionization potential for most atoms of interest. However, the high-energy electrons from the tail of Maxwell distribution play a very important role in the ionization process. In order to obtain the collisional cross section in the case of electron energy distribution, one have to integrate over the velocity distribution:

(1.33)

where k_i(T) is the ionization coefficient.

If Maxwellian distribution function can be assumed, we will arrive at the following expression:

(1.34)

In order to calculate the cross section of atom ionization σ(v), one can use the Thomson formula near E=mv²/2~E_i:

(1.35)

where

With this assumption, the above expression for ionization coefficient becomes the following simple form:

(1.36)

Below the ionization cross sections for electron impact with different materials are demonstrated. An example of total cross section for electron scattering on nitrogen molecules is shown in Figure 1.15 and ionization collision cross section for several gases [18] is shown in Figure 1.16A and B. One can see that total scattering cross section can have several maxima. Peak in ionization cross section is clearly recognizable in both cases.

1.2.1.6 Plasma equilibrium

One particular thermodynamic state of the plasma is widely used to describe plasma chemical composition, so called plasma equilibrium. Let us first define the degree of ionization in the plasma. Consider plasma consisting of electrons, ions, and atoms. In the case of a low-ionized plasma, i.e., n_en_a, the ionization degree is defined as

(1.37)

In the thermodynamic equilibrium, there are two reactions: direct reaction leading to ionization and a back reaction leading to recombination. In the case considered, the direct reaction is the electron impact ionization and back reaction is the recombination involving three particles. In other words, the stochiometric equation will have the form:

with the rate of ionization being k_i·n_a·n_e and the rate of recombination being k_r·n_i· where k_i and k_r are the ionization and recombination coefficient, respectively. We shall define the ionization equilibrium as a state in which ionization and recombination rates are equal.

The equilibrium constant K(T) can be defined as follows:

(1.38)

This equation can be solved for the electron density provided that the equilibrium constant dependent on temperature is known as a function of the pressure in the system. If equation of state for the plasma can be specified, this equation can be used to calculate the electron density as a function of the temperature.

Degree of ionization can also be determined from the thermodynamic equilibrium of plasma. We shall illustrate this below. This relationship that can be used to determine the plasma composition in equilibrium is called the Saha equation.

Let us derive the Saha equation from the statistical mechanics. We assume that plasma is in thermodynamic equilibrium, so that we can use a single temperature for all plasma components. The approach that will be used is quasi-classical taking into account statistical weights and quantum phase space. While being an approximation, this approach gives physically illustrative picture of the phenomena.

According to the statistical mechanics, the probability of a particle to be at the energy level ε is determined by the Boltzmann relation:

(1.39)

where T is the plasma temperature.

The number of particles can be calculated by multiplying the probability and number of states. The number of states assuming that particles do not have internal degrees of freedom is equivalent to the number of cells in the phase space h³, where h is the Planck’s constant. The volume in the momentum space for electrons having momentum between p and p+dp equals:

(1.40)

Thus, the number of electrons in the volume V can be calculated as

(1.41)

where g_e is the electron degeneracy.

We are considering the equilibrium state of electrons and neutrals and as such let us consider zero energy level corresponds to electrons in atom. It follows that free electron will have the energy

(1.42)

where E_i is the ionization potential. The total number of electrons can then be calculated by integrating over the entire spectrum:

(1.43)

After integration one can find that the electron density has the following expression:

(1.44)

Such quasi-classical approach cannot lead to the complete expression without considering additional assumptions. First of all, recall that in the integral (1.43) it was taken into account the number of electrons per a single atom. In order to calculate the full number of electrons, we have to multiply N_e by number of atoms in one quantum state, i.e., N_a/g_a. The resulting expression for the number of electrons is thus

(1.45)

Secondly, equilibrium state consists of electrons, neutrals, and ions. Thus, ionization balance must account for ions as well. This can be done by assuming that the volume corresponds to an ion in a given quantum state, i.e.,

(1.46)

By supplementing this expression into Eq. (1.45), we arrive at the final expression for the Saha equation:

(1.47)

Equation (1.47) describes plasma that consists of electrons, atoms, and ions of a single type. However, it can also be used to calculate the composition of gas mixture. Let us demonstrate this on the following example.

Let us start with equations for equilibrium for each individual species. In the case considered, these equations can be written as

(1.48)

(1.49)

where n_Hei is the density of helium ions and n_Ci is the density of carbon ions.

Arc plasma consists of the electrons, neutrals C and He, and ions C+ and He+. Thus, we have five unknown densities. In order to calculate an equilibrium composition, we can invoke two additional equations. One is the plasma quasi-neutrality:

(1.50)

Second equation is the equation for atom conservation, i.e., the ratio of total density of atoms and ions of helium to the total density of atoms and ions of carbon:

(1.51)

Finally, the total pressure is equal to the sum of partial pressures of all species.

Plasma composition calculation using Saha equilibrium is shown in Figure 1.17.

One can see that significant ionization starts at temperatures exceeding 12,000 K and it is dominated by ionization of carbon. Ionization of helium atoms starts at about 20,000.

1.3.1.1 Conservation law for electric charge and current: electromagnetic waves

Each charged particle in plasma has an electric charge. Electron has charge of about 1.6×10⁻¹⁹ C and ion charge being either single or multiple electron charges. While typically ions have a positive charge, negatively charged ions can be created due to electron attachment to atom or molecule. Good example is the oxygen that has very large cross section for electron attachment.

The electrostatic force F_ij is determined by the interaction of charges q_i and q_j with distance r_ij between the charges in form:

(1.52)

Charge is the additive property, i.e., the total charge in a volume can be expressed as a sum of number particles of type N_i with charges q_i:

(1.53)

where q_i is the charge of a single particles.

Density of charges per unit volume:

(1.54)

Considering uniform charge distribution in the volume V, current density can be defined as

(1.55)

where v is the average velocity of charges. When charges are distributed nonuniformly in the volume, the total charge can be calculated by integrating over the considered volume:

(1.56)

By employing the Gauss theorem:

(1.57)

one can arrive at the following equation which is the conservation of charges:

(1.58)

Electric and magnetic fields in the plasma must satisfy the Maxwell equations. These equations were proposed by James Maxwell to describe electromagnetic wave propagation in a media:

(1.59)

(1.60)

(1.61)

(1.62)

Introducing the electric potential, one can define the electric field as E=−∇φ, as a result the last equation will have the following form:

(1.63)

Equation (1.63) is called the Poisson equation. If n_e=n_i, i.e., quasi-neutrality is fulfilled, the Poisson equation will reduce to Δφ=0, which is equivalent to assuming the vacuum electric field. Note that the Poisson equation in reduced form can be used to calculate potential distribution only if plasma is uniform and in the absence of a magnetic field. In a general case, potential distribution in plasma can be calculated invoking the generalized Ohm’s law as will be described in Chapter 4.

1.3.1.2 Electromagnetic wave propagation

Waves can transport energy to and from a media. In order to describe the wave propagation, we will use Eqs (1.59) and (1.60). Two limited cases can be considered, namely, a media with high conductivity and a media with low conductivity.

1. Streaming instability: The beam of energetic particles travels through the plasma. The drift energy is utilized to excite waves. An example of this instability is the two-stream instability.

2. Rayleigh–Taylor instabilities: In this case, instability is driven by presence of the density gradient and external nonelectromagnetic forces.

3. Universal instabilities: Even if plasma is in the equilibrium, plasma pressure gradient leads to the expansion. All confined plasmas defuse slowly due to particle collisions. This expansion energy can drive the instability. This kind of instability always present in plasma and this is why they called universal. A known example is the drift instability.

4. Kinetic instability: If the velocity distribution of plasma particle is not Maxwellian, there is always deviation from equilibrium and the instability can be driven by energy distribution anisotropy.

1.3.7.1 Two-stream instability

Consider cold plasma with T_e=T_i=0 and without a magnetic field, i.e., B=0.

The linearized equations of motion for ions (Eq. (1.93)) and electrons (Eq. (1.100)) will have the following form:

(1.126)

where v₀ is the velocity of electron stream. Electrostatic wave is considered:

(1.127)

Employing the form (1.127) for the electric field in Eq. (1.126), we arrive at

(1.128)

Solving this equation for v_i1 yields

(1.129)

After employing the electric field in the form (1.127), the equation for electron motion will have the following form:

(1.130)

Solving this equation for the electron velocity

(1.131)

Under considered conditions, the ion continuity equation is

(1.132)

One can solve this equation for the ion density:

(1.133)

Electron continuity equation is

(1.134)

After linearization, this equation becomes

(1.135)

Solving this equation for the electron density yields

(1.136)

We can now plug Eqs (1.133) and (1.36) into the Poisson’s equation:

(1.137)

which looks like:

(1.138)

The dispersion relation is thus

(1.139)

Let us now examine whether these oscillations are stable or they can become unstable. To this end we will find a root of ω assuming that it is a complex variable, i.e.,

(1.140)

Taking into account Eq. (1.140), the electric field becomes

(1.141)

Let us analyze the dispersion equation (1.139) using the following normalized variables:

The dispersion equation becomes

(1.142)

The solution of Eq. (1.142) can be shown graphically in Figure 1.24.

The function F(x₁,x₂) is shown with two possible cases considered. It can be seen that in Figure 1.24A, there are four real roots, while in Figure 1.24B, there are two real and two imaginary roots with one of the roots being positive. Thus for sufficiently small kv_o plasma becomes unstable. In the case of a given v₀, plasma is unstable for long-wave oscillations.

The maximum growth rate can be estimated from Eq. (1.139) for m/M1:

(1.143)

This instability is sometimes called the “Buneman” instability. Physics of this instability can be explained as follows. The natural electron frequency is ω_p, while the ion frequency is Ω_p=(m/M)^0.5ω_p. Because of the Doppler shift of oscillations moving with electrons, these two frequencies could be equal for given kv_o. As a result, the energy can be transferred from electrons to ions providing conditions for wave growth.

1.3.7.2 Kinetic instabilities

So far we considered instabilities and waves using the fluid approximation and as such assuming that particles in plasma obey equilibrium (Maxwellian) velocity distribution. However, in many cases, the energy distribution function of the particles in plasmas can deviate from Maxwellian distribution due to imbalanced conditions. The deviation from the equilibrium distribution can be maintained due to rare collisional events in rarefied plasmas.

Let us consider small deviation from equilibrium as follows:

(1.144)

where f₀(r,v,t) is the Maxwellian distribution function. Linearized, the collisionless Boltzmann equation leads to

(1.145)

In accordance with previous analysis, we assume

(1.146)

By substituting Eq. (1.146) into Eq. (1.145), we arrive at

(1.147)

Solving Eq. (1.147) for f₁ yields the following:

(1.148)

Using the Poisson equation, the electric field E₁ can be expressed as

(1.149)

After substituting f₁ and some algebra, dispersion relation can be obtained:

(1.150)

One can see that integral (1.150) has a singularity. Landau treated this integral using the contour integration. By doing that, one can arrive at

(1.151)

where v_ϕ=(ω/k) is the phase velocity.

Taking into account that the f_o is the Maxwellian distribution function, one can obtain expression for the decay rate of this oscillations of distribution function.

(1.152)

Since imaginary part is negative, wave is damping. This effect is called Landau damping.

Note: This is very surprising result because the wave is damped without collisions! It was first discovered by Landau [23] theoretically and then was confirmed experimentally by Malmberg and Wharton [24].

In order to understand the underlying physical mechanism, let us consider the electron velocity distribution function as shown schematically in Figure 1.25. In particular we are interested in particles with velocities near the phase velocity as shown in Figure 1.25. The particles with velocity close to the phase velocity of the wave travel with wave and thus can interact efficiently with the wave. If decreasing branch of the velocity distribution function is considered as shown in Figure 1.25, it can be seen that more particles have velocity smaller than the phase velocity (slow electrons). In other words energy, more particles have energy smaller than the wave energy than particles with energy larger than the wave energy. In this case, the wave energy is transferred into particles and thus wave is damping.

Using analogy with waves in the sea can provide intuitive explanation of the Landau damping. In this case, we consider the particles as surfers trying to catch the wave as shown in Figure 1.26. If the surfer is moving on the water surface at a velocity slightly less than the waves, he will eventually be caught and pushed along the wave (and will gain the energy), while a surfer moving slightly faster than a wave will be pushing on the wave as he moves uphill (and will lose the energy to the wave).

In summary, in this section, we considered wave propagation in the plasma and plasma–wave interactions. In particular, we considered Langmuir wave, ion sound wave, as well as waves in a magnetic field. Plasma instability associated with either drift or gradients was considered for conditions relevant for some aerospace applications. In addition, we consider kinetic effect such as Landau damping which is relevant for rarefied plasmas considered in subsequent chapters.

1.4.1.1 Condition for stable sheath: Bohm criterion

Let us discuss the necessity of certain ion velocity at the sheath edge to have a monotonic potential distribution in the sheath. To this end, a collisionless sheath with quasi-neutral presheath can be considered schematically presented in Figure 1.27. The ion velocity at the sheath edge can be calculated using the equations of energy particle flux conservation[69]:

(1.153)

where φ(x) is the potential distribution function, M, V, and C_s are the ion mass, velocity, and sound velocity, respectively. The left-hand side corresponds to the ion kinetic energy evaluated in the sheath at boundary x=0.

Ion continuity equation can be written as

(1.154)

where n(x) is the density distribution function. From Eqs (1.153) and (1.154), one can derive:

(1.155)

According to Eqs (1.153) and (1.154), the ion density distribution is therefore:

(1.156)

Electrons obey the Boltzmann distribution:

(1.157)

Plasma is quasi-neutral at the presheath–sheath, i.e., n_es=n_is=n_s, where n_es and n_is are, respectively, the electron and ion densities at the sheath edge.

The Poisson’s equation:

(1.158)

Substituting expressions (1.156) and (1.157) for the ion and electron densities in the Poisson’s equation:

(1.159)

Considering the one-dimensional approach, Eq. (1.159) can be expressed as

(1.160)

Equation for the electric field can be obtained by first integration of Eq. (1.160):

where and C is the constant of integration.

Using the following boundary conditions at x=0:

it can be obtained:

(1.161)

Using Eq. (1.161), the equation for the electric field becomes

(1.162)

The Taylor expansion near the sheath edge reads

(1.163)

To reach the stable monotonic potential distribution in entire sheath region, the following condition at the external boundary of the sheath should be fulfilled:

(1.164)

Consider potential near the sheath edge, eφ/T_e<1. Thus high-order terms (such as (eφ/T_e)² and higher) can be neglected. After some algebra, Eq. (1.162) becomes

(1.165)

This leads to the following condition, which is known as Bohm criterion:

(1.166)

1.4.1.2 Monotonic solution for sheath–presheath region

Here we illustrate the sheath and in a more general sense plasma–wall interface by considering solutions of a quasi-neutral plasma and a sheath separately. By employing the patching technique, we will illustrate how to obtain the monotonic solution for entire plasma–wall transition region.

The fluid-based approach is considered below to illustrate the sheath problem and its solution. Schematically the transition regions are shown in Figure 1.27. According to Langmuir’s analysis [25], the ions are accelerated toward the Bohm velocity in a plasma region with a relatively weak electric field and this region called as presheath extends from the sheath into the quasi-neutral plasma volume. The electric field, thus, becomes a continuous function increasing in the sheath from a boundary value up to a maximal value at the wall. This boundary field can be obtained considering a model for the presheath region. The position of the sheath–plasma interface is determined by patching the electric field, ion velocity, and potential. Thus a relationship between the ion velocity and the electric field at the presheath–sheath interface should be determined to choose these parameters as boundary conditions for a sheath region solution.

The procedure described involves the following conditions: (i) the degree of the quasi-neutrality violation near the presheath edge that may affect the solution will be estimated; (ii) potential distribution is spatially monotonic in all regions, and (iii) continuity of the electrical field and potential at the sheath–presheath interface.

1.4.1.3 Mathematical formulation

Let us consider the governing equations to describe the problem including the plasma, the sheath, and the sheath–presheath interface. The plasma flow in the quasi-neutral plasma region is considered in the fluid approximation. Ion mass and momentum conservation taking into account and assuming that the electrons obey the Boltzmann distribution:

(1.167)

(1.168)

(1.169)

where n_e and n_i are electron and ion densities, respectively, ν_i is the ionization collision frequency, and ν_c is the charge-exchange collision frequency. In order to obtain main dependencies, we assume that collision frequency is constant, Z_i is the ion charge state (only single ionized ions will be considered, Z_i=1), P_i is the ion pressure (cold ions are assumed, so P_i=0) and E=−dφ/dx is the electric field and φ is the potential. The potential is distributed according to Poisson’s equation:

(1.170)

The system of equations (1.167)–(1.170) has following boundary conditions:

The above system of equations is solved separately for the quasi-neutral collisional plasma region (presheath) and for collisionless space charge region (sheath). Finally it will be described how to patch these two solutions.

1.4.1.4 Monotonic potential distribution in the sheath

Consider Poisson equation for the sheath potential distribution in dimensionless form:

(1.181)

The first integration of Eq. (1.181) in the vicinity of the sheath boundary with nonzero electric field at the sheath boundary, E_o=−(dη/dχ)_o, gives

(1.182)

where η_s is the potential at some point in the sheath including region close to the sheath edge. For the purpose of simplifying analyses, we assumed here that potential at the sheath boundary equals zero, η₀=0. All parameters (density, velocity, and potential) are normalized by their values at the sheath edge. Near the sheath edge, one can make a series expansion in the small parameter η_s1:

(1.183)

Neglecting small terms in the expansion with order higher than reduces Eq. (1.182) to the following:

(1.184)

The condition for the monotonic potential distribution in the sheath would be that the right-hand side of Eq. (1.184) is positive. This condition has the following form:

(1.185)

Obviously, Eq. (1.185) has no real solution in the sheath if V_o<1 and E_o=0. Therefore, for E_o=0, the potential distribution in the sheath is monotonic only when V_o≥1, i.e., with Bohm condition.

Equation (1.185) has real solution (the monotonic potential distribution) in the sheath for V_o<1 when E_o>0. The requested E_o that supports the monotonic potential distribution was obtained from the numerical solution of the sheath equations and shown below.

To get more understanding of the above points, let us analyze the solutions of Eq. (1.182) as a function of two variables η and dη/dχ, i.e., potential–electric field plane. In this case, the solution of Eq. (1.182) can be represented as dependences with parameter of Mathematical analysis of Eq. (1.182) in (η–dη/dχ) plane shows that the solutions could be centers or saddles. Centers solution corresponds to the periodic behavior of the mathematical system (oscillations), while saddles solution corresponds to spatially monotonic behavior.

The contours of directly computed from Eq. (1.182) are shown in Figure 1.29A. It can be seen that both saddles and centers curves of the same level of initial electric field E_o are possible and those solutions depend on the initial velocity V_o. Figure 1.29B shows that smaller corresponds to centers curves, while large corresponds to saddles curves.

Recall that dependent on the boundary conditions at the sheath edge (i.e., electric field and ion velocity), spatially monotonic or oscillatory behavior of potential distribution is possible.

1.4.1.5 Solutions in plasma and sheath regions: procedure of patching

Firstly, the quasi-neutral (presheath) region (i.e., Eqs (1.171), (1.172)) is considered. The distribution of the velocity, electric field, and quasi-neutrality deviation is shown in Figure 1.30A and Figure 1.30B. The ion velocity variation in this scale length (fraction of a mean free path) is small, while the electric field and quasi-neutrality deviation noticeably increase in the presheath with distance toward the presheath edge and become singular (Figure 1.30A). As shown in Figure 1.30B, both the electric field and the quasi-neutrality deviation significantly increase when the ion velocity approaches to the Bohm velocity. This deviation becomes stronger with coefficient ψ decreasing.

Now let us consider the procedure of patching the presheath and the sheath solutions. The existence of the solution with spatially monotonic potential distribution in the sheath region is examined in Section 1.4.1.4. The procedure of finding the monotonic solution is as follows. Having the boundary velocity V_o and corresponded electric field E_o, the potential distribution in the sheath (Eqs (1.174)–(1.176)) is calculated. Two types of potential distribution are possible as shown in Figure 1.31. When electric field E_o is small and V_o<1, the only oscillating solution of the Poisson equation is possible. A combination of the nonzero electric field E_o and velocity V_o (if V_o<1) provides the spatially monotonic solution. These conditions are the boundary condition for the sheath.

In the above described approach, the velocity V_o is searched as a minimal velocity for which spatially monotonic potential distribution in the sheath is possible.

The calculations show that when ψ increases, the ion velocity at the sheath edge is smaller than the Bohm velocity and at the same time the obtained electric field increases with ψ (Figure 1.32). However, the electric field increase coexists with the quasi-neutrality violation, as it is also shown in Figure 1.32. In the opposite limit, i.e., when ψ is small, the required velocity approaches the Bohm velocity and at the same time the electric field approaches zero. One can see that up until ψ=10⁻², the level of deviation from quasi-neutrality is small (about few percent) and can be tolerated in many cases. In the case when the parameter ψ increases and approaches unity, the collisions become important in the sheath and therefore Eq. (1.174) should be modified taking into account the change of momentum by the collisions. The tendency is shown in Figure 1.32; when parameter ψ increases, the velocity significantly decreases and the Bohm criteria requirement is relaxed. Also there is an intermediate range of ψ (where 10⁻⁴<range<10⁻²) when the sheath can still be considered collisionless, but the sheath and presheath solutions can be patched with velocity smaller than the Bohm velocity and a nonzero electric field.

The accuracy of the proposed patching procedure for ψ=10⁻⁴ is compared with the continuous direct numerical solution (1.177)–(1.180) as shown in Figure 1.33, where the electric field as a function of the potential in the presheath and the sheath is shown. One can see a good agreement between continuous (direct) solution and solution using above developed patching approach.

The density distribution across the sheath region shown in Figure 1.34 illustrates that two approaches (collisionless sheath-solid curve (i.e., patching) and dotted curve—direct numerical solution) produce very close results. The patching solution is obtained using the appropriate boundary conditions for ion velocity and electric field necessary for continuous distribution of these parameters at the sheath–presheath interface. The difference between those solutions is due to the fact that the collisions were neglected in the sheath solution. This implies that collisional effects in the sheath are not significant.

1.4.1.6 Typical electrostatic sheath

1.4.1.7 Sheath in a magnetic field

Magnetic field affects the plasma–sheath transition. Schematically plasma–wall interface in a magnetic field is shown in Figure 1.36.

Daybelge and Bein [42] were the first to completely analyze a collisionless sheath, in which a magnetic field exists between fully ionized plasma and an infinite plane metal wall. The potential distribution in the sheath was obtained from the solution of Poisson equation, using the wall potential as a parameter and assuming a Maxwellian distribution for the charged particle velocities at the sheath–plasma edge. They obtained that sheath thickness is comparable to the ion Larmor radius in the case of negative wall potentials, whereas for positive wall potential it is reduced to the electron Larmor radius. However, Daybelge and Bein considered in their model the electrostatic sheath region only for the case L_Dr_Le, where L_D is the Debye length and r_Le is the electron Larmor radius. Therefore, the ionic presheath acceleration was not considered. Later, the effect of the magnetic field on the plasma presheath was studied by Chodura [43] in the collisionless limit. Plasma quasi-neutrality in the presheath and a Boltzmann distribution for the electron density were assumed in order to determine the electric field in the sheath. Chodura found that in order to satisfy the usual Bohm condition at the sheath–plasma edge, a new double-layer presheath was required. In the first double presheath region, the ions are accelerated to supersonic velocities parallel to the magnetic-field lines, which may be oblique with respect to the wall. In the second presheath region, which has the thickness of approximately the ion Larmor radius, the ion velocity vector is rotated so that the velocity normal to the wall is supersonic at the entrance to the sheath. Unlike the above-mentioned models, Riemann’s analysis of the presheath region in a plasma placed in a magnetic field also accounted for collisions[70]. The assumptions of Riemann’s fluid model are similar to those of Chodura, except that an ion collision term of constant collision frequency was added. A dependence of the potential distribution in the presheath on the magnetic field and collision frequency was found. The main effect of a strong magnetic field is to reduce the collisional presheath thickness, as it approximates now the ion Larmor radius and not the ion mean free path. Riemann also showed that in the collisional case, additional plasma presheath is not required, in contrast to the case studied by Chodura.

In all of the above-mentioned magnetic presheath models, a Boltzmann distribution for the electron density was assumed, and the magnetic field acted only on the ions. It should be noted that the Boltzmann distribution for the electron density is physically significant in the case of plasma flow parallel to the magnetic field. However, for plasma transport across a magnetic field, it is not obvious that this distribution will occur. It was shown by Beilis and coworkers [44,45] that the assumption about the electron distribution function influences strongly the presheath thickness. Effect of the angle of the magnetic field with respect to the wall on the presheath thickness is illustrated in Figure 1.37. One can see that when magnetic field is changed from being parallel to the wall (0°) to small incidence angle a significant change in the presheath thickness is occurred.

It was also shown that while density was calculated from the general system of equation (i.e., electron momentum equation that takes into account the magnetic field), the density profile does not deviate significantly from the one obtained from the Boltzmann equation as shown in Figure 1.38. However, neglecting the magnetic field term in the electron momentum equation affects the potential, density, velocity distribution in the presheath and the presheath thickness.

1.5.1.1 Thermionic emission

When the metal is heated by an energy source, the surface temperature increases allowing the electrons to overcome the potential barrier at the surface. Integrating the distribution function taking the minimal energy electron as W_f:

(1.219)

where

Presence of the electric field changes the potential distribution near the surface and lowers the potential barrier (shown in Figure 1.40) by the so-called Shottky effect reducing the work function by

(1.220)

Taking this into account the electron reflection due to quantum mechanical effect of tunneling, the expression for the current density by thermionic electron emission will have the following form:

(1.221)

where D is the electron reflection coefficient. In general, electron reflection coefficient depends on material, surface conditions. As a result, effective thermoionic current varies significantly.

1.5.1.2 Field emission

The electron emission by electric field action is modeled assuming that surface temperature is zero. Presence of the electric field near the surface transforms the infinite thick potential distribution into a potential barrier of a finite width (Figure 1.41). As a result, electrons can escape the metal with probability determined by quantum mechanical tunneling mechanism. The current density due to the field emission, one needs to account for this quantum mechanical effect. Such treatment was done by Fowler–Nordheim, and the emission current density is given by the equation:

(1.222)

where E is the electric field,

1.5.1.3 T–F emission

When the metal temperature and electric field are sufficiently high, a combined effect of thermoionic and field action was takes place. This mechanism is known as T–F emission:

(1.223)

where eε_B is the lowest potential inside the metal. According to Eq. (1.223), electron emission current is determined by two functions, namely, density of states N and emission probability D. Murphy and Good [46] performed the T–F emission analysis for some limited regions of T and E. The electron emission for wide region of T and E appropriate for conditions in a cathode spot was calculated in Ref. [47]. Results can be presented in the form of emission current density distribution as a function of energy (relative to Fermi level) of emitted electrons as shown in Figure 1.42. One can see that electrons can be emitted with energy that is lower or higher than the Fermi level dependent on electric field and surface temperature. Such general approach can recover both limits of thermoionic emission in the case of a small electric field and field emission in the case of smaller temperature.

1.5.1.4 Secondary electron emission

SEE is defined as a phenomenon in which primary electrons of sufficient energy induce the emission of secondary electrons by hitting the surface. The number of secondary electrons emitted per incident particle is called secondary emission yield. If the primary incident particles are ions, the effect is termed secondary ion emission. The coefficient of the SEE depends on the energy of incident electrons and the incident angle. Primary electron collisions with the surface of a target generate emissions of two main types of secondary electrons, namely, “true” secondary electrons and elastically reflected backscattered electrons. Measured electron emission yield is due to contribution of all secondary electrons. In general, SEE is higher for dielectric materials in the low-energy range (up to 100 eV) and high in the case of metal target material in the case of a kV energy range. Dawson [48] measured yields for some dielectric matrials as shown in Figure 1.43. Lucalox alumina and sapphire are both alumina, but in the preparation of this Lucalox alumina, a small amount of magnesia was added. As shown in Figure 1.2, the yields from the highly polished sapphire were a little greater at all energies. However, those from the polished Lucalox had become similar to the sapphire in their variation with primary energy.

SEE yield δ(E_p) as a function of energy of the primary electrons was measured for several dielectric materials [49] as shown in Figure 1.44. One can see that SEE yield increases with energy and reaches unity at the energy of primary electrons of about 25–30 eV in the case of boron nitride.

1.5.2.1 Langmuir model

This is the simplified model that is based on the equilibrium condition near the surface. Consider a rate at which molecules condense on a surface κν, where κ is the sticking coefficient (ratio of the flux condensed/flux striking the surface) and ν is the ablated flux. Langmuir showed that for the case of vapor for same metal, the coefficient κ=1.

On the other hand, the flux of evaporated molecules noted as μ equal to ν in the equilibrium. Taking into account that ν=(1/4)nV_a, one arrives to the following rate of vaporization:

(1.224)

The main Langmuir’s assumptions:

Despite the fact that this model is not exact in case of P_∞≠0, it is widely used [58]. In the next section, more general kinetic model will be described.

1.5.2.2 Kinetic models

Anisimov [59] considered a case of vaporization of a metal exposed to laser radiation using a bimodal velocity distribution function in the nonequilibrium (kinetic) or Knudsen layer. The main result of this work is the calculation of the maximal flux of returned atoms to the surface, which was found to be about 18% of the flux of vaporized atoms. This result was obtained under the assumption that the atom flow velocity is equal to the sound velocity at the external boundary of the kinetic layer. Schematically this layer is shown in Figure 1.45. In many physical situations, however, the expansion of the vapor is not by the sound speed since there is dense plasma in the volume discharge. The vaporization phenomena in case of plasma presence at the metal surface with velocity lower than the sound speed were developed by Beilis [60–62] who considered metal vaporization into discharge plasmas in the case of a vacuum arc cathode spot. He concluded that the parameters at the outer boundary of the kinetic layer are close to their equilibrium values and that the velocity at the outer boundary of the kinetic layer is much smaller than the sound velocity.

The nonequilibrium layer close to the evaporating surface can be also modeled using the particle method known as direct simulation Monte Carlo (DSMC) [63]. This method solves numerically the Boltzmann equation allowing calculation of the thickness of the nonequilibrium layer. Below the numerical results will be compared with the results of above-mentioned analytical approach for the case when the vapor velocity at the outer boundary of the kinetic layer is given as a parameter.

1.5.2.3 Model of the nonequilibrium layer

1. Debye length
A conductive sphere of radius a is immersed in an infinite uniform quasi-neutral plasma having density n_o, electrons in thermal equilibrium at temperature T_e, and ions in thermal equilibrium at temperature T_i. Only single charged ions are presented. A small negative DC potential |V₀|T_i, T_e is applied to the sphere.

a. Start from Poisson’s equation in spherical coordinates and using Boltzmann’s relation for both the electrons and ions derive expression for potential distribution.

b. Find expression for Debye length and show that Debye length is determined mainly by the temperature of the colder species.

c. Calculate Debye length for arc discharge plasma having electron density of about 10¹⁴ m⁻³ and electron temperature of about 1 eV and ion temperature of 0.1 eV.

Note 1: In spherical symmetry: ∇²φ=(1/r)d²(rφ)/dr²

2. Plasma oscillation
Compute plasma frequency for the following cases:

a. Vacuum arc with electron density of about 10²¹ m⁻³.

b. Hall thruster with electron density of about 10¹⁷ m⁻³.

c. Cold plasma with electron density of about 10²⁰ m⁻³.

3. Estimate whether following ionized gases are plasmas according to conventional definition (i.e., quasi-neutrality) if characteristic size of the system is:

a. Interstellar space area (10 km)

b. Magnetic fusion device (1 m)

c. Internal confinement fusion (1 mm).

4. Compute the pressure (in atmospheres) exerted by arc plasma on its chamber. Assume that plasma is in equilibrium, i.e., T_e=T_i=T_a=3000 K, n=10²¹ m⁻³. Ideal gas assumption can be employed.

1. Ionization cross section
Plot ionization coefficient as a function of electron temperature (in eV) for nitrogen plasma. Ionization potential for nitrogen is 14.5 eV. Consider T_e range 1–10 eV.

2. Plasma equilibrium
Calculate electron density of helium plasma in arc discharge at 0.1 atm as a function of electron temperature. Assume that g_ig_e/g_a=2. Consider range of electron temperature 1–5 eV.
Note: LTE can be assumed.

3. Plasma composition
Calculate electron density of nitrogen–argon mixture. Pressure is 1 atm and ratio of partial pressures of nitrogen to argon is 0.5. Consider conditions for problem #2.

1. Calculate the skin depth of electromagnetic wave in copper and iron material as a function of frequency. Consider frequencies in 10 kHz–1 MHz range.

2. Waves in plasmas. Consider plasma with density of about 10¹⁸ m⁻³ and electron temperature of about 5 eV. Plot the dispersion relation for electron plasma wave and ion plasma wave. Comment on results.

3. Calculate upper-hybrid and low-hybrid frequencies of arc discharge plasma having electron temperature 2 eV and electron density of about 10²¹ m⁻³. Plot dependence of both frequencies as a function of a magnetic field which is less than 1 T. Comment on results. What is the lower limit of magnetic field for low-hybrid frequency?
Note: Characteristic size of arc is 0.01 m.

4. A plasma consists of two uniform streams of ions with velocities V_x and ⁻V_x and densities 1/3n₀ and 2/3n_o. The neutralizing unmobile electron fluid has density n_o. Derive a dispersion relation for instabilities in this system.

1. Calculate the floating potential on the probe in contact with Argon plasma with electron temperature of about 10 eV. How floating potential will change in presence of secondary electron emission (yield is about 0.4).

2. Consider hydrogen ion diode with interelectrode distance of about 2 mm. The applied voltage is −1000 V. Calculate the ion current density in the diode.

3. Consider sheath between the quasi-neutral helium plasma having electron density of about 10¹⁸ m⁻³ and electron temperature of about 2 eV and the floating wall. Calculate the electron density at the wall. What is the ion velocity at the wall.

1. Calculate the electron current density due to thermoionic emission from the copper electrode as a function of surface temperature in the range of 1500–4000 K.

2. Equilibrium vapor pressure for copper (in mmHg) can be estimated as:
P=exp(A−(B/T)), where A=9.232; B=1.726×10⁴.
Calculate ablation rate using the Langmuir formula in the case of surface temperature of about 2500 K.

3. Compare the ablation rate calculated by Langmuir formula with the one predicted by the kinetic model. Use assumption that the normalized velocity at the edge of the Knudsen layer is about 1. Consider copper surface and surface temperature in the rage of 2000–4000 K.

4. Calculate contribution of the Shottky effect on the thermionic current in the case of titanium electrode. Consider surface temperature in the rage of 2500–4000 K and electric field in the rage of 10⁷–10⁹ V/m.