Renovations! Please excuse the mess, but we are in the process of overhauling the site.

KINETIC PHENOMENA IN THE PLASMA, say, "thermodynamic equilibrium" Edit, click, copy, click.

Another Gary Search Page, say, "HotBot", click,

Then, "HotBot", Search, click, paste for "thermodynamic equilibrium" , click, then Find

Searching Entries for Find (Edit, then Find)

Seven types of string searches are available, 1) search for an author, (your name) 2) search for a title (or part of a title), 3) search for a journal, 5) search by year, or 6) search by keyword.

81

During the passage of an electric current, a plasma is in non-equilibrium. The deviation from thermodynamic equilibrium in a TIC is most pronounced near the electrodes and is comparatively slight deeper within the plasma. With small deviations from equilibrium, the particle and energy fluxes in the plasma are determined by the gradients of the plasma parameters, i.e., by the gradients of the particle densities, the particle temperatures, and the potential.

In this chapter expressions are derived which relate these fluxes to the gradients of the plasma parameters. The case of a weakly ionized plasma is considered, in particular, where the state of the neutral gas is not dependent on the current, on the density and temperature of the charged particles, or on the electric field. Equations are formulated, however, which are subsequently used to find the distribution of these variables in the TIC interelectrode space.

The chapter is concerned primarily with the calculation of the state of a weakly ionized TIC plasma. More detail on the kinetics of plasmas in general can be obtained from monographs devoted specifically to this problem. In particular, a survey of the most recent investigations of plasma kinetic scan be found in [1].

The state of a gas is totally characterized by the distribution function f(v@, r@, t), which is equal to the average number of particles @ which have coordinates from x to x + dx, from y to y + dy, and from z to z + dz; and velocity components from v_x to v_x + dv_x, from v_y to v_y + dv_y , and from v_z to b. These particles are therefore in a volume element of coordinate space d³r = dxdydz and of velocity space d³v = dv_xdv_ydv_z, i.e.,

Using the distribution function, one can determine any characteristic of the gas. For example, the particle density n(r@, t) is expressed as

Any particle function of velocity X(v) at a given point of space can be averaged over velocity by the relation

Since the distribution function f(v, r, t) is the particle density in a space of six dimensions with coordinates v_x, v_y, v_z, x, y, and z, then, in the absence of collisions, it should satisfy the continuity equation

where I@ is the flux of particles in the six-dimensional space.

Equation (1.4) is a generalization of the ordinary continuity equation for a three-dimensional space. The flux of particles with a given velocity v@ in ordinary space, as is well known, is given by ir@ = fdr@ / dt = fv@. The flux in velocity space is accordingly iv@ = fdv / dt = Ff@ / m, where F@ is the external force acting on the particles. Vector i@ is the sum

Calculating div i@ in the usual manner as the sum of the partial derivatives of projections ia to the coordinate axes, we find that

The subscripts r and v denote differentiation by r and v, respectively.

The force ± eE@ (due to the electric field E@) and the Lorentz force ± e(v@ x H@)/c (due to the magnetic field H) act on a particle with charge ± e. Therefore, in general, the force on a particle is

where the upper sign is taken for ions and the lower sign is taken for electrons. When calculating the right side of (1.6), we use the fact that the velocity v@, coordinate r@, and time t are independent variables in the continuity equation (1.4).

It is easy to show from (1.7) that div_vF@ = 0. As a result, from (l.4) and (1.6), we obtain the following equation for the distribution function f(v@, r@, t):

Equation (1.8) is valid if the particles considered do not collide with each other or with other particles. In the presence of collisions, the particle flow is no longer continuous, because particles change their velocity sharply with each collision , i.e., they disappear at specific points of velocity space and appear at other points. Therefore, the change of the distribution function due to collisions must be taken into account separately. By denoting the change of the distribution function due to collisions by I{f}, we obtain the following equation for the distribution function:

Equation (1.9) is called the kinetic equation or the Boltzmann equation. The right hand term I{f} is called the collision term of the kinetic

We note that the effects of "external" forces and of collisions on particle motion are taken into account in the kinetic equation by essentially different methods. It is assumed that collisions occur instantaneously at a given point of space (with coordinates x, y, and z) and that the only result of a collision is a change in the particle velocity (and in the general case, of the internal state as well). Therefore, the kinetic equation is valid for those cases where the collision time D t is much less than the time T during which external forces vary appreciably, and where the change of the particle coordinate during collision, D r, is much less than l (where l is the distance over which there is significant variation of the external fields).

When calculating the collision term, it is also usually assumed that the external fields do not affect the collision process, i.e., that the interaction forces between the particles are considerably greater during collision than the external field F@. (This effect, however, can be taken into account if necessary.)

The kinetic equation is a classical equation. For a particle moving in a gas to be regarded classically, it is necessary that its de Broglie wave length be considerably less than the mean free path of the particles, i.e., that the condition h/mv << l be fulfilled, or accordingly, that energy E >> h@/t . If this condition is fulfilled, one may assume that the particle moves between collisions according to the laws of classical mechanics. As for the collision term, generally speaking, it must be calculated using quantum mechanics.

As already noted above, Coulomb interactions in the plasma, strictly speaking, are not binary interactions, because very many charged particles are included in a Debye sphere (radius L_D). Therefore, a specific particle at any given moment of time interacts with many charged particles rather than just one.

However, when the temperature of the electrons and heavy particles are close to each other, and oscillations in the plasma do not differ strongly from the thermal noise, the results of precise theory (see, for example, [2-4]) coincide with the results of the theory of binary collisions, in which the Coulomb interaction is "cut-off" at a distance equal to L_D.*

During Coulomb collisions, the main interaction between particles occurs within a distance of the order of L_D. In this case, the interaction energy is of the order of e ²/L_D. This value is usually small compared to the kinetic energy mv²/2 of a colliding particle, so that the particle velocity v@ usually changes insignificantly as a result of the collision. Therefore, during Coulomb collisions, the charged particles seem to diffuse within velocity space. This concept of diffusion for the electron motion in velocity space is used extensively (see §§5 and 6 of this chapter). The condition for this concept to be applicable

is the fulfillment of the inequality (mv^2/e²)L_D >> 1. We note in this case the Coulomb logarithm ln@ = ln{(mv^2/e²)L_D}, which figures in equation (3.2.10) for the cross section Q_p for Coulomb collisions, is large compared to unity.

The kinetic equation, including the effect of collisions, is an integral-differential equation, and in general, can not be solved analytically. The simplest kinetic equation to be solved is the case of a weakly ionized, inhomogeneous plasma, where the density and temperature of the particles hardly change over the length of a mean free path (l dn/dx « n and l dT/dx « T), and where the electric field E@ in the plasma is sufficiently weak so that, first, the energy acquired by a particle in the field over the length of a mean free path is small compared to the thermal energy of the particle (eEl « kT), and second, the charged particles manage to effectively exchange the energy acquired by them from the field.*

If the particle density and temperature gradients and the potential gradients are weak, and if the energy acquired by charged particles from the field is dissipated rapidly, the particle velocity distribution function f(v@) does not differ greatly from the Maxwellian distribution function. This permits a solution of the problem by iterative approximations. To do this, let f(v@) be represented as a sum

where f⁽¹⁾(v@) is a minor non-equilibrium addition to the distribution function and f_M(v) is a Maxwellian distribution function:

Assume also that the addition f⁽¹⁾(v@) to the distribution function contains the electric field E@ and the terms in grad_r only to first order.

In zero approximation, i.e., disregarding small values, only the term (e/mc)(v@ x H@) grad_v f_M(v), which is equal to zero, remains on the left side of the kinetic equation (1.9), because grad_v f_M(v) = (v@ /v)df_M(v)/dv and v@ (v@ x H@) = 0. Substitution of f_M(v) into the right side of the kinetic equation also yields zero, because the collisions do not disrupt the Maxwellian distribution. @

*This condition is more strict for electrons, which lose a very insignificant fraction of their energy during collisions with heavy particles as a result of their small mass. In a weakly ionized TIC plasma, if the electron concentration is adequate (n ³ 10¹² cm^-3), the energy obtained by them from the field is dissipated primarily by electron-electron collisions. In this case, the electric field leads to a variation in the temperature of the electron gas, whereas the electron distribution function itself deviates comparatively little from the Maxwell distribution (see §6).

@ For the electron—atom collisions considered in this section, this condition is fulfilled, because the atoms are assumed to be at rest during collisions. If, however, the motion of the atoms were taken into account, the collision

Continuing the approximation, retaining terms of first order, we obtain an equation for calculating f⁽¹⁾(v@):*

By substituting f_M(v) into the left side of (2.2), equation (2.2) transforms to

where p = nkT is the pressure, n is the density of particles of a given specie, and w _c@ = eH@/mc is a vector, equal in modulus to the cyclotron frequency w _c.

To solve equation (2.3), it is necessary to have an explicit expression for the collision term I{ f⁽¹⁾(v@)}. In general, the collision term is in the form of an integral of the desired distribution function f⁽¹⁾(v@). Therefore, equation (2.3) is an integral equation. Only in some cases it can be solved exactly. Of these cases, those of greatest present interest are those where charged particles collide with each other only rarely (compared to collisions with constant scattering centers) and where scattering centers may be assumed immobile in view of their large mass. @ This is the case, in particular, for a weakly ionized plasma with low electron densities, where electron collisions with neutral atoms are the main scattering mechanism.

where N is the density of the scattering centers, s (v, q ) is the differential scattering cross section, v@' is the particle velocity after scattering, q is the scattering angle, and dW = 2p sinq dq is the element of solid angle. The collision term in (2.4) is represented in the form of the difference of two values.

The term vNf(v@ ) ò s (v, q ) dW is the decrease in the number of particles in a state with velocity v@ as a result of collisions, and in this case, the value of vNs (v, q ) is the particle scattering probability for deflection by angle q from the initial direction of motion (Fig. 4.1). Integration over dW denotes integration over all directions of electron velocity v@' after scattering.

term I_ea would be distinct from zero if the electron and atomic temperatures were different.

*Expression (2.1) represents the first two expansion terms in the Chapman-Enskog method [7]. Integral equations of the same form as (2.2) are obtained for expansion terms of higher order.

@ A gas of non-interacting particles, scattering from fixed centers, is usually called a Lorentz gas. Therefore, the approximation used is also called the Lorentz approximation.

The second term vNò f(v@')s (v,q ) dW corresponds to the increase of the number of particles with velocity v@' due to collisions. Integration over dW denotes integration over all velocities v@' prior to collision, while the value of vNs (v,q ) is the probability that the particle will move from a state with velocity v@' to a state with velocity v@. As can be seen from Fig. 4.l, the same scattering angle and, consequently, the same value of cross section @ corresponds to the transition from state s (v,q ).*

Here c @(v) is a vector dependent only on the velocity modulus v, which is calculated as a result of solving the kinetic equation, and q is the angle between the direction of particle velocity v@ and the direction of vector c @. The function f_M(v) is singled out on the right side of (2.5) for convenience in writing the subsequent expressions.

To calculate the integral in (2.6), we can use the polar coordinate system and direct the polar axis x along the particle velocity vector v@ before collision (Fig. 4.2). Then

By substituting (2.7) into (2.6) and taking into account the fact that the terms containing sinq and cosq do not contribute to the result during integration over angle q , we obtain

*The equivalence of the differential cross section s (v,q ) for direct and inverse transitions is a result of the principle of detailed balance (see §5, Chapter 3).

we substitute expression (2.8) for the collision term into equation (2.3). In this case, using (2.5) and the relation (v@ x w _c)gradvf(1)(v@) = - f_M(v@) v@ [c @ (v) x w _c ], which follows from (2.5) (see, for example, [5]), we find

where the + and - signs, as before, correspond to the positive and negative charges on the particle.

Formula (2.10) determines vector c @. In particular, it is obvious that vector c @ is dependent only on the velocity modulus v, which justifies the assumption made above.

The electron current density j_e@ is equal to the directional average of the electron velocity multiplied by the electron density and charge. The kinetic energy flux density S_e^k@ of the electrons is equal to the average energy transferred by one electron (averaged as in formula (1.2)) multiplied by the electron density.

Expressions for electron current and electron energy flux. Electron current j_e@ and kinetic energy flux S_e^k@, transferable by electrons, are calculated in the following manner using the electron velocity distribution function:

With this definition the forward direction of electric current is assumed to be in the direction of motion of the electrons, rather than that of the ions. In this case, j_e@ = +ei_e@ and j_i@ = - ei_i@, where i_e@ and i_i@ are the flux densities of the particles, electrons and ions respectively.

In this section we derive expressions for j_e@ and S_e^k@ in the absence of a magnetic field.

Calculating j_e@ and S_e^k@ only the non-equilibrium component to the distribution function f_e⁽¹⁾(v@), instead of f_e(v@), need be substituted into (3.1) and (3.2), because naturally there is no current or energy flux in the equilibrium state.

Since the non-equilibrium component to the distribution function is proportional, in this case, to both (en_eE@ + gradP_e) and n_ekgradT_e, there are two contributions to the current j_e@ that are proportional to these quantities. Denoting the proportionality constants by m _e and K_e^(T)m _e, respectively, we write the expression for the electric current transported by the electrons in the following form:

Coefficients m _e and K_e^(T) are called the mobility and the thermal-diffusion ratio, respectively. The value of s = em _en_e, as indicated above, is the electrical conductivity of the plasma.

When calculating the kinetic energy flux S_e^k@ it is convenient to separate terms proportional to electron flux j_e@/e and to the electron temperature gradient:

Here k _e is the thermal conductivity of an electron gas. Equations (3.3) and (3.4) are called the transport equations for electron current and electron kinetic energy flux, and the values of m _e, K_e^(T) and k _e are the kinetic coefficients. In the relaxation time approximation, the kinetic coefficients are calculated using the following expressions:

In formulas (3.5)—(3.7), t _p is the momentum relaxation time and the symbol < > denotes averaging over the Maxwellian distribution function, according to (1.2). Expressions (3.5) and (3.6) for mobility and the thermal-diffusion ratio are obtained directly by substituting (2.10) into (3.1) (with w _c@) and by integration by angular variables, i.e., over dW . To obtain expression (3.7) for the electron thermal conductivity, it is convenient to use the fact that the flux density of thermal conduction k _e grad T_e is the transport of energy by the particles in the absence of current, i.e., where j_e@ = 0. In this case, as follows from(3.3),

Having substituted this relation into (2.10), and the result into (3.2), and having performed integration over angles, we obtain (3.7). Typically the integrand in (3.7) is a function that changes sign: If j_e@ = 0, two counter particle-flows pass through the plasma which counteract each other. The kinetic energy flux S_e^k@ is non-zero in the presence of a temperature gradient, because the faster particles moving in the direction of lower temperature transfer greater kinetic

energy than slow particles moving in the other direction. The explicit expressions for the kinetic coefficients are determined by the dependence of the relaxation time t _p on velocity v.

Expressions for the kinetic coefficients and for the fluxes with various dependencies of relaxation time and of mean free path on energy. First, consider the case where the relaxation time t _p is not dependent on v. In this case,

At t _p = const, the electron distribution function f_e(v@ ) in the electric field is a Maxwellian distribution, shifted in velocity space an amount equal to the average drift velocity v_de@, the same for all electrons:

Actually, taking into account the low value for the drift velocity v_de@, from (3.9) we find that

By comparing (3.9a) to (2.1) and (2.10) (at w _c@ = 0 and gradT_e = 0) we find

i.e., v_de@ is actually not dependent on v. If there is a temperature gradient, the distribution function is no longer represented in the form of (3.9).

The value of thermal conduction k _e at t _p = const is obtained from (3.7), using the fact that in this case K_e^(T) = 0:

If the relaxation time t _p is dependent on v, the thermal-diffusion ratio is distinct from zero and the thermal-diffusion current

According to (3.6), the sign of the thermal-diffusion ratio and the direction of the thermal-diffusion current are determined by the nature of the dependence of t _p on v: K_e^(T) is positive if t _p is an increasing function of v, and negative in the opposite case.

When relaxation time is proportional to a power of v, i.e., when the mean free path is given by

(3.14)

where v_e@ = <v_e> = (8kT_e/p m)^1/2 is the average electron velocity of the Maxwellian distribution, r_e = 1/2 corresponds to constant collision time, and r_e = 0 corresponds to a constant mean free path. In the latter case

since G (2) = 1.

The approximation of a constant mean free path, in particular, corresponds to a model in which particle collisions are treated as the elastic impact of two hard spheres. In this case l_p = l = 1/NQ, where N is the density of the scattering particles and Q is the total elastic scattering cross section, equal to p (r_a + r_b)², where r_a and r_b are the radii of the spheres. This approximation is the one usually used to describe electron scattering from neutral Cs atoms.

The transport equations are often written in a somewhat different form: the gradient of pressure p = knT can be expressed in terms of density and electron temperature gradients, and the energy flux includes not only the flux of kinetic, but also that of the potential energy of the particles Vj_e@. In this case,

is the diffusion coefficient, and S_e@ = S_e^(k)@ + Vj_e@ is the total energy flux. The relationship (3.20) between the diffusion coefficient and mobility is called the Einstein relation. It is valid if the particle flux is

As indicated above, the Lorentz force F@ = ± e(v@ x H@ )/c acts on a charged particle moving in a magnetic field. Since F@ ~ (v@ x H@ ) the magnetic field affects particle motion only in the plane perpendicular to H@, not particle motion in a direction parallel to H@. With free motion in a plane perpendicular to the magnetic field, a charged particle would describe a circle with a radius r_c = cmv/eH@. If the particle has a velocity vector parallel to the magnetic field, this circle is transformed into a spiral wound on the magnetic lines of force (Fig. 43).

Collisions distort this spiral motion. With collisions, the nature of motion is determined by the relationship between r_c and the mean free path l_p, i.e., by the parameter l_p/r_c = w _{ct p} where w _c = eH/mc, already calculated above, is the cyclotron frequency (the particle rotation frequency along the circumference in the plane perpendicular to H@).

In a weak magnetic field, where w _{ct p} « 1, the particle trajectory is only slightly curved between collisions, because the particle manages to cover only a small fraction of the cyclotron circumference during time t _p (see Fig. 4.3a). Curving of the trajectory causes the particle to travel a shorter distance from the scattering center during the free path than if there were no magnetic field. As a result, the equivalent mean free path is decreased. Consequently, the transport coefficients in the plane perpendicular to the magnetic field also are decreased. In a strong magnetic field, where w _{ct p} >> 1 the particle manages to describe many turns of the spiral within time t _p. After each collision, the particle begins to describe a new spiral,

whose axis does not coincide with that of the preceding spiral (Fig. 4.3b).* Therefore, a particle in a strong magnetic field is shifted in a direction perpendicular to H@ only by collisions. Each collision is accompanied by a displacement of the particle a distance of the order of r_c (Fig. 4.3b). Since the extent of the magnetic effect is determined by the parameter w _{ct p}, which is (M/m)^1/2 time greater for electrons than for ions, the magnetic field acts on the electrons much more intensively than on ions.

Consider the effect of the magnetic field on the electron mobility and the diffusion coefficient for a weakly ionized plasma in the relaxation time approximation. Since the mobility and the diffusion coefficient are related by the Einstein relation, it is sufficient to consider the dependence of only one of these on the magnetic field, for example, the mobility m _e.

According to (2.10), the equation for calculation of c @ (v), in the relaxation time approximation, has the form

Multiplying (4.l) vectorially by w _c@ on the right side and using the property of the dual vector product, (A@ x (B@ x C@ )) = B@ (A@ · C@ ) - C@ (A@ · B@),

Upon vector multiplication of (4.1) by w _c@, we obtain

From (4.4), (2.5), and (3.1), we obtain the electron current j_e@:

*Note the characteristic feature of Coulomb collisions in a magnetic field. Since interaction during Coulomb collision occurs at a distance of the order of the Debye length L_D, the interaction time during collision is of the order of w _p^-1 (where w _p = Ö 4p nee2/m@ is the electron plasma frequency).

It is obvious from (4.5) that in a magnetic field the vectors j_e@ and - E@ cease to be parallel. Therefore, mobility m _{ea b} (H) is a tensor and has both diagonal and non-diagonal components in a magnetic field.

If the magnetic field is directed along the z-axis, then, if relaxation time t _p is not dependent on velocity, the current components perpendicular to the magnetic field are equal to

Here m _e = et _p/m_e is the electron mobility in the absence of a magnetic field (see (3.8)). The current component j_ex, as with H = 0, is expressed by the formula j_ex = - en_{em e}E_z. Therefore, for the mobility tensor components m _{ea b} (H), we obtain

If the electric and magnetic fields are mutually perpendicular, current j_e@ consists of two components: current components j_Ee@, directed along the vector -E@, and the so-called Hall current j_Eh@, directed perpendicular to both the electric and magnetic fields:

In crossed electric and magnetic fields, it is convenient to separate the electric field component E₀@ (parallel to current) from the so-called Hall field E_H@ (the field component perpendicular to both current and the magnetic field). Components E₀@ and E_H@ are calculated from (4.5). If the relaxation time is constant,

In this case, the average electron drift velocity in the direction of E₀@ is equal to v_de = -m _e E₀ and, consequently, is the same as in the absence of a magnetic field. Taking this relationship for E₀ and

If w _c » w _p, then the electron describes a circle between collisions whose radius is of the order of L_D. In this event, the magnetic field begins to affect the nature of the interaction between the particles. Consideration of this circumstance usually leads to a change in the maximum impact parameter for Coulomb collisions.

Fig. 4.4 explains the occurrence of the Hall field in a plasma placed between non-conducting electrodes in an external electric field E₀@. The Hall field occurs due to the accumulation of charged particles near the electrodes, which are shifted toward the electrodes during charged particle motion in crossed electric and magnetic fields. In this case the Hall field E_H@ exactly compensates the Lorentz force F@.*

In general, when t _p is dependent on v, the electron mobility in the plane perpendicular to H@, according to (4.5), is given by the expressions

Thus, it follows in particular, that if the length of the mean free path is constant (l_p = vt _p = const), then we have

where expression (3.17) is used for the electron mobility m _e with a constant mean free path. Function F(q )@ is shown graphically in Fig. 4.5.

In the presence of a magnetic field, the electron diffusion coefficient, like mobility, is a tensor. The diffusion tensor components D_{ea b} (H) are related to the mobility tensor components m _{ea b} (H)

*To be more general, if the relaxation time t _p is dependent on the electron velocity v, then the Hall field still compensates the Lorentz force-on the average-so that the total current in the direction of the Hall field is zero. In this case, the slow electron flow is directed in one direction, and fast electron flow is directed in another.

by the relation D_{ea b} (H) = (kT_e/e)m _{ea b} (H).

As the degree of plasma ionization increases, electron-ion and electron-electron collisions become important. Electron-ion collisions are an additional mechanism of electron scattering and, therefore, lead to a decrease in the electron kinetic coefficients. Electron-electron collisions play a different role. The total momentum and total energy of the electron system do not change as a result of electron-electron collisions. Nevertheless, electron-electron collisions do affect the magnitude of the electron current and energy flux, because the electron energy distribution depends on the extent of electron-electron collisions. Therefore, if the relaxation time t _p, which characterizes collisions of electrons with atoms and ions, is dependent on energy, the electron-electron collisions change the average value of relaxation time and, consequently, the value of mobility m _e.

However, electron-electron collisions have a stronger effect on the values of heat conduction k _e and the thermal diffusion ratio K_e^(T). Intensive electron-electron collisions lead to a Maxwellian distribution in the coordinate system moving with average drift velocity v_de@ of the electron flow (see (3.9)). For this distribution function, k _e and K_e^(T) are generally equal to zero.* Therefore, electron-electron collisions sharply reduce the electron heat conduction and thermal-diffusion ratio.

To see the effect of electron-electron collisions on the kinetic coefficients, consider the values of the mobility m _e, the thermal-diffusion ratio K_e^(T), and the thermal conductivity k _e for a completely ionized plasma in the absence of a magnetic field:

*This can be shown by substituting (3.9) into (3.2) and by comparing @ the result with (3.14) where j_e@ = env_de@ and v_de@ << Ö kT_e/m@.

The values of m _e, K_e^(T) and k _e are written so that the coefficients g _u, g _k, and g k , which are equal to unity without electron-electron collisions, are a measure of this effect. In the latter case, the kinetic coefficients for electrons in a completely ionized plasma are calculated by formulas (3.13)—(3.16), if the mean free path l_p(E) = 1/nQp(E) from (3.2.10) is substituted into them. When taking into account electron—electron collisions this way, we obtain g m = 0.582, g _k = 0.469, and g k = 0.235 [6]. Therefore it is clear that electron-electron collisions have an important affect on the values of the kinetic coefficients.

The first two terms, which describe electron scattering from atoms and ions, were written in the relaxation time approximation (t _p^ea and t _p^ei are the electron relaxation times with atoms and ions). The last term, which corresponds to electron-electron collisions, requires special consideration.

The characteristic feature of electron—electron collisions is that its analysis requires the motion of both particles to be taken into account. In general, the change of the distribution function for particles a due to collisions with particles b is given by the expression

Here v_a@ and v_b@ and v_a ^'@ and v_b '@ are the velocities of interacting particles before and after collision, respectively; g@ = v_a@ - v_b@ is the relative particle velocity, s (g, q ₀) is the scattering cross section (defined as a function of relative velocity g@ and angle q ₀ of the vector g@ as a result of the collision), and dW ₀ = sinq ₀dq ₀df is an element of solid angle about the vector g@ after collision. We note that the velocities v_a ^'@ and v_b '@ in (5.3) are dependent on v_a@, v_b@ and q ₀.

Expression (5.3) can be obtained in the following manner. Consider the number of collisions between particles a in an element of volume d³ v_a with particles b in an element of volume d³ v_b, as a result of which vector g is changed into the element of solid and dW ₀. The number of these collisions is equal to

After collision, the particles considered are in the element of volume d³ v_a' d³ v_b' and have a relative velocity of g'@ = v'@ - v'. The number of reverse collisions is equal to

Taking into account relations d³ v_a d³ v_b = d³ v_a' d³ v_b', g = g', q ₀ = q ₀', dW ₀ = dW ₀' (see, for example [7]), and integrating over all scattering angles, (i.e., over all impact parameters) and over all particle b velocities, we can calculate the net change in the number of particles in an element of volume d³ v_a:

Dividing the result by d³ v_a, we obtain the expression for the change in the number of particles a per unit volume of velocity space, i.e., the expression for the collision terms I_ab in (5.3). Expression (2.4), which we used previously, is a special case of (5.3). Actually, if particles b can be assumed to be fixed, then g@ = v_a@ and v_b@ = v_b'@. In this case, by integrating over d³v_b in (5.3) and by denoting ò f_b(v_b@ )d³v_b = N, we obtain (2.4).

We note that if the Maxwellian distribution functions f_aM(v_a) and f_aM(v_b@ ) with identical temperature (T_a = T_b) are substituted into I_ab{f_a(v_a@ ), f_b(v_b@ )}, the collision term naturally goes to zero, because m_a v_a² + m_b v_b² = m_a v_a' ² + m_b v_b' ², i.e., the collisions do not change the equilibrium energy distribution of the particles.

If we write the distribution function f_e(v@ ) in the form of (2.1) and (2.5) and if we disregard the product of small values in the integrand, the expression for the collision term that describes electron-electron collisions assumes the form

Here q ' and q _b' are the angles which indicate the directions of the velocity vectors v'@ and v_b'@ after collision.

We note that not only the direction but the value of the velocity v changes during electron-electron collisions, which does not permit the use of the relaxation time approximation of §3. To find the distribution function in this case, it is necessary to solve the integral equation (2.3) with the right side of (5.2). Some methods for calculating the kinetic coefficients are considered below.

The method of expansion by Sonine polynomials. To solve the integral equation (2.3), a method of solving the integral equation was employed in [7] that includes expansion of the desired function c (v) into a series by Sonine polynomials S_3/2 ⁽ⁿ⁾(x):

where x = m_ev²/2kT_e. Sonine polynomials are orthogonal over the range of values of x from 0 to ¥ with a weighting function x^3/2exp(-x)

The zeroth and first Sonine polynomials are given by S₃⁽⁰⁾_'2(x) = 1 and S₃⁽¹⁾_'2(x) = 5/1 - x. Therefore, if the scattering mechanism leads to a relaxation time t _p not dependent on energy, c (v) is expressed only by the zeroth and first Sonine polynomials (see formula (2.10)).

In the general case, if the scattering mechanism is arbitrary, the desired function c (v) is expanded into the series (5.6), but terminated in the term n = N. The expansion is substituted into equation (2.3) to obtain the coefficients a p @. For this, both sides of equation (2.3) are multiplied by S₃⁽¹⁾_'2(x) (n = 0, 1, 2, ..., N) and are integrated over x. As a result, a system of N + 1 linear equations is obtained with respect to the expansion coefficients a p @, and the left hand sides of these equations, because of (5.7), include only coefficients a ₀ and a ₁.

Thus, the problem of calculating the kinetic coefficients reduces to the solution of a system of equations for the values of a _n, where, as follows from (2.1), (2.5), (3.1), (3.2), (5.6), and (5.7), the kinetic coefficients in the expressions for j_e and S_e are dependent only on the expansion coefficients a ₀ and a ₁. The accuracy of finding the non-equilibrium distribution function and, consequently, the accuracy of calculating the kinetic coefficients is determined by the number of polynomials used in the expansion.*

The exact solution of the kinetic equation for the case where the relaxation time approximation is applicable (i.e., in the absence of electron-electron collisions) was compared [8] with the approximate solution of the same equation (obtained by the expansion of c (v) in Sonine polynomials) to evaluate the accuracy of this expansion method. The comparison showed that, in the case of electron-atom and electron-ion collisions (t _p^ea ~ E^1/2 and t _p^ei ~ E^1/2), use of the first two Sonine polynomials (N = 1) for calculation of mobility yields an error of only several percent (see curves 1 and 2 in Fig. 4.6)@. A much worse result is obtained, however, when the relaxation time decreases rapidly as the electron energy E increases (see curves 3 and 4 and II and III in Fig. 4.6).@ In this case c decreases rapidly as E increases, and c (E) is difficult to approximate by Sonine polynomials -- which are increasing functions of E.

The electrical conductivity of a completely ionized plasma in a magnetic field was calculated in [9] using an expansion by Sonine polynomials, where four polynomials (N = 3) were used for the expansion of the distribution function. Results from the calculation of the electron kinetic coefficients for a completely ionized plasma in a magnetic field, using a large number of polynomials (N = 5), are presented in [10]. The electron kinetic coefficients for a partially ionized plasma were calculated in [11, 12].

*To calculate k _e and K_e^(T), it is necessary to take into account a greater number of expansion terms to achieve the same accuracy in view of the sign-changing of function c (v).

@The poor convergence of the approximations for electron scattering from Argon atoms is explained by the presence of a strong Ramsauer minimum in the electron scattering cross section at low energies, for this introduces a rapid decrease of t _p with increase of E.

In order to write the collision term (5.3) in Fokker-Planck approximation, consider the variation < D F(v_a@ ) >_ab of the arbitrary function F(v_a@ ), averaged by the particle velocities a, as a result of collision with particles b. This variation is equal to

Here F(v_a@ ) and F(v_a'@ ) are the values of function F before and after collision, and the integration is carried out for the various types of collisions.

Since the collisions involve only slight changes of velocity, the function F(v_a@ ) can be expanded as a Taylor series

where D v_a@ = v_a'@ - v_a@ is the change of velocity during collision, while the subscripts a and b denote the projections of the velocity to the corresponding coordinate axes. Substituting expansion (5.9) into (5.8) and integrating over d³v_a by parts, we obtain

The variation < D F(v_a@ ) >_ab resulting from collisions of particles a with particles b can also be written in a different form:

where I_ab is the collision term of the kinetic equation, which describes the change of the distribution function f_a(v_a@ ) of particles a as a result of collision with particles b. Since (5.10) and (5.13) in the present approximation should be equal regardless of the specific form of function F(v_a), the integrands in (5.10) and (5.13) should also be equal, i.e.,

When using the Fokker-Planck approximation, we proceeded on the basis of collision term (5.3), which takes into account only binary collisions. However, the range of applicability of equation (5.14) is actually much broader. Formula (5.14) gives the variation of the distribution function in general, if this variation is caused by many small deflections. Expression (5.14), therefore, is valid for multiple Coulomb collisions and for describing the interaction of an electron with wave fields (if the change of momentum and energy during absorption and emission of field quanta are sufficiently small). However, in these cases, the values of {D v_aa }_ab and {D v_{aa D} v_ab }_ab should be calculated for the specific type of electron interaction with the medium.

We note that, in the Fokker-Planck approximation, the effect of small angle, multiple Coulomb collisions on the distribution function is similar to the effect of external fields, because in both cases the particle velocities vary in a continuous manner. This observation shows how a description of these collisions can be included in the kinetic equation (1.9), since the flow i_vc@ due to the effects of multiple Coulomb collisions can be added to the particle flow in velocity space, i_v = f(v)F@/m, that results from an external force F@. In this way the collision term can be written as

The particle flow i_vc@ is represented in (5.16) in the form of the sum of two terms. The term proportional to f_a(v_a@ ), by analogy with particle flow i_v@, is denoted as f_a(v_a@ )F_c@ /m_a. The value

is called the dynamical friction force. The last term in (5.16), which contains the derivatives of the velocity distribution function, is the diffusion flow in velocity space. The tensor D_{va b} is called the diffusion-in-velocity coefficient.

With binary Coulomb collisions, the values of < D v_aa >_ab and < D v_{aa D} v_ab >_ab are expressed in the following manner:

where m_ab = m_a m_b/( m_a + m_b ) is the reduced mass, m_a and m_b are the masses of colliding particles, Q_p(g) is the momentum transfer cross section,* and d a b is the Kronecker delta (d a b = 1 for a = b and d a b = 0 for a ¹ b ).

To derive (5.19), we substitute v'@ - v@ = D v@ = (m_ab/m_a)D g@ into (5.11) (see (3.1.8)) and obtain

Formula (5.21) expresses the change of the velocity, < D v_a@ >_ab of particles a in collisions with particles b, due to variation in the relative velocity, D g@. Upon integration over W ₀, only the vector component D g@ parallel to the initial direction g@, equal to g@ (1 - cosq ₀ ), makes a contribution to the integral distinct from zero. Taking this into account and using the definition of (3.1.15), we obtain (5.19). Similar but more cumbersome calculations (see [2]) lead to expression (5.20).

The Fokker-Planck approximation for binary Coulomb collisions was used in [6, 13] when calculating the electron kinetic coefficients for a completely ionized plasma. We note that although the Fokker-Planck approximation also simplifies the analysis of electron-electron collisions, nevertheless, the unknown function is still retained under the integral, because the values of < D va >_ee and < D va D vb >_ee are integrally dependent on the electron distribution function and, in particular, on the unknown component to the distribution function f_e⁽¹⁾(v@ ). Therefore, the solution of the integral-differential equation obtained in the Fokker-Planck approximation also requires numerical calculations, Sonine polynomial expansions being used in a number of cases (see, for example, [14]).

*Cross section Q_p(g) is obtained from (3.2.10), if the particle velocity v and mass m in that equation are replaced by the relative velocity g and the reduced mass m_ab.

where x = E /kT_e = mv²/2kT_e and F(x) is some function (dependent on t _p(x), w _{ct p}(x), and the type of kinetic coefficient being considered).

where x_k^(N) are the nodes of the quadrature formula and A_R@^(N) are the weight factors. The values of the nodes and weights are usually selected so that (5.23) yields a precise result if F(x) is a polynomial. Therefore, the formula with N nodes is equivalent to an approximation of the integrand with a polynomial of power 2N - 1. The accuracy of the quadrature formula (5.23) is high even for small values of N.

It is obvious from (5.23) that the non-equilibrium distribution function need be calculated only at some (generally speaking, small) number of nodal points x_k in order to determine the kinetic coefficients. This advantage can be used in solving the kinetic equation when the relaxation time approximation is inapplicable, for example, when self-scattering of free electrons is important.

In the Fokker-Planck approximation, the collision term for electron-electron collisions has the form [6]

where lnL is the Coulomb logarithm,

As can be seen from (5.24), the collision term in the Fokker-Planck form contains the derivatives and integrals of the desired function D₀(x). They can be expressed from values of D₀(x) at nodal points by interpolating D₀(x) in some manner, for example, by a Lagrangian interpolation. The

brace in (5.24), for this case, has the form at the k-th nodal point

where m _ks^(N) forms a general square matrix of rank N.

The matrix coefficients m _ks^(N) with the number of nodal points N = 2, 3, 4, 5, and 6 were calculated in [15]. It was found, for a completely ionized plasma (1/t _p^ea = 0), that the approximation with three nodal points leads to values of the kinetic coefficients which coincide with those obtained by a precise solution of the integral-differential equation. The latter solutions were presented above (see (5.1)).

One of the advantages of the nodal point method is the increased accuracy in the presence of additional scattering mechanisms, electrons with heavy particles, whereas the term in (5.2) proportional to 1/t _p^ea can be the main source of error in a solution of the integral equation by expansion in Sonine polynomials.

where m _ea, k _ea, m _ei, and k _ei are values of kinetic coefficients in the limiting cases of a weakly ionized and of a completely ionized plasma.

The correction coefficients fm _e, fk _e and K_e^(T), calculated by the nodal point method, for cases where t _p^ea = const and l_p = 1/NQ_p^ea = const, are depicted in Fig. 4.7 as a function of the ratio of the effective electron collision frequencies v_ea and v_ei with atoms and ions respectively. The effective collision frequencies v_ea and v_ei are essentially averaged inverse relaxation times and are calculated by the formulas

Here m _ea and m _ei@ are the electron mobilities for electron scattering by atoms and ions, obtained by disregarding electron-electron collisions. The values of m _ea and m _ei@ are calculated by formulas (3.14) and (5.1), respectively, where _{g u} = 1.

The values of m _ea, K_e^(T), and v_ea for the limiting case of a weakly

ionized, plasma are calculated by formulas (3.l4)-(3.l6). The values m _ei, K_ei^(T) and k _ei for a completely ionized plasma are calculated by expressions (5.1).

fm _e and fk _e from unity is only » 25% at intermediate values of the ratio v_ei/v_ea. With regard to K_e^(T), the dependence on v_ei/v_ea is very smooth, which allows an approximation of the results by simple formulas. We note also that, because of the large value of the Coulomb cross section, the transition from dominance by the one scattering mechanism to dominance by the other occurs in a weakly ionized plasma, that is, when n_e « N_a.

Electrons lose their directed momentum and energy by collisions with heavy particles (atoms and ions). The rates of momentum and energy losses are determined by the relaxation times t _p and t _E, respectively. With elastic collisions of electrons and heavy particles, t _E/t _p ~ M/m, where M is the mass of the heavy particle. As a result, the electrons in the plasma lose their directed velocity quite rapidly but lose their energy rather slowly. This leads to the fact that the electron distribution function has a low anisotropy in velocity space, whereas the electron energy distribution in a number of cases may differ considerably from the equilibrium distribution.

In a TIC plasma, it is possible to establish a Maxwellian energy distribution as a result of the intensive electron-electron collisions among the bulk of the electrons. However, the average energy and temperature T_e of the electrons usually differs appreciably from the average energy and temperature T of the heavy particles, because of the weak exchange of energy between the electrons and heavy particles. We now turn to the calculation of the momentum and energy imparted by the electrons to the heavy particles during collisions.

are acted on by the force R_e@, equal to

The values of R_ea@ < D p_e@ >_ea and R_ei@ < D p_e@ >_ei are the force components of R_e@, determined by the change of the electron momentum during collisions with atoms and with ions. They are a function of the momentum relaxation times t _p^ea and t _p^ei, respectively.

When the Maxwellian distribution function f_eM(v) is substituted into (6.1), the force R_e@ goes to zero. Therefore, only the non-equilibrium component f_e⁽¹⁾(v@ ) to the distribution function should be substituted into the integrals of (6.1)

It is possible to show that, with low electron drift velocity (v_de « v_e), and in the absence of a magnetic field, the effect of force R_e is equalized by the force - en_eE@ - gradp_e associated with the electric field and the pressure gradient, i.e.,

By comparing (3.3) and (6.2), we find the following expression for force R_e@:*

It is obvious that force R_e consists of two components: the friction force R_e^{(m )}@ = - j_e@ /m _e, caused by a relative velocity between the plasma components, and the so-called thermal force R_e^(T)@ = -K_e^(T)n_ekgradT_e, generated by the presence of the electron temperature gradient. If the relaxation time t _p is not dependent on energy, then the thermal-diffusion ratio K_e^(T), and therefore, the thermal force, approaches zero. In this case, the momentum transport of the electrons due to the heavy component is proportional to current j_e@, which can also be seen directly on the basis of expression (6.1).

The occurrence of the thermal force is related to the dependence of relaxation time t _p on electron energy E. As already indicated in §3, at j_e@ = 0 and in the presence of an electron temperature gradient, two mutually compensating counter flows of electrons pass through the plasma. If t _p increases as E increases (K_e^(T) > 0), the flow of slower electrons moving in the direction of the temperature increase loses greater momentum during collisions than a flow of faster electrons moving in the opposite direction. Therefore, the resulting force acting on the electrons is directed in the direction opposite to the flow of slow electrons, i.e., in the direction of the decrease of electron temperature. On the other hand, if t _p decreases as E increases (K_e^(T) < 0), the thermal force is directed toward the increase of electron temperature.

*Expression (6.3) can be obtained in a more accurate way if both sides of (2.3) at w _c = 0 are multiplied by m_ev@ and integrated over velocities. In this case, the right side of (2.3) should be represented in the form (5.2) (where ò I_ee{f_e(v@ )}mv@ d³v = 0, because electron-electron collisions do not alter the total momentum of the electron subsystem), and - en_eE@ - gradp_e in the left side of (2.3) must be expressed in j_e@ and grad T_e by using (3.3).

In general, the energy transferred from particles a (electrons) to particles b (ions or atoms) during collisions can be expressed as

The value of D S_ab is assumed to be positive if energy passes from particles a to particles b.

If the heavy particles have a Maxwellian distribution, only the spherically symmetrical part of the electron distribution function f_a⁽⁰⁾(v_a) contributes to the integral (6.4). Therefore, the Fokker-Planck equation for a spherically symmetric electron distribution function must be formulated in order to calculate D S_ab In this case, the particle flow i_vc@ in velocity space is directed along the electron velocity v_a@.

To obtain i_vc@, consider first electron-ion collisions. In calculating the dynamic friction force F_c@ by formulas (5.18)-(5.20), disregard for now the ion velocities v_b@ compared to v_a@ in (5.19) and (5.20). Then, since we assume g = v_a, we obtain

where N_b = ò f_b(v_b@ )d³v_b is the ion density. By substituting (6.5) and (6.6) into (5.18), and by using the definition of the reduced mass (3.1.4), we obtain

The diffusion tensor D_{va b} , given by formula (5.17), can be calculated in the coordinate system with the z-axis parallel to v_a@. Tensor D_{va b} is diagonal if the particle distribution over velocities in this coordinate system is spherically symmetric. By denoting the tensor component D_vzz by D_{v| |} , we obtain from (5.16)

In the approximation used here (disregarding the velocities v_b of slow particles), the tensor component D_{v| |} , as can be seen from (5.17) and (5.20), is equal to zero.* Therefore, when calculating D_{v| |} , it is necessary to retain the terms of order v_b/v_a in (5.20). However, if the

*This result is related to the fact that electrons do not acquire energy during scattering from fixed centers. They only lose energy-this process is described by the dynamical frictional force. Diffusion in velocity space occurs only in the direction perpendicular to v_a@, The corresponding diagonal components D_vxx = D_vyy are distinct from zero.

particles b have a Maxwellian distribution (characterized by temperature T_b), the diffusion coefficient D_v (we shall now omit the subscript | | ), can be found by a simpler method, namely from the condition that i_vc@ = 0 when particles a have a Maxwellian distribution with temperature T_a = T_b. In this case, thermodynamic equilibrium occurs between particles a and b. Then, by assuming f_d⁽⁰⁾(v_a) = f_M(v_a) and i_vc = 0, we obtain

Expression (6.10) and (6.11) can be used not only for electron-ion collisions, but also for electron—atom collisions, if Q_p(v_a) is understood as the momentum transfer cross section corresponding to these collisions. Note that only the condition v_a » v_b was used when deriving expression (6.10). The condition of the smallness of the mass m_a compared to mass m_b was not used. Therefore, these derived expressions can also be used when considering fast electron collisions with slow electrons with a Maxwellian distribution, entering into these expressions m_a = m_b = m and T_b = T_e (see §5, Chapter 5).

To calculate the energy lost by the electrons during collisions with heavy particles, substitute (6.10) and (6.11) into (6.4). Assuming that f_a(v_a) = f_e⁽⁰⁾(v), m_a = m, T_b = T (the temperature of the heavy component of the plasma), and d³ v = 4p v²dv, we obtain

where the subscript b corresponds to atoms or ions.

If the electrons have a Maxwellian distribution and f_e⁽⁰⁾(v) = f_eM(v), then, by integrating (6.12) by parts, we obtain

where the symbol <...> denotes averaging over the distribution function, and n_e = _{0ò ¥ 4 p} v²f_eM(v)dv is the electron density. The energy relaxation time t _E in (6.13) can be expressed in terms of the momentum relaxation time t _p using the following relation which results from (3.1.13) and (3.1.14)

By disregarding the electron mass m compared to the atomic mass, we obtain the following expression from (6.13) and (6.14) for electron-atom collisions:

where M_a = m_b is the atomic mass.

The effective energy exchange time t _Q^ea is so determined that if the relaxation time t _p is not dependent on electron energy, then t _Q^ea » t _p. If t _p and the mean free path (l_p = vt _p) are proportional to the r-th power of energy (3.13), then t _Q^ea = 3l₀/2v_e@ G (3 - r). In particular, if the mean free path is constant (r = 0), t _Q^ea = 3l₀/4v_e@.

The result for electron-ion collisions can be described in the same way. For this case, from (6.13), (6.14), (3.1.13), and (3.2.10), disregarding m compared to the ion mass M_i, we obtain

The variable n_i is the ion density and M_i is the ion mass (n_i = N_b,

M_I = m_b).

As can be seen from (6.15) and (6.16), the relaxation time of the electron temperature T_e to the temperature T of the heavy particles is of the order (M/m)t _p, i.e., much greater than the momentum relaxation time t _p. An exception to this occurs for molecular gases in which vibrational and rotational levels can be excited. The relaxation time of the electron temperature in these gases is several orders less.

The electron distribution function in a strong electric field. In the above discussion of kinetic phenomena, the electric field and density gradient were assumed to be everywhere small. Under these conditions the electron distribution function hardly differed from the Maxwellian distribution. Consider now the case where the electron distribution function differs strongly from the Maxwellian distribution. This may occur in the presence of strong electric fields or of large density gradients. Large density gradients are considered primarily in the discussion of electrode sheaths (see §6, Chapter 6). A homogeneous gas located in a strong electric field will be considered in this section.

It is interesting to note that even in a strong electric field the electron distribution function is characterized by weak anisotropy in velocity space. The energy acquired by the electron from the field per unit time is D E_E = v_deeE » (et _pE/m)eE, and the energy imparted to the heavy component of the plasma during collisions is D E_c » (mv²/t _p)m/M. In the steady state we have D E_E = D E_c and accordingly, v » (et _pE/m)Ö M/m@

» v_deÖ M/m@ >> v_de. * Thus, the directed electron velocity v_de in a homogeneous plasma should be considerably less than the random velocity v, even in a strong electric field.

Weak anisotropy of the electron distribution function f_e(v@ ) in a strong field leads to the fact that f_e(v@ ) may be represented in the form

where f_e⁽⁰⁾(v) is the isotropic part of the distribution function and f_e1(v) cosq is the small anisotropic component, which (as in the case of a weak electric field) is represented in the form of the product of cosq and the function f_e1. The latter is dependent only on the velocity modulus (compare (2.1)). The difference from the case of a weak field is in the fact that f_e⁽⁰⁾(v) now no longer coincides with the Maxwellian distribution function f_eM(v), so that, along with the equation for calculation of f_e1(v), it is necessary to formulate the equation to find f_e⁽⁰⁾(v).

The equations for the calculation of f_e⁽⁰⁾(v) and f_e1(v) can be obtained by the following method [16]. First we integrate the kinetic equation (1.9) term wise over dW . Using (6.18) and the relation

and having divided the result obtained above by 4p , we obtain

where I₀ = 1/4p ò I{ f_e(v@ )}dW . In like manner, having multiplied both sides of (1.9) by cosq , having integrated over W , and having divided the result by 4p /3, we obtain

where I₁ = 3/4p ò I{ f_e(v@ )}cosq dW .

The electron distribution function, for a homogenous plasma located in a steady electric field of arbitrary strength, @ is calculated from equations (6.19) and (6.20). The values of I₀ and I₁ characterize the rate of change due to collisions of the isotropic and anisotropy parts, respectively, of the distribution function. To an order of magnitude, I₀ » f_e⁽⁰⁾(v)/t _E and I₁ » f_e1(v)/t _p, i.e., the relaxation rate

*This result was obtained with the assumption that the mean electron energy greatly exceeds the mean energy of the heavy particles kT. If this is not true and v » Ö kT/m@, then the average energy transferable during each collision decreases compared to the value (mv²/t _p)m/M, and accordingly, v may exceed (et _pM/m)Ö M/m@.

@Equations (6.19) and (6.20) are easily generalized for the case of an inhomogeneous and transient plasma. However, in this case, variation of the distribution function in time and space should be sufficiently slow [17].

of the isotropic part of the distribution function to the Maxwellian function f_eM(v) is determined by the energy relaxation time t _E while the relaxation rate of the anisotropy part of the distribution function to a spherically symmetric distribution is determined by the momentum relaxation time t _p.

Consider now in which cases the application of an electric field can significantly distort the isotropic part of the distribution function f_e⁽⁰⁾(v) from the Maxwellian function. Having calculated f_e1(v) from (6.20) and having substituted it into (6.19), we find that the ratio of the left side of (6.19) to the right side (to an order of magnitude) is determined by the parameter b = e²E^2t _pt E/m²v².

If b « 1, then the left side of (6.19) can be set equal to zero in the first approximation. Then f_e⁽⁰⁾(v) is calculated from the equation I₀ {f_e⁽⁰⁾(v)} = 0. This equation is satisfied if f_e⁽⁰⁾(v) is a Maxwellian distribution function with a temperature equal to the temperature T of the heavy particles. In this case, we have mv² » kT and b » e²E^2t _pt E/mkT. Thus, f_e⁽⁰⁾(v) » f_eM(v) and T_e » T, if e²E^2t _pt E/mkT « 1. This is for the case of a weak electric field.

Consider the opposite case, of a strong field, where e²E^2t _pt E/mkT » 1. In this case, the average electron energy considerably exceeds kT, and as was indicated above, v » (et _pE/m)Ö M/m@, since b » (t _E/t _p)m/M. If the plasma is very weakly ionized, so that not only the momentum relaxation but the energy relaxation as well are provided by electron-atom collisions, then t _E/t _p » M/m and b » 1. In this case, the isotropic part of the distribution function may differ strongly from a Maxwellian distribution.

A different situation arises if the degree of ionization of the plasma increases, so that energy relaxation is provided by electron-electron collisions, i.e., when t _E^ee « t _E^ea » (M/m) t _p^ea. Under these conditions, b » (t _E^ee/t _E^ea)m/M « 1. The condition b « 1 is fulfilled to an even greater extent if the degree of ionization of the plasma is so great that momentum relaxation is provided by electron-ion rather than by electron-atom collisions. In this case t _E^ee » t _p^ei, and therefore, b » m/M. Under these conditions, the electron distribution function relaxes rapidly to a Maxwellian distribution as a result of electron-electron collisions. An increase of field intensity now leads to an increase of electron temperature T_e compared to the heavy particle temperature T,* rather than to deviation of the isotropic part of the distribution function from a Maxwellian distribution.

Consider now in more detail the case of a weakly ionized plasma, where the energy relaxation is provided by electron-atom collisions. In this case, I₀ is expressed by formulas (6.10) and (6.11), in which it is necessary to set t _E^ab(v) = t _E^ea(v) and I₁ = - f_e1(v)/t _p^ea(v). Having obtained f_e1(v) from (6.20) and having substituted it into (6.19), we find the equation for the distribution f_e⁽⁰⁾(v):

*The electron temperature is calculated from the condition D S_eb = j_eE(b = a, where D S_eb is the energy transferred from the electrons to the heavy particles (see (6.15) and (6.16)), and j_e is the electron current.

The expression in brackets in (6.21) is the total particle flow i_ov in velocity space (with the opposite sign): the particle flow due to the effects of the electric field is added to the particle flow due to the effects of collisions i_vc. The essence of formula (6.21) is that the divergence of the total particle flow in velocity space is zero. Integrating (6.21), we obtain v²i_ov = const. Assuming in this last expression that v = 0, we find that the integration constant is zero. But since the expression is true in general, we find that the flow itself, i_ov, is equal to zero.

The integration constant C is calculated by the normalization condition (1.1) and is therefore a function of the electron density n_e. In a weak electric field, by assuming E = 0 in (6.22), we obtain the Maxwellian distribution in which the electron temperature is equal to the atom temperature T. In a strong electric field, on the other hand, @ can be disregarded in the exponent. And the form of the distribution function f_e⁽⁰⁾(v) is determined by the nature of the function t _p^ea(v).

By assuming a constant mean free path (l = vt _p^ea(v) = const), and

by assuming t _E^ea(v) = (M/2m) t _p^ea(v), we obtain from (6.22) the well known Druyvesteyn distribution [18]

For f_e1(v), we obtain from

and calculating current j_e according to (3.1), we obtain the electron mobility

Expressions (6.25) and (6.26) are for strong fields, where e²E^2t _pt E/mkT » 1.

force R_ie@ with which the electron flow acts on the ions as a result of electron-ion collisions. The effect of force R_ie@ on the ions is similar to the effect of external forces. Therefore, force R_ie@ is added to force en_iE@, with which the electric field acts on the ions. It is obvious that R_ie@ = -R_ei@. The force R_ei@ can be accounted for in the same way that Coulomb collisions are treated in the calculation of m _e and K_e^(T).

The value of R_ei@ can be calculated easily if the degree of ionization of the plasma is sufficient that Coulomb collisions predominate over electron-atom collisions (u _ei@ » u _ea@). In this case, R_ei@ is calculated by formula (6.3), into which the values of mobility Pe and of the thermal-diffusion ratio K_e^(T) that correspond to a completely ionized plasma (calculated by formulas (5.1)) must be substituted.

If the scattering mechanism is mixed, when u _ei@ » u _ea@, the force R_ie@ may no longer be expressed so simply by the electron kinetic coefficients. In this case, R_ei@ may be written in the form

where g₁ and g₂ are the coefficients which are dependent on the ratio u _ei@/u _ea@. Expression (7.1) follows from the fact that the non-equilibrium component f_e⁽¹⁾(v) in the electron distribution function is proportional to the gradients p_e and T_e and to the electric field E@.

The dependence of g₁ and g₂ on the ratio of the effective collision frequencies u _ei@/u _ea@ is shown in Fig. 4.8 [12]. In the limiting case of a weakly ionized plasma, when u _ei@/u _ea@ ® 0, force R_ei@ does not contribute to R_e and the coefficients g₁ and g₂ approach zero. In the opposite case, when u _ei@/u _ea@ ® ¥ , R_e@ = R_ei@. Then, by comparing (3.3) and (6.3), we find g₁ = 1 and g₂ = 0.

The quasi-neutrality of the plasma is taken into account in (7.3), and the charged particle density is denoted by n(n_e = n_i = n).

We note that the thermal-diffusion ratio K_i^(T) for ions is small (see [12]), where K_i^(T) ® 0 as the frequency of ion-ion collisions increases, due to the fact that during intensive ion-ion collisions the ion distribution function begins to approach a Maxwellian distribution f_iM(ï v@ - v_di@ ï ) shifted by the value v_di@ = - j_i@ /en_i of the ion drift velocity (see page 95).

Expressions (7.2) and (7.3) are appropriate for weak gradients of potential, density, and temperature, when the ion distribution function does not differ significantly from the Maxwellian distribution function (see §2). However, ion mobility in moderate and strong electric fields will also be of subsequent interest (see §2, Chapter 6).

Ion mobility in a foreign gas. Ion interaction with foreign neutral molecules reduces primarily to the polarization of these molecules and to the attraction of the ion by the polarized molecules. The force of ion attraction to the polarized molecule is equal to [19]

where a is the polarizability of the molecule and r is the distance between the molecule and the ion. The interaction (7.4), for which the force is inversely proportional to the fifth power of the distance, has the remarkable property that the product of the relative velocity g and cross section s (g, q ₀) is not dependent on g. Because of this, the momentum transferred to the molecules during ion collisions, < D p_i@ > _im, can be calculated without finding the ion distribution function.

We may use expression (5.8) to calculate < D p_i@ > _im, and to do so, we turn to the center of mass system for the ion and molecule. The change of ion momentum during collision with a molecule is then expressed as D p_i@ = p_I'@ - p_im@ = mi(g'@ - g@), where m_im = M_iM_m/(M_I + M_m) is the reduced mass, M_i is the ion mass and M_m is the mass of the molecule (see (3.1.9)). Substituting D p_i@ into (5.8), we find

where f_m(v_m@ ) and f_i(v@ ) are the molecule and ion velocity distribution

functions, respectively, and R_im@ is the frictional force acting on the ions as a result of collisions with the molecules.

If the value of 1/t _p^im = gN_mò (1 - cosq ₀)s (g, q ₀)dW ₀ is not dependent on the relative velocity of the colliding particles, then, by substituting g@ = v@ - v_m@ into (7.5), we obtain

where N_m = ò f_m(v_m@ )d³v_m is the molecular density. Performing the integration over ion and molecule velocities in (7.6), we find

where v_di@ and v_dm@ are the average drift velocities of the ions and molecules, respectively. Since the frictional force R_im@ in a homogenous electric field in steady state is equalized by the force of the electric field en_iE@, it follows from (7.7) that the relative ion velocity is equal to

We note that the area of validity of expressions (3.8) and (7.9) is broader than that of the remaining results obtained in §§2 and 3 of this chapter or in equation (7.3): expressions (3.8) and (7.9) are valid for mobility (given a constant relaxation time) not just for a weak electric field but for an arbitrary electric field, because formula (7.8), as can be seen from its derivation, is valid for any form of the velocity distribution function.

Formula (7.4) describes the ion interaction with atoms and molecules at large distances. At small distances, when the exchange forces become important, it is no longer valid. For ion collision with atoms and molecules, one can consider an approximate model of elastic spheres. In this model, ion mobility is given by the following expression [7]:

where Q_p^im = p (r_i + r_m)² (r_i and r_m are the radii of the ion and molecule).

Langevin developed theories of ion mobility in a weak electric field by treating both the polarization interaction and ion-molecule collisions as collisions between elastic spheres [19 - 21].

where v@ and v_a@ are the ion and atomic velocities prior to collision and s (g, q ₀) is the differential cross section of the charge exchange. Expression (7.11) follows from (5.3), if we take into account that the atom and ion after collision seem to have exchanged velocities, i.e., v'@ = v_a@ and v_a'@ = v@.

In the simplest case of constant relaxation time, when g ò s (g, q ₀)dW ₀ is not dependent on the relative velocity of the colliding particles

where n_i and N_a are the densities of the ions and atoms and t ^ia is the time of the free path:

Here Q^ia(g) = ò s (g, q ₀)dW ₀ is the total cross section for charge exchange. If the atoms have a Maxwellian distribution, then (n_i/N_a)f_a(v) = f_iM(v), where f_iM(v) is the Maxwellian distribution function for ions at a temperature equal to the gas temperature T. In this case, the kinetic equation for ions is written in a form similar to (1.9) and (2.8):

However, unlike the case considered in §2, the non-equilibrium part of the distribution function f_i(v@ ) - f_iM(v) may not be quite as small.

Multiplying both sides of (7.14) by ev@ and integrating over d³v, we obtain the expression for current j_i@ = - ei_i' = - em _in_iE@, where m _I = et ^ia/M_i is the ion mobility. Thus, the approximation of a constant relaxation time permits a very simple calculation of the ion mobility in an electric field of arbitrary intensity.

However, the approximation of a constant relaxation time presupposes the specific dependence of the charge-exchange cross section on relative velocity: Q^ia(g) ~ g ^-1. Actually, the cross section @ is weakly dependent on g, so that the approximation of a constant cross section Q^ia, and accordingly, of a constant mean free path l_i = 1/Q^iaN_a is more accurate. In this case, the solution of the kinetic equation with the collision term (7.11) is very complicated. Results from a numerical solution of the kinetic equation for ions in a homogeneous electric field, obtained in [22], are presented below.*

The dependence of the ion drift velocity v_di@ on the electric field E@ is presented in Fig. 14.9. Here the drift velocity and the field are given in dimensionless units: z@ = v_di/(2kT/M_i)^1/2 and g = p eEl_i/4kT.

*When calculating ion mobility, it is convenient to turn from the integral-differential kinetic equation to the integral equation, which will be formulated in §2 of Chapter 6 for the case of a strong field (in this regard also see [23]).

The dashed lines denote the limiting values of ion drift velocity in weak and strong electric fields. The points and the solid line plotted through them are the results of calculations with intermediate values for the field intensity. For weak fields (g « 1), ion mobility with charge exchange was calculated independently by an expansion of the non-equilibrium component to the distribution function into a series of Sonine polynomials [24] and was found to be equal to

For the case of strong fields (g » 1), ion mobility in the approximation of a constant mean free path, as indicated below (see §2, Chapter 6), is equal to m _i = Ö 2el_i/p M_iE@ (see (6.2.11)).

117

References

1. I.P. Shkarofsky, T.W. Johnston and M.P. Bachynski, The Particle Kinetics of Plasmas, Addison-Wesley, Reading, Mass. , 1966,.Plasma kinetic, Language, English , eng,, Call Number, QC718 .S49, LCCN, 65014744 //r83, Dewey Decimal, 537.16, ISBN/ISSN, -

2. B.A. Trubnikov, Reviews of Plasma Physics, 1, 105, Consultants Bureau, New York , 1965,.Plasma kinetic equation, Language, English , eng,, Call Number, , LCCN, , Dewey Decimal, , ISBN/ISSN,

3. V.D. Shafranov, Reviews of Plasma Physics, 3, 1, Consultants Bureau, New York , 1967,.Plasma kinetic equation, Language, English , eng,, Call Number, , LCCN, , Dewey Decimal, , ISBN/ISSN,

4. V.N. Tsytovich, Nelineynyye effekty v plazme, "Nauka" , 1967,.Plasma kinetic equation,

5. A.I. Ansel’m, Vvedeniye v teoriyu poluprovodnikov, Fizmatgiz , 1962,.Solution of the kinetic equation in the relaxation time approximation,

6. L. Spitzer and R. Harm, Physical Review, 89, 977 , 1953,The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons, Language, English , eng,, Call Number, QC1.P4, LCCN, , Dewey Decimal, 530/.05, ISBN/ISSN, 0031-899X 0096-8250 0096-8269

7. S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd ed., Cambridge Univ. Press , 1970, 1990,.The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons, Language, English , eng,, Call Number, QC175 .C5 1970, LCCN, 70077285 //r962, Dewey Decimal, 533/.7, ISBN/ISSN, 0521075777, Language, English , eng,, Call Number, QC175 .C5 1990, LCCN, 90002685 //r96, Dewey Decimal, 533/.7, ISBN/ISSN, 052140844X

8. C.B. Kruger, J.R.M. Mitchner and U. Daybeige, Proc. IEEE, 56, 1458 , 1968,.The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons, Language, English , eng,, Call Number, , LCCN, , Dewey Decimal, , ISBN/ISSN,

9. B. Landshoff, Physical Review, 82, 1442 , 1951,.The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons, Language, English , eng,, Call Number, QC1.P4, LCCN, , Dewey Decimal, 530/.05, ISBN/ISSN, 0031-899X 0096-8250 0096-8269

10. B.J. Kaneko, Proc. Phys. Soc., Japan, 15, 1685 , 1960,.The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons, Language, English , eng,, Call Number, , LCCN, , Dewey Decimal, , ISBN/ISSN,

11. I.P. Stakhanov, et.al., Plazmennoye Termoymissionnoye Preobrazovaniye Energii, Atomizdat , 1968,.The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons,

12. D.R. Wilkins and E.P. Gyftopoulos, Journal of Applied Physics, 37, 3533 , 1966,.The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons, Language, English , eng,, Call Number, QC1.J83, LCCN, , Dewey Decimal, 530.5, ISBN/ISSN, 0021-8979

13. R. S. Cohen, L. Spitzer, and P. Routly, Physical Review, 80, 230 , 1950,.The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons, Language, English , eng,, Call Number, QC1.P4, LCCN, , Dewey Decimal, 530/.05, ISBN/ISSN, 0031-899X 0096-8250 0096-8269

14. S.I. Braginskii, Reviews of Plasma Physics, 1, 205, Consultants Bureau, New York , 1965,.The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons, Language, English , eng,, Call Number, , LCCN, , Dewey Decimal, , ISBN/ISSN,

15. O.S. Gryaznov and B. Ya. Moizhes, Sov. Phys. - Tech. Phys., 16, 1110 , 1972,.The effect of interelectrode collisions and ion scattering on the kinetic coefficients for electrons,

16. B.I. Davydcv, Sh. Eksp. Teor. Fiz., 2, 1069 , 1937,.Exchange of energy and momentum between electrons and the heavy components of a plasma,

17. V.L. Ginzburg and A.V. Gurevich, Sov. Phys. - Usp., 3, 115 , 1960,.Exchange of energy and momentum between electrons and the heavy components of a plasma,

18. M.J. Druyvesteyn, Physica, 10, 69 , 1930,.Exchange of energy and momentum between electrons and the heavy components of a plasma, Language, English , eng,, Call Number, , LCCN, , Dewey Decimal, , ISBN/ISSN,

19. Earl W. McDaniel, Collision Phenomena in Ionized Gases, John Wiley and Sons, New York , 1964,. ,.The kinetic coefficients for ions in a weakly ionized plasma, Language, English , eng,, Call Number, QC721 .M126, LCCN, 64013219 //r83, Dewey Decimal, 537.532, ISBN/ISSN, -

20. P. Langevin, Ann. Chem. Phys., 28, 289 , 1903,.The kinetic coefficients for ions in a weakly ionized plasma, Language, English , eng,, Call Number, , LCCN, , Dewey Decimal, , ISBN/ISSN,

21. H.R. Hasse, The Philosophical magazine, a journal of theoretical, experimental and applied physics, 1, 139 , 1926,.The kinetic coefficients for ions in a weakly ionized plasma, Language, English , eng,, Call Number, Q1 .P5, LCCN, 19004495 //r522, Dewey Decimal, - , ISBN/ISSN, -

22. V.I. Perel’, Sov. Phys. - JETP, 5, 440 , 1957,.The kinetic coefficients for ions in a weakly ionized plasma,

23. G. Ecker, et.al., Proc. 7th Conference on Phenomena in Ionized Gases, Belgrade, p. 557 , 1965,.The kinetic coefficients for ions in a weakly ionized plasma, Language, English , eng,, Call Number, , LCCN, , Dewey Decimal, , ISBN/ISSN,

24. Ia. M. Kagan and V.I. Perel’, Sov. Phys. - JETP, 2, 761 , 1956,.The kinetic coefficients for ions in a weakly ionized plasma,

25. H.R. Brode, Physical Review, 34, 673 , 1927,. Language, English , eng,, Call Number, QC1.P4, LCCN, , Dewey Decimal, 530/.05, ISBN/ISSN, 0031-899X 0096-8250 0096-8269

26. D.R. Golden, Physical Review, 151, 48 , 1966,. Language, English , eng,, Call Number, QC1.P4, LCCN, , Dewey Decimal, 530/.05, ISBN/ISSN, 0031-899X 0096-8250 0096-8269

Hosted by www.Geocities.ws