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119

Only elastic collisions were considered in the calculation of kinetic coefficients in Chapter 4. Actually, along with elastic collisions, inelastic collisions occur in a plasma. In particular, inelastic electron collisions with atoms in a TIC plasma lead to ionization of cesium atoms.

Inelastic electron collisions with atoms in a low-temperature TIC plasma are less frequent than elastic electron collisions. Therefore, when calculating the electron kinetic coefficients, inelastic electron collisions can usually be disregarded. The role of inelastic processes in a TIC plasma reduces mainly to the fact that these processes lead to a variation of the particle density and energy in the plasma. This is a major effect in the arc mode of TIC operation, and therefore, finding the rate of ionization-recombination in a plasma is one of the most important problems in TIC theory.

This chapter is devoted to the calculation of the ionization-recombination rate in a low-temperature plasma. The main contents of the chapter relate specifically to the gas-discharge TIC plasma. However, a number of the results may also be of interest for other applications of a low-temperature plasma.

Before turning to the description of the transport and ionization phenomena in a plasma, consider the parameters of the plasma under conditions of thermodynamic equilibrium. In this case, the whole description can be obtained by statistical physics, and there is no need to resort to the more complicated methods of kinetic theory.

The electron velocity distribution in an equilibrium plasma is described by the Maxwellian distribution function, while the population distribution over the excitation levels for atoms is according to the Boltzmann distribution function, i.e.,

where g_k and E_k are the statistical weight and energy of the k-level of excitation. Energy is calculated from the ground state to which corresponds the index k = 0.

The total number of atoms N_a(T) is related to the number of atoms N₀(T) in the ground state by the relation

The sum (1.2) is divergent at large values of k for an isolated atom. This follows from the fact that at higher levels all atoms are hydrogen-like, and in this case, the main quantum number of the hydrogen atom n can be used as the index k, where g_n = 2n² and E_ion - E_n ~ 1/n². Larger values of n correspond to very weak binding energy and to classical electron motion in orbit around the atom core at a very large radius. Therefore, even slight interaction with other particles of the

plasma leads to electron ionization, and in reality, there are no states in the plasma with large values of n. As a result of this, the sum (1.2) is terminated at some value of n_max

The disappearance of high excitation levels is related primarily to the effect on the atom of electric fields created by the electrons and ions of the plasma. Since the electric field intensity at the atom is determined by the density n_e of the charged particles, the number of levels associated with the discrete spectrum is also dependent on density. Calculation of the number of these levels, carried out with these assumptions, leads to the Inglis—Teller formulas [1]

where n_max is the number of the last level of the hydrogen spectrum, and a₀ = e²/2Ry = h@ ²/me² is the Bohr radius.

This symbolic notation denotes that n ₁ of particles of variety A₁, n ₂ of particles of variety A₂, ..., and n _m of particles of variety A_m are transformed to n ₁' particles of variety B₁, n ₂' particles of variety B₂, ..., and n _k' particles of variety B_m as a result of the reaction. The particle concentrations under conditions of chemical equilibrium for this reaction are calculated using the condition of steady state for the thermodynamic potential F [2]:

1) ionization of a Cs atom during collision with an electron, with the formation of an atomic ion: Cs + e ¬ ® Cs⁺₂ + 2e;

2) ionization of a Cs₂ molecule during collision with an electron, with formation of a molecular ion: Cs₂ + e ¬ ® Cs⁺₂ + 2e;

3) formation of a Cs₂ molecule during collision of two atoms with a third particle: 2Cs + A ¬ ® Cs₂ + A.

where N_a(T), n_e(T), n_i(T), N_Cs2(T), and N_Cs+2(T) are the concentrations and Z_a, Z_e, Z_i, Z(Cs₂), and Z(Cs⁺₂) are the statistical sums of the internal states (partition functions) for atoms, electrons, atomic ions, molecules and molecular ions, respectively.

where Z = å g_k exp(-E_k/kT) is the partition function or statistical sum of all the internal states with energy E_k, and the factor (2p kmT/h²)^3/2 is the statistical sum for the positive degrees of freedom of the particle.

Consider the partition function Z for different particles of a cesium plasma. For electrons we have Z_e = 2 (taking into account the two possible spin orientations). For atomic ions, the excitation of atomic electrons require very high energy and therefore does not contribute to Z_i in the range of temperature of interest. The value of Z_i is determined by the number of spatial ion orientations and is not dependent on temperature.

For atoms, the partition function should include a sum over excited levels, similar to (1.2). The sum of spatial orientations of the atomic core, which coincides with the corresponding sum for an ion, i.e., equal to Z_i, should also be included. As a result, we find

Since the ion energy is assumed to be zero, the energy of the excited state is equal to E_k - E_ion, i.e., to the binding energy of the valence electron with the opposite sign.

If the temperatures are sufficiently low and the contribution of excited levels to (1.10) is small (i.e., N_a(T) » N₀(T) according to (1.2)) then

and the equation of ionization equilibrium (1.6), called the Saha equation, may be written for a quasi-neutral plasma (n_e(T) = n_i(T)) in the form

The equation of ionization equilibrium can also be obtained by considering the reaction Cs + e ¬ ® Cs⁺ + e only with regard to those atoms at a specific level k. In this case N_k(T) instead of N_a(T) and the value of g_k exp(E_ion - E_k)/kT instead of the partition function Z_a

Calculation of the partition functions Z(Cs₂) and Z(Cs⁺₂) for molecules and molecular ions is complicated by the need to account for the degrees of freedom due to rotation of the molecule or of the molecular ion, and due to the atomic vibrations with respect to the average position. The internal energy of the molecule can be represented approximately [2] in the form of the sum of the atomic vibrational energy E_vib = h@ w (s + 1/2), the molecular rotational energy E_rot = (h@ ²/2I)k· (k + 1), and the electron energy E_el. Here s and k are the vibrational and rotational quantum numbers, w is the atomic vibrational frequency, I is the molecular moment of inertia, and E_el is calculated from the state of the valence electrons of the atoms. In this case, the partition function Z(Cs₂) can be represented in the form of the product of the corresponding partition functions:

The approximate expression (1.14) follows from an expansion of the potential energy of interaction of two atoms and the moment of inertia I into a Taylor series in the distance r between the atoms near the equilibrium value r₀. During subsequent approximation the terms, dependent on k and s simultaneously, occur in the expression for energy

[2].

The values of Z(Cs₂) are presented in the book [3]. Harris [4] calculated Z(Cs⁺₂) on the assumption that the vibrational and rotational partition functions for the molecular ion do not differ from the corresponding sums for the cesium molecule. The temperature dependence of the chemical equilibrium constants (p_Cs2/p²_Cs)10², (n_e(T) n_i(T)/p_Cs)^1/210^-16, and N_Cs+2(T)/n_i(T)p_Cs, calculated by Harris, are presented in Fig. 5.1. The first two constants are expressed by the right sides of (1.8) and (1.6), and the third constant is expressed by the right side of the equation obtained by multiplying (l.7) and (1.8) and dividing by (1.6).

According to [4], the ratios of N_Cs+2(T)/n_i(T) and N_Cs2(T)/N_a(T) do not exceed 9× 10^-2 and 2× 10^-4 in the temperature range of 1500 - 2000° K and pressures of 0.1 - 20 torr. Therefore, the presence of Cs₂ and Cs₂⁺ has a weak effect on the properties of an equilibrium cesium plasma.*

*In [4] the value reported for the dissociation energy of molecular ions was D^(mi) = 1.06 eV. Later data (see pp. 140-141) has given a smaller value for D^(mi), and therefore, small densities for molecular ions are indicated.

To do this, we number, in order of increasing energy, all the levels, beginning with the ground level, using the numbers 0, 1, 2,..., g_m (Fig. 5.2). The last excited level g_m is determined from the Inglis-Teller formula (1.3).

Having limited in this manner the number of levels, we can calculate the level populations N_l from the particle balance equations. For some k-th level, the particle balance equation has the following form:

Here a _kn_e³ = a _kn_e²n_i and b _kn_e² = b _kn_en_i are the rates of population of a given level due to collisional and optical recombination processes,* N_k[n_ew^(e)_k, ion + w^(ph)_k, ion] are the rates for the reverse processes: ionization of a given level by electron impact and photo-ionization, and n_{eå l¹ k}N_lwlk - n_{eå l¹ k}w_kl

*In this chapter a plasma will be assumed everywhere to be quasi-neutral, i.e., n_e » n_i.

The probability of ionization w^(e)_k, ion and the probability of excitation w_lk(l > k), by electron impact, are determined from the cross sections for the corresponding processes Q_kion(E) and Q_kl(E), and from the electron velocity distribution function f_e(v@ ), by the formulas

where E = mv²/2. Either the classical Thomson and Grizinski cross sections or the quantum-mechanical cross section, obtained in Bethe-Born approximation, with subsequent correction (see §5, Chapter 3), are usually employed as the ionization and excitation cross sections for electron impact.

*In this case, processes of induced emission are not taken into account, which may be significant only in the upper region of the spectrum where the distance between the levels is close to kT_e. However, transitions occur in the upper regions of the spectrum more frequently by electron impact, since optical processes usually do not affect the populations of upper excited states.

Q_rec(E ', E₂ ® k) and Q_lk(E ' ) by the formulas

where E ' = mv'²/2 and E₂ = mv₂²/2, and where the cross sections of direct and reverse processes are related to each other by the principle of detailed balance (see §5, Chapter 3).

The probability of photoionization w^(ph)_k, ion is also expressed by the cross section for this process:

where r (w )/h@ w is the density of electromagnetic radiation quanta of frequency w , and Q^(ph)_k, ion(w ) is the photoionization cross section from level k.

where Q^(ph)_k, ion ¬ (E ' ) is the radiative recombination cross section to level k. The expression for the radiative recombination cross section was presented in §7, Chapter 3. The photoionization and radiative recombination cross sections are related to each other by the principle of detailed balance, just as the ionization recombination cross sections for electron impact and collision recombination are related.* The rate of ionization—recombination is determined from the populations of the discrete levels of the atom and from the ionization and recombination cross sections by the formula

*The principle of detailed balance for cross sections Q^(ph)_k, ion and Q^(ph)_k, rec is described in the following manner: g_kk_ph²Q^(ph)_k, ion = g_ionk_e²Q^(ph)_k, rec, where k_ph and k_e are the wave vectors of the photon and electron, and g_k and g_ion are the statistical weights of the atom at the k—th level of excitation and of the ion, respectively. Since the photon and electron momentum are equal to h@ k_ph and h@ k_e, respectively, then the derived relation is similar to (3.5.6). Taking into account that E = h@ k/2m and that w = k_phc , we find that for atoms with a single optical electron

(g_ion = 1),

Therefore, to calculate the rate G of ionization—recombination, it is necessary to calculate the free electron distribution function f_e(v@ ) and the electromagnetic radiation density r (w ). Since inelastic electron collisions with atoms during ionization and recombination processes affect the free electron distribution function, and since photon emission and absorption processes affect the populations of excited states, the populations of the excited states must be taken into account together with the electron distribution function f_e(v@ ) and the electromagnetic radiation density r (w ). This approach to the problem was examined in [5].

However, in some cases, there is no need to solve the problem in its more general form, because the rates of different processes differ strongly from each other. In particular, if the density of free electrons is great enough, the development of a Maxwellian distribution of the free electrons due to mutual interaction occurs much more rapidly than the deviation of the distribution function f_e(v@ ) due to different inelastic processes. In this case, the free electron velocity distribution is the Maxwellian distribution, and the probabilities w_kl and w^(e)_kion for inelastic processes, and also the recombination coefficient a _k, are determined by the concentration n_e and temperature T_e of the free electrons. We consider this case in more detail, because it is of greatest interest for the TIC.

The rate of impact ionization and impact excitation for a Maxwellian distribution of free electrons. For a Maxwellian distribution of free electrons, coefficients w_k and a _k can be related to the probabilities of the corresponding direct processes w^(e)_kion and w_kl without turning directly to relations (3.5.6) and (3.5.7). Taking into account that in a state of thermodynamic equilibrium n_eN_l(T_e)w_kl = n_eN_k(T_e)w_kl where N_k(T_e) and N_l(T_e) are the equilibrium values of the populations of the excited levels, calculated by formula (1.1), we obtain for the de-excitation probability

where E_kl = hw _lk = E_l - E_k. The total number of transitions between levels k and l is then equal to

is the population of the k-th level of the atom, if this level is in thermodynamic equilibrium with the ground state, i.e.,

(2.14)

is the Maxwellian energy distribution,* and Q_kl(E) is the excitation cross section.

The coefficient a _k for collisional recombination to level k can be expressed in like fashion in terms of the probability of ionization from level k. Since the rate of ionization from a level in a state of thermodynamic equilibrium (due to the effect of electron collisions) is equal to the rate of collisional recombination to this same level, we have

where n_e(T_e) is the equilibrium density of free electrons according to Saha, n_e(T_e) is expressed in terms of the equilibrium concentration of the atoms N_a(T_e), and the temperature T_e of the free electrons is given by formula (1.11).

Here N₀(T_e) » N_a(T_e) is the equilibrium concentration of atoms in the ground state and N₀ » N_a is the actual concentration of atoms in the ground state,

(Q_kion(E) is the ionization cross section from level k). The values of W_kl and W_kion which determine the rates of excitation and ionization of the atoms, can be rewritten, separating out the dependence on electron temperature. Having denoted x = (E - E_kl)/kT_e, we find from (2.13)

*The electron energy distribution function n(E) is related to the velocity distribution function f_e(v) by the relation f_e(v). 4p v²dv = n(E)dE. Using the expression for f_eM(v) and taking into account that mvdv = dE, we obtain (2.14).

It is obvious from equations (2.19) and (2.20) that the transition intensities due to the effect of electron impacts are very strongly dependent on electron temperature T_e, where the main dependence on T_e is determined by the exponential factor exp(- E_l/kT_e) and exp(- E_ion/kT_e).

Relationships between the rates of collisional and radiative processes. When determining the rates of photoexcitation and photoionization of atoms, a knowledge of the spectral density of the electromagnetic radiation r (w ) is essential. Unlike the electron energy distribution, the photon spectral distribution is usually not in equilibrium. The non-equilibrium of the radiation is related to the non-equilibrium population of the excited levels of the atoms, and also to the fact that the plasma is optically transparent to many lines in the emission spectrum and to much of the recombination continua. The plasma transparency leads to the fact that events of photoemission during optical dc—excitation of atoms and photo-recombination occur much more frequently than the reverse processes of photoexcitation and photoionization. This is usually the situation for emission accompanied by transitions to the upper energy levels, where the populations of excited states are low and, at the same time, the absorption coefficients are small.* On the other hand, in the lower energy levels, where the absorption coefficients are large, events of photoemission in the lines are approximately compensated for by events of absorption. The emission processes then cease to affect the populations of the excited states, i.e., N_lA_lk - r (w _lk)N_kB_kl » 0.

The role of radiation in determining the populations of excited states in an optically transparent plasma is determined by the ratio of the rate of optical de-excitation, N_lA_lk, or of optical recombination, b _kn_e², to the rates for de-excitation by electron impact, n_eN_lw_lk, and of collisional recombination, a _kn_e³. The probabilities for collisional processes, w_lk, have the greatest values in the upper energy levels, where the levels are densely distributed, and the probabilities of spontaneous optical transitions A_lk are greatest in the lower region of the spectrum, where the levels are more separated: A_lk ~ w_lk³ (see (3.7.11)). As a result, the ratio A_lk/n_ew_lk increases in going to the lower energy levels.

The values for the transition probabilities n_ew_n,n-1 between two adjacent levels of a hydrogen atom due to the effect of electron impacts, calculated in [6] using Grizinski excitation cross section, and the probability of optical de-excitation of the level with the main quantum number n equal to A_n = å _n = 1ⁿ - 1A_nn are presented in Fig. 5.3 as an example. It is obvious that with the values of the free electron density used in the calculation, collision de-excitation is prevalent at the upper levels and emission de-excitation is prevalent at the lower levels.

The effect of plasma inhomoqeneity on the rate of ionization recombination. Ionization and recombination under steady-state and transient conditions. Equation (2.1) is written with the assumption of plasma homogeneity. In a heterogeneous plasma, the derivatives dN_k/dt should be replaced by the relations

where i_k@ is the flow of atoms excited to the k-th level. Having substituted (2.21) into (2.1), we find the natural result that change in the density of excited atoms at a given location in the plasma, i.e., ¶ N_k/¶ t, is determined by the change of the population of the given excited state because of the different inelastic and emission processes (the left side of (2.1)) plus the flow divergence i_k@.

In steady-state, when the densities of excited atoms do not vary in time (¶ N_k/¶ t = 0), change in N_k because of inelastic and emission processes is compensated for by an influx (or outflow) of excited atoms into the region of the plasma being considered. Similarly, change in the electron and ion density due to ionization-recombination processes is compensated for by an influx (or outflow) of charged particles to a given point of the plasma.

The flow of atoms, electrons, and ions in the plasma is determined by the heterogeneous distribution of the plasma parameters (n_e, T_e, N_a, and T) and also by any electric field present. In general, the presence of these flows can have an appreciable effect on the rates of the ionization-recombination processes. For example, if ionization is a multi-stepped process, then some time should pass from the beginning of excitation of the atom to the moment of its ionization. During this time, the excited atom may have diffused to another region of the plasma,

where the values of n_e, T_e, N_a, and consequently, the rates of ionization-recombination G will be different. Therefore, the value of may not be regarded as a local variable in a heterogeneous plasma, i.e., it is not possible to assume that G is dependent only on the values of n_e, T_e, and N_a a given point of the plasma.

However, if the rate of ionization is sufficiently high, the excited atom can be assumed to be immobile during ionization, and the expression for the rate of ionization retains its local nature. Disregarding the diffusion of excited atoms means that the processes leading to the depopulation and population of each excited level in the steady-state mode should be mutually compensated for, and the right sides of equations (2.1) for k = 1, 2, ..., g_m should be set equal to zero.

is considerably greater than the inverse of the time which characterizes the duration of the recombination process itself. In this case the concentration of excited atoms is determined only by the instantaneous value of the plasma parameters, and the term dN_k/dt (k = 1, 2, ..., g_m) may be omitted in equations (2.1) for the balance of particles at excited levels of atoms even in the transient case.

The variation of charged particle density n_e in a decaying plasma is described by the equation dn_e/dt = G (G < 0). If optical recombination occurs, then G = - b n_e². During collisional recombination G = - a n_e³. However, in most cases, the experimentally observed value of G is written in the form - b _effn_e². In this case, the recombination coefficient b _eff may itself be dependent on the electron density n_e, where b _eff is proportional to n_e in the case of collisional recombination.

The calculated values of the recombination coefficient b _eff for an optically transparent hydrogen plasma (using the Grizinski cross sections of excitation by electron impact)* are presented in Fig. 5.4 [6]. Using the value of b _eff, the time t , which characterizes the length of the recombination process, can be calculated to an order of magnitude: t ^-1 = b _effn_e. It is obvious from the comparison of the values obtained with the transition probabilities in the hydrogen atom, presented in Fig. 5.3, that the transition probabilities between levels exceed t ^-1 by an order of magnitude.

Basic description of the recombination process. The reason that the recombination time t greatly exceeds the deexcitation time of individual discrete levels is as follows. In the case of photorecombination, because of the smallness of the photorecombination cross section Q^(ph)_krec, the probability for electron transition from the continuum to the

*One may become acquainted with these calculations from [7]. Similar calculations, for radiation re-absorption, are reported in [8].

discrete levels is very small compared to the probabilities for optical de-excitation of discrete levels. As a result, the populations of the discrete levels rapidly relax to their steady-state values, and the recombination time is determined by the time of the slowest process, i.e., by the time of electron transition from the continuum to the discrete levels. In particular, the maximum value of b _eff at small values of n_e (where recombination is by photoemission (see Fig. 5.4)) is close to the coefficient of direct photorecombination from the continuum to the ground state.

When recombination occurs by a mixture of collisional and emission processes, we can again point out those processes in the sequence (for the electron going from the continuum to the ground state) which proceed slowly. As can be seen from Fig. 5.3, de-excitation occurs most slowly (for an optically transparent plasma) in that part of the level sequence where the probability of collisions of second kind n_ew_n,n-1 is comparable to the probability of optical de-excitation A_n.

The populations of the levels in equilibrium with the continuum are given by the Saha formula (1.13), using the density n_e and temperature T_e of the free electrons:

We note that formula (2.17) can now be used to calculate the rate of recombination, because slight deviations of the relative populations from the values u _n^cont@, which correspond to thermodynamic equilibrium of the discrete levels with the continuum, would have to be taken into account to obtain a result distinct from zero, Instead, it is simpler to consider the transitions at the "narrow point" of the level sequence, where the rate of the de-excitation processes appreciably exceeds the rate of the reverse processes. In this case, an approximate

expression can be obtained for the recombination rates by substituting the number of atomic electron transitions to levels with n < n⁴ [6]:

Here n*@ is the number of the discrete level in the narrow region of the level sequence, where A _n4 » n_ew_n4,n4-1@ (see Fig. 5.3) and A_n^n* = å _n'=1^n*-1A_nn'@ is the probability of optical de-excitation from level n to levels n' < n*. The recombination time is calculated by the following expression:

In most of the cases considered below, the populations of the atomic excited states, N_n*^cont(T_e), are small compared to the density of free electrons n_e. As can be seen from (2.25), under these conditions the inverse recombination time is actually small compared to the probabilities of de-excitation n_ew_n,n-1.

In purely collisional recombination, which occurs at large values of n_e, and also in an optically dense plasma, a "narrow point" may also be identified which limits the rate of this recombination process. In collisional recombination, the number of de-excitation events during atomic electron transitions between adjacent upper levels is calculated by the value of n_eNn^(cont)(T_e)w_n,n-1. This value has a minimum near some value of n = n⁴.

The occurrence of a minimum is related to the fact that in the range of small values of n, as can be seen from (2.22), Nn^(cont)(T_e) begins to increase sharply as n decreases because of the increase in the function exp(E_ion - E_n)/kT_e, whereas in the upper region of the spectrum n_eNn^(cont)(T_e)w_n,n-1 increases as n increases because of an increase in the transition probabilities w_n,n-1 and because of the increase in the statistical weights of the states g_n = 2n². As a result, the rate of collisional recombination can be calculated approximately, by analogy with (2.24), as

*A similar approach in the calculation of the recombination coefficient was used in [10].

passing in sequence through a number of excited states. For the lower energy excited levels, the de-excitation processes are almost totally balanced by the excitation processes. Actually, as can be seen from (2.9), the probability for collisions of second kind w_lk is (g_k/g_l)exp(E_kl/kT_e) times greater than the probability for collisions of first kind w_kl. Moreover, the probability of de-excitation in an optically transparent plasma, increases rapidly, because of photon emission. Because of all this, an atom excited to a lower level is usually de-excited back, and an atomic electron is held for a rather long time at lower excitation states in the process of ionization.

If the plasma is optically dense, the balance of forward and reverse processes at the lower levels causes the lower excited levels to be in equilibrium with the ground state. In this case, the number of electron excitations to a higher level n_eN_k⁽⁰⁾(T_e)w_kl, proportional according to (2.19) to exp(- E_l/kT_e) in the lower region of the spectrum, rapidly decreases exponentially as the energy level E_l increases. However, in the upper region of the spectrum, the levels are densely concentrated and the change of the value of n_eN_k⁽⁰⁾(T_e)w_kl during transition to each subsequent higher level is determined by the value of the statistical weight of the level g_k and of the transition probability w_kl rather than by a decrease of the exponential exp(- E_l/kT_e). As a result, the number of excitations again begins to increase and a " narrow point" forms in the level sequence, in the same manner as in recombination.

The rate of ionization in an optically dense plasma may be calculated most conveniently as the number of transitions at the narrow point" [6,10]. Assuming that the levels located in the lower region of the spectrum are in equilibrium with the ground state, we obtain the following approximate expression for the rate of ionization:

An approximate expression for the rate of ionization-recombination. We can now derive a general formula which takes into account both the ionization and recombination processes. To do this, we distinguish two adjacent levels n* - 1 and n* such that the level n* - 1 is directly below the "narrow point," and level n* is directly above the "narrow point." The number of transitions between these levels is expressed by formula (2.10). By assuming that levels 0, 1, 2, ..., n* - 1

are in thermodynamic equilibrium with the ground state, and that levels n*, n* + 1, ..., g_m are in equilibrium with the continuum,* and assuming that the transitions occur only between two adjacent levels in the level sequence, we obtain

The right side of (2.29) is nothing more than the difference of G _ion - G _rec. If there is a strong depletion of free electron density below equilibrium and u _n*^(cont) « 1@, expression (2.28) for the rate of ionization follows from (2.29). On the other hand, if the density of charged particles considerably exceeds the equilibrium value, and u _n*^(cont) » 1@, expression (2.26) for G _rec follows from (2.29)

where G _ion is calculated by expression (2.28). It follows from this approximate analysis that the value of the rate of ionization is determined by the value of the transition probability w_n*- 1,n* at the "narrow point" of the level sequence, and the temperature dependence of G _ion is determined by the exponential exp(- E_n*/kT_e) according to (2.13), (2.19), and (2.28).

It is an essential point that the rate of ionization G _ion, calculated by formula (2.29), greatly exceeds in value the rate of direct ionization from the ground state N₀n_ew^(e)_{0 ion}. In like manner the rate of recombination G _rec greatly exceeds the rate of direct recombination a ₀n_e³. For this reason, the process of collision ionization-recombination in a low-temperature plasma is a multi-stepped process. However, in a high-temperature plasma, and also in a plasma with a non-equilibrium energy distribution for the free electrons, the situation may of course be quite different.

In this case, the relative population of the ground state is n ₀ - 1.

As can be seen from expressions (3.1), (2.19), and (2.20), the relative populations of excited states n _k are determined by the parameter n_e²N₀(T_e)/n_e²(T_e)N₀, and also by the electron temperature T_e. Moreover, the rate of ionization-recombination G is also dependent on n_e. To determine the nature of the dependence of n _k and G on these parameters, we introduce the new unknowns

Since all the right hand terms in the linear system of equations (3.4) are proportional to the quantity of n_e²N₀(T_e)/n_e²(T_e)N₀ - 1, the solutions to the system are also proportional to this quantity. Denoting the proportionality constants by p_k, we can write

Since all the values of W_kl and W_kion are proportional to N₀, the coefficients p_k are not dependent on N₀, but are dependent only on electron temperature T_e.

The factor n_e{W_0ion + å [ … ] W_kion} on the right side of (3.6) is proportional to n_eN₀ and is dependent on electron temperature T_e. This factor may be written in the form n_ev_e@ s ₀(T_e)N₀, where v_e@ is the mean velocity of electrons in the Maxwellian distribution, and s ₀(T_e) has the dimensions of a cross section, which we shall call the effective ionization cross section from the ground state. As a result, the following expression is obtained for the rate of ionization-recombination

Expression (3.7) coincides in form with (2.30), if the rate of ionization G _ion is understood as the quantity

Note in passing that the concepts "rate of ionization" and "rate of recombination," calculated by expressions (3.8) and (3.9), are arbitrary, because each of these quantities separately is not equal respectively to the number of ionization events n_{eå k}=0^?mn _kW_kion and to the number of recombination events n_e^3å _k=0^?ma _k.

By using the relationship established between the values of G _ion and G _rec, we need only make a detailed calculation of one of these quantities to obtain both. For example, we can set n_e²N₀(T_e)/n_e²(T_e)N₀ = 0, calculate the value of G _ion, i.e., calculate the effective ionization cross section s ₀(T_e), and then calculate the rate of recombination G _rec. We can proceed in the same manner to calculate the population of excited levels. If the values of n _k for n_e²N₀(T_e)/n_e²(T_e)N₀ = 0 are denoted by n _k', then, according to (3.3) and (3.5),

Therefore, having calculated p_k, from (3.3) and (3.5), we obtain

It is obvious from (3.11) that all values of n _k increase as recombination appears and as n_e²N₀(T_e)/n_e²(T_e)N₀ increases. In a state of thermodynamic equilibrium, when n_e = n_e(T_e) and N₀ = N₀(T_e), we have for energy level n _k = 1, and naturally, for equilibrium, G _ion = G _rec. It is obvious from (3.7)—(3.11) that the relative role of recombination is determined by the parameter n_e²N₀(T_e)/n_e²(T_e)N₀, which, according to (1.13), is equal to (n_e²N₀)(g₀/N_e(T_e))exp(E_ion/kT_e). If this parameter is small compared to unity, i.e.,

The rate of ionization-recombination in a cesium plasma. With an electron density of n_e » 10¹³ - 10¹⁴ cm^-3, which usually occurs in the cesium plasma of, a TIC, the rate of radiative de-excitation of the levels is relatively large in the lower region of the spectrum. However, the corresponding emission lines are strongly absorbed and emission transitions are usually not taken into account when calculating the populations of the excited states.

Numerical calculations of the ionization rate and of the populations of excited states were made in [5, 11-15]. Although various approximate formulas were used for the scattering cross sections, the final results of these calculations do not differ greatly from each other. The method for the calculation reported in [11] is described in detail below.

The transition probabilities between the levels in the first group were calculated in the Bethe-Born approximation. Since the calculated values of the cross sections are an order greater than the experimental values, the theoretical values must be reduced by approximately an order to bring the theoretical cross sections into agreement with the experimental values [16].

The oscillator strengths f_kl@, averaged over the states of the lower, i.e., the k-th level of the atom, were used to calculate the transition probabilities in a cesium plasma [17] (see Chapter 3). It is these oscillator strengths that are contained in expression (3.5.14) for the excitation cross section. The doublet levels were combined in pairs into a single equivalent level with a total statistical weight, which reduces the number of unknowns in the system of particle balance equations. In this case, the oscillator strengths f_kl@ were averaged over the states of the lower level which correspond to the different values of the quantum number j.

The levels of the second group correspond to large values for the main quantum number n. Therefore, Thomson’s classical formula* was used, which can be used both to calculate the ionization probability and to calculate the excitation probability of the atom to a discrete level (see §5, Chapter 3).

The results from calculating the relative level populations in a weakly ionized Cs plasma under conditions where the charged particle concentration n_e and the excited atom concentration are low compared to the concentration N₀ of atoms in the ground state are presented in Fig. 5.5. In this case the first group of levels included levels 6S -- 9S, 6P -- 9P, 5D -- 8D, 4F, 5F, and 5G. The remaining levels were included

The probabilities for transitions between the levels of the first and second group were calculated using Thomson cross sections. The last level in the discrete spectrum was calculated by the Inglis-Teller formula (1.3), where under conditions typical for a TIC plasma, n_max = 14. The populations of the individual levels of the spectrum are shown in Fig. 5.5, where the calculated points are connected by the curves. The curves are plotted for fixed values of n_e and N₀ and at different values of T_e. The values of n ^(cont), which correspond to the thermodynamic equilibrium between the discrete levels and the free electrons, are also noted in Fig. 5.5.

Recombination dominates ionization (and n _k > l)at a low free electron temperature T_e, where n_e²N₀(T_e)/n_e²N₀(T_e) > 1. If T_e increases, ionization increases rapidly, and the inequalities change signs. It is obvious that a sharp break occurs on the curves at an energy of

E_k » 3 eV. As one could expect from the approximations used for the ionization process, the break is at the "narrow point" of the spectrum, where the exponential decrease (-E_k/T_e) begins to be compensated for by the increase of the level statistical weight g_k and the transition probability w_{k, k}+1, i.e., where the distance between adjacent levels E_{k, k}+1 @ kT_e@. As T_e increases, the break shifts to a lower region of the level sequence, where the distance between the levels is greater. Gradually, the break disappears.

If ionization dominates recombination, i.e., G _ion » G _rec, then the levels located below the "narrow point" are actually in approximate equilibrium with the ground state. However, for levels located directly above the "narrow point," equilibrium with the continuum generally does not occur. The relative populations of these levels, n _k, considerably exceed n ^(cont). Therefore, the value of G is actually less than the value which follows from the approximate formula (2.29). On the other hand, if recombination is dominant, i.e., where G _rec » G _ion, the levels located above the "narrow point" are in approximate equilibrium with the continuum, i.e., for them n _k » n ^(cont). However, for levels located below the "narrow point," n _k > 1 and equilibrium with the ground state does not occur. In this case calculation by formula (2.29) also raises the value of ï G ï .

The absence of thermodynamic equilibrium at the upper levels leads to the fact that the electron temperature calculated from the relative intensities of spectral lines associated with the upper energy levels differs from the free-electron temperature. In this case if ionization is prevalent in the plasma, then the level populations decrease more rapidly as energy increases than by the law N_k ~ g_kexp(- E_k/kT_e), and the electron temperature, calculated from the relative line intensities, is less than the true electron temperature. On the other hand, if recombination is dominant, an increased value of the free-electron temperature is obtained from the relative intensities.

The results from calculating the effective ionization cross section s (T_e) are presented in Fig. 5.6. For T_e £ 3000° K, when the lower excited levels are close to thermodynamic equilibrium with the ground state, and when the ionization rate G _ion is calculated to an order of magnitude by the number of transitions at the "narrow point" of the spectrum, the value of s (T_e) is easily approximated by the formula

where, according to data of [11], s _ion = l.44× 10^-12 cm² and E₀ = 3.21 eV, and from data of [12], s _ion = 0.603× 10^-12 cm² , and E₀ = 2.98 eV.

The lower excited levels are stripped rapidly as temperature increases to T_e > 3000° K K. Then G _ion and s (T_e) begin to increase more slowly as T_e increases than by exponential law (see curve 1 in Fig. 5.6).

The values of the effective ionization cross section s (T_e), calculated from the results of [13,14], are also presented in Fig. 5.6.

The results of a calculated experimental rate of ionization in the cathode layer of the low-voltage arc discharge in a TIC are presented in Fig. 5.7 from the data of [18]. The effective ionization time

is presented in the figure as a function of electron temperature T_e. The calculated values of t _i from data of [11, 12] are also shown here. Good agreement is seen between the experimental and theoretical results. (See also [19-22]).

We may note in concluding this section that, besides the so-called collisional or collisional-radiative mechanism of ionization-recombination in a cesium plasma considered in this chapter, associative—dissociative ionization-recombination may also occur. Associative ionization is a two-step process. Excitation of the cesium atom by electron impact occurs during the first stage. During the second stage, the excited Cs* atoms are combined into an excited Cs₂* molecule, which then decays to a molecular Cs₂⁺ ion and an electron. One of the possible schemes for this reaction was suggested in [23]:

1) Cs + e « Cs* + e

2) Cs* + Cs* ® Cs₂* ® Cs₂⁺ + e

The energy D E, required to accomplish this reaction is given by D E = E^(m)_ion - D^(m), where E^(m)_ion is the ionization energy of the Cs₂ molecule, and D^(m) is the dissociation energy for the break-up of the Cs₂ molecule into two Cs atoms. If the data of [24] are used, according to which E^(m)_ion = 3.17 eV and D^(m) = 0.45 eV, then D E = 2.72 eV. Therefore, to accomplish the reaction by the scheme considered above, it is sufficient that the Cs* atoms be excited to the resonance level (the excitation energy of each cesium atom to the resonance level is E₁ » 1.41 eV).* The reciprocal process to the above ionization scheme is called dissociative recombination.

*The value E^(m)_ion = 3.28 eV is reported in [4]. In this case D E = 2.83 eV, so that D E essentially coincides with 2E₁.

As already indicated in §1, at plasma parameters typical for the TIC, and in the presence of total thermodynamic equilibrium, the concentration of molecular ions in the plasma is always low. According to theoretical estimates [25] and to experimental data [36-28], a similar situation also occurs in the non—equilibrium cesium plasma of a TIC. The low concentration of molecular ions generally does not mean that associative ionization and dissociative recombination are not important compared to impact ionization-recombination. Molecular ions have a large enough recombination coefficient in comparison to atomic ions, but little energy of dissociation.* Measurements of the rate of dissociative recombination, carried out in [4], indicated that dissociative recombination plays essentially no role at P_Cs » 1-20 torr and T_e > 2000° K. The presence of molecular ions under TIC conditions apparently may affect the properties of the plasma in the comparatively cold pre-anode region. There, recombination is dominant over ionization in only some cases. However, the rate of recombination near the anode is usually low (see §6, Chapter 9) and usually has a weak effect on the distribution of the TIC parameters.

Calculating the ionization-recombination rate using the approximation of electron diffusion through the upper excited levels. We shall briefly discuss some of the other possible approaches to the problem of multi-stepped ionization. As was indicated above, one can assume in a sufficiently low-temperature plasma that the lower excited levels are approximately in equilibrium with each other. In this case, one needs to take into account the deviation from equilibrium only in the upper excited levels. Since the upper excited levels are comparatively close to each other, the transition probabilities and relative populations hardly vary in going from one level to another. Because of this, the diffusion approximation can be used where the electron flow through the excited levels is proportional to the gradient of the relative population of these levels, i.e., to the gradient in the n _k. In this case, since the upper atomic levels are hydrogen-like, an analysis appropriate for all atoms can be carried out. The situation is also facilitated by the fact that the electron may be regarded classically in upper excited levels, i.e., the discreteness of the levels can be disregarded. This approach of electron diffusion in energy space was carried out in [29-31].@

The following expression was obtained in [29] for the recombination coefficient a of multi-charged ions in a dense low-temperature plasma:

Here z is the ion charge. Since the value of a is weakly dependent on z, the expression obtained can be extrapolated for the case of a singly charged ion, z = 1. Using relation (3.9) and the Saha formula (1.11), by the known value of the recombination coefficient, we can calculate the ionization cross section s (T_e):

*The dissociation energy for molecular ions is D^(mi) = E_ion + D^(m) - E^(m)_ion.

@Electron diffusion, keeping the discreteness of the levels, was considered in [5, 32].

The values of the effective ionization cross section, calculated by this method from formula (3.15) for a Cs atom (E_ion = 3.89 eV), are shown in Fig. 5.6. It is obvious that this calculation yields a satisfactory result for Cs.

Expression (3.15) was obtained on the assumption that the electron diffuses in energy space as a result of collisions with free electrons. In a very weakly ionized plasma, electron diffusion in energy space is no longer by electron-electron, but by electron-atom collisions. This recombination mechanism was considered in [30].

Radiation does not play a significant role in determining the populations of excited states in a sufficiently optically dense plasma. However, radiation may affect the energy balance of the plasma and even the value of the electron temperature T_e. Since the ionization rate G _ion is strongly dependent on electron temperature, then, in a number of cases, consideration of radiation may be required for correct calculation of the ionization rate.

The expression for the radiant flux from the plasma. The shape of the output emission line. Since the populations of excited levels in a low-temperature plasma decrease rapidly as the level energy E_k increases, then emission from the lower levels, and in particular, resonance emission, plays the main role in the energy balance. To illustrate the main characteristics of energy transfer during emission in narrow lines, we can calculate the energy radiated from a plane layer of plasma of thickness L [33-35]. To do this, we introduce Einstein’s coefficients for photons of a given frequency w :

where L_lk(w ) is the shape of the line, normalized to unity (see §8, Chapter 3). Because of the narrowness of the emission line, A_lk(w ) and B_lk(w ) are related, as previously, according to relations (3.7.5) and (3.7.8).

We can now calculate the emission absorption coefficient k(w ). Let a plane of radiant flux be propagated in the medium with spectral density of S(w , x) = cr (w , x), where c is the speed of light, and r (w , x) is the energy density. The amount of energy absorbed per unit volume per unit time, according to (3.7.2), is equal to dS(w , x)/dx = - h@ w _lkN_kB_kl(w , x)r (w , x). By definition,

an element of volume d³r, is equal to

where q is the angle of incidence of emission to the surface and exp{- ò ₀^x (w , x')/cosq )dx'} is the fraction of photons which have traversed the path

If the plasma is homogenous, and if the populations of excited states N_k and N_l, and also the shape of lines, L_lk(w ), are not dependent on x, then, by substituting dr for dx/cosq in (4.5) and by introducing the new integration variable u = 1/cosq , we obtain

where Ei(z) is the tabulated exponential integral Ei(z) = ò _z¥ t^-1e^-tdt.

Expression (4.7) determines the shape of the emission line emerging from the plasma. If the plasma is optically transparent to photons of a given frequency (k(w )L « 1), then

In the opposite limiting case, when the plasma is optically dense (k(w )L » 1), only the first term can be retained in the right side of (4.7). If the ratio of the populations of the l - and k-levels is calculated by the Boltzmann formula (3.7.6), then from (4.3), (4.7), (3.7.5), and (3.7.8) we obtain for a dense plasma (k(w )L » 1),

where r _Te(w ) is the density of equilibrium Planck emission, given by formula (3.7.7).*

Thus, the spectral emission density in the center of the emission line, so long as k(w )L » 1, is calculated by the Planck formula. For k(w )L » 1,S_lk(w ) decreases (Fig. 5.9), and in the range of k(w )L « 1, becomes equal to expression (4.8) for an optically transparent plasma.

The shape of the emission line for an equilibrium plasma layer can be calculated by the principle of detailed balance: we merely assume that an equilibrium, Planck radiant flux impinges on the plasma and calculate the difference between the incident and exit flux. Since a plasma in a state of thermodynamic equilibrium emits as much energy as it absorbs, the shape of the emission line should coincide with the shape of the absorption line. It is obvious that at k(w )L » 1 the emission is completely absorbed, whereas at k(w )L « 1 the plasma is transparent to emission. In general, the shape of the emission line obtained is that depicted in Fig. 5.9.

The width of the emission line, emerging from the plasma, W _lk is given to an order of magnitude by the condition k(w ± W _lk/2)L = 1. If we assume that the shape of the emission line of an individual atom is given by expression (3.8.1), then

where k₀ is the absorption coefficient in the center of the line. If k₀L » 1, then W _lk » g ₀. In this case, disregarding the value of g ₀²/4 in the numerator of (4.10), we find that W _lk = g _0Ö k₀L@. The total energy carried in the line is calculated by the expression

where D w = w - w _lk. The main contribution to S_lk is made by the values of D w » g ₀/2. Therefore, the first term in the numerator of (4.10), as before, can be disregarded. Then by using (4.3) and (4.6), and also (3.7.5), (3.7.7), and (3.7.8), disregarding the induced emission, and assuming that the levels are populated in a Boltzmann distribution,

Performing an integration over @ in (4.12), we obtain

The characteristics of propagation of line emission. The probability of a photon emerging from the plasma. It is certain that the energy carried in a line is dependent on the thickness of the plasma. The dependence on the thickness is explained by the fact that the photon mean free path in the distant wings of the atomic emission line, where the absorption coefficient is small, may be very large and may even be an order or more larger than the dimensions of the emitting system. This is the principal difference of propagation of resonance emission from particle diffusion. In the latter, free particle motion from a given point occurs only a distance on the order of the mean free path. It is for this reason, in calculating the resonance emissive flux (or generally, emission in a line), that integration must be distributed over the entire system rather than be limited by distances on the order of some average mean free path. Therefore, the photon flux for resonance energy may not be expressed by the values of the concentration of excited atoms in direct proximity to the considered point, and a diffusion coefficient may not be introduced, as is done when calculating the particle flow in a slightly heterogeneous plasma (for this problem see [33, 34]).

The absence of a local relationship between the emissive flux in a line and the gradient of the excited atom concentration leads to the fact that a self-consistent calculation of the populations of excited states for a heterogeneous plasma with simultaneous consideration of emission and absorption of photons in a spectral line is a very complicated problem. To simplify this problem, the concept of the effective probability of a photon emerging from the plasma, w^(ph), is introduced. The parameter w^(ph) is understood as the ratio of the number of photons leaving the plasma to the number of photons emitted in the entire volume. Since emission emerges from the entire volume of the plasma, rather than only from the surface layer, and since the emissive flow in a spectral line is rather weakly dependent on the thickness of the emitting

*When calculating S_lk is more convenient to use formula (4.6) than (4.7), because integration of individual terms over w in the latter leads to divergent expressions.

layer of the plasma (see (4.14)), we can assume, approximately, that the value of w^(ph) is identical for all the emitting atoms. As a result a probability for spontaneous emission, equal to w^(ph)_lk× A_lk is attributed to each atom.

The energy emitted in a line for a layer of a homogeneous plasma can be calculated as 2S_lk (the multiplier 2 takes into account that emission emerges in both directions). The total energy emitted by a plasma layer of thickness L, according to (3.7.15), is equal to S₀ = L× D S_lk. Then, by using (3.7.5)-(3.7.8), (3.7.15), (3.8.1), (4.1), and (4.3), we find from (4.14) that

The absorption coefficient k₀ for the center of the line occurs in (4.15). The value of k₀ is calculated from (4.3). By substituting (3.7.5), (3.7.8), and (3.8.1) into (4.3), we obtain

where l _lk is the emission wavelength.

Thus, in an optically dense plasma, i.e., where k₀L » 1, there is an appreciable increase in the lifetime of an atom in an excited state, because of reabsorption of the emission. In particular, under the conditions found in the cesium plasma of a TIC, reabsorption of emission causes de-excitation by electron collision to be more probable than optical de-excitation, even in the lower region of the level sequence where optical de-excitation of atoms occurs more frequently in a number of cases than electron de-excitation.

In the plasma layer located between the electrodes, an increase of the effective lifetime of the atom in an excited state may also occur due to reflection of the emission from the electrodes. If we denote the energy flux incident to the electrode and experiencing n reflections prior to this by S⁽ⁿ⁾_lk, then in a manner similar to (4.5), we obtain

Here r is the reflection coefficient of the electrode (which gives the fraction of energy reflected from the electrode), and the factor exp[-nk(w )Lu] takes into account attenuation of the emission after n reflections of the path between the electrodes. The quantity L/cosq is the path covered by the photon between the electrodes (Fig. 5.10).

With reflection from the electrodes, the effective probability of a photon emerging from the plasma w^(ph) is equal to

Using the probability of a photon emerging from the plasma, w^(ph), we can approximately calculate the energy lost by emission per unit volume of plasma:

It is interesting to compare the radiation energy losses per unit volume of plasma, D S_rad, to ionization energy losses, D S_ion = E_ionG . In a low-temperature plasma the main contribution to D S_rad is resonance radiation. Then, if the lower excited levels of atoms are populated in a Boltzmann distribution (N_i = N⁽⁰⁾_i(T_e)), and recombination is insignificant at a given point of the plasma, then, from (3.8) and (2.12), we can find the condition of the smallness of D S_rad compared to D S_ion:

Here t _eff is the averaged effective lifetime of the first excited state (for a Cs atom 1/t _eff has the value of A₁₀w^(ph)₁₀, averaged for the states 6P_1/2 and 6P_3/2).*

In the developed arc mode of a TIC, condition (4.22) is fulfilled in most cases, so that the relative contribution of D S_rad to the energy balance is comparatively small.

*The main broadening mechanism which determines the value of t _eff in a TIC plasma is resonance broadening [36].

In a low-temperature plasma, the greater energy loss is from the flow of atoms excited to the first level. This flow is denoted below as i₁. A loss of excited atoms leads to a decrease in their concentration at the interface with the electrode compared to their concentration within the plasma. As a result a layer forms near the electrode which is depleted of excited atoms. If the de-excitation processes occur slowly, so that the atom is scattered many times during de-excitation, the emergence of an excited atom from the plasma is by diffusion. In this case, the flow of excited atoms is proportional to their concentration gradient and is equal to

where D₁ is the diffusion coefficient of excited atoms.

When calculating the diffusion coefficient of excited atoms, we must take into account, along with the ordinary scattering of excited atoms, processes of transfer of excitation to adjacent atoms, which are equivalent to scattering [37]. The transfer of excitation may occur by optical de-excitation with absorption of the emitted photon by the adjacent atom or by resonance transfer of excitation. In the cesium plasma of a TIC, with a pressure P_Cs » 1 torr, the second process is more probable. In this case, the diffusion coefficient of excited atoms is calculated in the constant relaxation time approximation and (according to (4.3.13), (4.3.14), and (4.3.20)) is equal to D₁ = (kT/M_a)t . The parameter t is the lifetime of the atom in an excited state, which is related to the probability of resonance transfer of excitation by the expression l/t = g ₀/2.*

Variation of the flow of excited atoms in the layer next to the electrode is related to collisions of first and second kind and to radiative transitions. If collisions of second kind are prevalent over radiative de-excitation, as a result of low probability w_(ph) of a photon emerging from the plasma, or due to a large density of electrons n_e, then in the presence of an equilibrium energy distribution for electrons in the electrode layer, we have

The right side of (4.24) determines the variation of the population N₁ of the first excited level as a result of transitions of the atomic electron to the ground state (k = 0) and to other excited states (k = 2, 3, . . ., g_m) as a result of collisions with free electrons. If transitions between the first excited and ground states occur most frequently @ then

*An oscillator with amplitude A(t), damped as A(t) ~ exp(-t/t ), may be compared to the emitting atom (see chapter 3, §8) if g ₀ = 2/t .

@This occurs with a Boltzmann distribution of excited atoms if E₁₂ » kT (see (2.19)).

Here n ₁ is the relative population of the first excited state, calculated by formula (2.11), and L₁ = Ö D_{1t 1}@ is the de-excitation length. (t ₁ = (n_ew₁₀)^-1 is the de-excitation time of the first level by electron collision.) To an order of magnitude, L₁ is the distance which an excited atom first covers before it is de-excited due to a collision of second kind.

As the distance from the electrode in a low-temperature plasma increases, n ₁ ® 1 (see the preceding section). At the interface with the electrode, we have n ₁ « 1, that is, if the de-excitation time by electron impact, t ₁, is much greater than t (the time of resonance transfer of excitation) and if the emission of excited atoms from the electrode is insignificant. The solution of equation (4.26) with the boundary condition n ₁ = 0 at x = 0 has the form

The free-electron distribution function was assumed above to be Maxwellian when determining the populations of excited states and during calculation of the rate of ionization-recombination. As already indicated in §2, inelastic collisions between electrons and atoms, which accompany ionization and recombination processes, generally affect the nature of the free-electron distribution function. If inelastic collisions between electrons and atoms occur more intensively than the processes which lead to a Maxwellian distribution among the free electrons, then the electron distribution function begins to deviate from the Maxwellian function [11, 38-45]. In turn, the absence of a Maxwellian electron distribution alters the expression for the rate of ionization-recombination in a plasma. Deviations from the free-electron Maxwellian distribution will be taken into account in this section, and a more general expression will be presented for the rate of ionization-recombination in a plasma which includes the effect of these deviations.

Since ionization and recombination in a low-temperature plasma are multi-step processes, in order to calculate the value of G , one must know the electron energy distribution function n(E) up to values of E_max » E_k,k+1 + kT_e where E_k,k+1 is the distance between the most widely separated adjacent levels in the atomic spectrum. For atoms of alkali metals, hydrogen, and most other elements, the ground state and

first excited state are separated farthest form each other and E_max » E₁ + kT_e.

The extent of the deviation of the free-electron distribution function n(E) from the Maxwellian distribution function is determined by the ratio of G to the intensity of the process for free electron Maxwellization. The intensity of this process is equal, to an order of magnitude, to E n(E)/t _E, where t _E is the effective Maxwellization time. Since the electron concentration n(E) decreases exponentially as energy E increases, deviations of the distribution function due to the ionization-recombination process are more significant in the range of higher energies E » E₁ and decrease exponentially as energy E decreases. Therefore, if the electron temperature T_e is sufficiently low, we need to take into account only deviations from the Maxwellian distribution related to resonance transition, i.e., transition of an atom from the ground state to the first excited state. We note that the emergence of resonance radiation from a plasma, the same as ionization, causes a prevalence of processes of electron excitation of the atom to the first level or of de-excitation processes and, therefore, reduces the density of fast electrons compared to the equilibrium value.

Variation of the electron distribution function due to interaction with discrete levels. The rate of variation of the free-electron distribution function n(E) due to interaction with two discrete levels k and l is expressed, in the general case, in the following manner:

The right side of (5.1) is the sum of four terms. The first term corresponds to the excitation of an atom from state k to state l and to transition of a free electron from a state with energy E to a state with energy E - E_kl. The second term corresponds to the de-excitation of an atom from state l to state k with a change of electron energy from E - E_kl to E. The third term corresponds to the de-excitation of an atom from state l to state k with the transition of an electron from a state with energy E to a state with energy E + E_kl. The

*In many cases, the deviation of the fast electron distribution from a Maxwellian distribution is also related to the presence of an electric field (see §6, Chapter 4 and [46-48]). However, in a TIC plasma, the electric fields in the plasma column are usually weak and do not significantly affect the form of the electron distribution function.

In the case of a resonance transition, when E_kl = E₁ » kT_e, the interaction of a fast electron with energy E » E₁ with an excited atom, as with the reverse process, is hardly probable, and only the first two terms in (5.1) need to be taken into account.

Let the ratio of the distribution function n(E) to its equilibrium value n_M(E) be denoted by n (E):

For the resonance transition, we have n _k = n · = 1. Moreover, since the distribution function in the range of thermal energies is close to the Maxwellian function, then n (E - E_kl) = 1. Therefore, the variation of the distribution function due to resonance transitions is equal to

where n ₁ is the relative population of the first excited level, calculated by formula (2.11).

We note that each atomic excitation event is equivalent to the loss of a fast electron. On the other hand, de-excitation of an atom is equivalent to generation of a fast electron. Therefore, the resulting number of atomic transitions from the ground state to the first excited states is equal to the loss of the number of fast electrons with energy E > E₁:

When calculating the populations of excited states, transition from the ground state to only the first excited state needs to be taken into account. Then @ should be set equal to the value of G . To illustrate this circumstance, let us consider a specific scheme of the levels of a cesium atom (see Fig. 3.6). Level 5D is the next level after resonance level 6P. Level 5D is populated due to transitions from the ground state (6S « 5D) and due to transitions from the first excited state (6P « 5D).

However, if the Bethe-Born approximation is used to calculate the transition probabilities, then the transitions 6S « 5D, like those prohibited in the dipole approximation, will generally have zero probability. If the classical Thomson or Grizinski formulas are used, then the probability of the transition 6S Õ 5D, although distinct from zero, will be very small compared to the probability of transitions 6P Õ 5D as a result of a decrease of the cross section as the transition energy increases. For example, with a Maxwellian free-electron distribution function and Boltzmann population of the lower excited states, the ratio of the number of transitions 6S Õ 5D to 6P Õ 5D, according to (2.19), is equal, to an order of magnitude, to

For Cs, E₁₂ = 0.36 eV, E₁ = 1.4 eV, and W₀₂/W₀₁ « 1 (a similar situation also occurs for hydrogen, where the excited levels are closely adjacent to each other).

If resonance radiation processes also have an effect, along with ionization, on the form of the fast electron distribution function, then F₀₁ should be set equal to the sum of G and the number of resonance emission events per unit volume of the plasma (see below).

The equation for the fast electron distribution function. In steady state, the loss of fast electrons with energy E » E₁, due to ionization and emission processes, is compensated for by electron flow in energy space from the region of thermal energy as a result of collisions. If the energy exchanged during each collision is low compared to kT_e, then the motion of the electron in this energy space may be regarded as diffusion. We are referring to electron-electron collisions, which are the main mechanism for electron motion in energy space in a TIC plasma.

The change in the energy distribution n(E) due to electron-electron collisions is expressed as

where i_Ec(E) is the collisions (compare formula (4.5.15)). The flux i_Ec is related to the electron flux i_vc in the velocity space by the relation

By substituting (4.6.10) into (5.7) and by converting from the velocity distribution function f_e⁽⁰⁾ (v) to the energy distribution function n(E) (see footnote on p. 127), we obtain

The quantity D_E(E) is the fast electron diffusion coefficient in energy space (compare to (4.6.9)). Expression (5.9) for t _E(E) follows from (3.1.14) and (3.2.10) if in (3.2.10) m is replaced by the reduced mass for two electrons, m/2. When converting from (4.6.10) to (5.8), the derivative with respect to E of the fast electron velocity v was disregarded, because the velocity v, compared to n(E), is a very slow function of energy E. According to this, by substituting the Maxwellian distribution function n_M(E) into (5.8), the derivative with respect to E of E^1/2 should also be disregarded on the right side of (5.8). In this case, we have i_Ec = 0 if the electron distribution function is equilibrium, i.e., if n(E) = n_M(E). If the distribution function n(E) is not equilibrium, then, by using (5.2) and (5.9), we obtain from (5.8) the following expression for the electron flux in energy space:

For E > E₁, the derivative di_Ec(E)/dE is equal to the fast electron losses due to processes of atomic excitation to the first level. For E < E₁, the electrons cannot excite atoms to the first level. In this lower energy range, the flux i_Ec should be constant, if, of course, the effect of transitions between other atomic levels, and also the flux of electrons from region E > E₁ to region E < E₁ during the excitation of atoms to the first level are not taken into account.*

*These electrons arrive primarily into the region of thermal energies E » kT_e. This energy range will actually not be considered henceforth.

By substituting expressions (5.3) and (5.10) for I₀₁{n(E)} and i_Ec(E) into equations (5.11) and by disregarding the derivative dD_E/dE, we obtain the differential equation for calculating n (E):

The dependence. of v and D_E on energy may be disregarded on the right side of (5.12) for kT_e £ E₁, and we may assume that E = E₁, since the significant range of energies has a width of about kT_e near the excitation threshold. Further, we assume that the excitation cross section near the threshold is a linear function of energy.*

When solving equation (5.12), the regions E ³ E₁ and E £ E₁ must be considered separately. If we convert to new variables

The dimensionless parameter g is the ratio of the energy relaxation time t _E to the free path time for electrons which excite atoms to the first level:

which is the time for an energy relaxation equal to kT_e, is in the numerator of (5.16) rather than the total energy relaxation time t _E. The appearance of t _E^* instead of t _E comes from the fact, as will be shown below, that deviation of the distribution function from a Maxwellian distribution, caused by ionization, occurs only in an energy range the order of kT_e near the excitation threshold E₁, and therefore, that the electron need exchange only a fraction ( » kT_e/E₁) of its energy for Maxwellization.

*According to [49], s₀₁ = 1.5× 10^-14 cm₂/eV (see also [50-52]).

Then, for z(x) for x ³ 0, we obtain the following equation:

Since n (E) should be limited as E ® ¥ , we require

Expression (5.21) is one of the boundary conditions to equation (5.20). The second boundary condition, the value of z(0), should be obtained as a result of sewing together the solutions for x £ 0 and x ³ 0 at the point x = 0.

The rate of resonance excitation of atoms. The form of the distribution function near the excitation threshold. Solution of equation (5.20) with boundary condition (5.21) is expressed by the MacDonald function K_1/3 (see, for example, [53]):

For x £ 0, having set i_Ec(E) = const, we obtain from (5.2) and (5.10) the relationship between the rate of atomic excitation to the first level, F₀₁, and n (E):

For E » kT_e, expression (5.23) can be simplified:

Formulas (5.22) and (5.23) are solutions of equation (5.12) for E ³ E₁ and E £ E₁, respectively. The integration constants z(0) and F₀₁ follow from the continuity conditions n (E) and dn (E)/dE at E = E₁.

On the basis of expressions (5.10), (5.14), (5.19), and (5.22), we can calculate the value of i_Ec(E) for E ³ E₁. Having set the value of i_Ec(E) at E = E₁ equal to the rate of excitation of atoms to the first level, we find the expression for z(0), or which is the same thing, for n (E₁):

Using the properties of the MacDonald functions [53]:

The graph of function C(g ) is presented in Fig. 5.12. If the plasma is sufficiently optically dense and if the emission yield does not affect the fast electron distribution function, then (5.30) is also the rate G of ionization-recombination.

From (5.14), (5.19), (5.22), and (5.28), we obtain the expression which gives n (E) for E ³ E₁:

It is obvious that n (E) is distinct from unity, i.e., the distribution function n(E) is distinct from the Maxwellian distribution n_M(E).

At E £ E₁, the expression for n (E) is obtained from (5.25) and (5.30):

The nature of the function n (E) is illustrated by Fig. 5.13. It is obvious that n (E) decreases from 1 to n ₁ near the excitation threshold of the first level, with the variation in n (E) limited to a comparatively small energy range D E of the order of several kT_e. It is this circumstance that permitted us to disregard the dependence on E in the coefficients of equation (5.12).

We may note, at small values of g , when t * « t ₀₁, Maxwellization of free electrons occurs rapidly and inelastic processes do not distort the free electron distribution function near the excitation threshold. Actually, in this case, by using the asymptotic representation of Kn (z) = (p /2z)^1/2e^-z(z » 1) [53], we find from (5.31) that in the some energy range near the threshold, where g x = (t _E/t ₀₁)× (E - E₁)/E₁ « 1,

i.e., n (E) = 1 and, according to (5.2), n(E) = n_M(E).

On the other hand, at large values of g , when Maxwellization occurs slowly compared to the rate of atomic excitation distortion of the electron distribution function is stronger. In this case, the range of values of E, where the transition occurs from n (E) = 1 to n (E) = n ₁, is shifted toward lower energies and is localized near the excitation threshold. Actually, by assuming E = E₁ (i.e., x = 0) in (5.31), and taking into account that K_1/3(z) ~ z^-1/3 and K_2/3(z) ~ z^-2/3 for z « 1 [53], we find from (5.31) that

i.e., the value of n (E) = n ₁ (see Fig. 5.13) is reached in this case at E = E₁.*

Change in the free electron distribution function as a result of non-resonance transitions. Electron diffusion in energy space during interaction with discrete levels. Besides resonance transitions, the distribution function may also be affected by non-resonance atomic transitions-transitions between excited states [11, 12, 45]. In cases of practical interest to the TIC, it is sufficient to consider the cesium transitions 6P ¹ ? 5D in order to calculate the effect of non-resonance transitions on the fast electron distribution function. Because of the wide energy gap above level 5D, levels 6P and 5D may be assumed to be in equilibrium with each other; therefore, their relative populations are given by the Boltzmann relation (3.7.6). We note that the energy separation of levels 6P and 5D is small compared to the fast electron energy E.

The rate of change I_kl{n(E)} of the distribution function during the interaction with two discrete levels is determined by expression (5.1) for E_kl « E, by using (3.7.6) and (3.5.6), we obtain from (5.1)

*Expressions (5.33) and (5.34) agree with formula (5.32), because C(g ) ® 0 as g ® 0, and C(g ) ® 1 as g ® ¥ (see Fig. 5.12).

To simplify it, the case where levels k and l are located close to each other, so that E_kl « kT_e, will be considered in this section. The right side of (5.35) may then be expanded in powers of the small value E_kl/kT_e. In this case, the least non-disappearing terms of the expansion are of the order of (E_kl/kT_e)²n(E), because dn(E)/dE » n(E)/kT_e, and d²n(E)/dE² » n(E)/(kT_e)². As a result we obtain

If the dependence of coefficient vQ_kl(E) is disregarded, then the right side of (5.36) can be represented as the energy derivative from some flux i_Ec(E) in energy space (see (5.6)). In this case i_Ec(E) and D_E are expressed as before by formulas (5.8) and (5.9), in which t _E is now the time required for energy exchange due to the interaction of free electrons with closely adjacent discrete levels:

Thus, if the discrete levels of an atom are located near each other, the motion of a free electron in energy space due to interaction with these levels, as previously, may be regarded as diffusion.* With simultaneous consideration of inelastic interactions and electron-electron collisions in the approximation being considered, the effective energy relaxation time l/t _E is the sum of both types of energy relaxation times:

To be more general, other energy transfer mechanisms may also be taken into account similarly on the right side of (5.38), if the value of the energy exchanged during each collision is small compared to kT_e. For example, this is the case for elastic collisions between electrons and atoms.

The rate of multi-stepped ionization-recombination and the deviation of the electron distribution function from equilibrium. The expressions found for the fast electron distribution function are dependent on the relative population of the first level n ₁. To calculate n ₁, the rate of ionization G from the first excited level must be calculated. The transitions between the excited levels involve mainly electron transitions

*If the levels being considered are not in equilibrium with the free electrons, then T_e should be replaced by the effective excitation temperature T^(kl)_eff = E_kl/k ln(g_lN_k/g_lN_l) in (5.9) and (5.37).

for energy E < E₁ Since the electron distribution function for E < E₁ according to (5.32), is close to a Maxwellian distribution, then the derivation of the expression for the ionization rate from the first level is similar to the derivation of (3.7), and the following expression is obtained for G :

where s ₁(T_e) is the equivalent ionization cross section for the resonance level.

The optical de-excitation of the levels was disregarded both for the derivation of expression (5.3.7) and for the derivation of (5.39), because the effective optical lifetimes of the excited states in a TIC plasma (due to radiation reabsorption) considerably exceed the reverse probabilities of de-excitations by electron impact. However, if the process of fast electron Maxwellization proceeds slowly and if non-equilibrium of the fast electron distribution function occurs, then consideration of resonance radiation may still be significant when calculating the populations of the excited states and the rate of ionization, that is, if the number of effective optical de-excitations from the first level (with radiation reabsorption) is comparable to the rate of generation of fast electrons in the plasma. The latter is given by the right side of (5.30).* Under these conditions the population of the first excited level n ₁ is calculated from the equation

where t _eff is the effective lifetime of the first excited state (see §4).

g₀ and g₁ are the statistical weights of the ground and first excited states. Formulas (5.39) and (5.41) give the value of the ionization-recombination rate G .

The factor C(g )/g in (5.42) takes into account the final velocity acquired by the fast electrons. For g « 1, C(g )/g » 1. In this case, Maxwellization occurs rapidly, and the rate of de-excitation of atoms to the first level may be calculated by assuming that the free electrons have a Maxwellian energy distribution.

*See, for example, [11, 18, 36, 42, 44, 45].

For g » 1, C(g ) » 1. Then, by using formulas (5.16)-(5.17) and expression (2.14) for n_M(E), we find from (5.42) that

In this case, F is the ratio of the rate for the fast electron energy relaxation process to the rate for ionization of excited atoms.

The results from calculations for a cesium plasma. If the free electron concentration n_e is sufficiently high, when intensive Maxwellization of fast electrons occurs, F is dependent only on T_e. Under these conditions, radiation loss from a cesium plasma of a TIC is usually not reflected in the value of n ₁, i.e., [n_ev_e@ s ₁(T_e) t _eff]^-1 « 1 + F. By substituting (5.41) into (5.39) and disregarding n_ev_e@ s ₁(T_e) t _eff, we obtain expression (3.7) for the ionization-recombination rate G , in which

Formula (5.45) determines the relationship between cross sections s ₀(T_e) and s ₁(T_e). The calculated value of s ₁(T_e) for a cesium plasma is presented in Fig. 5.6. As indicated by calculations, for T_e < 3000° K, the value of F^-1(T_e) can be disregarded compared to 1. Then the following simple relation is obtained between cross sections s ₀(T_e) and s ₁(T_e):

Since the ionization rate increases more rapidly than the rate of fast electron formation as electron temperature increases, the deviation of the free electron distribution function from a Maxwellian distribution becomes more significant as the electron temperature T_e increases. For example, this is obvious from Fig. 5.14, where the values of the fast electron distribution function with energy of E = E₁ and the populations of the first excited level are presented in relative units (that is, n (E₁) and n ₁) as a function of the free electron density T_e at two values of electron temperature T_e.

The values of T_e are selected so that the de-excitation probability for an atom excited to the first level is significantly greater than

the ionization probability. In this case, the equilibrium population of the first excited level (n ₁ = 1) corresponds to the equilibrium free electron distribution function (n (E₁) = 1). Actually, it is obvious from Fig. 5.14 that the population of the first level varies slightly and n ₁ » 1 initially, as n_e decreases compared to n_e(T_e), although a non-equilibrium process-ionization-occurs in the plasma. As n_e decreases, Maxwellization subsequently becomes inadequate and a fast electron depletion occurs, which reduces the population of the first excited level. For n_e > 10¹² cm^-3, the main mechanism of Maxwellization in a cesium plasma is electron-electron collision.* With a further decrease of n_e, however, the role of electron-electron collisions weakens and the interaction of fast electrons with the excited levels of the atoms can emerge to primary importance. The energy relaxation time t _E must then be calculated taking into account both terms in formula (5.38). This complicates the calculations, because parameter g and (together with it) the right sides of (5.41) and (5.42) now begin to depend on n ₁. In this case, formula (5.41) is a transcendental equation in n ₁.

The values of n (E₁) and n ₁, depicted in Fig. 5.14, are calculated for the case where the resonance radiation loss has no effect on the populations of the excited states (t _eff = ¥ ). The role of resonance radiation is illustrated by Fig. 5.15, where the dependence of n ₁ on free electron concentration n_e is presented with and without regard for resonance radiation in a cesium plasma.

It is obvious that the resonance radiation loss leads to an appreciable decrease of n ₁ as n_e decreases.@

Using (5.41), it is possible, in particular, to calculate n ₁ in the case of the lowest possible electron concentration n_e, where electron diffusion in energy space is determined exclusively by interaction with discrete levels. In this case t _E = t ^(1,2)_E , where the superscripts 1 and 2 refer to the 6P and 5D levels, respectively, of Cs. Substitution of (5.37) into (5.17) yields

we obtain, in the absence of recombination (n_e²N₀(T_e)/ n_e² (T_e)N₀ « F), the transcendental equation

*This can be seen in particular, by comparing the curves in Fig. 5.14 with similar curves in Fig. 6 of [11], constructed on the assumption that the only mechanism of Maxwellization is electron-electron collisions. The results of the calculations hardly differ from each other for n_e > 10¹² cm^-3.

to calculate n ₁. Since C(g ) £ 1, in order that equation (5.49) have a solution, it is necessary to fulfill the condition

(in this respect, see [11, 45]). However, W(T_e) decreases as T_e increases in a cesium plasma.

If the value of T_e at which W(T_e) is equal to the right side of (5.50) is denoted by T_e^(cr), then equation (5.49) has no solution for T_e > T_e^(cr).* This means that the populations of the excited states can decrease as much as desired for T_e > T_e^(cr) in the absence of recombination and as the free electron concentration n_e decreases. The populations of excited states approach specific finite limits as n_e decreases for T_e < T_e^(cr). In this case, the limiting value of n ₁ is found from equation (5.49).

For a cesium plasma, T_e^(cr) » 3000° K. Therefore, for T_e > 3000° K, equation (5.49) has no solution even in a sufficiently optically dense plasma, where the resonance radiation loss is insignificant. Taking resonance radiation into account aggravates the situation and reduces the value of T_e^(cr). We note that the consideration of inelastic processes carried out here is approximate, since for cesium E₁₂ » T_e. However, because of the absence of adequately precise data for the cross sections of elementary processes, the approximation used here is totally within the limits of the accuracy of ordinary calculations.

*Condition (5.50) can be obtained from (5.40) if we require that the rate of stripping of the first level by ionization and emission, i.e., G + N/t _eff = N₁[n_ev_e@ s ₁(T_e) + 1/t _eff], does not exceed the maximum rate of fast electron production due to the interaction with discrete levels, i.e., the value D_En_M(E₁)/kT_e = E₁n_M(E₁)/t _E⁽¹²⁾(E₁) (see (5.30), where t _E⁽¹²⁾ is calculated by expression (5.37). Since the rate of fast electron production increases more slowly than s ₁(T_e) as T_e increases, equation (5.40) may no longer have a solution if T_e increases.