Following the initial stage, a sequence of plasmochemical processes occur in the plasma, tending to return it to its equilibrium state. An important place among such processes is held by charged particle recombination reactions, as a result of which plasma neutralization and the formation of charged atoms occur.

Kinetics of Plasma Processes for a Single-Component Mixture

For a single-component gas medium A at atmospheric pressure, the basic recombi­nation processes are:

1. collisional-radiative recombination

A+ + e + e! A * +e, (4.6)

2. three-body recombination

A++ e + A! A *+A, (4.7)

3. dissociative recombination

A++ e! A *+A. (4.8)

Molecular ions A+ are formed as a result of ion conversion:

A++ 2A! A++ A, (4.9)

and the rate constants of this process differ insignificantly for all ions of rare gases and amount to kic = (0.6-3.5) x 10~31 cm6/s at a gas temperature of Tg = 300 K [44].

As a result of recombination processes (4.6) and (4.7), initially highly excited atom states are formed, which are stabilized through collisions with a third particle (electron or atom). Further relaxation of excited states occurs as a result of colli­sions with plasma electrons and gas atoms, and through spontaneous decay. A special place among the recombination processes is held by the dissociative recom­bination reaction (4.8), which at a high pressure often is not only the basic process of neutralization of charged particles, but also one of the chief channels for the formation of excited particles, including populating of the upper lasing levels of NPLs [45]. In the case of dissociative recombination, the binding energy of the recombination electron is converted into the kinetic energy of recession of the atoms.

Values of the rate constants of recombination processes differ greatly and depend first of all on the electron temperature Te. For recombination processes (4.6) and (4.7) of atomic ions A+, the rate constants equal:

• for collisional-radiative recombination kcr = 4 x 10~9Te~45cm6/s (Te, K) for any atomic ions of rare gases [46];

• for three-body recombination ktrк(0.5-30) x 10~22Te~25cm6/s (Te, K) for A = Xe, Kr, Ar, Ne, He [47].

Rate constants of the dissociative recombination (4.8) for molecular ions of rare gases are shown in Table 4.8 [48]. The dependence on electron temperature for kdr is significantly weaker than for kcr and ktr, so with an increase in Te, the influence of dissociative recombination grows sharply.

For clarity, let us examine an argon plasma and estimate the characteristic times of plasma processes for q к 0.01-5 kW/cm3, which occur during operation of gas NPLs. Given an atmospheric pressure of the argon, Teк0.5-2.0 eV; ne к 5 x 1012-

2 x 1014 cm-3 and Z ~ 2 x 10-7-7 x 10-5. A comparison of the characteristic times of the recombination process Tcr = (kcrne2)-1« 1 s, Ttr = (ktrne[Ar])-1« 0.3 s ([Ar] = 2.7 x 1019 cm-3, the concentration of Ar atoms), the process of ion conversion Tic = (kic[Ar]2)-1« 5 x 10-9 s and dissociative recombination t^ = (kdrne)-1 « 1 x 10-7-2 x 10-6 s shows that in argon plasma, neutralization of charged particles occurs exclusively by the channel of dissociative recombination of molecular ions.

A similar conclusion can be drawn for plasma based on Ne, Kr, and Xe. First of all, for helium plasma, the constant kdr is two or three orders of magnitude lower; secondly, the process of electron thermalization occurs more efficiently, and accordingly, Te is significantly less. In this case, one cannot entirely neglect the recombination processes (4.6) and (4.7) with the participation of atomic He+ ions.

Balance equations of charged particles in plasma for the conditions examined above, without taking into account recombination processes (4.6) and (4.7), can be written as follows

XF1 = f+- к’сИ+1М2.

ІИ = ксИ+ЦД]2 — kdr [A+ ]ne,

ne = [A+] + [A+],

where [A+] and [A2+] are the concentrations of atomic and molecular ions, f+ = q/wi is the rate of ion formation (w, is the energy cost of forming an ion-electron pair). For Eq. (4.10), the solution has the simple form [2, 49]:

[A+] = f+Tic [1 — exp( tjTic)], Tic = k;c [A]2 , (4.13)

which was obtained given the condition that [A+] = 0 for t = 0.

In the plasma of gas NPLs, the characteristic times of all the main plasma processes are considerably less than the duration of the excitation pulse (minimal duration of reactor pulse is ~50 ps), so that for q « 0.01-5 kW/cm3, a quasi-stationary mode is established in the plasma over a time on the order of (f+kdr)-1/2 « 5 x 10-8­2 x 10-6 s. The concentration of electrons and ions can be found from the solution to Eqs. (4.10)-(4.12), if we equate the values of the derivatives with zero:

ne = f-T-c (vS+T + 1); [A+] = f+Tic;

[Д+]f (vffiT-1),S=f+rpd;. (4Л4)

The characteristics of the plasma being examined depend significantly on the dimensionless parameter S, the insertion of which makes it possible to carry out a
convenient classification of the excitation conditions. With so-called “weak” exci­tation, the condition S > 1 is fulfilled, and from the expression (4.14) we obtain:

n — [A+] = f >>[A+]. (4.15)

In the case of “weak” excitation, the non-stationary system of Eqs. (4.10)-(4.12) has an analytical solution [49]:

In the case of “strong” excitation (S ^ 1) we have:

If we assume S = 1, it is possible to obtain the expression for the degree of plasma ionization, Zws, which separates the “weak” and “strong” excitation regions:

For example, for argon at atmospheric pressure, — 5 x 10~4 (Te — 0.5-2 eV). In experiments with pulsed reactors, q < 5 kW/cm3 and Z < 10~5 ^ Zws, so that the estimate of parameters of the plasma of gas NPLs can be made in an approximation of the “weak” excitation.

For the conditions in which atmospheric pressure gas NPLs were studied with using of pulsed reactors, there is “weak” excitation (S > 103), so that for evaluative calculations of the plasma characteristics, one can use the formula (4.15). The validity of Eq. (4.15) is confirmed, for example, in experiment [50], where the electron concentration was measured for excitation of neon (PNe = 0.24 atm) by a proton beam with an energy of 20 MeV in a range q — 2 x 10~5-5 x 10~2 W/cm3 (S > 104).

The balance Eqs. (4.10)-(4.12) are written on the assumption that neutralization of the charged particles occurs exclusively owing to the dissociation recombination of molecular ions. This assumption is based on estimates of the rates of recombi­nation processes, and it was assumed that the values Te are known, and for gas NPLs are 0.5-2.0 eV.

To determine Te, it is necessary to consider the energy balance of plasma electrons. Studies [2, 49] analyzed the basic processes leading to establishment of some average energy of plasma electrons e. There are four such processes: (1) for­mation, in the interval 0 < £e < Im, of electrons with an average energy 7m/2, exceeding ee; (2) dissociative recombination resulting in the disappearance of the

Z	1	2	4	6	8	10	20
F(Z)	0.53	0.40	0.28	0.23	0.20	0.17	0.11

Table 4.9 Values of the function F(Z) [2, 49]

slowest electrons; (3) inelastic electron-atom processes of atom excitation; (4) elas­tic electron-atom collisions. As a result of the first two processes, electrons are heated, and as a result of the latter two they are cooled.

For the region of “weak” excitation (S ^ 1), the electron energy balance equa­tion has the form [2]:

0

Table 4.9.

As a result of transformations, from expression (4.19) it is possible to obtain a rather simple transcendent equation:

Te = 300 K). The dimensionless constant C is equal to 0.015, 0.03, 0.053, and 0.06, respectively, for Ne, Ar, Kr, and Xe. For large values Z ^ 1, Eq. (4.20) is simpli­fied, since in this case F(Z) ^ 1 (Table 4.9):

e 1/g = Cg,0.5 — g,.

From solving of the approximate Eq. (4.21) given the condition that ne = [A^] (S ^ 1), it follows that the values g = Te/1m for Ne, Ar, Kr, and Xe differ insignif­icantly and are found in the range of 0.14-0.17. In particular, the calculation for Ar (C = 0.03; 1m « 11.5 eV) yields g « 0.16 and Te« 2.2 x 104 K.

From the cited results, it follows that in the case of “weak” excitation for Z ^ 10-4, the electron energy balance and Te scarcely depend on Z. In the opinion of the authors of study [2], this is explained by the fact that with an increase in the gas pressure, there is a sharp drop in the energy distribution function in the region of єе > 1m and consequently a reduction in the percentage of plasma electrons expending their energy on the excitation of atoms.