Gas ionization in the initial stage occurs directly by charged particles and secondary electrons. A picture of the electron energy distribution in the gas is shown in

Fig. 4.2 [11], where fe is the energy distribution function of the electrons, normal-

1

0

electron energy range can be divided into three regions:

fe( Ee)

1. The region of the ionization cascade Vt < ee < e0 (e0 is the particle initial energy, Vi is the ionization potential of the gas atoms and molecules), in which the energy of the electrons is sufficient for ionization of gas particles, and is expended for ionization and excitation of particles.

2. The region Im < ee < Vi (Im is the minimal threshold of electron, or vibrational excitations) in which the energy of the electrons is reduced, primarily due to excitation of electron and vibrational states of the gas particles. In this region, as in the previous one, electrons lose energy in comparatively large “portions,” equal to or greater than Im (for helium, Im = 19.8 eV is the energy of the lowest 23S state; for molecular gases Im is substantially lower and can be ~1 eV).

3. In the subthreshold region (ee < Im), electrons lose energy in small “portions” due to elastic collisions with gas particles, and the share of energy lost by an electron in one collision, Sea = 2me/Ma (where me and Ma are the masses of the electron and atom, respectively). This process is sometimes called electron thermalization.

An estimate of the characteristic time of energy loss by electrons in the high — energy region (ee > Im) for gases at atmospheric pressure gives Tne1 ~ (Meffex[A])_1 ~0.1 ns [2], where ue ~108 cm/s is the electron velocity, aex ~10-17 cm2 is the characteristic cross section for inelastic processes, and [A] is the concentration of gas particles. In study [12], the Monte Carlo method was used to calculate the time required to slow primary electrons jointly with secondary electrons of an ionization cascade from an initial energy of e0 to Im. This time Tex was calculated for He, Ne, Ar, Kr, and Xe on the assumption that an electron beam pulse constitutes a S function at the moment of time t = 0 (Fig. 4.3). Calculations

Tex’P, ns^atm ex*P, ns^atm Fig. 4.3 Dependence of the product Tex-P (Р is the gas pressure) on the initial electron beam energy e0 for He, Ne, Ar, Kr, Xe: (a) 150 keV < e0 < 1 MeV; (b) e0 < 125 keV [12]. For helium, the values of Tex-P should be doubled

show that Tex increases with growth in £0 and for £0 = 1 MeV, at atmospheric pressure Tex varies from 25 ns (xenon) to 250 ns (helium). It should be noted that the time to slow the secondary electrons (ts), which have lower energies and accordingly higher cross sections of inelastic interaction with the gas atoms, is much lower than Tex, and amounts to ts > 0.01 ns atm [12]. These are the times that are realized when fast electrons excite a gas volume of small size (on the order of a few centimeters).

Calculations of the high-energy part of fe for argon (£e > Im = 11.6 eV) showed that stationary distribution is established rather quickly in times of <0.1 ns after switching on the electron beam [13]. The time to establish stationary distribution increases with the growth in the initial energy of the electrons: from ~0.001 ns at £0 = 1 keV to ~0.1 ns at £0 = 1 MeV. These results agree with calculations conducted in study [14] on temporal evolution fe(£e > Im) when £0 = 10 keV for neon (4 atm) and xenon (1 atm). The time required to reach stationary mode for xenon is roughly an order of magnitude less than for neon, and amounts to ~0.001 ns, which is explained by the higher cross sections of inelastic processes.

For electrons with energies £e < Im, the mechanism of energy loss is related to the processes of elastic scattering with gas particles. The characteristic time of this energy-loss process Tel~ (ueoea8ea[A])~l where ueoea ~10~9 cm3/s is the rate con­stant of the elastic process for rare gases [2]. With an average energy of plasma electrons £e~1 eV and at atmospheric pressure, Tel > 100 ns, which greatly exceeds Tnel. Measurements of the characteristic times of the electron thermalization pro­cesses at an initial energy of 0.15 eV and a gas temperature of Tg = 300 K, carried out using microwave methods in study [15] for Ar (0.4 atm), Kr (0.15 atm) and Xe (0.14 atm), showed that the products Tel • Р = 1,050 ns • atm, 240 ns • atm, and 260 ns • atm, respectively.

Experimental data for He and Ne obtained in study [16] with an initial electron energy of around 1 eV and Tg = 300 K are 4.8 ns • atm and 120 ns • atm, respectively.

The time to establish stationary distribution fe in the subthreshold region £e < Im considerably exceeds the analogous time for the high-energy region. From the calculations of fe(£e < Im), obtained as a result of a numerical solution of the Boltzmann non-stationary kinetic equation for He [6], Ne [17], Ar [18, 19], and Xe [19] in the case of ionization by electron beams, it follows that the time required to reach stationary mode at atmospheric pressure is 5-10 ns.

To a significant extent, the electron energy distribution function in the sub­threshold range is determined by the frequency of electron-electron collisions, which depends on the degree of plasma ionization Z = ne/[A] (ne is the electron concentration). From the data shown in Fig. 4.4 [17] for the mixture Ne-Xe-HCl, it follows that the influence of electron-electron collisions on the formation of the electron energy distribution function is great even for a comparatively low specific power deposition q = 2 kW/cm3 (Z < 10~6). The complicated structure of the distribution shown in Fig. 4.4, calculated without taking electron-electron collisions into account, is explained by the influence of vibrational excitation of HCl molecules.

Fig. 4.4 Calculation of stationary electron energy distribution in the low-energy region for a Ne-Xe-HCl mixture (596:3:1) at a pressure of 4.9 atm, excited by an electron beam, without taking into account (1) and with taking into account (2) for electron-electron collisions [17]

For a low degree of plasma ionization, the electron energy distribution in the subthreshold region (ee < Im) can differ from the Maxwell distribution. To establish the Maxwell energy distribution for slow electrons at temperature Te, it is necessary for the frequency of the electron-electron collisions to greatly exceed the frequency of energy loss in elastic electron-atom collisions. This constraint can be written as follows [2]:

where oea is the cross section of elastic electron scattering on the atoms; e is the electron charge; and Л is the Coulomb logarithm. For helium, which is frequently used in NPL gas media as the buffer gas, uea ~5 x 10~15 cm~2; ee~0.15eV; Л ~10; me/Ma = 1.4 x 10~4; and consequently, Z ^ 10~6. We note that in experiments with pulsed reactors for atmospheric pressure NPLs, q < 5 kW/cm3 and the degree of ionization Z < 10-5, so that the condition (4.1) is fulfilled only in the case of maximal possible power deposition for NPLs.

Methods of calculating electron energy distributions are considered, for exam­ple, in studies [11,20]: the use of the continuous slowing down model, in which it is presumed that a fast electron loses energy in the degradation process continuously; modeling of the process of electron energy exchange by the Monte Carlo method, as

well as numerical solution of the Boltzmann kinetic equation. The continuous slowing down model is applicable primarily to calculate the electron energy distribution with energies ee > Im. For precise determination of electron energy distribution in the region of subthreshold and thermal energies, numerical solution of the Boltzmann kinetic equation is necessary.

For analysis of the kinetics of plasma processes in NPLs, information about the electron energy distribution for He, Ne, or Ar is the most interesting, because these are the gases that are used as the basic component (buffer gases) in nearly all known gas NPLs. It should be noted that all the data relating to electron energy distribu­tions in rare gases were obtained by computation except for study [21], where such distributions were found using spectroscopic and probing measurements for He and Ar during excitation by 235U fission fragments (Figs. 4.5 and 4.6) in a pressure range of 0.03-1 atm (q~0.01 W/cm3 at a pressure of 1 atm, ne~1011 cm~3, Z ~ 10~8). For comparison, Fig. 4.5 also shows the results of the calculation [22] for helium excited by a particles.

Information about the electron energy distributions for helium with excitation by fission fragments at an energy of 85 MeV and protons with an energy of 0.6 MeV (PHe = 0.4 atm; q = 1 kW/cm3) is contained in study [6], where these data were obtained as a result of solving the non-stationary kinetic equation. It follows from Fig. 4.7 that the electron energy distributions for two methods of excitation in the region ee < Im virtually coincide and are close to the Maxwell distribution.

Fig. 4.5 The electron energy distribution in helium excited by fission fragments [21]:

Р = 0.07 atm (1);

0.H3 atm (2) and 1 atm (3); (4) calculation results [22] with excitation of helium by a particles; dotted line is the Maxwell distribution

Fig. 4.6 Electron energy distribution in argon excited by fission fragments [21]: PA = 0.4 atm (1); 0.6 atm (2); 0.8 atm (3) and 1 atm (4); dotted line is the Maxwell distribution

Fig. 4.8 Electron energy distribution in argon (PAr = 3 atm) excited by an electron beam [18]. The dotted line is the Maxwell distribution when Te = 1.5 eV

The authors are not aware of any computational or experimental data on the electron energy distributions in neon during excitation by nuclear particles.

Information on electron energy distributions for plasma excited by electron beams is more extensive. Calculations by the Monte Carlo method for He, Ar and Xe in a range of initial electron energies £0 = 20-10,000 eV showed that the electron energy distribution in the subthreshold region (ee < 20 eV) does not depend on £0 for £0 > 300 eV [23].

The electron energy distribution for Ar (PAr = 3 atm) during excitation by an electron beam with a duration of 5 ns and £0 = 200 keV (q = 1.5 MW/cm3; ne = 2 x 1015 cm~3, Z = 2.5 x 10~5) is shown in Fig. 4.8 [18]. From the data cited, it follows that for electrons with £e < 10 eV, the electron distribution roughly corresponds with the Maxwell distribution. This circumstance is more clearly illustrated by the data in Fig. 4.9, which shows the ratio of the electron distribution to the Maxwell distribution for three different pressures of Ar [18]. With higher £e, these functions differ: the number of electrons in the region £e = 10-30 eV is less, while for £e > 30 eV it is greater than the Maxwell distribution. From a comparison of the data of Figs. 4.6 and 4.8, one can conclude that the shape of the distribution for argon under significantly differing conditions of excitation is roughly equal.

Calculation of the electron energy distribution for Ne was carried out in the studies [14, 17] for the following conditions: an electron beam with £0 = 200 keV, PNe = 4 atm, q = 200 kW/cm3, ne « 5 x 1014 cm~3, Z ~ 5 x 10~6. For neon (as for argon), the electron distribution function in the low-energy region (£e < Im) also

Fig. 4.9 Ratio afm of the electron energy distribution to the Maxwell distribution at argon pressures of 1 (1), 3 (2), and 10 (3) atm [18]. The excitation conditions are the same as in Fig. 4.8

coincides with the Maxwell distribution, while for ee > 40 eV the number of electrons significantly exceeds the equilibrium value (Fig. 4.10).

From the data cited it follows that in the range of conditions in which atmospheric-pressure gas NPLs were studied in experiments using pulsed reactors with excitation by nuclear reaction products (q < 5 kW/cm3, Z < 10~5), the electron energy distribution consists of two parts. The chief quantity of plasma electrons are concentrated in the subthreshold region, ee < Im, and their energy distribution is

similar to the Maxwell distribution, with a certain characteristic temperature Te = 2jsefe(ee)de/3k = 2 ee/3k. The electron energy distribution function in the region ee < Im is virtually independent of the type of charged particles. The low-energy electrons of this region participate in processes of great importance in the kinetics of populating and relaxation of lasing levels of NPLs: electron-ion recombination, “quenching” of excited states, the processes of attachment to electronegative atoms and molecules, and excitation and ionization in collisions with gas particles that are in excited states.

For ee > Im, the distribution function decreases as a consequence of the effective inelastic electron-atom collisions and differs from the Maxwell distribution. The electrons in this region scarcely participate in the recombination processes, and this region itself is a source of electrons for the subthreshold region.