Electron mean energy

Figure C2.13.2. Electron energy distributions/(U) for a mean electron energy of 4.2 eV, Maxwell distribution (M), Dmyvesteyn distribution (D) and a calculated distribution (Ar) for an Ar plasma [12].

Top typical saturation curve and variation of mean electron energy with applied field. Middle fraction of the electron swarm exceeding the specific energy at each field strength. Calculated assuming constant collision cross-section and Maxwell-Boltzman distribution. Bottom variation of products typical of involvement of ionic precursors (methane) and excited intermediates (ethane) with applied field strength... 

In the cross modulation experiments (Mentzoni and Row, 1963 Mentzoni and Rao, 1965), an electron plasma is briefly heated by a microwave pulse while a weak microwave signal probes the mean electron energy. Assuming no electron loss and insignificant ambient gas heating, these authors derived the following equation for the relaxation of electron Maxwellian temperature T.toward the ambient temperature T ... 

The determination of electron concentration by the frequency shift method is limited to time resolution greater than a few hundred nanoseconds and is therefore not applicable to liquids. The microwave absorption method can be used virtually down to the pulse width resolution. Under conditions of low dose and no electron loss, and assuming Maxwellian distribution at all times, Warman and deHaas (1975) show that the fractional power loss is related to the mean electron energy (E) by... 

Here the left-hand side is the ratio of power loss at time t, when the mean electron energy is (E), to that at thermalization, and C and n are determinable constants. This idealized equation is not expected to be valid in presence of the Ramsauer effect, but Warman and deHaas apply it anyway to N2, Ar, and He at atmospheric pressure. The method relates the gradual decrease of collision frequency to an increase in conductivity, which finally rides to a plateau interpreted to be the thermal conductivity. The time needed to reach 90% of the thermal conductivity is called the thermalization time (see Table 8.1). 

Early theoretical models were based on fractional energy loss 2m/M per elastic collision (for details, see LaVeme and Mozumder, 1984, Sect. 3, and references therein). Thus, frequently, the energy loss rate was written as —d (E)/dt = (2m/M)((E)-3feBT/2)vc, where vc is the collision frequency and (E) is the mean electron energy over an unspecified distribution. The heuristic inclusion of the term 3feBT/2 allowed the mean energy to attain the asymptotic thermal... 

Two other attempts, without the use of a distribution function, are worth mentioning, as these are operationally related to experiments and serve to give a rough estimate of the thermalization time. Christophorou et al. (1975) note that in the presence of a relatively weak external field E, the rate of energy input to an electron by that field is (0 = eEvd, where vd is the drift velocity in the stationary state. Under equilibrium, it must be equal to the difference between the energy loss and gain rates by an electron s interaction with the medium. The mean electron energy is now approximated as (E) = (3eD )/(2p), where fl = vd /E is the drift mobility and D is the perpendicular diffusion coefficient (this approximation is actually valid for a Maxwellian distribution). Thus, from measurements of fl and D the thermalization time is estimated to be... 

The mean energy loss in an elastic collision may be taken as <5(m/M) [(e) — (3/2)fegT] where (e) is the mean electron energy, nt/M is the ratio of electron mass to that of the rare gas atom, and S is a numerical parameter. The collision rate may be approximated by A0 1 (2(e)/m)m. The equation for the rate of energy loss may now be given as follows ... 

In ionization processes, a positive ion is created every time an electron is created, and the net rate of creation of positive ions equals that of electrons. The equation for mean electron energy (referred to as electron temperature) is... 

Figure 7. Total electron attachment rates as a function of mean electron energy, and swarm-unfolded electron attachment cross sections as a function of electron energy for CcFe (30) and CeFsCF, (19)...

If some such adjustment were admitted, it is plausible to infer that the corresponding trend in P/p is from about 2.0 to 1.0 that would be roughly consistent with the relatively small decrease observed in from about 12 X 10 to 10 X 10 2. Such trends in P/p and in rj would be expected if the mean electron energy were to fall correspondingly with X/p, as it is known to do at much lower current densities (33). 

When the current density is so low that the gas consists almost exclusively of molecules unchanged by the discharge, so that p, the total pressure, is sensibly the same as Pry the partial pressure of the reactant, a special case arises when IF, and F, the mean electron energy, are functions of X/p as in the experimental conditions adopted by the Townsend school for their measurements 73), For then S is a function of X/p only, and is computable when /(F) and Q(F) are also known. Relation 3 then becomes... 

Fig. 10. Mean electronic energy loss for H incident on He at 200 keV as a function of the projectile scattering angle. Closed squares with error bars experimental results from Ref. [61]. Solid line (three-body) Eikonal-AO results dashed-line (two-body) AO results for mean-field projectile trajectories.

From Formula 2 it is apparent that aXK depends on the overlap between QXK and E1/2f(E). For a particular gas, however, f(E) and E, the mean electron energy, are functions of X/n, the Townsend parameter. The mean electron energy is given by... 

For mixtures of molecular gases a rule is lacking to estimate the dependence of the mean electron energy and drift velocity on X/n. For... 

It should be noted that an electron temperature is defined properly only when the EEDF is Maxwellian. For non-Maxwellian distributions, an equivalent electron temperature may be defined based on the mean electron energy, by using Eq. (10). Another popular form of the EEDF which can be expressed analytically is the Druyvesteyn distribution [42, 43, 48]. 

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