Maxwellian electron energy distribution

For low-pressure plasmas containing mainly inert gases the electrons can be characterized by a Maxwellian electron energy distribution function (EEDF). How-... 

The use of maxwellian or of non maxwellian electron energy distribution functions is clearly reflected in the values of the rate constants. 

Electron energy distribution function The distribution function of electrons in a plasma. That of a low-pressure radiofrequency plasma generally consists of two Maxwellian distributions, that is, fast and slow electrons. 

Plasmas typical of C02 laser discharges operate over a pressure range from 1 Torr to several atmospheres with degrees of ionization, that is, nJN (the ratio of electron density to neutral density) in the range from 10-8 to 10-8. Under these conditions the electron energy distribution function is highly non-Maxwellian. As a consequence it is necessary to solve the Boltzmann transport equation based on a detailed knowledge of the electron collisional channels in order to establish the electron distribution function as a function of the ratio of the electric field to the neutral gas density, E/N, and species concentration. Development of the fundamental techniques for solution of the Boltzmann equation are presented in detail by Shkarofsky, Johnston, and Bachynski [44] and Holstein [45]. 

Tg and ng were calculated from the relation between and according to the equivalent resistance method developed by Dote ( ), in which it is assumed that the electron energy distribution is Maxwellian. The value of Ug was calculated according to Maiter and Webster ( 5). ... 

Specified electron energy distribution function The EEDF is specified, normally assumed Maxwellian (Eq. 9). The electron energy balance (Eq. 31) is solved assuming an adiabatic condition for electron temperature at the wall. The Maxwellian assumption is very common in the literature [100, 125, 126, 130, 133, 135-137]. Measured EEDFs in ICPs, however, have a Maxwellian bulk (due to electron-electron collisions), and a depleted tail due to inelastic losses and escape of fast electrons to the walls. Thus a bi-Maxwellian distribution may be more appropriate [154]. A Maxwellian distribution is not expected to have a great effect on ion densities since the ionization rate is self-adjusted to balance the loss rate of ions to the walls and the latter depends only very weakly on the EEDF. The good agreement with experimental data [101, 130, 148, 152] is an indirect evidence that the Maxwellian EEDF is reasonable for obtaining species densities and their distributions. Other forms of... 

Useful formulas for estimations of the direct ionization cross sections can be also found in Barnett (1989). The ionization rate coefficient A i(7 ) canthenbe calculated by integration of the cross section cri (e) over the electron energy distribution function (see (2-7)). Assuming the Maxwellian EEDF, the direct ionization rate coefficient can be presented as... 

Druyvesteyn Electron Energy Distribution Function. Calculate the average electron energy for the Druyvesteyn distribution. Define the effective electron temperature of the distribution and compare it with that of the Maxwellian distribution frmction. 

Fig. 3.31. Distributions (i)/(Ee) dEe of electron energy (E ) for a low-pressure HF-plasma (suffix pi, Maxwellian with temperature = 80000 K) and an electron beam (suffix eb, simplified to Gaussian shape with 40 eV half-width) (ii) rTx (Ej) ofthe Ej dependent electron impact ionization cross-section for X=Ti...

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... 

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... 

Additional information concerning the fast electron distribution can be obtained in such experiments by direct measurement of forward escaping electrons using a calibrated stack of radiochromic films [52] that can provide information of the angular and energy distribution of fast electrons. These measurements were performed in the same experiment described above [31] and the summary of those measurements is reported in Fig. 7.10 and reveal a distribution that is consistent with a relativistic Maxwellian distribution with a characteristic temperature of 160keV. 

It should be noted that a Maxwellian form of fie) is a reasonable approximation to the actual distribution at low electron energies. This observation is indicated in Figure 5 i24). However, the first ionization potential of most atoms and molecules is above eV. Thus, many of the important homogeneous processes that occur in glow discharges, such as ionization, take place as a result of high energy electrons in the "tail" of the distribution. These electrons are precisely the ones that are not adequately described by a Maxwellian distribution function. 

Central to the categorization of plasmas are electron temperature and electron density. Electrons have a distribution of energies, so it is useful to assume a Maxwellian distribution, in terms of electron energy, E, such that... 

Figure 5.6 The left panel shows storage ring results [100] in the form of rate coefficients (shaded area) for Si3+ recombining into Si2+. The thick line is the calculated cross section [100] folded with the electron beam temperature. The energies of the resonant states (doubly excited states) in Si2+ are shown in the form of a level scheme. It is obvious that the rate coefficients map out the energy level scheme. The right panel shows the rate coefficients as a function of temperature. The upper black line shows the storage ring results folded with a Maxwellian energy distribution, see [99], while the lower grey curve shows a theoretical prediction [107].

---