Electron, density Maxwellian

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... 

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]. 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

Here it was assumed that there are no ions other than the fuel ions present (i.e., no impurities and no helium ash), and that D and T are present in the optimal 50—50% mixture. Md, Mx, and e are the deuterium, tritium, and electron densities, respectively, and (crv) is the Maxwellian-averaged fusion reaction rate at the temperature T as given inO Chap. 6 of Vol. 1. 

An important specific feature of the present experiment is worth noting. The X-ray photons have energies that are several orders of magnitude larger than those of optical photons. The pump and probe processes thus evolve on different time scales and can be treated separately. It is convenient to start with the X-ray probing processes, and treat them by Maxwellian electrodynamics. The pumping processes are studied next using statistical mechanics of nonlinear optical processes. The electron number density n(r,t), supposed to be known in the first step, is actually calculated in this second step. 

In the above Maxwellian description of X-ray diffraction, the electron number density n r, t) was considered to be a known function of r,f. In reality, this density is modulated by the laser excitation and is not known a priori. However, it can be determined using methods of statistical mechanics of nonlinear optical processes, similar to those used in time-resolved optical spectroscopy [4]. The laser-generated electric field can be expressed as E(r, f) = Eoo(f) exp(/(qor— flot)), where flo is the optical frequency and qo the corresponding wavevector. The calculation can be sketched as follows. 

The kd (v) rate coefficients have been obtained by using the cross sections of Fig. 12 and the non-Maxwellian electron distribution functions of Fig. 13. The edfs have been obtained by a numerical solution of the Boltzmann equation (BE) which includes the superelastic vibrational collisions involving the first three vibrational levels, and the dissociation process from all vibrational levels (see Ref.9) for details). The vibrational population densities inserted in the BE are self-consistent with the quasi-stationary values reported in Figs. 8 and 10. It should be noted that the DEM rates (Fig. 14) depend on E/N as well as on the vibrational non equilibrium present in the discharge, which affects the electron distribution functions, as discussed in Sect. 2.1. 

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... 

As long as the radiation density is low (which is the case for the d. c. arc) the plasma can be assumed to operate under local thermal equihbrium. This is not the case for low-pressure discharges where both collisions with electrons and radiative de-excitation are very important. Also, for low-pressure plasmas, the assumption of a Maxwellian velocity distribution of the particles is no longer valid. 

In Eq. (75), Da Eo, E, hv) is the density of the electronic acceptor state in solution, which depends on the ground electronic energy, Eq, of the ion corresponding to the ground vibrational-rotational state of the ion-solvent bond. They can be expressed in terms of a Maxwellian distribution, i.e.. 

Figures 15(a) and 15(b) show the calculation results of EEPF and VDF of N2 X, respectively. The calculations were run at Tg = 1200 K, Ne = 5.0 x IQU cm-3 and Te = 2.5 - 4.5 eV. These parameters were chosen to correspond to our experimental results at P = 1.0 Torr and 2 = 60 mm obtained in the experimental apvparatus shown in Fig. 3. It should be repeated that we choose a reduced electric field so that the electron mean energy s) equals (3/2)/cTe when we compare the numerical calculation with the number densities obtained experimentally by OES measurement. Obviously, the EEPF is not like Maxwellian. It has a dip in the range from 2 to 3 eV owing to frequent consumption of electrons with this energy range due to inelastic collisions to make vibrationally excited molecules. Meanwhile, Fig. 15(b) shows that the VDF is also quite far from the Maxwellian distribution. The number density of the vibrational levels shows rapid decrease first, then moderate decrease, and rapid decrease again as the vibrational quantum number increases. This behaviour of the VDF of N2 X state has been frequently reported, and consequently, our model is also considered to be appropriate. If we can assume corona equilibrium of some excited states of N2 molecule, for example, N2 C state, we can calculate the number density of the vibrational levels of the excited state that can be experimentally observed. This indicates that we can verify the appropriateness of the calculated VDF of the N2 X state as shown in Fig. 15(b).

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