Vibrational distribution functions

E.E.Nikitin, Vibrational distribution functions of polyatomic molecules in thermal decomposition, Doklady Akad.Nauk SSSR, 135, 1442 (1960)... 

If diatomic molecnles are considered harmonic oscillators, the vibrational distribution function follows the same Boltzmann formnla (3-12) even when > To. Interesting non-eqnilibrinm statistical phenomena take place, however, when we take into account anharmonicity. Then vibrational-vibrational (W) exchange is not resonant and translational degrees of freedom become involved in the vibrational distribution, which results in a strong deviation from the Boltzmann distribntion. Considering vibrational quanta as qnasi-particles, and nsing the Gibbs distribution with a variable number of quasi-particles, V, the relative population of vibrational levels can be expressed (Kuznetzov, 1971) as... 

Figure 3-3. Comparison of Treanor and Boltzmann vibrational distribution functions.

Fokker-Plank Kinetic Equation for Non-Equilibrium Vibrational Distribution Functions... 

Vibrationally and electronically excited atoms and molecules, which make a significant contribution to plasma-chemical kinetics, can be described in terms of distribution functions. We already introduced the vibrational distribution functions (population of vibrationally... 

The W-exchange process involves two excited molecules (energies E, E ), which makes the W flux non-linear with respect to the vibrational distribution function (Demma, Macheret, Fridman, 1984) ... 

This plateau-Boltzmann transitional energy corresponds to the equahty of probabilities of the VT relaxation and the resonant W exchange. The vibrational distribution function in the regime of strong excitation in non-thermal plasma is illustrated in Fig. 3-7. 

The population of vibrationally excited states at the Treanor minimum E = Exr) is large in this regime, and the non-linear resonant W exchange dominates and provides a plateau at > i Tr even though < 5yy. At low levels E < E-y ), the linear non-resonant W exchange dominates over the non-linear one. It does not change the vibrational distribution function, however, because both non-resonant and resonant components of the W exchange result in the same Treanor distribution at < Ej. ... 

Vibrational distribution function (3-141) can include a domain of inverse population... 

The vibrational distribution function /i (E) for the heavier isotope (small additive) is determined by W exchange between the isotopes, which is slower than W exchange because of the defect of resonance. As a result, VT relaxation makes the vibrational distribution/i( ) decrease at lower energies = 1) with respect to the distribution... 

Figure 3-14. Vibrational distribution functions of two diatomic molecular isotopes (1,2) in non-equilibrium conditions T S> To).DashedlinesrepresentrelevantBoltzmanndistribution functions for the isotopes E and E2 indicate vibrational energies eorresponding to a strong exponential decrease of the distribution funetions related to VT relaxation from highly exeited vibrational levels.

Macrokinetic reaction rates of vibrationaiiy excited molecules are self-consistent with the influence of the reactions on vibrational distribution functions / E), which can be taken into account by introducing into the Fokker-Planck kinetic equation (3-130) an additional flux related to the reaction ... 

The function - ME) determines the flux of molecules in the energy spectrum, which are going to react when they have enough energy E > E ). At relatively low energies E < a), where chemical reaction can be neglected, -ME) = kR E )nof E )dE = Jo = const. As a result, perturbation of the vibrational distribution function M E) at E < E hy chemical reaction can be presented as... 

Figure 3-16. Influence of a chemical reactionon the vibrational distribution function in the weak excitation regime of non-equilibrium plasma Ty To).

The slow reaction limit corresponds to an inequality opposite that of (3-177). In this case the population of highly reactive states (E > E ) by W exchange is faster than the elementary chemical reaction itself. The vibrational distribution function in this case is almost not perturbed by chemical reaction / E) f ° E), and the total macroscopic reaction rate coefficient can be found as... 

Gordiets Vibrational Distribution Function in Non-Thermal Plasma. Compare the discrete Gordiets vibrational distribution with continuous distribution function (3-141). Pay special attention to the exponential decrease of the vibrational distribution functions at high vibrational energies in the case of low translational temperatures (To < ftoS). 

Numerically, E (15-20) x quasi continuum actually takes place at very high levels of excitation of the asymmetric vibrational mode, close to the dissociation energy (see Fig. 5-14). Thus, most of the vibrational distribution function relevant to CO2 dissociation in this case, in contrast to the one-temperature approach, is not continuous but discrete. The discrete distribution function /(Va, Vs) over vibrational energies (5-16) can be presented analytically according to Licalter (1975a,b, 1976) in the Treanor form ... 

Treanor Effect for Low Discrete Leveis of CO2 Symmetric Osciiiations. Explain the vibrational distribution function related to low discrete levels of CO2 symmetric modes, presented in Fig. 5-15. Why does this vibrational distribution clearly combine two exponential functions with two separate temperatures - a vibrational one and a rotational-translational one ... 

With some additional assumptions the transfer rate for this can be expressed more explicitly. If the transfer is so slow that it occurs between thermally equilibrated molecules, the total transfer rate nj contains certain combinations of vibrational distribution functions and Franck-Condon factors. These are exactly the same combinations which determine the intensities of the vibronic transitions in the emission spectrum of the donor and in the absorption spectrum of the acceptor. Under the further assumption of pure dipole-dipole interaction this leads to the following expression, which can be obtained either by quantum-mechanical theory or even from classical arguments ... 

On the other hand, it is also quite important to study reaction kinetics in nitrogen plasmas to understand quantitative amount of various excited species including reactive radicals. Many theoretical models have been proposed to describe the number densities of excited states in the plasmas. Excellent models involve simultaneous solvers of the Boltzmann equation to determine the electron energy distribution function (EEDF) and the vibrational distribution function (VDF) of nitrogen molecules in the electronic ground state. Consequently, we have found noteworthy characteristics of the number densities of excited species including dissociated atoms in plasmas as functions of plasma parameters such as electron density, reduced electric field, and electron temperature (Guerra et al, 2004 Shakhatov Lebedev, 2008). 

In the present analysis, the EEDF is determined by solving the Boltzmann equation as a fimction of the reduced electric field E/N so that the electron mean energy equals 3/2 times the electron temperature experimentally measured by the probe. The Boltzmann equation is simultaneously solved with the master equations for the vibrational distribution function (VDF) of the N2 X iZg+ state, since the EEDF of N2-based plasma is strongly affected by the VDF of N2 molecules owing to superelastic collisions with vibrationally excited N2 molecules. A more detailed account of obtaining the EEDF is given in the next section. 

Table 4.7. Vibrational distribution functions for the OH radical in the reaction...

As can be seen, the surprisal parameter for the vibrational distribution function of all reactions in Table 4.8 is negative. This implies that hydroxyl radicals formed in these reactions are much more vibrationally excited than it is expected from the statistical distribution. This fact indicates the predominant vibrational excitation of the... 

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