Laser Diagnostics of a Hydrogen Discharge

Abstract

When modelling a partially ionized plasma, one is working with a set of rate equations, describing the production and destruction of the particle species occurring in the discharge (Hiskes, 1987; Gorse et al., 1987). Depending on the level of detail of the description, the number of interacting species can increase to a large number. An elementary description of a hydrogen plasma would involve molecules, H₂, metastable molecules, H₂ ^*, atoms, H^o, one type of ion, H_i ⁺, and electrons. When more precision is required, one would have to replace the ion by H^-, H⁺, H₂ ⁺ and H₃ ⁺, and successively add more and more excited states, including both electronic and molecular excitations. The reaction rates occurring in the equations are expressions of the following form:

$$ R = \iint {\sigma (\left| {{{v}_{1}} - {{v}_{2}}} \right|)(\left| {{{v}_{1}} - {{v}_{2}}} \right|){{F}^{a}}({{v}_{1}}){{F}^{b}}({{v}_{2}}){{d}^{3}}{{v}_{1}}{{d}^{3}}{{v}_{2}},} $$

((1))

where, σ, which is a function of the relative velocity |v1 - v2|, is the cross section describing the interaction between particles of species a and b, and F^a(v1) and F^b(v2) are their distribution functions. Equation (1) clarifies that an ideal diagnostic technique measures the complete distribution function of each particle species. Only then, a precise comparison between modelling and experimental results is possible. In the majority of discharges, however, the particles make a sufficient number of collisions for the distribution function to approach a Maxwellian. Then, a measurement of density and temperature would suffice. For two species of the same temperature T, Eq. (1) attains the much simpler form,

$$ R = {{n}_{a}}{{n}_{b}}\surd (8kT/\pi \mu )\smallint \sigma (E)\exp ( - E)EdE, $$

((2))

where, n gives the densities of species a and b, respectively, E = (µv²/2kT) is the relative energy of motion with v the relative velocity, and µ is the reduced mass, µ = (m_a + m_b)/m_am_b.