Abstract

Previously, in this journal, it has been shown that the atomic state populations in low-pressure ICP systems can be modeled with the use of Fermi-Dirac counting statistics. In these works the relative population of the upper state of an emission transition, <i>n</i>_<i>i</i>, is set proportional to the average occupation number from Fermi-Dirac counting: <i>n</i>_<i>i</i> = <i>I</i>λ/<i>gA</i> = <i>C</i>*[exp((ε_<i>i</i> − μ)/<i>kT</i>]⁻¹ (1) where <i>n</i>_<i>i</i> is the relative population, <i>I</i> is the intensity of the transition corrected for spectral response, λ is the wavelength, <i>g</i> is the orbital degeneracy, <i>A</i> is the Einstein coefficient for spontaneous emission, <i>C</i> is the proportionality constant, ε_<i>i</i> is the energy of the upper level, μ is the chemical potential for an electron in the atom, <i>k</i> is Boltzmann's constant, and <i>T</i> is the absolute temperature. Since relative populations are usually expressed as logarithms, Eq. 1 becomes ln(<i>n</i>_<i>i</i>) = ln <i>C</i> + ln[exp[((ε_<i>i</i> − μ)/<i>kT</i>) + 1]⁻¹. (2) In this expression there are three variable quantities: <i>C</i>, μ, and <i>T</i>. All other quantities are known or measured experimentally. In previous works, the variable quantities were determined in a cumbersome and somewhat arbitrary manner. This method consisted of equating the most populous state to an occupation number of one and solving for <i>C</i>, followed by a "hand optimization" of μ and <i>T</i> to minimize the deviation between experimentally determined and calculated populations.

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