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A Simplex Optimization Program for the Determination of Temperatures in Reduced-Pressure ICPS

Volume 46, Number 12 (Dec. 1992) Page 1929-1930

Turner, D.E.; Fannin, H.B.

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, ni, is set proportional to the average occupation number from Fermi-Dirac counting:

ni = Iλ/gA = C*[exp((εi − μ)/kT]−1 (1)

where ni is the relative population, I is the intensity of the transition corrected for spectral response, λ is the wavelength, g is the orbital degeneracy, A is the Einstein coefficient for spontaneous emission, C is the proportionality constant, εi is the energy of the upper level, μ is the chemical potential for an electron in the atom, k is Boltzmann's constant, and T is the absolute temperature. Since relative populations are usually expressed as logarithms, Eq. 1 becomes

ln(ni) = ln C + ln[exp[((εi − μ)/kT) + 1]−1. (2)

In this expression there are three variable quantities: C, μ, and T. 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 C, followed by a "hand optimization" of μ and T to minimize the deviation between experimentally determined and calculated populations.