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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, *n*_{i}, is set proportional to the average occupation number from Fermi-Dirac counting:

*n*

_{i}=

*I*λ/

*gA*=

*C**[exp((ε

_{i}− μ)/

*kT*]

^{−1}(1)where

*n*

_{i}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, ε

_{i}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 becomesln(

*n*

_{i}) = ln

*C*+ ln[exp[((ε

_{i}− 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.