@article {Turner:1992-12-01T00:00:00:0003-7028:1929,
author = "Turner, D. E. and Fannin, H. B.",
title = "A Simplex Optimization Program for the Determination of Temperatures in Reduced-Pressure ICPS",
journal = "Applied Spectroscopy",
volume = "46",
number = "12",
year = "1992-12-01T00:00:00",
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, *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 becomes

ln(*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.",
pages = "1929-1930",
url = "http://www.ingentaconnect.com/content/sas/sas/1992/00000046/00000012/art00028",
doi = "doi:10.1366/0003702924123566",
keyword = "Spectroscopic temperature, Reduced-pressure ICP, Simplex"
}