tang_1972

K. T. Tang, “One-Term Sellmeier Formula for Dispersion of Dilute Gases”, J. Opt. Soc. Am. 62, 644 (1972)

Source record · Catalogue · Review register

DOI: 10.1364/josa.62.000644

Source kind: theory

Recorded technique: Theory: the two parameters of the one-term Sellmeier formula n - 1 = c / (eta^2 - omega^2) chosen from the static polarizability, the electron count (the Thomas-Reiche-Kuhn sum) and the first excitation energy so that the formula is a rigorous upper or lower bound on the Kramers-Heisenberg polarizability below the first resonance, or matched at one frequency so that it crosses the true curve there only; illustrated on atomic hydrogen (pp. 644-648).

Materials in structured entries: None recorded.

Dataset quantities: None recorded.

Contents: 0 datasets; 0 formulas.

Curation notes

Identity VALIDATED_WITH_WARNINGS by the library (no DOI printed; found by title, author and year); the first page reads J. Opt. Soc. Am. 62, No. 5 (May 1972), p. 644, K. T. Tang (Pacific Lutheran University), received 7 July 1971. Single author. Journal pages 644-648 = pdf pages 1-5. Atomic units throughout; the refractivity is 2 pi N0 alpha (Eq. 2, the dilute limit of Lorentz-Lorenz). No gas is treated and no coefficient is tabulated: the illustration is atomic hydrogen at NTP against Karplus and Kolker’s exact curve (Figs. 1-3, not digitised; hydrogen atoms are not a species of the package). Cited for method (TC-723).

THE RESULTS. alpha_a(omega) = alpha0 eta^2 / (eta^2 - omega^2) (Eq. 6), exact at omega = 0, with alpha(omega) = sum_n f_n / (omega_n^2 - omega^2) (Eq. 3). In Cauchy moments S(k) = sum_n f_n omega_n^k, alpha0 = S(-2). - eta = omega_1, the first excitation energy (Eqs. 11-13, 36): an upper bound for omega < omega_1 (the difference g = alpha_a - alpha, Eq. 12, is positive). - eta^2 = N / alpha0 = S(0) / S(-2) (Eq. 14, with the sum rule N = sum f_n, Eq. 15; Eqs. 19, 37): a lower bound for omega <= omega_1 (Eqs. 16-18). This choice keeps both the total strength and the static polarizability. - eta^2 = alpha(omega0) omega0^2 / [alpha(omega0) - alpha0] (Eqs. 20, 38), matched to one value at omega0 < omega_1: an upper bound below omega0 and a lower bound between omega0 and omega_1, and the two curves cross only once below omega_1 (the theorem, Eqs. 22-35). - With f_1 known, splitting off the first term (Eqs. 39-45) gives bounds valid up to the second resonance omega_2. Derived here, not in the paper: as omega0 -> 0, Eq. (20) gives eta^2 = S(-2) / S(-4), the one-term formula that reproduces the first Cauchy coefficient (the [0/1] Pade approximant in omega^2); by the theorem it is a lower bound for 0 < omega < omega_1, and since omega_1^2 <= S(-2)/S(-4) <= S(0)/S(-2) (the second inequality is Cauchy-Schwarz) it is the tighter of the two lower bounds. The bounds need every f_n >= 0.

TEXT CORRECTIONS (no coefficient involved). Below Eq. (23) the text says ‘with eta given by Eq. (21)’; eta is defined by Eq. (20). Eq. (43) prints f_omega / (omega_1^2 - omega^2) for the first term; by Eqs. (39)-(42) it is f_1. Eq. (19) prints the numerator as ‘.V’ (a scan artefact of N; Eq. 37a has N0 N).

RELEVANCE. optmat.gas.infrared represents each HITRAN vibrational band by one pole with E^2 = sum S / sum (S / nu^2), which is Eq. (14) applied to the band’s lines: the band’s S(0) and S(-2) are kept. By Tang’s theorem that pole is a lower bound on the band’s contribution at frequencies below the band; the same argument (derived here) makes it a signed upper bound (a contribution too small in magnitude) above the band, where optical frequencies lie. See TC-723 for the size of the error.

Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets

No datasets are recorded.

Formulas

No formulas are recorded.

Corrections

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.