tang_1972
=========

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

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/tang_1972.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1364/josa\.62\.000644 <https://doi.org/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
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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
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No datasets are recorded.

Formulas
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No formulas are recorded.

Corrections
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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 :doc:`review register </register>` for the checks and decisions.