voronin_2017 ============ A\. A\. Voronin and A\. M\. Zheltikov, "The generalized Sellmeier equation for air", Sci\. Rep\. 7, 46111 \(2017\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1038/srep46111 `__ **Source kind:** fit **Recorded technique:** A 'full\-model analysis' \(FMA\) of the refractive index of air, Eq\. \(1\)\: a linear sum of Lorentz oscillators over more than 650,000 lines of HITRAN 2012 \(frequency, width, oscillator strength\) for N2, O2, H2O, CO2, O3, CH4, CO, Ar and Ne, with the electronic \(ultraviolet\) oscillators from Margoliash and Meath \(N2, O2, H2O\) and Eggarter, Chan et al\. and Wuilleumier \(Ar\)\. Far from the lines \(Eq\. 3\) the lines of each absorption band r are replaced by two effective terms \(Eq\. 4\) whose strengths sum to the band's \(A\_1r \+ A\_2r = sum\_q f\_rq / 2\) and whose wavelengths are fitted\: the 'generalized Sellmeier equation' \(GSE, Eq\. 5\), 15 bands, 30 poles, for one air mixture \(the densities of Fig\. 1\)\. No fitting procedure, weights or fitting range are stated \(pp\. 2\-6\)\. **Materials in structured entries:** Ar, CO2, N2, O2 **Dataset quantities:** None recorded. **Contents:** 0 datasets; 4 formulas. Curation notes -------------- Identity VALIDATED; the pdf is the version of record \(11 pages, article 46111; pages cited are pdf pages\)\. CC BY 4\.0\. THE FORM \(Eqs\. 4\-5, pp\. 5\-6\)\. n\(lambda\) \- 1 = \(1/N\_cr\) sum\_r N\_r \[A\_1r lambda\_1r^2 / \(lambda^2 \- lambda\_1r^2\) \+ A\_2r lambda\_2r^2 / \(lambda^2 \- lambda\_2r^2\)\], N\_cr = m eps0 omega^2 / e^2 the critical density at the light's frequency\. Since lambda\_j^2 / \(lambda^2 \- lambda\_j^2\) = omega^2 / \(omega\_j^2 \- omega^2\), each term is n \- 1 = N\_r e^2 A\_jr / \(m eps0 \(omega\_jr^2 \- omega^2\)\)\: a Lorentz oscillator of strength f = 2 A\_jr and energy E = hc / lambda\_jr, linear in the density of its species \(n \- 1, not n^2 \- 1; no Lorentz\-Lorenz, no real\-gas density\)\. Per molecule, alpha = sum\_j f\_j / \(E\_j^2 \- omega^2\) in atomic units, with E\_j = 45\.5634 nm / lambda\_j hartree\. The two terms of a simple parallel band are its P and R branches; for the other bands and the electronic terms they are placed near the band's edges \(p\. 6\)\. GVD follows analytically \(Eq\. 6\)\. PRINTING \(checked on the table image, tbl\_0899b125f94c\.jpg\)\. \(i\) Table 1 heads A\_1r and A\_2r 'ps^2', but the definitions make them dimensionless, and the electronic rows sum to half the electron count, A\_1r \+ A\_2r = 7\.000 \(N2\), 8\.000 \(O2\), 5\.000 \(H2O\) \(Ar 8\.149\: 16\.3 electrons\)\: the unit is a misprint\. \(ii\) Eq\. \(4\) prints the second term's numerator as A\_2r lambda\_1r^2; Eqs\. \(5\) and \(6\) have lambda\_2r^2, and only with lambda\_2r^2 do the rows reproduce Margoliash and Meath's S\(\-2\) \(below\)\: Eq\. \(4\) is misprinted\. \(iii\) The last column, N\_r, is printed once per species \(merged cells\), and for H2O reads 'Eq\. \(7\) at T = 296 K' \(the library's cell text is garbled\)\. No coefficient is corrected\. THE TERMS \(Table 1, p\. 6\)\. CO2 r = 1\-4 \(the 15, 4\.3, 2\.7 and 2\.0 um bands\) and the electronic terms of N2 \(r = 12\), O2 \(13\) and Ar \(14\) are recorded as formulas \(gse\_CO2\_infrared, gse\_N2, gse\_O2, gse\_Ar\); tools/extract/voronin\_2017\.py asserts that their coefficients equal the table's cells \(it writes no data file\)\. Water is not a curated species; its rows as printed \(r\: A\_1r, A\_2r, lambda\_1r nm, lambda\_2r nm\)\: 5 \(000\-000, the pure rotational band, 'terahertz'\)\: 2\.945e\-5, 6\.583e\-8, 47862, 16603; 6 \(000\-010\)\: 3\.273e\-6, 3\.094e\-6, 6719\.0, 5729\.9; 7 \(000\-020, 000\-100, 000\-001\)\: 1\.862e\-6, 2\.788e\-6, 2775\.6, 2598\.5; 8 \(000\-011, 000\-110\)\: 2\.544e\-7, 2\.181e\-7, 1835\.6, 1904\.8; 9 \(000\-021, 000\-200, 000\-101\)\: 1\.126e\-7, 2\.336e\-7, 1417\.6, 1364\.7; 10 \(000\-111, 000\-210\)\: 6\.856e\-9, 9\.479e\-9, 1145\.3, 1123\.2; 11 \(000\-201\)\: 1\.985e\-9, 2\.882e\-9, 947\.73, 935\.09; 15 \(electronic\)\: 0\.25787285, 4\.742131, 128, 34\.924\. No CO2 electronic term is given, nor any for O3, CH4, CO or Ne\. The water density is Eq\. \(7\), N = h p\_s / \(k\_B T\) with the Wagner\-Pruss saturation pressure \(p\. 6\)\. THE MIXTURE \(Fig\. 1 caption, p\. 3; p\. 6\)\: N\_N2 = 1\.987e19, N\_O2 = 5\.3291e18, N\_Ar = 2\.3763e17, N\_CO2 = 9\.4136e15 cm^\-3 \(370 ppm of the dry gas\), N\_H2O = 7\.0733e16 \(10 % humidity; 2\.8305e17 at 40 %, Fig\. 2\), 'T = 296 K'\. These densities total 2\.551e19 cm^\-3, the density of 1 atm near 15 degrees C \(2\.547e19\), not of 296 K \(2\.479e19\)\. ACCURACY \(pp\. 7\-8\)\: the GSE differs from the FMA by \|eta\| ~ 1e\-10 in n 'almost everywhere within the transparency region' from 0\.3 to 13 um \(larger only in narrow regions next to the strongest bands; below 1e\-10 over 3\.4\-4\.1 um\), and in GVD by less than 1 fs^2/cm below 4\.1 um away from the bands and 1\-2 fs^2/cm in most of the LWIR window\. These are agreements with the authors' own line sum, which shares the electronic terms; no comparison with measurements of air is made beyond Fig\. 5\. CHECKS \(computed 2026\-09\-29; TC\-720\)\. \(1\) The N2, O2 and H2O electronic terms are two\-pole condensations of Margoliash and Meath's pseudospectra \(margoliash\_1978, Table I\)\: they reproduce its S\(0\) and S\(\-2\) to 1e\-6 \(N2 13\.99935 and 11\.73772; O2 16\.00000 and 10\.59006; H2O 10\.00001 and 9\.64240\), S\(\-4\) to 1e\-4 \(N2\) and 1\.2e\-3 \(O2\), and depart at S\(\-6\) and beyond\. \(2\) The four electronic terms at the Fig\. 1 densities sum to Ciddor's standard\-air formula as the paper prints it \(dry, 15 degrees C\) within 1e\-5 of n \- 1 from 300 to 1500 nm \(29154\.7 against 29156\.9e\-8 at 300 nm, 27653\.2 against 27653\.0e\-8 at 633 nm\)\. \(3\) Per molecule, against the stored fits\: N2 \+185 ppm at 633 nm and \+174 ppm static \(11\.7377 against 11\.7357 au\), O2 \+178 and \+277 ppm; Ar \-6\.53 % at 633 nm and \-6\.50 % static \(10\.357 against 11\.077 au\)\. The Ar term reproduces Peck and Fisher's 0 degrees C formula \(peck\_1964/peck\_0C\) at 633 nm to 0\.12 % only at the density printed for argon in Fig\. 5, 2\.879e19 cm^\-3, 7\.1 % above the 0 degrees C density \(2\.687e19\) that formula refers to\. \(4\) The CO2 terms' strengths are those of HITRAN's principal isotopologue alone\: 2\(A\_1 \+ A\_2\) is within 0\.15 % of the 12C16O2 bands of optmat's HITRAN 2024 table in the 15 and 4\.3 um regions and 1\.6 % below all isotopologues together \(details in the formula's check\)\. OTHER FORMULAS PRINTED \(verbatim, not recorded as formulas\)\. The 'standard Sellmeier equation' for air, Eq\. \(2\) with B\_1 = 0\.05792105, B\_2 = 0\.00167917, C\_1 = 238\.0185, C\_2 = 57\.362 um^\-2 \(p\. 2\)\: Ciddor's \(1996\) standard\-air formula, cited to refs\. 11\-15; air is not a curated species\. Fig\. 5's single\-term formulas \(p\. 8\) are reproductions of curated ones\: N2, B\_1 = 3\.243157e\-2, C\_1 = 144\.0, D = 6\.8552e\-5 \(peck\_1966, 0 degrees C\); O2, 9\.708931e\-3, 75\.4, 1\.181494e\-4 \(kren\_2011/kren\_O2, 20 degrees C\); Ar, 3\.0182943e\-2, 144\.0, 6\.7867e\-5 \(peck\_1964/peck\_0C, whose constant 678\.6711e\-7 is rounded\)\. Fig\. 5 plots them at N\_N2 = 2\.688e19, N\_O2 = 2\.504e19 and N\_Ar = 2\.879e19 cm^\-3\. LATER USE\. egan\_2026 \(pdf p\. 19\) states that the GSE's water\-vapour refractivity is 2\.2 % too low at 0\.633 um and adjusts its electronic pair \(strengths \+3\.61 %, frequency \+0\.72 %\) to its own two\-colour ratio\. **Recorded bibliographic identity status:** VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- No datasets are recorded. Formulas -------- voronin\_2017/gse\_CO2\_infrared ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "gse_CO2_infrared", "material": "CO2", "equation": "Eq. (5) with Table 1, rows r = 1-4 (p. 6)", "printed": "CO2, N_r = 9.4136 x 10^15 cm^-3; r = 1: A_1r = 4.051 x 10^-6, A_2r = 1.010 x 10^-6, lambda_1r = 15131 nm, lambda_2r = 14218 nm; r = 2: 2.897 x 10^-5, 2.728 x 10^-5, 4290.9, 4223.1; r = 3: 8.573 x 10^-7, 6.620 x 10^-7, 2684.9, 2769.1; r = 4: 1.550 x 10^-8, 5.532 x 10^-9, 2011.3, 1964.6 (A headed 'ps^2', dimensionless in fact)", "variable": "lambda: vacuum wavelength (nm); n - 1 = (N_r / N_cr) sum_j A_j lambda_j^2 / (lambda^2 - lambda_j^2), N_cr = m eps0 omega^2 / e^2; per molecule alpha = sum_j f_j / (E_j^2 - omega^2) (atomic units), f_j = 2 A_j, E_j = 45.5634 / lambda_j[nm] hartree", "state": "none (per molecule; n - 1 linear in N_r)", "density": "none", "range": "0.3-13 um in the atmospheric transmission windows, outside the bands themselves (pp. 1, 7); fitted to the FMA of one air mixture (Fig. 1 densities)", "coefficients": { "A1_r1": "4.051e-6", "A2_r1": "1.010e-6", "lambda1_r1": "15131", "lambda2_r1": "14218", "A1_r2": "2.897e-5", "A2_r2": "2.728e-5", "lambda1_r2": "4290.9", "lambda2_r2": "4223.1", "A1_r3": "8.573e-7", "A2_r3": "6.620e-7", "lambda1_r3": "2684.9", "lambda2_r3": "2769.1", "A1_r4": "1.550e-8", "A2_r4": "5.532e-9", "lambda1_r4": "2011.3", "lambda2_r4": "1964.6", "N": "9.4136e15" }, "check": "Computed 2026-09-29 against optmat's HITRAN 2024 band table (data/external/hitran/CO2_bands.csv, one pole per band, all isotopologues) and fit/CO2 (TC-720). Total strength sum 2(A_1 + A_2) = 1.2570e-4 against 1.2787e-4 (-1.7 %); static vibrational polarizability sum f/E^2 = 2.0837 against 2.1212 au (-1.8 %). By region, against HITRAN's 12C16O2 bands alone: 15 um (r = 1, 0-1000 cm^-1) strength -0.15 %, static -0.14 %; 4.3 um (r = 2, 1500-2600 cm^-1) -0.10 %, -0.22 %; 2.7 um (r = 3) -2.5 %, -2.6 %; 2.0 um (r = 4) -35 %; against all isotopologues -1.7, -1.6, -3.9 and -36 %. The poles (cm^-1): 660.9 and 703.3 (optmat's one pole: 668.6), 2330.5 and 2367.9 (2346.2), 3724.5 and 3611.3 (3670.7), 4971.9 and 5090.1 (4982.0). The infrared polarizability against optmat's fixed bands in fit/CO2 (au; ppm of fit/CO2's total alpha): 633 nm -0.024791 against -0.025217 (+24 ppm); 1064 nm +72 ppm; 1550 nm +166 ppm; 3.5 um -2.0853 against -2.1077 (+1450 ppm); 4.0 um -8800 ppm (next to the 4.3 um band); 8 um -1055 ppm; 10 um +0.3266 against +0.3372 (-596 ppm); 10.6 um -536 ppm; 12 um -545 ppm. The infrared part of the GVD of pure CO2 (1 atm, 296 K, ideal gas, n - 1 linear): 3.5 um -16914 against -16831 fs^2/m; 10 um -4810 against -4900 fs^2/m; at 8, 10.6 and 12 um the two differ by 650-12000 fs^2/m because optmat's one-pole-per-band sum places weak bands (the 9.4 and 10.4 um 'laser' bands, hot bands) inside the window, which the GSE smooths over." } The terms are CO2's infrared contribution alone \(the GSE has no electronic CO2 term\)\. Fitted to the line sum of air with 370 ppm CO2 and 10 % humidity; how the wavelengths were fitted and over what range is not stated\. The strengths agree with HITRAN's principal isotopologue to 0\.1\-0\.2 % in the two strong bands, so the other isotopologues \(1\.6 % of the strength\) appear to be left out \(inferred, not stated\)\. Linear in density and per molecule with no temperature dependence \(HITRAN's 296 K strengths\)\. voronin\_2017/gse\_N2 ~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "gse_N2", "material": "N2", "equation": "Eq. (5) with Table 1, row r = 12 (p. 6)", "printed": "N2, electronic transitions (ref. 7): A_1r = 1.2029482, A_2r = 5.796725 (headed 'ps^2'), lambda_1r = 85 nm, lambda_2r = 24.546 nm, N_r = 19870 x 10^15 cm^-3", "variable": "as gse_CO2_infrared: per molecule alpha = sum_j 2 A_j / (E_j^2 - omega^2), E_j = 45.5634 / lambda_j[nm] hartree", "state": "none (per molecule; n - 1 linear in N_r)", "density": "none", "range": "0.3-13 um (the GSE's stated range for air)", "coefficients": { "A1": "1.2029482", "A2": "5.796725", "lambda1": "85", "lambda2": "24.546", "N": "19870e15" }, "check": "Computed 2026-09-29: S(0) = 13.99935, S(-2) = 11.73772, S(-4) = 30.11653, equal to margoliash_1978's N2 pseudospectrum's 13.99935, 11.73773, 30.11384 (S(-6) 101.70 against 101.82, S(-8) 353.0 against 384.6). Against fit/N2: alpha(633 nm) = 11.89654 against 11.89434 au (+185 ppm), 300 nm +236 ppm, 1550 nm +176 ppm, static +174 ppm (11.73772 against 11.73567)." } A two\-pole condensation of Margoliash and Meath's \(1978\) N2 DOSD, keeping its S\(0\) and S\(\-2\)\: the offset from fit/N2 is that DOSD's\. voronin\_2017/gse\_O2 ~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "gse_O2", "material": "O2", "equation": "Eq. (5) with Table 1, row r = 13 (p. 6)", "printed": "O2, electronic transitions (ref. 7): A_1r = 0.26507582, A_2r = 7.734925 (headed 'ps^2'), lambda_1r = 127 nm, lambda_2r = 29.469 nm, N_r = 5329.1 x 10^15 cm^-3", "variable": "as gse_CO2_infrared: per molecule alpha = sum_j 2 A_j / (E_j^2 - omega^2), E_j = 45.5634 / lambda_j[nm] hartree", "state": "none (per molecule; n - 1 linear in N_r)", "density": "none", "range": "0.3-13 um (the GSE's stated range for air)", "coefficients": { "A1": "0.26507582", "A2": "7.734925", "lambda1": "127", "lambda2": "29.469", "N": "5329.1e15" }, "check": "Computed 2026-09-29: S(0) = 16.00000, S(-2) = 10.59006, S(-4) = 34.707, against margoliash_1978's O2 pseudospectrum's 16.00000, 10.59007, 34.749. Against fit/O2: alpha(633 nm) = 10.77687 against 10.77495 au (+178 ppm), 300 nm +172 ppm, 1550 nm +252 ppm, static +277 ppm (10.59006 against 10.58713)." } A two\-pole condensation of Margoliash and Meath's \(1978\) O2 DOSD, keeping its S\(0\) and S\(\-2\)\. voronin\_2017/gse\_Ar ~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "gse_Ar", "material": "Ar", "equation": "Eq. (5) with Table 1, row r = 14 (p. 6)", "printed": "Ar, electronic transitions (refs. 8-10): A_1r = 0.93132145, A_2r = 7.217322 (headed 'ps^2'), lambda_1r = 87 nm, lambda_2r = 22.645 nm, N_r = 237.63 x 10^15 cm^-3", "variable": "as gse_CO2_infrared: per atom alpha = sum_j 2 A_j / (E_j^2 - omega^2), E_j = 45.5634 / lambda_j[nm] hartree", "state": "none (per atom; n - 1 linear in N_r)", "density": "none", "range": "0.3-13 um (the GSE's stated range for air)", "coefficients": { "A1": "0.93132145", "A2": "7.217322", "lambda1": "87", "lambda2": "22.645", "N": "237.63e15" }, "check": "Computed 2026-09-29: S(0) = 16.297 (argon has 18 electrons; Mathar 2004's sum to 500 eV is 13.3), static alpha = S(-2) = 10.357 au against fit/Ar's 11.077 (-6.50 %); alpha(633 nm) = 10.49186 against 11.22448 au (-6.53 %), -6.54 % at 500 nm, -6.50 % at 10 um. At N = 2.879e19 cm^-3 (Fig. 5c) the term gives n - 1 = 2.8124e-4 at 633 nm, 0.12 % above Peck and Fisher's 0 degrees C formula (peck_1964/peck_0C, 2.8090e-4, whose density is 2.687e19); at 2.687e19 it gives 2.6246e-4, 6.6 % below. Not argon's polarizability." } The GSE's argon term is 6\.5 % below argon's polarizability at every wavelength\: consistent with its having been matched to Peck and Fisher's 0 degrees C formula at the density printed for argon in Fig\. 5 \(2\.879e19 cm^\-3, 7\.1 % above 0 degrees C, 1 atm\)\. In air \(0\.93 % argon\) this is \-1\.6e\-7 in n \- 1 at 1 atm, which the four electronic terms together hide\: their sum reproduces Ciddor's formula for air \(source notes\)\. 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 :doc:`review register ` for the checks and decisions.