langhoff_1969#
P. W. Langhoff and M. Karplus, J. Opt. Soc. Am. 59, 863 (1969)
Source record · Catalogue · Review register
Source kind: fit
Recorded technique: Four-term Cauchy coefficients (alpha_k = S(-2k-2)) fitted to measured refractivity by a least-squares procedure constrained to a convergent series of Stieltjes (CREEP program, projection method), then summed by upper- and lower-bounding Pade approximants ([n,n]_f, [n,n-1]_f, [n,n]_k, ...) to extrapolate n - 1 into the ultraviolet.
Materials in structured entries: Ar, H2, He, Kr, N2, Ne, O2, Xe
Dataset quantities: cauchy_moment, refractivity
Contents: 31 datasets; 16 formulas.
Curation notes#
All values at NTP (0 degrees C, 1 atm), n - 1 = 2 pi N_0 alpha(omega) with 2 pi N_0 = 0.2502e-4 (omega in atomic units): N_0 is Loschmidt’s number, so the moments and predictions refer to the ideal-gas density (density ‘ideal’). The refractivity data fitted are those of the Landolt-Boernstein compilation (Ref. 1) over the intervals of Table I (e.g. Ar 2894-20586 A); for N2 the data span 20 587-4679 A (p. 8). No uncertainties are given for the moments.
Table III’s ‘present results’ are the [2,1]_alpha = [2,2]_f lower bounds ‘in every case but the last two in each column’, which are means of the [2,2]_f and [1,1]_k bounds (p. 6). Recomputing them (checked 2026-09-24) shows that for Ne only 1300 and 1200 A are means, for Ar every entry from 1800 A down and for Kr from 1800 A down differ from the lower bound by more than rounding (means), and for Xe all entries are lower bounds, as Tang and Poon 1974 found by repeating the calculation. Xe at 1600 A is misprinted (erratum). Table III’s ‘previous calculation’ columns and Table IV’s second column are Kingston’s (1964) values; Table IV’s measured column is Heddle et al.’s and Gill and Heddle’s (1963) and Marmo’s (via Liggett and Levinger 1968); Table V’s measured column is Wilkinson’s (1960) nitrogen data, not otherwise in the library. The Pade columns of Tables IV and V are extracted as checks; their +- is the spread over the measurement’s wavelength band (‘the uncertainties on the wavelength values correspond to the range used in the measurements’, Table IV note b), not an uncertainty.
Table VI (H and H2 Cauchy coefficients for N = 2-5 terms, fitted to 15 exact H points or Kirn’s 15 H2 measurements, with rms deviations) documents the convergence of the fit; its ‘Final’ H2 row is Table I’s and nothing else is extracted. Atomic hydrogen (Tables I, II, VI) is not a species of the package.
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
langhoff_1969/cauchy_He#
{
"id": "cauchy_He",
"material": "He",
"file": "He/langhoff_1969__cauchy_He.csv",
"table": "tbl_2b7dd49ddcb9",
"quantity": "cauchy_moment",
"unit": "au",
"abscissa": "order",
"abscissa_unit": "1",
"use": "check"
}
Table I, Cauchy coefficients alpha_0..alpha_3 = S(-2), S(-4), S(-6), S(-8) (footnote 19), rows by moment order, atomic units. Estimated from the refractivity data by the constrained fit, so not independent of them: a check.
langhoff_1969/cauchy_Ne#
{
"id": "cauchy_Ne",
"material": "Ne",
"file": "Ne/langhoff_1969__cauchy_Ne.csv",
"table": "tbl_2b7dd49ddcb9",
"quantity": "cauchy_moment",
"unit": "au",
"abscissa": "order",
"abscissa_unit": "1",
"use": "check"
}
Table I, Cauchy coefficients alpha_0..alpha_3 = S(-2), S(-4), S(-6), S(-8) (footnote 19), rows by moment order, atomic units. Estimated from the refractivity data by the constrained fit, so not independent of them: a check.
langhoff_1969/cauchy_Ar#
{
"id": "cauchy_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__cauchy_Ar.csv",
"table": "tbl_2b7dd49ddcb9",
"quantity": "cauchy_moment",
"unit": "au",
"abscissa": "order",
"abscissa_unit": "1",
"use": "check"
}
Table I, Cauchy coefficients alpha_0..alpha_3 = S(-2), S(-4), S(-6), S(-8) (footnote 19), rows by moment order, atomic units. Estimated from the refractivity data by the constrained fit, so not independent of them: a check.
langhoff_1969/cauchy_Kr#
{
"id": "cauchy_Kr",
"material": "Kr",
"file": "Kr/langhoff_1969__cauchy_Kr.csv",
"table": "tbl_2b7dd49ddcb9",
"quantity": "cauchy_moment",
"unit": "au",
"abscissa": "order",
"abscissa_unit": "1",
"use": "check"
}
Table I, Cauchy coefficients alpha_0..alpha_3 = S(-2), S(-4), S(-6), S(-8) (footnote 19), rows by moment order, atomic units. Estimated from the refractivity data by the constrained fit, so not independent of them: a check.
langhoff_1969/cauchy_Xe#
{
"id": "cauchy_Xe",
"material": "Xe",
"file": "Xe/langhoff_1969__cauchy_Xe.csv",
"table": "tbl_2b7dd49ddcb9",
"quantity": "cauchy_moment",
"unit": "au",
"abscissa": "order",
"abscissa_unit": "1",
"use": "check"
}
Table I, Cauchy coefficients alpha_0..alpha_3 = S(-2), S(-4), S(-6), S(-8) (footnote 19), rows by moment order, atomic units. Estimated from the refractivity data by the constrained fit, so not independent of them: a check. Tang and Poon 1974 show these Xe moments give ultraviolet refractivities incompatible with Kingston’s and with the measurements, and recommend another set (tang_1974/cauchy_recommended_Xe).
langhoff_1969/cauchy_H2#
{
"id": "cauchy_H2",
"material": "H2",
"file": "H2/langhoff_1969__cauchy_H2.csv",
"table": "tbl_2b7dd49ddcb9",
"quantity": "cauchy_moment",
"unit": "au",
"abscissa": "order",
"abscissa_unit": "1",
"use": "check"
}
Table I, Cauchy coefficients alpha_0..alpha_3 = S(-2), S(-4), S(-6), S(-8) (footnote 19), rows by moment order, atomic units. Estimated from the refractivity data by the constrained fit, so not independent of them: a check.
langhoff_1969/cauchy_N2#
{
"id": "cauchy_N2",
"material": "N2",
"file": "N2/langhoff_1969__cauchy_N2.csv",
"table": "tbl_2b7dd49ddcb9",
"quantity": "cauchy_moment",
"unit": "au",
"abscissa": "order",
"abscissa_unit": "1",
"use": "check"
}
Table I, Cauchy coefficients alpha_0..alpha_3 = S(-2), S(-4), S(-6), S(-8) (footnote 19), rows by moment order, atomic units. Estimated from the refractivity data by the constrained fit, so not independent of them: a check.
langhoff_1969/cauchy_O2#
{
"id": "cauchy_O2",
"material": "O2",
"file": "O2/langhoff_1969__cauchy_O2.csv",
"table": "tbl_2b7dd49ddcb9",
"quantity": "cauchy_moment",
"unit": "au",
"abscissa": "order",
"abscissa_unit": "1",
"use": "check"
}
Table I, Cauchy coefficients alpha_0..alpha_3 = S(-2), S(-4), S(-6), S(-8) (footnote 19), rows by moment order, atomic units. Estimated from the refractivity data by the constrained fit, so not independent of them: a check.
langhoff_1969/theory_Ne#
{
"id": "theory_Ne",
"material": "Ne",
"file": "Ne/langhoff_1969__theory_Ne.csv",
"table": "tbl_af4050e393d8",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table III, ‘Present results’: Pade predictions of n - 1 at nominal vacuum wavelengths; lower bounds except where noted in the source notes (means of bounds).
langhoff_1969/theory_Ar#
{
"id": "theory_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__theory_Ar.csv",
"table": "tbl_af4050e393d8",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table III, ‘Present results’: Pade predictions of n - 1 at nominal vacuum wavelengths; lower bounds except where noted in the source notes (means of bounds).
langhoff_1969/theory_Kr#
{
"id": "theory_Kr",
"material": "Kr",
"file": "Kr/langhoff_1969__theory_Kr.csv",
"table": "tbl_af4050e393d8",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table III, ‘Present results’: Pade predictions of n - 1 at nominal vacuum wavelengths; lower bounds except where noted in the source notes (means of bounds).
langhoff_1969/theory_Xe#
{
"id": "theory_Xe",
"material": "Xe",
"file": "Xe/langhoff_1969__theory_Xe.csv",
"table": "tbl_af4050e393d8",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table III, ‘Present results’: Pade predictions of n - 1 at nominal vacuum wavelengths; lower bounds except where noted in the source notes (means of bounds).
langhoff_1969/kingston_1964_Ar#
{
"id": "kingston_1964_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__kingston_1964_Ar.csv",
"table": "tbl_af4050e393d8",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "reproduced",
"reproduced_from": [
"kingston_1964"
]
}
Table III, ‘Previous calculation’: Kingston’s Table I, identical to kingston_1964.
langhoff_1969/kingston_1964_Kr#
{
"id": "kingston_1964_Kr",
"material": "Kr",
"file": "Kr/langhoff_1969__kingston_1964_Kr.csv",
"table": "tbl_af4050e393d8",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "reproduced",
"reproduced_from": [
"kingston_1964"
]
}
Table III, ‘Previous calculation’: Kingston’s Table I, identical to kingston_1964.
langhoff_1969/kingston_1964_Xe#
{
"id": "kingston_1964_Xe",
"material": "Xe",
"file": "Xe/langhoff_1969__kingston_1964_Xe.csv",
"table": "tbl_af4050e393d8",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "reproduced",
"reproduced_from": [
"kingston_1964"
]
}
Table III, ‘Previous calculation’: Kingston’s Table I, identical to kingston_1964.
langhoff_1969/heddle_1963_Ar#
{
"id": "heddle_1963_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__heddle_1963_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"heddle_1963"
],
"uncertainty": {
"column": true,
"level": "1 sigma"
}
}
Table IV, ‘Measured’ at 1850 +- 80, 1805 +- 80 and 1210 +- 30 A: Heddle et al.’s Cerenkov values, identical to heddle_1963.
langhoff_1969/gill_1963_Ar#
{
"id": "gill_1963_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__gill_1963_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"gill_1963"
]
}
Table IV, ‘Measured’ at 1216 A, first value: Gill and Heddle’s 5.65, identical to gill_1963.
langhoff_1969/marmo_Ar#
{
"id": "marmo_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__marmo_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"liggett_1968"
]
}
Table IV, ‘Measured’ at 1216 A, second value marked e: ‘F. Marmo, as listed in Liggett and Levinger’ (Rayleigh scattering, unpublished); identical to liggett_1968’s Table III value.
langhoff_1969/kingston_1964_table4_Ar#
{
"id": "kingston_1964_table4_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__kingston_1964_table4_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "reproduced",
"reproduced_from": [
"kingston_1964"
]
}
Table IV, ‘Previous calculation’ (first value at 1216 A): Kingston’s Table III calculated column, identical to kingston_1964/theory_table3_Ar; the +- (sigma column, not declared) is the spread over the wavelength band.
langhoff_1969/liggett_1968_Ar#
{
"id": "liggett_1968_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__liggett_1968_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"liggett_1968"
]
}
Table IV, ‘Previous calculation’ at 1216 A, second value marked f: Liggett and Levinger’s Kramers-Kronig value 4.98, identical to liggett_1968.
langhoff_1969/pade11f_Ar#
{
"id": "pade11f_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__pade11f_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table IV, present results, the [1,1]_f Pade approximant at Heddle’s and Gill’s wavelengths; the sigma column (not declared) is the spread over the wavelength band.
langhoff_1969/pade21f_Ar#
{
"id": "pade21f_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__pade21f_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table IV, present results, the [2,1]_f Pade approximant at Heddle’s and Gill’s wavelengths; the sigma column (not declared) is the spread over the wavelength band.
langhoff_1969/pade22f_Ar#
{
"id": "pade22f_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__pade22f_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table IV, present results, the [2,2]_f (lower bound) Pade approximant at Heddle’s and Gill’s wavelengths; the sigma column (not declared) is the spread over the wavelength band.
langhoff_1969/pade11k_Ar#
{
"id": "pade11k_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__pade11k_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table IV, present results, the [1,1]_k (upper bound) Pade approximant at Heddle’s and Gill’s wavelengths; the sigma column (not declared) is the spread over the wavelength band.
langhoff_1969/pade10k_Ar#
{
"id": "pade10k_Ar",
"material": "Ar",
"file": "Ar/langhoff_1969__pade10k_Ar.csv",
"table": "tbl_c1a45b3fe5fa",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table IV, present results, the [1,0]_k (upper bound) Pade approximant at Heddle’s and Gill’s wavelengths; the sigma column (not declared) is the spread over the wavelength band.
langhoff_1969/wilkinson_1960_N2#
{
"id": "wilkinson_1960_N2",
"material": "N2",
"file": "N2/langhoff_1969__wilkinson_1960_N2.csv",
"table": "tbl_6e6971f10d5e",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "unknown",
"use": "reproduced",
"reproduced_from": [
"P. G. Wilkinson, J. Opt. Soc. Am. 50, 1002 (1960)"
]
}
Table V, ‘Measured’: Wilkinson’s nitrogen refractivities from line shifts in a high-dispersion vacuum spectrograph, 2042.48-1649.19 A (vacuum wavelengths). The original is not in the library, so this reproduction may be fitted. Langhoff and Karplus: the values at about 1750-1650 A ‘are noticeably above the [1,1]_k upper bounds, a discrepancy that may be due to the presence of oxygen contamination, as suggested by Heddle et al.’ (p. 7). No uncertainty is given.
langhoff_1969/pade11f_N2#
{
"id": "pade11f_N2",
"material": "N2",
"file": "N2/langhoff_1969__pade11f_N2.csv",
"table": "tbl_6e6971f10d5e",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table V, present results, the [1,1]_f Pade approximant at Wilkinson’s wavelengths.
langhoff_1969/pade21f_N2#
{
"id": "pade21f_N2",
"material": "N2",
"file": "N2/langhoff_1969__pade21f_N2.csv",
"table": "tbl_6e6971f10d5e",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table V, present results, the [2,1]_f Pade approximant at Wilkinson’s wavelengths. The 1874.25 A value is misprinted (erratum).
langhoff_1969/pade22f_N2#
{
"id": "pade22f_N2",
"material": "N2",
"file": "N2/langhoff_1969__pade22f_N2.csv",
"table": "tbl_6e6971f10d5e",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table V, present results, the [2,2]_f (lower bound) Pade approximant at Wilkinson’s wavelengths.
langhoff_1969/pade11k_N2#
{
"id": "pade11k_N2",
"material": "N2",
"file": "N2/langhoff_1969__pade11k_N2.csv",
"table": "tbl_6e6971f10d5e",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table V, present results, the [1,1]_k (upper bound) Pade approximant at Wilkinson’s wavelengths.
langhoff_1969/pade10k_N2#
{
"id": "pade10k_N2",
"material": "N2",
"file": "N2/langhoff_1969__pade10k_N2.csv",
"table": "tbl_6e6971f10d5e",
"quantity": "refractivity",
"scale": 0.0001,
"abscissa_unit": "angstrom",
"wavelength_medium": "vacuum",
"state": "1atm_0C",
"density": "ideal",
"use": "check"
}
Table V, present results, the [1,0]_k (upper bound) Pade approximant at Wilkinson’s wavelengths.
Formulas#
langhoff_1969/cauchy_He#
{
"id": "cauchy_He",
"material": "He",
"equation": "Eq. (29), p. 4, with Table I",
"printed": "(n - 1) = 2 pi N_0 (alpha_0 + alpha_1 omega^2 + alpha_2 omega^4 + alpha_3 omega^6), 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "fitted to 2753-5462 A; convergent only below omega_1 = 0.7797 au",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"alpha_0": "1.3838",
"alpha_1": "1.550",
"alpha_2": "2.066",
"alpha_3": "2.95",
"omega_1": "0.7797"
},
"check": "At 5462.258 A (vacuum, omega = 0.08342 au) it gives 34.895e-6, Leonard's (1974) recommended value to the digit. Checked 2026-09-24."
}
Four-term Cauchy fit constrained to be a Stieltjes series (Eqs. (5), (28), (A2)-(A7)); alpha_k = S(-2k-2), extracted as dataset cauchy_He. 2 pi N_0 = 0.2502e-4 is 2 pi N_L a_0^3 with Loschmidt’s number (2.5016e-5), hence density ‘ideal’.
langhoff_1969/pade22f_He#
{
"id": "pade22f_He",
"material": "He",
"equation": "Eq. (30), p. 4, with Table II",
"printed": "(n - 1) > 2 pi N_0 [2,2]_f = 2 pi N_0 [f_a/(omega_a^2 - omega^2) + f_b/(omega_b^2 - omega^2)], 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "a lower bound to n - 1 below omega_1 = 0.7797 au; built from the Table I moments",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"omega_a": "0.8227",
"omega_b": "1.485",
"f_a": "0.6097",
"f_b": "1.065"
},
"check": "Its four moments f_a/omega_a^(2k+2) + f_b/omega_b^(2k+2) reproduce Table I's alpha_0..alpha_3 to 0.1 %. No tabulated values (Fig. 5 only). Checked 2026-09-24."
}
The [2,2]_f Pade approximant (identical to [2,1]_alpha), a two-term Sellmeier form with fictitious transition frequencies and strengths; the paper suggests it ‘is an accurate representation’ for H2 (p. 7).
langhoff_1969/cauchy_Ne#
{
"id": "cauchy_Ne",
"material": "Ne",
"equation": "Eq. (29), p. 4, with Table I",
"printed": "(n - 1) = 2 pi N_0 (alpha_0 + alpha_1 omega^2 + alpha_2 omega^4 + alpha_3 omega^6), 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "fitted to 2894-5462 A; convergent only below omega_1 = 0.6127 au",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"alpha_0": "2.6680",
"alpha_1": "2.863",
"alpha_2": "5.682",
"alpha_3": "13.4",
"omega_1": "0.6127"
},
"check": "At 5462.258 A it gives 67.259e-6, 0.03 % above Leonard's (1974) recommended 67.236. Checked 2026-09-24."
}
Four-term Cauchy fit constrained to be a Stieltjes series (Eqs. (5), (28), (A2)-(A7)); alpha_k = S(-2k-2), extracted as dataset cauchy_Ne. 2 pi N_0 = 0.2502e-4 is 2 pi N_L a_0^3 with Loschmidt’s number (2.5016e-5), hence density ‘ideal’.
langhoff_1969/pade22f_Ne#
{
"id": "pade22f_Ne",
"material": "Ne",
"equation": "Eq. (30), p. 4, with Table II",
"printed": "(n - 1) > 2 pi N_0 [2,2]_f = 2 pi N_0 [f_a/(omega_a^2 - omega^2) + f_b/(omega_b^2 - omega^2)], 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "a lower bound to n - 1 below omega_1 = 0.6127 au; built from the Table I moments",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"omega_a": "0.6401",
"omega_b": "1.672",
"f_a": "0.3755",
"f_b": "4.896"
},
"check": "Its four moments f_a/omega_a^(2k+2) + f_b/omega_b^(2k+2) reproduce Table I's alpha_0..alpha_3 to 0.1 %. Its value equals Table III's predicted Ne column to 0.0002e-4 from 2200 to 1400 A; the 1300 and 1200 A entries are means of the [2,2]_f and [1,1]_k bounds (0.7864, 0.8174 against 0.7858, 0.8158). Checked 2026-09-24."
}
The [2,2]_f Pade approximant (identical to [2,1]_alpha), a two-term Sellmeier form with fictitious transition frequencies and strengths; the paper suggests it ‘is an accurate representation’ for H2 (p. 7).
langhoff_1969/cauchy_Ar#
{
"id": "cauchy_Ar",
"material": "Ar",
"equation": "Eq. (29), p. 4, with Table I",
"printed": "(n - 1) = 2 pi N_0 (alpha_0 + alpha_1 omega^2 + alpha_2 omega^4 + alpha_3 omega^6), 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "fitted to 2894-20586 A; convergent only below omega_1 = 0.4272 au",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"alpha_0": "11.091",
"alpha_1": "28.16",
"alpha_2": "96.78",
"alpha_3": "387",
"omega_1": "0.4272"
},
"check": "At 5462.258 A it gives 282.52e-6, 0.02 % above Leonard's (1974) recommended 282.4655. Checked 2026-09-24."
}
Four-term Cauchy fit constrained to be a Stieltjes series (Eqs. (5), (28), (A2)-(A7)); alpha_k = S(-2k-2), extracted as dataset cauchy_Ar. 2 pi N_0 = 0.2502e-4 is 2 pi N_L a_0^3 with Loschmidt’s number (2.5016e-5), hence density ‘ideal’.
langhoff_1969/pade22f_Ar#
{
"id": "pade22f_Ar",
"material": "Ar",
"equation": "Eq. (30), p. 4, with Table II",
"printed": "(n - 1) > 2 pi N_0 [2,2]_f = 2 pi N_0 [f_a/(omega_a^2 - omega^2) + f_b/(omega_b^2 - omega^2)], 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "a lower bound to n - 1 below omega_1 = 0.4272 au; built from the Table I moments",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"omega_a": "0.4808",
"omega_b": "0.8899",
"f_a": "1.069",
"f_b": "5.123"
},
"check": "Its four moments f_a/omega_a^(2k+2) + f_b/omega_b^(2k+2) reproduce Table I's alpha_0..alpha_3 to 0.1 %. Reproduces Table IV's [2,2]_f column (3.321, 3.357, 4.964, 4.914 at 1850, 1805, 1210, 1216 A against 3.32, 3.36, 4.96, 4.91) and Table III's argon entries from 2200 to 1900 A to about 0.002e-4; from 1800 A down Table III prints means of bounds (3.364 ... 6.021 against 3.362 ... 5.053). Checked 2026-09-24."
}
The [2,2]_f Pade approximant (identical to [2,1]_alpha), a two-term Sellmeier form with fictitious transition frequencies and strengths; the paper suggests it ‘is an accurate representation’ for H2 (p. 7).
langhoff_1969/cauchy_Kr#
{
"id": "cauchy_Kr",
"material": "Kr",
"equation": "Eq. (29), p. 4, with Table I",
"printed": "(n - 1) = 2 pi N_0 (alpha_0 + alpha_1 omega^2 + alpha_2 omega^4 + alpha_3 omega^6), 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "fitted to 2025-6707 A; convergent only below omega_1 = 0.3687 au",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"alpha_0": "16.740",
"alpha_1": "55.53",
"alpha_2": "238.7",
"alpha_3": "1200",
"omega_1": "0.3687"
},
"check": "At 5462.258 A it gives 428.80e-6, 0.47 % below Leonard's (1974) recommended 430.84 (Leonard rescales Koch's and Kronjager's data, which the fit used, to his absolute value). Checked 2026-09-24."
}
Four-term Cauchy fit constrained to be a Stieltjes series (Eqs. (5), (28), (A2)-(A7)); alpha_k = S(-2k-2), extracted as dataset cauchy_Kr. 2 pi N_0 = 0.2502e-4 is 2 pi N_L a_0^3 with Loschmidt’s number (2.5016e-5), hence density ‘ideal’.
langhoff_1969/pade22f_Kr#
{
"id": "pade22f_Kr",
"material": "Kr",
"equation": "Eq. (30), p. 4, with Table II",
"printed": "(n - 1) > 2 pi N_0 [2,2]_f = 2 pi N_0 [f_a/(omega_a^2 - omega^2) + f_b/(omega_b^2 - omega^2)], 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "a lower bound to n - 1 below omega_1 = 0.3687 au; built from the Table I moments",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"omega_a": "0.4226",
"omega_b": "0.7272",
"f_a": "1.150",
"f_b": "5.448"
},
"check": "Its four moments f_a/omega_a^(2k+2) + f_b/omega_b^(2k+2) reproduce Table I's alpha_0..alpha_3 to 0.1 %. Reproduces Table III's krypton entries from 2200 to 1900 A to 0.0015e-4; from 1800 A down Table III prints means of bounds. Checked 2026-09-24."
}
The [2,2]_f Pade approximant (identical to [2,1]_alpha), a two-term Sellmeier form with fictitious transition frequencies and strengths; the paper suggests it ‘is an accurate representation’ for H2 (p. 7).
langhoff_1969/cauchy_Xe#
{
"id": "cauchy_Xe",
"material": "Xe",
"equation": "Eq. (29), p. 4, with Table I",
"printed": "(n - 1) = 2 pi N_0 (alpha_0 + alpha_1 omega^2 + alpha_2 omega^4 + alpha_3 omega^6), 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "fitted to 2144-6707 A; convergent only below omega_1 = 0.3100 au",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"alpha_0": "27.340",
"alpha_1": "116.2",
"alpha_2": "928.3",
"alpha_3": "8520",
"omega_1": "0.3100"
},
"check": "At 5462.258 A it gives 705.47e-6, 0.16 % below Leonard's (1974) recommended 706.60. Checked 2026-09-24."
}
Four-term Cauchy fit constrained to be a Stieltjes series (Eqs. (5), (28), (A2)-(A7)); alpha_k = S(-2k-2), extracted as dataset cauchy_Xe. 2 pi N_0 = 0.2502e-4 is 2 pi N_L a_0^3 with Loschmidt’s number (2.5016e-5), hence density ‘ideal’.
langhoff_1969/pade22f_Xe#
{
"id": "pade22f_Xe",
"material": "Xe",
"equation": "Eq. (30), p. 4, with Table II",
"printed": "(n - 1) > 2 pi N_0 [2,2]_f = 2 pi N_0 [f_a/(omega_a^2 - omega^2) + f_b/(omega_b^2 - omega^2)], 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "a lower bound to n - 1 below omega_1 = 0.3100 au; built from the Table I moments",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"omega_a": "0.3265",
"omega_b": "0.9318",
"f_a": "1.098",
"f_b": "14.80"
},
"check": "Its four moments f_a/omega_a^(2k+2) + f_b/omega_b^(2k+2) reproduce Table I's alpha_0..alpha_3 to 0.1 %. Table III's xenon entries (all lower bounds, as Tang and Poon 1974 note) follow the [2,1]_alpha approximant of Table I's moments (Eq. (12b); Tang and Poon's Eq. (51)) to 0.002e-4 from 2200 to 1500 A (8.7959, 9.1215, 9.5548, 10.1592, 11.0596, 12.5443, 23.9009 against 8.795 ... 23.898); these rounded Table II coefficients give 0.004-0.008e-4 more (0.04 more at 1500 A, near the pole at about 1396 A). At 1600 A the moments give 15.462 and this form 15.474 against the printed 15.640, corrected by an erratum to 15.460 (Tang and Poon's recomputation). Checked 2026-09-24."
}
The [2,2]_f Pade approximant (identical to [2,1]_alpha), a two-term Sellmeier form with fictitious transition frequencies and strengths; the paper suggests it ‘is an accurate representation’ for H2 (p. 7).
langhoff_1969/cauchy_H2#
{
"id": "cauchy_H2",
"material": "H2",
"equation": "Eq. (29), p. 4, with Table I",
"printed": "(n - 1) = 2 pi N_0 (alpha_0 + alpha_1 omega^2 + alpha_2 omega^4 + alpha_3 omega^6), 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "fitted to 1854-5462 A; convergent only below omega_1 = 0.4110 au",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"alpha_0": "5.4390",
"alpha_1": "20.02",
"alpha_2": "81.61",
"alpha_3": "350",
"omega_1": "0.4110"
},
"check": "At 5462.258 A it gives 139.67e-6. Table VI's 'Final' H2 row equals these coefficients. Checked 2026-09-24."
}
Four-term Cauchy fit constrained to be a Stieltjes series (Eqs. (5), (28), (A2)-(A7)); alpha_k = S(-2k-2), extracted as dataset cauchy_H2. 2 pi N_0 = 0.2502e-4 is 2 pi N_L a_0^3 with Loschmidt’s number (2.5016e-5), hence density ‘ideal’.
langhoff_1969/pade22f_H2#
{
"id": "pade22f_H2",
"material": "H2",
"equation": "Eq. (30), p. 4, with Table II",
"printed": "(n - 1) > 2 pi N_0 [2,2]_f = 2 pi N_0 [f_a/(omega_a^2 - omega^2) + f_b/(omega_b^2 - omega^2)], 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "a lower bound to n - 1 below omega_1 = 0.4110 au; built from the Table I moments",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"omega_a": "0.4736",
"omega_b": "0.7441",
"f_a": "0.8627",
"f_b": "0.8822"
},
"check": "Its four moments f_a/omega_a^(2k+2) + f_b/omega_b^(2k+2) reproduce Table I's alpha_0..alpha_3 to 0.1 %. No tabulated values (Fig. 6 only). Checked 2026-09-24."
}
The [2,2]_f Pade approximant (identical to [2,1]_alpha), a two-term Sellmeier form with fictitious transition frequencies and strengths; the paper suggests it ‘is an accurate representation’ for H2 (p. 7).
langhoff_1969/cauchy_N2#
{
"id": "cauchy_N2",
"material": "N2",
"equation": "Eq. (29), p. 4, with Table I",
"printed": "(n - 1) = 2 pi N_0 (alpha_0 + alpha_1 omega^2 + alpha_2 omega^4 + alpha_3 omega^6), 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "fitted to 4679-20586 A; convergent only below omega_1 = 0.4487 au",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"alpha_0": "11.743",
"alpha_1": "30.17",
"alpha_2": "99.21",
"alpha_3": "374",
"omega_1": "0.4487"
},
"check": "At 5462.258 A it gives 299.19e-6. Checked 2026-09-24."
}
Four-term Cauchy fit constrained to be a Stieltjes series (Eqs. (5), (28), (A2)-(A7)); alpha_k = S(-2k-2), extracted as dataset cauchy_N2. 2 pi N_0 = 0.2502e-4 is 2 pi N_L a_0^3 with Loschmidt’s number (2.5016e-5), hence density ‘ideal’.
langhoff_1969/pade22f_N2#
{
"id": "pade22f_N2",
"material": "N2",
"equation": "Eq. (30), p. 4, with Table II",
"printed": "(n - 1) > 2 pi N_0 [2,2]_f = 2 pi N_0 [f_a/(omega_a^2 - omega^2) + f_b/(omega_b^2 - omega^2)], 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "a lower bound to n - 1 below omega_1 = 0.4487 au; built from the Table I moments",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"omega_a": "0.4929",
"omega_b": "0.8531",
"f_a": "1.244",
"f_b": "4.820"
},
"check": "Its four moments f_a/omega_a^(2k+2) + f_b/omega_b^(2k+2) reproduce Table I's alpha_0..alpha_3 to 0.1 %. Reproduces Table V's [2,2]_f column at all 22 wavelengths to within +0.0002 to +0.0012 (x 1e-4), the rounding of the four-digit coefficients. Checked 2026-09-24."
}
The [2,2]_f Pade approximant (identical to [2,1]_alpha), a two-term Sellmeier form with fictitious transition frequencies and strengths; the paper suggests it ‘is an accurate representation’ for H2 (p. 7).
langhoff_1969/cauchy_O2#
{
"id": "cauchy_O2",
"material": "O2",
"equation": "Eq. (29), p. 4, with Table I",
"printed": "(n - 1) = 2 pi N_0 (alpha_0 + alpha_1 omega^2 + alpha_2 omega^4 + alpha_3 omega^6), 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "fitted to 4388-9224 A; convergent only below omega_1 = 0.2249 au",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"alpha_0": "10.600",
"alpha_1": "36.97",
"alpha_2": "132.0",
"alpha_3": "480",
"omega_1": "0.2249"
},
"check": "At 5462.258 A it gives 271.81e-6. Checked 2026-09-24."
}
Four-term Cauchy fit constrained to be a Stieltjes series (Eqs. (5), (28), (A2)-(A7)); alpha_k = S(-2k-2), extracted as dataset cauchy_O2. 2 pi N_0 = 0.2502e-4 is 2 pi N_L a_0^3 with Loschmidt’s number (2.5016e-5), hence density ‘ideal’.
langhoff_1969/pade22f_O2#
{
"id": "pade22f_O2",
"material": "O2",
"equation": "Eq. (30), p. 4, with Table II",
"printed": "(n - 1) > 2 pi N_0 [2,2]_f = 2 pi N_0 [f_a/(omega_a^2 - omega^2) + f_b/(omega_b^2 - omega^2)], 2 pi N_0 = 0.2502 x 10^-4",
"variable": "omega: photon energy (angular frequency) in atomic units (hartree)",
"state": "1atm_0C",
"density": "ideal",
"range": "a lower bound to n - 1 below omega_1 = 0.2249 au; built from the Table I moments",
"coefficients": {
"two_pi_N0": "0.2502e-4",
"omega_a": "0.5121",
"omega_b": "0.6200",
"f_a": "2.033",
"f_b": "1.095"
},
"check": "Its four moments f_a/omega_a^(2k+2) + f_b/omega_b^(2k+2) reproduce Table I's alpha_0..alpha_3 to 0.1 %. No tabulated values ('no short-wavelength data are available', p. 8). Checked 2026-09-24."
}
The [2,2]_f Pade approximant (identical to [2,1]_alpha), a two-term Sellmeier form with fictitious transition frequencies and strengths; the paper suggests it ‘is an accurate representation’ for H2 (p. 7).
Corrections#
[
{
"dataset": "theory_Xe",
"row": "1600",
"column": "value",
"printed": "15.640",
"corrected": "15.460",
"reason": "The xenon entries of Table III are the [2,1]_alpha lower bounds; at 1600 A that bound is 15.46, and 15.640 transposes two digits.",
"evidence": "The [2,1]_alpha approximant from Table I's xenon moments (Eq. (12b), in Tang and Poon's explicit form, Eq. (51)) gives 15.462 at 1600 A (and reproduces the other xenon entries to 0.002); the [2,2]_f form with Table II's coefficients gives 15.474; Tang and Poon 1974 (Table V, 'Lower bound Pade [2,1]_alpha') print 15.460 from the same moments. Table image tbl_af4050e393d8.jpg shows 15.640. Computed 2026-09-24."
},
{
"dataset": "pade21f_N2",
"row": "1874.25",
"column": "value",
"printed": "4.492",
"corrected": "3.492",
"reason": "The [2,1]_f bound lies between the [1,1]_f and [2,2]_f bounds at every other wavelength (it is a lower bound converging from below); 4.492 is 1.0 above both and above the neighbouring rows, 3.492 fits (3.464 < 3.492 < 3.496; neighbours 3.453 at 1930.04 A and 3.513 at 1845.87 A).",
"evidence": "Table image tbl_6e6971f10d5e.jpg shows 4.492; the leading digit is misprinted. The [2,1]_f bound can be computed from Eq. (16b) with Table I's N2 moments (N = 14; 11.743, 30.17, 99.21; 2 pi N0 = 0.2502e-4): it gives 3.492 at 1874.25 A and reproduces the other 21 rows of the column to +-0.001 (second read 2026-09-25; TC-102)."
}
]
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.