shelton_1986#

D. P. Shelton and V. Mizrahi, Phys. Rev. A 33, 72 (1986)

Source record · Catalogue · Review register

Source kind: measurement

Recorded technique: Periodic phase matching of electric-field-induced second-harmonic generation (ESHG): a cw dye or Ar+ laser (458-700 nm, about 1 W) is focused through a cell with an array of 150 wire-electrode pairs (N = 75, mean spacing 2.6924 mm) giving a spatially periodic static field; the gas density is varied until |Delta k| = |2 k_w - k_2w| equals |Delta k|_opt = 1161.5 m^-1 (+-0.05 %), found from the peak second-harmonic signal (parabola fit, 1e-4 precision in pressure). Temperature by a thermistor in the gas (+-0.2 C); pressure by Baratron (+-0.15 %) or Heise gauge (He, +-0.1 % of 160 bar); density from the virial equation of state (Dymond & Smith). Pressures up to 130 bar (He) and 10 bar (others); virial corrections to the refractivity below 0.05 % (p. 73). Dye-laser wavelengths by a 0.75 m grating spectrometer.

Materials in structured entries: Ar, D2, H2, He, N2, O2

Dataset quantities: phase_match_density_ratio

Contents: 2 datasets; 12 formulas.

Curation notes#

Observable (Eq. 3): Delta k = 2 k_w - k_2w = (4 pi / lambda_w)(n_w - n_2w) = -(2 pi / (lambda_w eps0)) (alpha_2w - alpha_w) rho, lambda_w the fundamental’s vacuum wavelength; periodic phase match occurs at |Delta k| = |Delta k|_opt. So the measured density rho (real gas, virial EOS) at phase match gives the dispersion n(2 omega) - n(omega) per unit density. The Ar+ lines are listed at their vacuum wavelengths (514.7, 496.6, 488.1, 476.6 nm), which confirms Table III is in vacuum wavelengths.

Uncertainty: ‘The estimated overall uncertainty of the measurements varies between +-0.2 % and +-0.4 % for He, but has a nearly uniform value of +-0.2 % for all the other gases’ (p. 74); the D2/H2 ratio is ‘about +-0.05 %’ because both gases were measured with the same gauge (p. 74). He differs by 0.5 % from the earlier Shelton & Buckingham (1982) measurement and by 0.3 % from Glover & Weinhold’s theory (‘consistent with a pressure scale error’, p. 74).

The individual phase-match densities of all six gases are shown only in Fig. 2 (lambda^3/rho against nu^2) and are not tabulated; the paper’s results are the fits of Table I (lambda^3/rho, Eq. 6) and their transformation to (n_w - n_0) at the ideal-gas density of 0 degrees C, 1 atm (Table II, Eq. 8), recorded as formulas. Only Table III, the D2/H2 density ratio, is data. The ‘calc’ column of Table III is Shelton’s evaluation of Rychlewski’s (1983) ab initio polarizabilities, kept as a reproduction. The D2 was 99.5 at.% (chief impurity HD), which would shift the ratio up ‘by at most 0.02 %’ (p. 75).

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

Datasets#

shelton_1986/ratio_D2_H2#

{
  "id": "ratio_D2_H2",
  "material": "D2",
  "file": "D2/shelton_1986__ratio_D2_H2.csv",
  "table": "tbl_0a217d3795df",
  "quantity": "phase_match_density_ratio",
  "abscissa_unit": "nm",
  "wavelength_medium": "vacuum",
  "density": "real",
  "use": "primary",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "reference_material": "H2",
    "harmonic": 2,
    "process": "ESHG (electric-field-induced second-harmonic generation), periodic phase matching",
    "definition": "rho_gas / rho_ref at which both gases give the same wave-vector mismatch |Delta k| = (4 pi / lambda_w) |n(2w) - n(w)| (the apparatus's |Delta k|_opt); to first order in density, rho_gas / rho_ref = [alpha_ref(2w) - alpha_ref(w)] / [alpha_gas(2w) - alpha_gas(w)]; wavelength = fundamental (vacuum)",
    "densities": "real gas, virial equation of state"
  }
}

rho_D2 / rho_H2 at equal ESHG wave-vector mismatch, measured together under the same conditions (Table III). Per-point errors are the printed parenthesised last-digit errors; the overall accuracy is ‘about +-0.05 %’ (p. 74). The zero-frequency limit of the fitted line (Fig. 3) is 1.0290.

shelton_1986/rychlewski_ratio_D2_H2#

{
  "id": "rychlewski_ratio_D2_H2",
  "material": "D2",
  "file": "D2/shelton_1986__rychlewski_ratio_D2_H2.csv",
  "table": "tbl_0a217d3795df",
  "quantity": "phase_match_density_ratio",
  "abscissa_unit": "nm",
  "wavelength_medium": "vacuum",
  "use": "reproduced",
  "reproduced_from": [
    "rychlewski_1983"
  ],
  "parameters": {
    "reference_material": "H2",
    "harmonic": 2,
    "process": "ESHG (electric-field-induced second-harmonic generation), periodic phase matching",
    "definition": "as ratio_D2_H2"
  }
}

Table III ‘calc’: the ratio evaluated by Shelton & Mizrahi from Rychlewski’s (ref. 23) ab initio polarizabilities; ‘the calculation is seen to overestimate the dispersion difference between H2 and D2 at high frequencies’ (p. 75).

Formulas#

shelton_1986/phase_match_He#

{
  "id": "phase_match_He",
  "material": "He",
  "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)",
  "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 0.05791, B' (10^-35 m^8/mol) = 0.0875, C' not given",
  "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1",
  "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A1": "0.05791e-21",
    "B1": "0.0875e-35"
  },
  "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A, B to the last digit (checked 2026-09-24)."
}

The fit to the measured phase-match densities of Fig. 2 (the points are not tabulated). With Eq. (7), (n_2w - n_w)_STP,i = rho_STP,i |Delta k|_opt lambda / (4 pi rho): the dispersion n(2 omega) - n(omega) of the ideal-gas-density gas at 0 degrees C, 1 atm. Accuracy +-0.2 % (He: +-0.2 to +-0.4 %, and 0.3 % from the Glover-Weinhold theory, possibly a pressure-scale error, p. 74).

shelton_1986/phase_match_Ar#

{
  "id": "phase_match_Ar",
  "material": "Ar",
  "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)",
  "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 1.0639, B' (10^-35 m^8/mol) = 3.386, C' (10^-48 m^10/mol) = 1.96",
  "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1",
  "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A1": "1.0639e-21",
    "B1": "3.386e-35",
    "C1": "1.96e-48"
  },
  "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A to 1e-5, B 0.93086 (0.9308), C 1.2829 (1.285) (checked 2026-09-24)."
}

shelton_1986/phase_match_H2#

{
  "id": "phase_match_H2",
  "material": "H2",
  "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)",
  "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 0.7662, B' (10^-35 m^8/mol) = 2.477, C' (10^-48 m^10/mol) = 2.48",
  "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1",
  "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A1": "0.7662e-21",
    "B1": "2.477e-35",
    "C1": "2.48e-48"
  },
  "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A exact, B 0.68097 (0.6808), C 1.6233 (1.626) (checked 2026-09-24)."
}

shelton_1986/phase_match_D2#

{
  "id": "phase_match_D2",
  "material": "D2",
  "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)",
  "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 0.7419, B' (10^-35 m^8/mol) = 2.513, C' (10^-48 m^10/mol) = 2.20",
  "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1",
  "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A1": "0.7419e-21",
    "B1": "2.513e-35",
    "C1": "2.20e-48"
  },
  "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A 1.01980 (1.0197), B 0.69086 (0.6908), C 1.4400 (1.442) (checked 2026-09-24)."
}

shelton_1986/phase_match_N2#

{
  "id": "phase_match_N2",
  "material": "N2",
  "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)",
  "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 1.1469, B' (10^-35 m^8/mol) = 3.528, C' (10^-48 m^10/mol) = 2.02",
  "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1",
  "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A1": "1.1469e-21",
    "B1": "3.528e-35",
    "C1": "2.02e-48"
  },
  "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A, B exact, C 1.3222 (1.320) (checked 2026-09-24)."
}

shelton_1986/phase_match_O2#

{
  "id": "phase_match_O2",
  "material": "O2",
  "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)",
  "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 1.3588, B' (10^-35 m^8/mol) = 5.469, C' (10^-48 m^10/mol) = 19.49",
  "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1",
  "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A1": "1.3588e-21",
    "B1": "5.469e-35",
    "C1": "19.49e-48"
  },
  "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A 1.86778 (1.8677), B 1.50351 (1.5036), C 12.7574 (12.758) (checked 2026-09-24)."
}

shelton_1986/dispersion_He#

{
  "id": "dispersion_He",
  "material": "He",
  "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)",
  "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 0.079 60, B (10^-8 um^4) = 0.024 06, C not given",
  "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)",
  "state": "1atm_0C",
  "density": "ideal",
  "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A": "0.07960e-6",
    "B": "0.02406e-8"
  },
  "check": "Recomputed from Table I by Eq. (8): A, B to the last digit (checked 2026-09-24); the differences in C follow from C' having three digits."
}

Derived from the Table I fit, not separately fitted: use one or the other. ‘The absolute values of n_w may be obtained from Eq. (8) if the refractive index is known at a single frequency’ (p. 74). STP,i: the density of an ideal gas at 273.15 K and 1 atm (p. 74). Accuracy +-0.2 % (He +-0.2 to +-0.4 %).

shelton_1986/dispersion_Ar#

{
  "id": "dispersion_Ar",
  "material": "Ar",
  "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)",
  "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.462 4, B (10^-8 um^4) = 0.930 8, C (10^-10 um^6) = 1.285",
  "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)",
  "state": "1atm_0C",
  "density": "ideal",
  "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A": "1.4624e-6",
    "B": "0.9308e-8",
    "C": "1.285e-10"
  },
  "check": "Recomputed from Table I by Eq. (8): A to 1e-5, B 0.93086 (0.9308), C 1.2829 (1.285) (checked 2026-09-24); the differences in C follow from C' having three digits."
}

shelton_1986/dispersion_H2#

{
  "id": "dispersion_H2",
  "material": "H2",
  "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)",
  "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.053 2, B (10^-8 um^4) = 0.680 8, C (10^-10 um^6) = 1.626",
  "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)",
  "state": "1atm_0C",
  "density": "ideal",
  "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A": "1.0532e-6",
    "B": "0.6808e-8",
    "C": "1.626e-10"
  },
  "check": "Recomputed from Table I by Eq. (8): A exact, B 0.68097 (0.6808), C 1.6233 (1.626) (checked 2026-09-24); the differences in C follow from C' having three digits."
}

shelton_1986/dispersion_D2#

{
  "id": "dispersion_D2",
  "material": "D2",
  "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)",
  "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.019 7, B (10^-8 um^4) = 0.690 8, C (10^-10 um^6) = 1.442",
  "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)",
  "state": "1atm_0C",
  "density": "ideal",
  "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A": "1.0197e-6",
    "B": "0.6908e-8",
    "C": "1.442e-10"
  },
  "check": "Recomputed from Table I by Eq. (8): A 1.01980 (1.0197), B 0.69086 (0.6908), C 1.4400 (1.442) (checked 2026-09-24); the differences in C follow from C' having three digits."
}

shelton_1986/dispersion_N2#

{
  "id": "dispersion_N2",
  "material": "N2",
  "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)",
  "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.576 5, B (10^-8 um^4) = 0.969 9, C (10^-10 um^6) = 1.320",
  "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)",
  "state": "1atm_0C",
  "density": "ideal",
  "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A": "1.5765e-6",
    "B": "0.9699e-8",
    "C": "1.320e-10"
  },
  "check": "Recomputed from Table I by Eq. (8): A, B exact, C 1.3222 (1.320) (checked 2026-09-24); the differences in C follow from C' having three digits."
}

shelton_1986/dispersion_O2#

{
  "id": "dispersion_O2",
  "material": "O2",
  "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)",
  "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.867 7, B (10^-8 um^4) = 1.503 6, C (10^-10 um^6) = 12.758",
  "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)",
  "state": "1atm_0C",
  "density": "ideal",
  "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'",
  "coefficients": {
    "A": "1.8677e-6",
    "B": "1.5036e-8",
    "C": "12.758e-10"
  },
  "check": "Recomputed from Table I by Eq. (8): A 1.86778 (1.8677), B 1.50351 (1.5036), C 12.7574 (12.758) (checked 2026-09-24); the differences in C follow from C' having three digits."
}

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.