shelton_1986_krxe#

D. P. Shelton, J. Chem. Phys. 84, 404 (1986)

Source record · Catalogue · Review register

Source kind: measurement

Recorded technique: Gas-phase electric-field-induced second-harmonic generation (ESHG) with periodic phase matching (electrode spacing 2.69 mm, phase-match pressures 0.4-6 atm), cw dye or argon-ion laser at 650.0, 590.0, 514.5 and 488.0 nm; Kr and Xe measured against Ar; densities from pressure and temperature by the virial equation of state (Dymond and Smith); refractive indices at phase match from tables (Bogaard and Orr; Leonard 1974) (p. 405).

Materials in structured entries: Ar, Kr, Xe

Dataset quantities: hyperpolarizability, hyperpolarizability_ratio

Contents: 13 datasets; 3 formulas.

Curation notes#

No datasets: the paper reports only hyperpolarizability data, the ratios gamma_Kr/gamma_Ar and gamma_Xe/gamma_Ar (Table I, ‘accuracy of the ratios is about +-0.3 %’, p. 405), the absolute gamma_Kr, gamma_Xe calibrated on gamma_Ar/gamma_He (Eq. (7), from Mizrahi and Shelton 1985) and Sitz and Yaris’s gamma_He (Eq. (8)), and other processes’ gamma values (Table II). The phase-match densities, which carry the linear dispersion, are neither tabulated nor quoted, and no density ratio or coherence length is given. The refractive indices used (from Leonard 1974 = leonard_1974, and Bogaard and Orr) only enter a small local-field correction.

Not extracted (third-order susceptibility, for later): Table I, gamma_Kr/gamma_Ar = 2.359(6), 2.402(11), 2.477(7), 2.502(7) and gamma_Xe/gamma_Ar = 6.761(26), 7.128(21), 7.704(29), 7.958(30) at 650.0, 590.0, 514.5, 488.0 nm; the fits gamma = A[1 + B nu^2], Kr A = 1.397e-61 C^4 m^4 J^-3, B = 18.93e-10 cm^2; Xe A = 2.999e-61, B = 39.44e-10 cm^2 (Figs. 1, 2 captions).

THIRD-ORDER DATA (added 2026-09-26; supersedes “No datasets” and “Not extracted” above). Theory (pp. 404-405): the single-resonance model gamma_zzzz(-omega_s; omega_1, omega_2, omega_3) = hbar^-3 K(-omega_s; omega_1, omega_2, omega_3) {S_1 P_1 - S_2 P_2} (Eq. 2), K the process factor, giving gamma(-omega_s; ...) = gamma(0; 0, 0, 0) [1 + C (omega_L/omega_0)^2 + ...] (Eq. 5) with omega_L^2 = omega_s^2 + omega_1^2 + omega_2^2 + omega_3^2 (Eq. 4): relative dispersion dc Kerr : ESHG : THG = 1 : 3 : 6 (p. 407; Sitz & Yaris’s He gives 1 : 3.1 : 6.1). gamma is the zzzz ESHG gamma of Shelton’s series (Taylor convention, SI C^4 m^4 J^-3), on the old He calibration (Sitz & Yaris; Eq. 8 converts with 1 au = 6.2353e-65 C^4 m^4 J^-3, against 6.2360e-65 in shelton_1986_co2). Parts: atoms, electronic. ‘The dispersion of gamma is large, yielding an optical hyperpolarizability about twice as large as the static value’ (p. 406). Datasets: Table I ratios (check) and absolute gammas (primary), Table II other processes (reproduced), and Lehmeier’s THG values from the note added in proof. The fits (formulas) are against nu^2 of the fundamental, not nu_L^2.

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_krxe/gamma_Kr_over_Ar#

{
  "id": "gamma_Kr_over_Ar",
  "material": "Kr",
  "file": "Kr/shelton_1986_krxe__gamma_Kr_over_Ar.csv",
  "table": "tbl_c9d92892bdc1",
  "quantity": "hyperpolarizability_ratio",
  "unit": "1",
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "check",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "reference_material": "Ar",
    "process": "eshg",
    "convention": "T",
    "part": "electronic"
  }
}

Table I (p. 405): gamma_Kr/gamma_Ar by ESHG, zzzz component, as printed; ‘The uncertainty of the last digits is given in parentheses’ (caption), turned into the sigma column; ‘The accuracy of the ratios is about +-0.3%’ (p. 405). Ratio by Eq. (6): gamma_A/gamma_B = (S_A/S_B)^(1/2) (rho_A n_A / rho_B n_B)^-1. The raw measurement behind gamma_Kr.

shelton_1986_krxe/gamma_Kr#

{
  "id": "gamma_Kr",
  "material": "Kr",
  "file": "Kr/shelton_1986_krxe__gamma_Kr.csv",
  "table": "tbl_c9d92892bdc1",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "primary",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "eshg",
    "convention": "T",
    "part": "electronic"
  }
}

Table I (p. 405): gamma_Kr in 1e-61 C^4 m^4 J^-3 at 650.0, 590.0, 514.5, 488.0 nm (air), note a: ‘Obtained using Refs. 15 and 18’, i.e. calibrated through Mizrahi & Shelton’s gamma_Ar/gamma_He = 25.85 [1 + 7.11e-10 nu^2] (Eq. 7, mizrahi_1985_arn2o2) and Sitz & Yaris’s gamma_He = 42.6 au [1 + 2.94e-10 nu^2] (Eq. 8, 1 au = 6.2353e-65 C^4 m^4 J^-3), nu the fundamental in cm^-1; ‘An uncertainty of +-0.4% due to the gamma_Ar/gamma_He calibration’ is included, the ~1 % of gamma_He is not (p. 406). Parenthesised last-digit uncertainties as sigma. shelton_1990 Table III(b) reproduces these on the new He calibration (e.g. 204.6e-63 against 2.024e-61 at 650.0 nm).

shelton_1986_krxe/gamma_Xe_over_Ar#

{
  "id": "gamma_Xe_over_Ar",
  "material": "Xe",
  "file": "Xe/shelton_1986_krxe__gamma_Xe_over_Ar.csv",
  "table": "tbl_c9d92892bdc1",
  "quantity": "hyperpolarizability_ratio",
  "unit": "1",
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "check",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "reference_material": "Ar",
    "process": "eshg",
    "convention": "T",
    "part": "electronic"
  }
}

Table I (p. 405): gamma_Xe/gamma_Ar by ESHG, zzzz component, as printed; ‘The uncertainty of the last digits is given in parentheses’ (caption), turned into the sigma column; ‘The accuracy of the ratios is about +-0.3%’ (p. 405). Ratio by Eq. (6): gamma_A/gamma_B = (S_A/S_B)^(1/2) (rho_A n_A / rho_B n_B)^-1. The raw measurement behind gamma_Xe.

shelton_1986_krxe/gamma_Xe#

{
  "id": "gamma_Xe",
  "material": "Xe",
  "file": "Xe/shelton_1986_krxe__gamma_Xe.csv",
  "table": "tbl_c9d92892bdc1",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "primary",
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "eshg",
    "convention": "T",
    "part": "electronic"
  }
}

Table I (p. 405): gamma_Xe in 1e-61 C^4 m^4 J^-3 at 650.0, 590.0, 514.5, 488.0 nm (air), note a: ‘Obtained using Refs. 15 and 18’, i.e. calibrated through Mizrahi & Shelton’s gamma_Ar/gamma_He = 25.85 [1 + 7.11e-10 nu^2] (Eq. 7, mizrahi_1985_arn2o2) and Sitz & Yaris’s gamma_He = 42.6 au [1 + 2.94e-10 nu^2] (Eq. 8, 1 au = 6.2353e-65 C^4 m^4 J^-3), nu the fundamental in cm^-1; ‘An uncertainty of +-0.4% due to the gamma_Ar/gamma_He calibration’ is included, the ~1 % of gamma_He is not (p. 406). Parenthesised last-digit uncertainties as sigma. shelton_1990 Table III(b) reproduces these on the new He calibration (e.g. 586.5e-63 against 5.800e-61 at 650.0 nm).

shelton_1986_krxe/table2_Kr_dc_kerr#

{
  "id": "table2_Kr_dc_kerr",
  "material": "Kr",
  "file": "Kr/shelton_1986_krxe__table2_Kr_dc_kerr.csv",
  "table": "tbl_e36736d86d09",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "reproduced",
  "reproduced_from": [
    "buckingham_1968"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "dc_kerr",
    "convention": "T",
    "part": "electronic"
  }
}

Table II (p. 406): ‘Hyperpolarizabilities of Kr and Xe measured by several nonlinear optical processes’, 1e-61 C^4 m^4 J^-3, note a: ‘Conversion factor 1 esu = 6 x 10^9/c^4 = 7.4279 x 10^-25 C^4 m^4 J^-3 has been used, where the extra factor of 6 converts from susceptibility to hyperpolarizability’ (Ward’s per-molecule esu chi = gamma/6). Reproduced (converted by the author). Note b: gamma(-omega; omega, 0, 0), omega_L^2 = 2 omega^2, ‘from Ref. 19’ (Buckingham & Dunmur, cited as Trans. Faraday Soc. 64, 1176 (1968); the paper is on p. 1776 = buckingham_1968), 632.8 nm.

shelton_1986_krxe/table2_Xe_dc_kerr#

{
  "id": "table2_Xe_dc_kerr",
  "material": "Xe",
  "file": "Xe/shelton_1986_krxe__table2_Xe_dc_kerr.csv",
  "table": "tbl_e36736d86d09",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "reproduced",
  "reproduced_from": [
    "buckingham_1968"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "dc_kerr",
    "convention": "T",
    "part": "electronic"
  }
}

Table II (p. 406): ‘Hyperpolarizabilities of Kr and Xe measured by several nonlinear optical processes’, 1e-61 C^4 m^4 J^-3, note a: ‘Conversion factor 1 esu = 6 x 10^9/c^4 = 7.4279 x 10^-25 C^4 m^4 J^-3 has been used, where the extra factor of 6 converts from susceptibility to hyperpolarizability’ (Ward’s per-molecule esu chi = gamma/6). Reproduced (converted by the author). Note b: gamma(-omega; omega, 0, 0), omega_L^2 = 2 omega^2, from Ref. 19 (buckingham_1968; cited with page 1176 for 1776), 632.8 nm.

shelton_1986_krxe/table2_Kr_cars#

{
  "id": "table2_Kr_cars",
  "material": "Kr",
  "file": "Kr/shelton_1986_krxe__table2_Kr_cars.csv",
  "table": "tbl_e36736d86d09",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "reproduced",
  "reproduced_from": [
    "lundeen_1983"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "cars",
    "convention": "T",
    "part": "electronic",
    "stokes_nm": 683.0
  }
}

Table II (p. 406): ‘Hyperpolarizabilities of Kr and Xe measured by several nonlinear optical processes’, 1e-61 C^4 m^4 J^-3, note a: ‘Conversion factor 1 esu = 6 x 10^9/c^4 = 7.4279 x 10^-25 C^4 m^4 J^-3 has been used, where the extra factor of 6 converts from susceptibility to hyperpolarizability’ (Ward’s per-molecule esu chi = gamma/6). Reproduced (converted by the author). Note c: gamma(-2 omega_1 + omega_2; omega_1, omega_1, -omega_2), omega_L^2 = 4 omega_1^2 + 2 (omega_1 - omega_2)^2, from Ref. 10 (Lundeen et al. 1983), pump 532 nm, Stokes 683 nm (the abscissa is the pump). FWM ‘is referenced to the N2 Raman Q-branch intensity’ (p. 406).

shelton_1986_krxe/table2_Kr_eshg#

{
  "id": "table2_Kr_eshg",
  "material": "Kr",
  "file": "Kr/shelton_1986_krxe__table2_Kr_eshg.csv",
  "table": "tbl_e36736d86d09",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "reproduced",
  "reproduced_from": [
    "R. S. Finn and J. F. Ward, Phys. Rev. Lett. 26, 285 (1971)"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "eshg",
    "convention": "T",
    "part": "electronic"
  }
}

Table II (p. 406): ‘Hyperpolarizabilities of Kr and Xe measured by several nonlinear optical processes’, 1e-61 C^4 m^4 J^-3, note a: ‘Conversion factor 1 esu = 6 x 10^9/c^4 = 7.4279 x 10^-25 C^4 m^4 J^-3 has been used, where the extra factor of 6 converts from susceptibility to hyperpolarizability’ (Ward’s per-molecule esu chi = gamma/6). Reproduced (converted by the author). Note d: gamma(-2 omega; omega, omega, 0), omega_L^2 = 6 omega^2, from Ref. 20 (Finn & Ward 1971), 694.3 nm.

shelton_1986_krxe/table2_Xe_eshg#

{
  "id": "table2_Xe_eshg",
  "material": "Xe",
  "file": "Xe/shelton_1986_krxe__table2_Xe_eshg.csv",
  "table": "tbl_e36736d86d09",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "reproduced",
  "reproduced_from": [
    "R. S. Finn and J. F. Ward, Phys. Rev. Lett. 26, 285 (1971)"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "eshg",
    "convention": "T",
    "part": "electronic"
  }
}

shelton_1986_krxe/table2_Kr_thg#

{
  "id": "table2_Kr_thg",
  "material": "Kr",
  "file": "Kr/shelton_1986_krxe__table2_Kr_thg.csv",
  "table": "tbl_e36736d86d09",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "reproduced",
  "reproduced_from": [
    "ward_1969"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "thg",
    "convention": "T",
    "part": "electronic"
  }
}

Table II (p. 406): ‘Hyperpolarizabilities of Kr and Xe measured by several nonlinear optical processes’, 1e-61 C^4 m^4 J^-3, note a: ‘Conversion factor 1 esu = 6 x 10^9/c^4 = 7.4279 x 10^-25 C^4 m^4 J^-3 has been used, where the extra factor of 6 converts from susceptibility to hyperpolarizability’ (Ward’s per-molecule esu chi = gamma/6). Reproduced (converted by the author). Note e: gamma(-3 omega; omega, omega, omega), omega_L^2 = 12 omega^2, from Ref. 21 (Ward & New 1969), 694.3 nm.

shelton_1986_krxe/table2_Xe_thg#

{
  "id": "table2_Xe_thg",
  "material": "Xe",
  "file": "Xe/shelton_1986_krxe__table2_Xe_thg.csv",
  "table": "tbl_e36736d86d09",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "reproduced",
  "reproduced_from": [
    "ward_1969"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "thg",
    "convention": "T",
    "part": "electronic"
  }
}

shelton_1986_krxe/text_thg_lehmeier_Kr#

{
  "id": "text_thg_lehmeier_Kr",
  "material": "Kr",
  "table": "text",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "reproduced",
  "reproduced_from": [
    "lehmeier_1985"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "thg",
    "convention": "T",
    "part": "electronic"
  }
}

1 inline points in the source record.

Note added in proof (p. 407): ‘The recent, accurate THG measurements by H. J. Lehmeier et al. [Opt. Commun. 56, 67 (1985)], give gamma_Kr = 1.74 +- 0.17 x 10^-61 C^4 m^4 J^-3 and gamma_Xe = 5.11 +- 0.51 x 10^-61 C^4 m^4 J^-3 at lambda = 1.055 um, in good agreement with the predicted dispersion curves’. Lehmeier’s values converted by the author to this gamma (the conversion is not stated; Lehmeier’s own chi is defined with P = 4 eps0 chi E^3, see lehmeier_1985). Medium of 1.055 um not stated (air assumed, as the other lines).

shelton_1986_krxe/text_thg_lehmeier_Xe#

{
  "id": "text_thg_lehmeier_Xe",
  "material": "Xe",
  "table": "text",
  "quantity": "hyperpolarizability",
  "unit": "SI",
  "scale": 1e-61,
  "abscissa_unit": "nm",
  "wavelength_medium": "standard_air",
  "use": "reproduced",
  "reproduced_from": [
    "lehmeier_1985"
  ],
  "uncertainty": {
    "column": true,
    "level": "not stated"
  },
  "parameters": {
    "process": "thg",
    "convention": "T",
    "part": "electronic"
  }
}

1 inline points in the source record.

Formulas#

shelton_1986_krxe/gamma_fit_Kr#

{
  "id": "gamma_fit_Kr",
  "material": "Kr",
  "equation": "Fig. 1 caption, p. 405",
  "printed": "gamma = A[1 + B nu^2]; A = 1.397 x 10^-61 C^4 m^4 J^-3 and B = 18.93 x 10^-10 cm^2, where nu is given in cm^-1",
  "variable": "nu: vacuum wavenumber of the fundamental in cm^-1 (not nu_L^2; for ESHG nu_L^2 = 6 nu^2)",
  "process": "eshg",
  "convention": "T",
  "part": "electronic",
  "range": "488.0-650.0 nm (Table I)",
  "coefficients": {
    "A": "1.397e-61",
    "B": "18.93e-10"
  },
  "coefficient_units": {
    "A": "C^4 m^4 J^-3",
    "B": "cm^2"
  },
  "calibration": "old He (Sitz & Yaris) via Ar (Eqs. 7, 8)",
  "check": "Gives 2.023 at 650.0 nm (15380 cm^-1) and 2.507 at 488.0 nm (20487 cm^-1) against 2.024 +- 0.010 and 2.506 +- 0.012. Checked 2026-09-26."
}

Least-squares straight line against nu^2. Superseded by shelton_1990 Table IV (162.1e-63, A = 1.389e-10, B = 3.465e-20 against nu_L^2).

shelton_1986_krxe/gamma_fit_Xe#

{
  "id": "gamma_fit_Xe",
  "material": "Xe",
  "equation": "Fig. 2 caption, p. 406",
  "printed": "gamma = A[1 + B nu^2]; A = 2.999 x 10^-61 C^4 m^4 J^-3 and B = 39.44 x 10^-10 cm^2, where nu is given in cm^-1",
  "variable": "nu: vacuum wavenumber of the fundamental in cm^-1 (not nu_L^2)",
  "process": "eshg",
  "convention": "T",
  "part": "electronic",
  "range": "488.0-650.0 nm (Table I)",
  "coefficients": {
    "A": "2.999e-61",
    "B": "39.44e-10"
  },
  "coefficient_units": {
    "A": "C^4 m^4 J^-3",
    "B": "cm^2"
  },
  "calibration": "old He (Sitz & Yaris) via Ar (Eqs. 7, 8)",
  "check": "Gives 5.797 at 650.0 nm and 7.964 at 488.0 nm against 5.800 +- 0.032 and 7.970 +- 0.044. Checked 2026-09-26."
}

Least-squares straight line against nu^2; its intercept (3.0e-61 = 4810 au) is the ‘visible-only linear fit’ static value that is too low (shelton_1990’s refit with 1064 and 1319 nm gives 429.5e-63 = 6888 au).

shelton_1986_krxe/calibration_Ar_over_He#

{
  "id": "calibration_Ar_over_He",
  "material": "Ar",
  "equation": "Eq. (7), p. 406",
  "printed": "gamma_Ar/gamma_He = 25.85 [1 + (7.11 x 10^-10 cm^2) nu^2]",
  "variable": "nu: vacuum wavenumber of the fundamental in cm^-1",
  "process": "eshg",
  "convention": "T",
  "part": "electronic",
  "range": "'Over the range of frequencies employed in the present measurements' (488-650 nm)",
  "coefficients": {
    "ratio0": "25.85",
    "B": "7.11e-10"
  },
  "coefficient_units": {
    "ratio0": "1",
    "B": "cm^2"
  },
  "reproduced_from": [
    "mizrahi_1985_arn2o2"
  ],
  "check": "Not checked against mizrahi_1985_arn2o2 here."
}

From Ref. 15 (Mizrahi & Shelton, Phys. Rev. Lett. 55, 696 (1985)). Eq. (8), gamma_He = 42.6 au [1 + 2.94e-10 cm^2 nu^2], is the same Sitz & Yaris line as shelton_1986_co2’s calibration_He, but converted here with 1 au = 6.2353e-65 C^4 m^4 J^-3.

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.