gentle_1989#
I. R. Gentle, D. R. Laver and G. L. D. Ritchie, “Second hyperpolarizability and static polarizability anisotropy of carbon dioxide”, J. Phys. Chem. 93, 3035 (1989)
Source record · Catalogue · Review register
DOI: 10.1021/j100345a033
Source kind: measurement
Recorded technique: Absolute dc Kerr effect (electric birefringence) of CO2 (> 99.8 %) at 632.8 nm (2 mW He-Ne laser; Fig. 1 caption, Table I caption), about 299-490 K and 100-1000 kPa: the gas cell (Rogowski-profiled SC23 steel electrodes, effective length 0.500 +- 0.001 m, d^2/l = (5.236 +- 0.015) x 10^-5 m, confirmed to +-0.5 % against a small reference assembly; distortion < 1.5 % at 413 K) and a reference cell of liquid cyclohexane in series, a 512 Hz ac voltage on both and a dc bias on each, nulled at the fundamental by a feedback loop (k1 V1 = -k2 V2, Eq. 5); the reference cell calibrated with Faraday coils against an analyser micrometer (reproducible to better than +-1 %). Pressures (Baratron, +-0.5 %), temperatures (+-0.2 K control, +-1 K accuracy); molar volumes from Dymond & Smith’s virial coefficients; mK0 = (2/27)(n_par - n_perp) V_m F^-2 fitted against 1/V_m at each temperature (Eq. 6) (pp. 3035-3037).
Materials in structured entries: CO2
Dataset quantities: hyperpolarizability, kerr_virial_coefficient, molar_kerr_constant, polarizability_anisotropy
Contents: 7 datasets; 1 formula.
Curation notes#
Pages are the journal’s (3035-3038; p. = pdf page + 3034). The PDF’s last page continues with the next article (Whittaker et al.), not used.
DEFINITIONS (p. 3035). mK = 6 n V_m [(n^2 + 2)^2 (eps_r + 2)^2]^-1 [(n_par - n_perp) F^-2]_(F=0) (Eq. 1); mK = A_K + B_K V_m^-1 + ... (Eq. 2); for a linear nondipolar molecule A_K = (N_A/(405 eps0))[5 gamma^K + (kT)^-1 Delta alpha Delta alpha^0] (Eq. 3), with gamma^K = [3 gamma_abab(-w; w, 0, 0) - gamma_aabb(-w; w, 0, 0)]/10 ‘the second Kerr hyperpolarizability’, Delta alpha = alpha_zz - alpha_xx (optical), Delta alpha^0 the static anisotropy. gamma^K is Buckingham’s dc Kerr gamma_K in the Taylor-series convention (SI: C m^4 V^-3 = C^4 m^4 J^-3; shelton_1994 Table XII reproduces it as 2005 +- 513 au). Eq. (6): mK0 = A_K + [B_K + A_K (2 A_eps + A_R/2)] V_m^-1, with A_eps and A_R from the static (3.241 x 10^-40) and optical (2.933 x 10^-40 C m^2 V^-1) polarizabilities of refs 9 and 14.
WAVELENGTH: 632.8 nm, stated in the Fig. 1 caption (‘L, 2-mW He-Ne laser (632.8 nm)’) and the Table I caption (‘Temperature Dependence of the Kerr Effect of Carbon Dioxide at 632.8 nm’): this settles the wavelength that shelton_1986_co2’s note added in proof left unstated (TC-233).
ACCURACY. ‘The overall accuracy of measurements of A_K is conservatively estimated at +-2% over the full temperature range’ (p. 3036); Table I: ‘the uncertainties quoted are based on the standard deviations derived from the least-squares fitting of straight lines to the density-dependence data, together with appropriate allowance for systematic errors’ (p. 3037): +-0.003 to +-0.006 (0.1-0.3 %) on A_K. The A_K dataset carries the stated 2 % (level not stated); the printed errors are kept in the file’s ‘printed’ column. ‘the values of B_K are too imprecise to draw any firm conclusions’ (p. 3037).
TABLE II AND ITS ERRORS (checked here, 2026-09-26). A weighted least-squares line of Table I’s A_K against 1/T, weighted by the printed Table I errors, gives intercept 1.056 x 10^-28 and slope 6.889 x 10^-25, with standard errors 0.080 and 0.029 that become 0.269 and 0.096 when scaled by sqrt(chi^2/dof) = sqrt(11.2): Table II’s 1.05 +- 0.27 and 6.89 +- 0.10 exactly. So gamma^K = 0.125 +- 0.032 and Delta alpha^0 = 2.41 +- 0.08 carry the scatter of the fit, not the +-2 % accuracy of A_K (Fig. 3’s caption nevertheless says ‘2% uncertainties’). Weighted instead by the stated 2 %, the same data give intercept (0.89 +- 0.79) x 10^-28 (standard error, unscaled; chi^2/dof = 0.13), i.e. gamma^K = 0.106 +- 0.094 x 10^-60, and slope 6.96 x 10^-25. The line reproduces Table I to within +-0.3 % except 370.9 K (-0.8 %), 348.8 K (+0.8 %) and 330.9 K (-1.4 %). Checks of Table II: 81 eps0/N_A x 1.05e-28 = 0.1250e-60 (gamma^K); 405 eps0 k/N_A x 6.89e-25 = 5.664e-80 (Delta alpha Delta alpha^0, printed 5.66 +- 0.08); 5.66/2.35 = 2.409 (Delta alpha^0). ‘the term (N_A/81 eps0) gamma^K contributes only 4.4%, rather than 17%, of the value of A_K at 300 K’ (p. 3037).
PARTS. gamma^K is the temperature-independent intercept: electronic plus vibrational (the authors: Shelton’s ESHG-based prediction 0.073 +- 0.002, which allows for the electronic dispersion and a vibrational contribution, suggests ‘the latter correction could be larger than was thought’, p. 3038).
EARLIER KERR STUDY (Buckingham et al. 1970, buckingham_1970), pp. 3037-3038: its gamma^K 0.56 +- 0.12 and Delta alpha^0 2.39 +- 0.18; ‘A reexamination of the original data indicates that the intercept has an uncertainty, expressed as the standard deviation, of 75% rather than the quoted value of 21%; furthermore, if the lowest temperature point (10^3 T^-1 = 3.97) is omitted, the value of gamma^K changes from +0.56 +- 0.42 to -0.06 +- 0.18’ (text_reanalysis_buckingham_1970). Extrapolation ratio T_min/(T_max - T_min): 3.0 there, 1.6 here. Laser Stark spectroscopy (Gough, Orr & Scoles 1983) gave Delta alpha^0 = 2.89 +- 0.20, ‘about 20% too high’, and from Rayleigh depolarisation and infrared intensities 2.57 +- 0.06 (text_anisotropy_static_literature).
Reproductions elsewhere: shelton_1986_co2/text_dc_kerr_gentle (the pre-publication 110 +- 39 x 10^-63, not this paper’s 125 +- 32 x 10^-63); shelton_1994/table12_CO2_dc_kerr (2005 +- 513 au = 0.125 +- 0.032 x 10^-60 / 6.2354e-65). The library copy has no validation warnings (identity reviewed: title, first author and year agree).
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
gentle_1989/table1_A_K_CO2#
{
"id": "table1_A_K_CO2",
"material": "CO2",
"file": "CO2/gentle_1989__table1_A_K_CO2.csv",
"table": "tbl_18459ce921c2",
"quantity": "molar_kerr_constant",
"unit": "m5 V-2 mol-1",
"scale": 1e-27,
"abscissa": "temperature",
"abscissa_unit": "K",
"use": "primary",
"uncertainty": {
"relative": 0.02,
"level": "not stated"
},
"parameters": {
"wavelength_nm": 632.8
}
}
Table I (p. 3037): the zero-density molar Kerr constant A_K of CO2 (including the orientation term) at 9 temperatures, 299.2-489.5 K, 632.8 nm; 7-12 pressures each, 150-1040 kPa. Uncertainty: the stated ‘overall accuracy ... +-2%’ (p. 3036); the printed errors (+-0.003 to +-0.006, ‘based on the standard deviations derived from the least-squares fitting ... together with appropriate allowance for systematic errors’) are in the ‘printed’ column only. table2_gamma_K_CO2 and table2_anisotropy_static_CO2 are derived from these values: fit these or those.
gentle_1989/table1_B_K_CO2#
{
"id": "table1_B_K_CO2",
"material": "CO2",
"file": "CO2/gentle_1989__table1_B_K_CO2.csv",
"table": "tbl_18459ce921c2",
"quantity": "kerr_virial_coefficient",
"unit": "m8 V-2 mol-2",
"scale": 1e-32,
"abscissa": "temperature",
"abscissa_unit": "K",
"use": "check",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"wavelength_nm": 632.8,
"order": 2
}
}
Table I (p. 3037): the second Kerr virial coefficient B_K of CO2 (Eqs. 2, 6), 9 temperatures, printed ‘10^32 B_K/m^8 V^-2 mol^-1’ (checked on the image): the unit is misprinted, mol^-2 being meant (B_K V_m^-1 has the unit of A_K, m^5 V^-2 mol^-1). The unit is recorded as m8 V-2 mol-2; the format has no erratum for a unit. ‘the values of B_K are too imprecise to draw any firm conclusions’ (p. 3037). sigma: the printed error (fit, with an allowance for systematic errors; level not stated).
gentle_1989/table2_gamma_K_CO2#
{
"id": "table2_gamma_K_CO2",
"material": "CO2",
"file": "CO2/gentle_1989__table2_gamma_K_CO2.csv",
"table": "tbl_1b1d9fc6dd0c",
"quantity": "hyperpolarizability",
"unit": "SI",
"scale": 1e-60,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"use": "primary",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "dc_kerr",
"convention": "T",
"part": "electronic_vibrational",
"component": "K"
}
}
Table II (p. 3037) and abstract: the second Kerr hyperpolarizability gamma^K = (0.125 +- 0.032) x 10^-60 C m^4 V^-3 (2005 +- 513 au) of CO2 at 632.8 nm, from the T^-1 = 0 intercept of A_K (Eq. 3), electronic plus vibrational. The error is the Birge-scaled standard error of a line weighted by Table I’s printed errors (reproduced here; see the record’s notes), so it does not include the +-2 % accuracy of A_K; weighted by the 2 % instead, the intercept gives 0.106 +- 0.094 x 10^-60 (computed here). A long extrapolation: T_min/(T_max - T_min) = 1.6 (p. 3037).
gentle_1989/table2_anisotropy_static_CO2#
{
"id": "table2_anisotropy_static_CO2",
"material": "CO2",
"file": "CO2/gentle_1989__table2_anisotropy_static_CO2.csv",
"table": "tbl_1b1d9fc6dd0c",
"quantity": "polarizability_anisotropy",
"unit": "SI",
"scale": 1e-40,
"abscissa_unit": "nm",
"use": "primary",
"uncertainty": {
"column": true,
"level": "not stated"
}
}
Table II (p. 3037) and abstract: the static polarizability anisotropy Delta alpha^0 = alpha_zz^0 - alpha_xx^0 = (2.41 +- 0.08) x 10^-40 C m^2 V^-1 of CO2, from the slope of A_K against 1/T (Delta alpha Delta alpha^0 = (5.66 +- 0.08) x 10^-80 C^2 m^4 V^-2) divided by the optical Delta alpha of ref 9 (2.35 +- 0.07 at 632.8 nm, table2_anisotropy_optical_CO2). Static limit (wavelength inf). Error: the scaled fit error propagated with Delta alpha’s assumed 3 %; level not stated.
gentle_1989/table2_anisotropy_optical_CO2#
{
"id": "table2_anisotropy_optical_CO2",
"material": "CO2",
"file": "CO2/gentle_1989__table2_anisotropy_optical_CO2.csv",
"table": "tbl_1b1d9fc6dd0c",
"quantity": "polarizability_anisotropy",
"unit": "SI",
"scale": 1e-40,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"use": "reproduced",
"reproduced_from": [
"M. P. Bogaard, A. D. Buckingham, R. K. Pierens and A. H. White, J. Chem. Soc., Faraday Trans. 1 74, 3008 (1978)"
],
"uncertainty": {
"column": true,
"level": "not stated"
}
}
Table II (p. 3037): the optical-frequency anisotropy Delta alpha = (2.35 +- 0.07) x 10^-40 C m^2 V^-1 of CO2 at 632.8 nm, note b: ‘Reference 9; lambda = 632.8 nm, 3% uncertainty assumed’ (the 3 % is the authors’ assumption, not ref 9’s).
gentle_1989/text_reanalysis_buckingham_1970#
{
"id": "text_reanalysis_buckingham_1970",
"material": "CO2",
"table": "text",
"quantity": "hyperpolarizability",
"unit": "SI",
"scale": 1e-60,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"use": "check",
"reproduced_from": [
"buckingham_1970"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "dc_kerr",
"convention": "T",
"part": "electronic_vibrational",
"component": "K"
}
}
3 inline points in the source record.
pp. 3037-3038: gamma^K of CO2 from the earlier temperature study (Buckingham et al. 1970), ‘as deduced ... from the original temperature-dependence study (0.56 +- 0.12)’; ‘A reexamination of the original data indicates that the intercept has an uncertainty, expressed as the standard deviation, of 75% rather than the quoted value of 21%; furthermore, if the lowest temperature point (10^3 T^-1 = 3.97) is omitted, the value of gamma^K changes from +0.56 +- 0.42 to -0.06 +- 0.18’. Rows 2-3 are these authors’ reanalysis of another group’s data (shelton_1986_co2 made the same point, -61 +- 200 x 10^-63). Row 1 is Buckingham et al.’s value as quoted here, in 10^-60 C m^4 V^-3.
gentle_1989/text_anisotropy_static_literature#
{
"id": "text_anisotropy_static_literature",
"material": "CO2",
"table": "text",
"quantity": "polarizability_anisotropy",
"unit": "SI",
"scale": 1e-40,
"abscissa_unit": "nm",
"use": "reproduced",
"reproduced_from": [
"buckingham_1970",
"T. E. Gough, B. J. Orr and G. Scoles, J. Mol. Spectrosc. 99, 143 (1983)"
],
"uncertainty": {
"column": true,
"level": "not stated"
}
}
3 inline points in the source record.
p. 3038: static anisotropies Delta alpha^0 of CO2 compared, in 10^-40 C m^2 V^-1: ‘the same as the result of the original investigation (2.39 +- 0.18); however, the uncertainty in the latter quantity appears, once again, to have been underestimated. Both are lower than the value derived by Gough et al. by laser Stark spectroscopy (2.89 +- 0.20), which now must be considered to be about 20% too high. In the same study, Delta alpha^0 was also independently determined from considerations of the frequency dependence of the gas-phase Rayleigh depolarization ratio and the intensities of the infrared-active vibrational bands. The result (2.57 +- 0.06) ...’. Levels not stated.
Formulas#
gentle_1989/table2_A_K_line#
{
"id": "table2_A_K_line",
"material": "CO2",
"equation": "Eq. (3), Table II, p. 3037",
"printed": "A_K = (N_A/405 eps0)[5 gamma^K + (kT)^-1 Delta alpha Delta alpha^0]; Table II: 10^28 (intercept)/m^5 V^-2 mol^-1 = 1.05 +- 0.27, 10^25 (slope)/m^5 V^-2 K mol^-1 = 6.89 +- 0.10",
"variable": "T: temperature in K; A_K = intercept + slope/T in m^5 V^-2 mol^-1",
"range": "299.2-489.5 K (Table I)",
"coefficients": {
"intercept": "1.05",
"slope": "6.89"
},
"coefficient_units": {
"intercept": "1e-28 m^5 V^-2 mol^-1",
"slope": "1e-25 m^5 V^-2 K mol^-1"
},
"check": "Reproduced (2026-09-26): a line through Table I's A_K against 1/T weighted by the printed Table I errors gives 1.056 and 6.889, standard errors x sqrt(chi^2/dof = 11.2) = 0.269 and 0.096, as printed. The line reproduces Table I to +-0.3 % except 370.9 K (-0.8 %), 348.8 K (+0.8 %) and 330.9 K (-1.4 %). Intercept gives gamma^K = 81 eps0/N_A x 1.05e-28 = 0.1250e-60; slope gives Delta alpha Delta alpha^0 = 405 eps0 k/N_A x 6.89e-25 = 5.664e-80, both as printed."
}
The weighted least-squares line of Table II (note a: ‘From weighted least-squares straight line fitted to data in Table I’); Fig. 3 plots it. Its errors are the scatter of the fit (see the record’s 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 review register for the checks and decisions.