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 <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/gentle_1989.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1021/j100345a033 <https://doi.org/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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "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 :doc:`review register </register>` for the checks and decisions.