tammer_1990#
R. Tammer and W. Hüttner, “The anisotropy of the second hyperpolarizability of molecular hydrogen from the pressure and temperature dependence of the Kerr effect”, Chem. Phys. 146, 155 (1990)
Source record · Catalogue · Review register
DOI: 10.1016/0301-0104(90)90014-z
Source kind: measurement
Recorded technique: Absolute dc Kerr effect of H2 and D2 at 632.8 nm (25 mW He-Ne laser): a temperature-stabilised Senarmont-type ellipsometer whose Faraday compensator current, held at null by a PID loop with 600 Hz Faraday modulation and phase-sensitive detection, measures the phase difference; Kerr condenser of two stainless-steel plates 0.03 x 0.63 m, 6.00 +- 0.01 mm apart on glass-ceramic spacers, in a pressure cell to 125 bar, +-0.1 C from -100 to 200 C; about 12 kV; 0.5 % edge correction (pp. 157-158). H2 (99.999 %) at 199 (p, T) settings at 13 temperatures, 196.1-411.7 K, pressures to 120 bar; D2 (99.7 %) at 65 pressures, 3-16 bar, 24.9 C. Densities from the Beattie-Bridgeman relation; K = a rho + b rho^2 per temperature (Eq. 12); molar Kerr constants with two local-field functions and extrapolation to zero density, mK = A_K + B_K N (Eqs. 3-4) (pp. 158-159).
Materials in structured entries: D2, H2
Dataset quantities: hyperpolarizability, kerr_virial_coefficient, molar_kerr_constant
Contents: 11 datasets; 2 formulas.
Curation notes#
SUPERSEDED CALIBRATION. The field integral of this apparatus was wrong (‘a machining fault in manufacturing the Kerr plates’, tammer_1992 p. 154); Tammer, Löblein, Peting & Hüttner (tammer_1992) calibrated it by the Stark effect of OCS and state that ‘all values K, mK, and A_K given in ref. [3] should be multiplied by’ 1.0637 +- 0.0031, re-evaluating this paper’s Table 1 in their Table 1 (the refit is not a uniform factor: see tammer_1992). Every value here is therefore recorded as printed with use = ‘check’ (or ‘reproduced’ for other authors’ values): the H2 gamma of this paper, 4.15(12) x 10^-62, is 4.68(14) x 10^-62 after the correction. The D2 value was not re-evaluated in tammer_1992; with the same factor its A_K would be 7.802 x 10^-29 and gamma about 4.17 + 0.0637 x 7.335 x 1.1909 = 4.73 x 10^-62 (computed here, not a value of either paper). Pages are the journal’s (155-163; p. = pdf page + 154).
DEFINITIONS (pp. 155-156). gamma = (1/10) sum (3 gamma_ghgh - gamma_gghh) = (3/2)(gamma_zzzz - gamma_xxzz) = (3/2)(gamma_par - gamma_perp) (Eq. 1), the anisotropy of the hyperpolarizability, with the field-induced polarizability (1/2) sum gamma_ghij E_i E_j (p. 155): Buckingham’s gamma_K in the Taylor-series convention. Delta n = n_z - n_x = lambda K E^2 (Eq. 2); K = (27/2)(f(n, eps)/(lambda n)) mK N (Eq. 3), N in mol per volume; mK = A_K + B_K N (Eq. 4); A_K = (N_A/(81 eps0))[gamma + (Delta alpha Delta alpha^0)_eff / 5 k_B T] (Eq. 5), with the quantum statistical average over rotational states of Eqs. (6)-(8) (spin weights 1:3 for H2, 2:1 for D2; hG(H2) = 59.340 cm^-1, hG(D2) = 29.908 cm^-1, Table 1 note a). Local-field functions f^(2) = ((n^2 + 2)/3)((eps + 2)/3) (Eq. 13) and f^(4) = ((n^2 + 2)/3)^2((eps + 2)/3)^2 (Eq. 14); ‘A_K as a function of temperature was found unaffected by the choice of the local-field function while two different sets of B_K were obtained’ (p. 159). gamma is A_K minus the orientation term computed from the J-dependent ab initio anisotropies (Rychlewski 1980, 1983; Bishop & Cheung 1980): an electronic plus vibrational gamma_K.
UNCERTAINTY (p. 158): reproducibility after reassembly ‘better than +-0.5%’ (compensation current calibration included); ‘An additional systematic error caused by the combined uncertainties in determining p, T, and E of at most +-1.1% must be taken into account in the mK and A_K values’. Table 1: ‘The errors given in parentheses in units of the least significant figures are standard deviations’. Mean gamma = (4.149 +- 0.013) x 10^-62 SI (‘the error means one standard deviation’, Eq. 15, p. 159); with the 1.1 % systematic part of A_K the final 4.15(12) of Table 2 (the systematic contribution ‘+-0.11 x 10^-62 SI’, p. 162). D2: ‘one standard deviation and an additional systematic error of 1.6% in A_K(D2)’ (p. 159). Fitting B_K = a0 + a1/T + a2/T^2 (Eq. 24) with gamma to the original data gave gamma = 4.160(15) x 10^-62 SI (p. 163).
TABLE 1 (p. 160). Taken: A_K, B_K^(4), B_K^(2), gamma (H2, 13 temperatures; D2 at 298.1 K: A_K and gamma only). Not taken: 10^19 a (C^2 m^4 J^-2 kg^-1) and 10^21 b (C^2 m^7 J^-2 kg^-2) of Eq. (12), and 10^29 Delta ‘Delta = A_K - K_K^calc’ (note b: K_K^calc is a slip for A_K^calc) with A_K^calc from Eq. (5) and the mean gamma. Footnote c: ‘The mean value is gamma = 4.149(13) x 10^-62 C^4 m^4 J^-3’. The 376.8 K row is printed 376.5 K in tammer_1992.
TABLE 2 (p. 161): ‘Comparison of hyperpolarizability anisotropies of H2 and D2 obtained in this work with literature values’, with an esu column (‘1 esu corresponds to 1.2379 x 10^-25 C^4 m^4 J^-3’, note a; not taken). Rows: this work (note b: ‘Error includes one standard deviation (see table 1) and systematical contributions’); [3] Buckingham & Orr 1968 (buckingham_1968b; 3.43(37) = its 0.277 esu for H2); [4] Carusotto et al. 1985 (carusotto_1985), ‘Transferred to lambda = 632.8 nm, see text’ (note c): the authors subtracted the orientation terms from Carusotto’s room-temperature A_K and moved gamma from 514.5 to 632.8 nm with Bishop, Pipin & Rérat’s electronic dispersion and vibrational terms (p. 161); [1] Bishop, Pipin & Rérat 1990 ab initio (note d: ‘Error is estimated’). ‘They all refer to 632.8 nm.’ The authors’ result is 6.8 % below theory (p. 161).
TEXT (pp. 162-163): ESHG and ac Kerr results transferred to the dc Kerr gamma at 632.8 nm. From Mizrahi & Shelton’s ESHG ratio (Eq. 20, fitted by Shelton & Lu 1988) and Bishop & Pipin’s He: Eq. (21) gamma_par,ESHG^e and Eq. (22) gamma^e (formulas), plus the vibrational correction -0.045 x 10^-62 SI for He-Ne light: ‘gamma = (4.42 +- 0.03) x 10^-62 SI’ (the 0.6 % experimental error of Mizrahi & Shelton); from Rosasco & Hurst’s ac Kerr (4.92 +- 0.24) x 10^-62 SI at 676.6 and 563.3 nm: gamma^e = (4.57 +- 0.23) and gamma = (4.52 +- 0.23) x 10^-62 SI (text_transferred_H2). B_K^(4) are ‘smaller by a factor of two than those found by Buckingham and Orr [3], but have the same (positive) sign’ (p. 162).
Reproductions elsewhere: bishop_1991/expt_dc_kerr_H2 (row tammer_huttner_1990, 666 +- 19 au = 4.15(12) x 10^-62 / 6.2354e-65) and expt_dc_kerr_D2 (669 +- 26 au = 4.17(16)); shelton_1993/gamma_R_tammer (685.3 au: Shelton & Rugar’s evaluation of this paper’s orientation expression, Eq. 6, not a value printed here). 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#
tammer_1990/table1_A_K_H2#
{
"id": "table1_A_K_H2",
"material": "H2",
"file": "H2/tammer_1990__table1_A_K_H2.csv",
"table": "tbl_1f9c20083bf3",
"quantity": "molar_kerr_constant",
"unit": "C2 m5 J-2 mol-1",
"scale": 1e-29,
"abscissa": "temperature",
"abscissa_unit": "K",
"use": "check",
"uncertainty": {
"column": true,
"level": "1 sigma"
},
"parameters": {
"wavelength_nm": 632.8
}
}
Table 1 (p. 160): the first molar Kerr virial coefficient A_K of H2 (Eq. 4, including the orientation term) at 13 temperatures, 196.1-411.7 K, 632.8 nm. Standard deviations; a systematic +-1.1 % (p, T, E) applies (p. 158). Superseded calibration: tammer_1992/table1_A_K_H2 holds these values re-evaluated (about x 1.0637).
tammer_1990/table1_A_K_D2#
{
"id": "table1_A_K_D2",
"material": "D2",
"file": "D2/tammer_1990__table1_A_K_D2.csv",
"table": "tbl_1f9c20083bf3",
"quantity": "molar_kerr_constant",
"unit": "C2 m5 J-2 mol-1",
"scale": 1e-29,
"abscissa": "temperature",
"abscissa_unit": "K",
"use": "check",
"uncertainty": {
"column": true,
"level": "1 sigma"
},
"parameters": {
"wavelength_nm": 632.8
}
}
Table 1 (p. 160): A_K of D2 at 298.1 K (24.9 C in the text, p. 159), 65 pressures between 3 and 16 bar, extrapolated linearly to zero density. Standard deviation; the text adds a 1.6 % systematic error (p. 159). Superseded calibration (the tammer_1992 factor 1.0637 applies to ‘all values K, mK, and A_K’; D2 was not re-evaluated there).
tammer_1990/table1_B_K4_H2#
{
"id": "table1_B_K4_H2",
"material": "H2",
"file": "H2/tammer_1990__table1_B_K4_H2.csv",
"table": "tbl_1f9c20083bf3",
"quantity": "kerr_virial_coefficient",
"unit": "C2 m8 J-2 mol-2",
"scale": 1e-34,
"abscissa": "temperature",
"abscissa_unit": "K",
"use": "check",
"uncertainty": {
"column": true,
"level": "1 sigma"
},
"parameters": {
"wavelength_nm": 632.8,
"order": 2
}
}
Table 1 (p. 160): the second molar Kerr virial coefficient B_K^(4) of H2 (with the local-field function f^(4), Eq. 14), 13 temperatures. Standard deviations. Superseded calibration (tammer_1992/table1_B_K4_H2).
tammer_1990/table1_B_K2_H2#
{
"id": "table1_B_K2_H2",
"material": "H2",
"file": "H2/tammer_1990__table1_B_K2_H2.csv",
"table": "tbl_1f9c20083bf3",
"quantity": "kerr_virial_coefficient",
"unit": "C2 m8 J-2 mol-2",
"scale": 1e-34,
"abscissa": "temperature",
"abscissa_unit": "K",
"use": "check",
"uncertainty": {
"column": true,
"level": "1 sigma"
},
"parameters": {
"wavelength_nm": 632.8,
"order": 2
}
}
Table 1 (p. 160): B_K^(2) of H2 (with the local-field function f^(2), Eq. 13), 13 temperatures. Standard deviations. Superseded calibration (tammer_1992/table1_B_K2_H2).
tammer_1990/table1_gamma_H2#
{
"id": "table1_gamma_H2",
"material": "H2",
"file": "H2/tammer_1990__table1_gamma_H2.csv",
"table": "tbl_1f9c20083bf3",
"quantity": "hyperpolarizability",
"unit": "SI",
"scale": 1e-62,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"columns": {
"temperature": "K"
},
"use": "check",
"uncertainty": {
"column": true,
"level": "1 sigma"
},
"parameters": {
"process": "dc_kerr",
"convention": "T",
"part": "electronic_vibrational",
"component": "K"
}
}
Table 1 (p. 160), last column: gamma = (3/2)(gamma_par - gamma_perp) of H2 at 632.8 nm per temperature (extra column), A_K minus the calculated orientation term; electronic plus vibrational. Standard deviations; mean 4.149(13) (note c). Superseded calibration (tammer_1992/table1_gamma_H2).
tammer_1990/table1_gamma_D2#
{
"id": "table1_gamma_D2",
"material": "D2",
"file": "D2/tammer_1990__table1_gamma_D2.csv",
"table": "tbl_1f9c20083bf3",
"quantity": "hyperpolarizability",
"unit": "SI",
"scale": 1e-62,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"columns": {
"temperature": "K"
},
"use": "check",
"uncertainty": {
"column": true,
"level": "1 sigma"
},
"parameters": {
"process": "dc_kerr",
"convention": "T",
"part": "electronic_vibrational",
"component": "K"
}
}
Table 1 (p. 160): gamma of D2 at 298.1 K and 632.8 nm, 4.17(2), with Rychlewski’s (1983) and Bishop & Cheung’s anisotropies for the orientation term (p. 159). Standard deviation. Superseded calibration.
tammer_1990/table2_gamma_H2#
{
"id": "table2_gamma_H2",
"material": "H2",
"file": "H2/tammer_1990__table2_gamma_H2.csv",
"table": "tbl_01dc04374704",
"quantity": "hyperpolarizability",
"unit": "SI",
"scale": 1e-62,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"use": "check",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "dc_kerr",
"convention": "T",
"part": "electronic_vibrational",
"component": "K"
}
}
Table 2 (p. 161) and abstract: gamma(H2) = (4.15 +- 0.12) x 10^-62 C^4 m^4 J^-3 (666 +- 19 au), the dc Kerr gamma_K at 632.8 nm, the mean over 196-412 K, electronic plus vibrational. ‘Error includes one standard deviation (see table 1) and systematical contributions’ (note b): 0.013 + 0.11 (the 1.1 % of A_K); level not stated. Superseded calibration: tammer_1992 gives 4.68(14).
tammer_1990/table2_gamma_D2#
{
"id": "table2_gamma_D2",
"material": "D2",
"file": "D2/tammer_1990__table2_gamma_D2.csv",
"table": "tbl_01dc04374704",
"quantity": "hyperpolarizability",
"unit": "SI",
"scale": 1e-62,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"use": "check",
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "dc_kerr",
"convention": "T",
"part": "electronic_vibrational",
"component": "K",
"temperature_k": 298.1
}
}
Table 2 (p. 161) and abstract: gamma(D2) = (4.17 +- 0.16) x 10^-62 C^4 m^4 J^-3 (669 +- 26 au) at 632.8 nm and 298.1 K, electronic plus vibrational; the error ‘includes one standard deviation and an additional systematic error of 1.6% in A_K(D2)’ (p. 159); level not stated. Superseded calibration, not re-evaluated in tammer_1992.
tammer_1990/table2_literature_H2#
{
"id": "table2_literature_H2",
"material": "H2",
"file": "H2/tammer_1990__table2_literature_H2.csv",
"table": "tbl_01dc04374704",
"quantity": "hyperpolarizability",
"unit": "SI",
"scale": 1e-62,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"use": "reproduced",
"reproduced_from": [
"buckingham_1968b",
"carusotto_1985",
"bishop_1990b"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "dc_kerr",
"convention": "T",
"part": "electronic_vibrational",
"component": "K"
}
}
Table 2 (p. 161), literature rows for H2 at 632.8 nm: ‘ref 3’ Buckingham & Orr 1968 (buckingham_1968b), 3.43(37); ‘ref 4’ Carusotto et al. 1985 (carusotto_1985), 4.90(30), ‘Transferred to lambda = 632.8 nm’ by the authors (orientation term subtracted, electronic dispersion and vibrational terms of Bishop, Pipin & Rérat; not Carusotto’s own number); ‘ref 1’ Bishop, Pipin & Rérat 1990 ab initio, 4.452(62) (‘Error is estimated’). Levels not stated.
tammer_1990/table2_literature_D2#
{
"id": "table2_literature_D2",
"material": "D2",
"file": "D2/tammer_1990__table2_literature_D2.csv",
"table": "tbl_01dc04374704",
"quantity": "hyperpolarizability",
"unit": "SI",
"scale": 1e-62,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"use": "reproduced",
"reproduced_from": [
"buckingham_1968b",
"carusotto_1985",
"bishop_1990b"
],
"uncertainty": {
"column": true,
"level": "not stated"
},
"parameters": {
"process": "dc_kerr",
"convention": "T",
"part": "electronic_vibrational",
"component": "K"
}
}
Table 2 (p. 161), literature rows for D2 at 632.8 nm: ‘ref 3’ Buckingham & Orr 1968 (buckingham_1968b), 3.0(6); ‘ref 4’ Carusotto et al. 1985 (carusotto_1985), 4.81(28), transferred to 632.8 nm by the authors; ‘ref 1’ Bishop, Pipin & Rérat 1990, 4.371(62) (‘Error is estimated’). Levels not stated.
tammer_1990/text_transferred_H2#
{
"id": "text_transferred_H2",
"material": "H2",
"table": "text",
"quantity": "hyperpolarizability",
"unit": "SI",
"scale": 1e-62,
"abscissa_unit": "nm",
"wavelength_medium": "standard_air",
"use": "check",
"reproduced_from": [
"mizrahi_1985_h2d2",
"rosasco_1986"
],
"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. 162-163: other experiments transferred by the authors to the dc Kerr gamma_K of H2 at 632.8 nm. ESHG: Eq. (22) plus ‘the vibrational correction of -0.045 x 10^-62 SI, valid for HeNe radiation [1], we arrive at gamma = (4.42 +- 0.03) x 10^-62 SI ... The given uncertainty is based on the experimental error of 0.6% quoted in the original work [23]’. Ac Kerr: Rosasco & Hurst’s gamma^e_acKE = (4.92 +- 0.24) x 10^-62 SI ‘can be transferred in a similar way ... The result for 632.8 nm is gamma^e = (4.57 +- 0.23) x 10^-62 SI which leads, after adding the vibrational contribution, to gamma = (4.52 +- 0.23) x 10^-62 SI’. Rows 1 and 3 are electronic plus vibrational; row 2 electronic only (the part parameter is that of rows 1 and 3). Derived values, not measurements of this paper.
Formulas#
tammer_1990/eq21_eshg_par_H2#
{
"id": "eq21_eshg_par_H2",
"material": "H2",
"equation": "Eq. (21), p. 162 (from Eq. 20)",
"printed": "gamma^e_par,ESHG = 4.067 x 10^-62 (1 + 1.8393 x 10^-10 nu_L^2 + 0.7845 x 10^-20 nu_L^4) SI",
"variable": "nu_L^2 = nu_sigma^2 + nu_1^2 + nu_2^2 + nu_3^2 (Eq. 19), wavenumbers in cm^-1",
"process": "eshg",
"convention": "T",
"part": "electronic",
"range": "not stated (Shelton & Lu's fit to Mizrahi & Shelton's ESHG data)",
"coefficients": {
"gamma0": "4.067",
"A": "1.8393",
"B": "0.7845"
},
"coefficient_units": {
"gamma0": "1e-62 C^4 m^4 J^-3",
"A": "1e-10 cm^2",
"B": "1e-20 cm^4"
},
"reproduced_from": [
"mizrahi_1985_h2d2",
"bishop_1989",
"D. P. Shelton and Z. Lu, Phys. Rev. A 37, 2231 (1988)"
],
"check": "Constructed by the authors from Eq. (20), gamma^e_par,ESHG(H2)/gamma_par,ESHG(He) = 15.13 (1 + 1.385 x 10^-10 nu_L^2) (Shelton & Lu's fit to Mizrahi & Shelton), and Bishop & Pipin's He, gamma_par(0) = 0.2688 x 10^-62 SI, A = 0.4544 x 10^-10, A' = 0.1552 x 10^-20. Reproduced here (2026-09-26): 15.13 x 0.2688 = 4.0669; 1.385 + 0.4544 = 1.8394; 1.385 x 0.4544 + 0.1552 = 0.7845."
}
The electronic ESHG gamma_par of H2 on Bishop & Pipin’s He scale, an intermediate of the transfer to the dc Kerr gamma (Eq. 22). A derived formula, not a fit of this paper’s data.
tammer_1990/eq22_dc_kerr_H2#
{
"id": "eq22_dc_kerr_H2",
"material": "H2",
"equation": "Eq. (22), p. 162",
"printed": "gamma^e = (3/2)(gamma^e_par - gamma^e_perp) = 4.067 x 10^-62 x (1 + 1.928 x 10^-10 nu_L^2 + 0.947 x 10^-20 nu_L^4) SI",
"variable": "nu_L^2 = nu_sigma^2 + nu_1^2 + nu_2^2 + nu_3^2, wavenumbers in cm^-1; dc Kerr: nu_L^2 = 2 nu^2",
"process": "dc_kerr",
"convention": "T",
"part": "electronic",
"range": "not stated",
"coefficients": {
"gamma0": "4.067",
"A": "1.928",
"B": "0.947"
},
"coefficient_units": {
"gamma0": "1e-62 C^4 m^4 J^-3",
"A": "1e-10 cm^2",
"B": "1e-20 cm^4"
},
"reproduced_from": [
"mizrahi_1985_h2d2",
"bishop_1989",
"D. P. Shelton and Z. Lu, Phys. Rev. A 37, 2231 (1988)"
],
"check": "Reproduces the text's 4.42 x 10^-62 SI (checked 2026-09-26): at 632.8 nm (air; vacuum wavenumber 15798.5 cm^-1) nu_L^2 = 2 nu^2 = 4.992e8 cm^-2, gamma^e = 4.067 x 1.0986 = 4.468, plus the vibrational -0.045 gives 4.423 x 10^-62 SI."
}
The electronic dc Kerr gamma_K of H2 from Eq. (21) with Bishop’s Eqs. (16)-(18) (gamma_par/gamma_perp = 3(1 + C nu_L^2), C_KE = -C_ESHG) and Shelton & Lu’s C_ESHG = -0.177 x 10^-10 (p. 162). A derived formula.
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.