stone_2004 ========== J\. A\. Stone and A\. Stejskal, Metrologia 41, 189 \(2004\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1088/0026\-1394/41/3/012 `__ **Source kind:** theory **Recorded technique:** Helium\: the refractive index computed ab initio \(Lorentz\-Lorenz with the molar polarizability from Bhatia and Drachman's dynamic polarizability, its static term replaced by Lach, Jeziorski and Szalewicz's QED value, plus the density and refractivity virial coefficients\)\. Nitrogen\: measured with two Fabry\-Perot refractometers \(Zerodur spacers of 452 and 94 mm, lasers locked to transmission maxima at 633 nm\), their pressure distortion calibrated with helium's calculated index \(pp\. 189\-194\)\. **Materials in structured entries:** He, N2 **Dataset quantities:** molar\_refractivity, refractivity **Contents:** 9 datasets; 4 formulas. Curation notes -------------- Formulas \(pp\. 191\-192\)\: Eq\. \(5\) alpha\(omega\) in reduced atomic units, Eq\. \(6\) A\_R = 0\.37395692 alpha \(cm^3/mol, 4He, 2002 CODATA\), Eq\. \(7\) omega = 91\.13919792/lambda \(lambda in nm\), Eq\. \(8\) A\_R\(lambda\), Eq\. \(9\) B\(T\) \(Hurly and Moldover\), Eq\. \(10\) B\_R near 633 nm, Eq\. \(12\) n = 1 \+ 3z/2 \+ 3z^2/8 with z = \(n^2\-1\)/\(n^2\+2\) = A\_R rho \+ B\_R rho^2\. The paper's A\_R is \(4 pi/3\) N\_A alpha in Gaussian units, i\.e\. the package's molar refractivity\. Wavelengths are vacuum wavelengths \(omega from lambda through a0 and the fine\-structure constant\)\. The uncertainty of Eq\. \(8\) is 'an estimated relative uncertainty of 1e\-6 at optical frequencies' \(p\. 191\), dominated by an arbitrary allowance for QED in the dynamic terms\. The paper argues \(p\. 191\) that the two direct He measurements \(Achtermann et al\. 1991, Birch 1991\) are 'seriously inconsistent' with the theory, high by 7e\-8 and 1\.1e\-7 in n at 1 atm, and that indirect evidence \(dielectric\-constant measurements, Mansfield and Peck's and Velsko and Eimerl's dispersion\) supports the theory to 4e\-9 at 633 nm\. Nitrogen \(p\. 195\)\: two sets, 105\-109 kPa at 19\.97 degrees C and 97\-101 kPa at 23\.4 degrees C, reduced to the molar refractivity 'from our measurements, plus known values of the virial coefficients B\(T\) \[15\] and B\_R \[10\]' \(the paper's \[15\] is Hurly and Moldover's helium calculation; for nitrogen this looks like a wrong reference\); 'expanded \(k = 2\) uncertainty \(0\.0007 cm^3 mol^\-1\) arises primarily from the measurement of the pressure and temperature'\. Nitrogen of 99\.9993 % purity \(Scott\)\. **Recorded bibliographic identity status:** VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- stone\_2004/theory\_molar\_refractivity\_He ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "theory_molar_refractivity_He", "material": "He", "table": "text", "quantity": "molar_refractivity", "unit": "cm3/mol", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "use": "check", "uncertainty": { "column": true, "level": "1 sigma" } } 1 inline points in the source record. Eq\. \(8\) evaluated at 633 nm, '0\.520 263 4\(5\) cm3 mol\-1' \(p\. 191\); the formula itself is recorded \(eq8\)\. stone\_2004/theory\_refractivity\_He ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "theory_refractivity_He", "material": "He", "table": "text", "quantity": "refractivity", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "state": { "pressure_pa": 101325, "temperature_c": 20 }, "density": "real", "use": "check", "uncertainty": { "column": true, "level": "1 sigma" } } 1 inline points in the source record. 'the refractive index of helium for 633 nm radiation, at 101 325 Pa pressure and 20 degrees C, is 1\.000 032 426 00\(8\)' \(p\. 192\), standard uncertainty 8e\-11; entered as n \- 1 = 3\.242600e\-5\. Recomputed with the paper's procedure and today's CODATA k\: 1\.00003242604 \(the 4e\-11 difference is the change in k since 2002\)\. stone\_2004/achtermann\_1991\_He ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "achtermann_1991_He", "material": "He", "table": "text", "quantity": "molar_refractivity", "unit": "cm3/mol", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "use": "reproduced", "reproduced_from": [ "H. J. Achtermann, G. Magnus and T. K. Bose, J. Chem. Phys. 94, 5669 (1991)" ], "uncertainty": { "column": true, "level": "not stated" } } 1 inline points in the source record. '0\.5213\(1\)' \(p\. 191\); the paper judges it high by 7e\-8 in n at 1 atm\. The uncertainty's level is not stated here\. stone\_2004/birch\_1991\_He ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "birch_1991_He", "material": "He", "table": "text", "quantity": "molar_refractivity", "unit": "cm3/mol", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "use": "reproduced", "reproduced_from": [ "K. P. Birch, J. Opt. Soc. Am. A 8, 647 (1991)" ], "uncertainty": { "column": true, "level": "not stated" } } 1 inline points in the source record. '0\.5220\(3\)' \(p\. 191\); judged high by 1\.1e\-7 in n at 1 atm\. stone\_2004/experiment\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "experiment_N2", "material": "N2", "table": "text", "quantity": "molar_refractivity", "unit": "cm3/mol", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "use": "primary", "uncertainty": { "column": true, "level": "2 sigma" } } 1 inline points in the source record. 'our best estimate of the molar refractivity at 633 nm is 4\.4454\(7\) cm3 mol\-1', expanded \(k = 2\) uncertainty \(p\. 195\)\. 633 nm is the nominal He\-Ne \(and diode\) laser wavelength\. stone\_2004/birch\_1991\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "birch_1991_N2", "material": "N2", "table": "text", "quantity": "molar_refractivity", "unit": "cm3/mol", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "use": "reproduced", "reproduced_from": [ "K. P. Birch, J. Opt. Soc. Am. A 8, 647 (1991)" ], "uncertainty": { "column": true, "level": "not stated" } } 1 inline points in the source record. '4\.4457\(8\)' \(p\. 195\)\. stone\_2004/hou\_thalmann\_1994\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "hou_thalmann_1994_N2", "material": "N2", "table": "text", "quantity": "molar_refractivity", "unit": "cm3/mol", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "use": "reproduced", "reproduced_from": [ "W. Hou and R. Thalmann, Measurement 13, 307 (1994)" ], "uncertainty": { "column": true, "level": "not stated" } } 1 inline points in the source record. '4\.4456\(6\)' \(p\. 195\)\. stone\_2004/montixi\_1983\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "montixi_1983_N2", "material": "N2", "table": "text", "quantity": "molar_refractivity", "unit": "cm3/mol", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "use": "reproduced", "reproduced_from": [ "G. Montixi, G. Coulon and R. Occelli, Can. J. Phys. 61, 473 (1983)" ], "uncertainty": { "column": true, "level": "not stated" } } 1 inline points in the source record. '4\.4454\(5\)' \(p\. 195\)\. stone\_2004/achtermann\_1991\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "achtermann_1991_N2", "material": "N2", "table": "text", "quantity": "molar_refractivity", "unit": "cm3/mol", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "use": "reproduced", "reproduced_from": [ "H. J. Achtermann, G. Magnus and T. K. Bose, J. Chem. Phys. 94, 5669 (1991)" ], "uncertainty": { "column": true, "level": "not stated" } } 1 inline points in the source record. '4\.4464\(5\)' \(p\. 195\); 'at best only marginally consistent' with this paper\. Formulas -------- stone\_2004/eq5 ~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq5", "material": "He", "equation": "Eq. (5), p. 191", "printed": "alpha = 1.3831916 + 0.385530216 omega^2 + 0.12753895 omega^4 + 0.04573114 omega^6", "variable": "omega: photon energy in reduced Rydberg units, omega = 91.13919792/lambda with lambda the vacuum wavelength in nm (Eq. 7); alpha in reduced atomic units (4He)", "state": "none (per atom)", "density": "none", "range": "visible and near infrared (the series in omega^2 is for omega well below the first resonance; checked by the authors at 532-1064 nm)", "coefficients": { "a0": "1.3831916", "a2": "0.385530216", "a4": "0.12753895", "a6": "0.04573114" }, "check": "With Eqs. (6) and (7) its four terms become 0.51725407, 1197.5410, 3.290677e6 and 9.800874e9, Eq. (8)'s coefficients to all printed digits. Checked 2026-09-24." } Static term from Lach, Jeziorski and Szalewicz \(QED included\); the dynamic terms from Bhatia and Drachman, Phys\. Rev\. A 58, 4470 \(1998\)\. Reduced units carry the \(1 \+ m\_e/m\_alpha\) factors of Eqs\. \(6\)\-\(7\)\. stone\_2004/eq8 ~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq8", "form": "molar_refractivity_series", "terms": { "a": [ "0.51725407", "1197.5410", "3.290677e6", "9.800874e9" ] }, "range_nm": [ 300.0, "inf" ], "material": "He", "equation": "Eq. (8), p. 191", "printed": "A_R = 0.51725407 + 1197.5410/lambda^2 + 3.290677 x 10^6/lambda^4 + 9.800874 x 10^9/lambda^6", "variable": "lambda: vacuum wavelength in nm; A_R in cm^3/mol", "state": "none (molar refractivity)", "density": "none", "range": "optical frequencies ('estimated relative uncertainty of 1e-6 at optical frequencies')", "coefficients": { "a0": "0.51725407", "a2": "1197.5410", "a4": "3.290677e6", "a6": "9.800874e9" }, "check": "Gives 0.5202634 at 633 nm, the paper's 0.520 263 4(5); with Eqs. (9), (10), (12) at 101325 Pa, 293.15 K it gives n = 1.00003242604 against the printed 1.000 032 426 00(8). Checked 2026-09-24." } stone\_2004/eq9 ~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq9", "material": "He", "equation": "Eq. (9), p. 192", "printed": "B(T) = 13.028 - 0.0041 T", "variable": "T in K; B in cm^3/mol", "state": "none", "density": "none", "range": "275-325 K (within 0.001 cm^3/mol of Hurly and Moldover; their uncertainty 0.03)", "coefficients": { "b0": "13.028", "b1": "-0.0041" }, "check": "Used in the n(633 nm) check above." } Second density virial coefficient of helium, a linear fit to J\. J\. Hurly and M\. R\. Moldover, J\. Res\. NIST 105, 667 \(2000\)\. stone\_2004/eq10 ~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq10", "material": "He", "equation": "Eq. (10), p. 192", "printed": "B_R = -0.032 - 0.0001 T", "variable": "T in K; B_R in cm^6/mol^2, at 633 nm", "state": "none", "density": "none", "range": "273-323 K; uncertainty 0.007 cm^6/mol^2", "coefficients": { "b0": "-0.032", "b1": "-0.0001" }, "check": "Used in the n(633 nm) check above; changes n by less than 2e-10." } Refractivity virial coefficient, based on Koch et al\., J\. Chem\. Phys\. 111, 10108 \(1999\)\. 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 ` for the checks and decisions.