shelton_1986 ============ D\. P\. Shelton and V\. Mizrahi, Phys\. Rev\. A 33, 72 \(1986\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1103/physreva\.33\.72 `__ **Source kind:** measurement **Recorded technique:** Periodic phase matching of electric\-field\-induced second\-harmonic generation \(ESHG\)\: a cw dye or Ar\+ laser \(458\-700 nm, about 1 W\) is focused through a cell with an array of 150 wire\-electrode pairs \(N = 75, mean spacing 2\.6924 mm\) giving a spatially periodic static field; the gas density is varied until \|Delta k\| = \|2 k\_w \- k\_2w\| equals \|Delta k\|\_opt = 1161\.5 m^\-1 \(\+\-0\.05 %\), found from the peak second\-harmonic signal \(parabola fit, 1e\-4 precision in pressure\)\. Temperature by a thermistor in the gas \(\+\-0\.2 C\); pressure by Baratron \(\+\-0\.15 %\) or Heise gauge \(He, \+\-0\.1 % of 160 bar\); density from the virial equation of state \(Dymond & Smith\)\. Pressures up to 130 bar \(He\) and 10 bar \(others\); virial corrections to the refractivity below 0\.05 % \(p\. 73\)\. Dye\-laser wavelengths by a 0\.75 m grating spectrometer\. **Materials in structured entries:** Ar, D2, H2, He, N2, O2 **Dataset quantities:** phase\_match\_density\_ratio **Contents:** 2 datasets; 12 formulas. Curation notes -------------- Observable \(Eq\. 3\)\: Delta k = 2 k\_w \- k\_2w = \(4 pi / lambda\_w\)\(n\_w \- n\_2w\) = \-\(2 pi / \(lambda\_w eps0\)\) \(alpha\_2w \- alpha\_w\) rho, lambda\_w the fundamental's vacuum wavelength; periodic phase match occurs at \|Delta k\| = \|Delta k\|\_opt\. So the measured density rho \(real gas, virial EOS\) at phase match gives the dispersion n\(2 omega\) \- n\(omega\) per unit density\. The Ar\+ lines are listed at their vacuum wavelengths \(514\.7, 496\.6, 488\.1, 476\.6 nm\), which confirms Table III is in vacuum wavelengths\. Uncertainty\: 'The estimated overall uncertainty of the measurements varies between \+\-0\.2 % and \+\-0\.4 % for He, but has a nearly uniform value of \+\-0\.2 % for all the other gases' \(p\. 74\); the D2/H2 ratio is 'about \+\-0\.05 %' because both gases were measured with the same gauge \(p\. 74\)\. He differs by 0\.5 % from the earlier Shelton & Buckingham \(1982\) measurement and by 0\.3 % from Glover & Weinhold's theory \('consistent with a pressure scale error', p\. 74\)\. The individual phase\-match densities of all six gases are shown only in Fig\. 2 \(lambda^3/rho against nu^2\) and are not tabulated; the paper's results are the fits of Table I \(lambda^3/rho, Eq\. 6\) and their transformation to \(n\_w \- n\_0\) at the ideal\-gas density of 0 degrees C, 1 atm \(Table II, Eq\. 8\), recorded as formulas\. Only Table III, the D2/H2 density ratio, is data\. The 'calc' column of Table III is Shelton's evaluation of Rychlewski's \(1983\) ab initio polarizabilities, kept as a reproduction\. The D2 was 99\.5 at\.% \(chief impurity HD\), which would shift the ratio up 'by at most 0\.02 %' \(p\. 75\)\. **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/ratio\_D2\_H2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ratio_D2_H2", "material": "D2", "file": "D2/shelton_1986__ratio_D2_H2.csv", "table": "tbl_0a217d3795df", "quantity": "phase_match_density_ratio", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "density": "real", "use": "primary", "uncertainty": { "column": true, "level": "not stated" }, "parameters": { "reference_material": "H2", "harmonic": 2, "process": "ESHG (electric-field-induced second-harmonic generation), periodic phase matching", "definition": "rho_gas / rho_ref at which both gases give the same wave-vector mismatch |Delta k| = (4 pi / lambda_w) |n(2w) - n(w)| (the apparatus's |Delta k|_opt); to first order in density, rho_gas / rho_ref = [alpha_ref(2w) - alpha_ref(w)] / [alpha_gas(2w) - alpha_gas(w)]; wavelength = fundamental (vacuum)", "densities": "real gas, virial equation of state" } } rho\_D2 / rho\_H2 at equal ESHG wave\-vector mismatch, measured together under the same conditions \(Table III\)\. Per\-point errors are the printed parenthesised last\-digit errors; the overall accuracy is 'about \+\-0\.05 %' \(p\. 74\)\. The zero\-frequency limit of the fitted line \(Fig\. 3\) is 1\.0290\. shelton\_1986/rychlewski\_ratio\_D2\_H2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "rychlewski_ratio_D2_H2", "material": "D2", "file": "D2/shelton_1986__rychlewski_ratio_D2_H2.csv", "table": "tbl_0a217d3795df", "quantity": "phase_match_density_ratio", "abscissa_unit": "nm", "wavelength_medium": "vacuum", "use": "reproduced", "reproduced_from": [ "rychlewski_1983" ], "parameters": { "reference_material": "H2", "harmonic": 2, "process": "ESHG (electric-field-induced second-harmonic generation), periodic phase matching", "definition": "as ratio_D2_H2" } } Table III 'calc'\: the ratio evaluated by Shelton & Mizrahi from Rychlewski's \(ref\. 23\) ab initio polarizabilities; 'the calculation is seen to overestimate the dispersion difference between H2 and D2 at high frequencies' \(p\. 75\)\. Formulas -------- shelton\_1986/phase\_match\_He ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "phase_match_He", "material": "He", "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)", "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 0.05791, B' (10^-35 m^8/mol) = 0.0875, C' not given", "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1", "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A1": "0.05791e-21", "B1": "0.0875e-35" }, "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A, B to the last digit (checked 2026-09-24)." } The fit to the measured phase\-match densities of Fig\. 2 \(the points are not tabulated\)\. With Eq\. \(7\), \(n\_2w \- n\_w\)\_STP,i = rho\_STP,i \|Delta k\|\_opt lambda / \(4 pi rho\)\: the dispersion n\(2 omega\) \- n\(omega\) of the ideal\-gas\-density gas at 0 degrees C, 1 atm\. Accuracy \+\-0\.2 % \(He\: \+\-0\.2 to \+\-0\.4 %, and 0\.3 % from the Glover\-Weinhold theory, possibly a pressure\-scale error, p\. 74\)\. shelton\_1986/phase\_match\_Ar ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "phase_match_Ar", "material": "Ar", "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)", "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 1.0639, B' (10^-35 m^8/mol) = 3.386, C' (10^-48 m^10/mol) = 1.96", "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1", "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A1": "1.0639e-21", "B1": "3.386e-35", "C1": "1.96e-48" }, "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A to 1e-5, B 0.93086 (0.9308), C 1.2829 (1.285) (checked 2026-09-24)." } The fit to the measured phase\-match densities of Fig\. 2 \(the points are not tabulated\)\. With Eq\. \(7\), \(n\_2w \- n\_w\)\_STP,i = rho\_STP,i \|Delta k\|\_opt lambda / \(4 pi rho\)\: the dispersion n\(2 omega\) \- n\(omega\) of the ideal\-gas\-density gas at 0 degrees C, 1 atm\. Accuracy \+\-0\.2 % \(Table I caption\)\. shelton\_1986/phase\_match\_H2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "phase_match_H2", "material": "H2", "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)", "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 0.7662, B' (10^-35 m^8/mol) = 2.477, C' (10^-48 m^10/mol) = 2.48", "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1", "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A1": "0.7662e-21", "B1": "2.477e-35", "C1": "2.48e-48" }, "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A exact, B 0.68097 (0.6808), C 1.6233 (1.626) (checked 2026-09-24)." } The fit to the measured phase\-match densities of Fig\. 2 \(the points are not tabulated\)\. With Eq\. \(7\), \(n\_2w \- n\_w\)\_STP,i = rho\_STP,i \|Delta k\|\_opt lambda / \(4 pi rho\)\: the dispersion n\(2 omega\) \- n\(omega\) of the ideal\-gas\-density gas at 0 degrees C, 1 atm\. Accuracy \+\-0\.2 % \(Table I caption\)\. shelton\_1986/phase\_match\_D2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "phase_match_D2", "material": "D2", "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)", "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 0.7419, B' (10^-35 m^8/mol) = 2.513, C' (10^-48 m^10/mol) = 2.20", "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1", "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A1": "0.7419e-21", "B1": "2.513e-35", "C1": "2.20e-48" }, "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A 1.01980 (1.0197), B 0.69086 (0.6908), C 1.4400 (1.442) (checked 2026-09-24)." } The fit to the measured phase\-match densities of Fig\. 2 \(the points are not tabulated\)\. With Eq\. \(7\), \(n\_2w \- n\_w\)\_STP,i = rho\_STP,i \|Delta k\|\_opt lambda / \(4 pi rho\)\: the dispersion n\(2 omega\) \- n\(omega\) of the ideal\-gas\-density gas at 0 degrees C, 1 atm\. Accuracy \+\-0\.2 % \(Table I caption\)\. shelton\_1986/phase\_match\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "phase_match_N2", "material": "N2", "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)", "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 1.1469, B' (10^-35 m^8/mol) = 3.528, C' (10^-48 m^10/mol) = 2.02", "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1", "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A1": "1.1469e-21", "B1": "3.528e-35", "C1": "2.02e-48" }, "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A, B exact, C 1.3222 (1.320) (checked 2026-09-24)." } The fit to the measured phase\-match densities of Fig\. 2 \(the points are not tabulated\)\. With Eq\. \(7\), \(n\_2w \- n\_w\)\_STP,i = rho\_STP,i \|Delta k\|\_opt lambda / \(4 pi rho\)\: the dispersion n\(2 omega\) \- n\(omega\) of the ideal\-gas\-density gas at 0 degrees C, 1 atm\. Accuracy \+\-0\.2 % \(Table I caption\)\. shelton\_1986/phase\_match\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "phase_match_O2", "material": "O2", "equation": "Eq. (6), p. 73, with Table I (tbl_f88beb50f6dd)", "printed": "lambda^3/rho = A' + B' lambda^-2 + C' lambda^-4; A' (10^-21 m^6/mol) = 1.3588, B' (10^-35 m^8/mol) = 5.469, C' (10^-48 m^10/mol) = 19.49", "variable": "lambda: fundamental vacuum wavelength in m; rho: real-gas density (mol/m^3) at optimum ESHG phase match in this apparatus, |Delta k|_opt = 1161.5 m^-1", "range": "458-700 nm (the lasers used); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A1": "1.3588e-21", "B1": "5.469e-35", "C1": "19.49e-48" }, "check": "Transformed by Eq. (8) (rho_STP,i = 44.6150 mol/m^3, |Delta k|_opt = 1161.5 m^-1) it gives Table II: A 1.86778 (1.8677), B 1.50351 (1.5036), C 12.7574 (12.758) (checked 2026-09-24)." } The fit to the measured phase\-match densities of Fig\. 2 \(the points are not tabulated\)\. With Eq\. \(7\), \(n\_2w \- n\_w\)\_STP,i = rho\_STP,i \|Delta k\|\_opt lambda / \(4 pi rho\)\: the dispersion n\(2 omega\) \- n\(omega\) of the ideal\-gas\-density gas at 0 degrees C, 1 atm\. Accuracy \+\-0\.2 % \(Table I caption\)\. shelton\_1986/dispersion\_He ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dispersion_He", "material": "He", "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)", "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 0.079 60, B (10^-8 um^4) = 0.024 06, C not given", "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)", "state": "1atm_0C", "density": "ideal", "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A": "0.07960e-6", "B": "0.02406e-8" }, "check": "Recomputed from Table I by Eq. (8): A, B to the last digit (checked 2026-09-24); the differences in C follow from C' having three digits." } Derived from the Table I fit, not separately fitted\: use one or the other\. 'The absolute values of n\_w may be obtained from Eq\. \(8\) if the refractive index is known at a single frequency' \(p\. 74\)\. STP,i\: the density of an ideal gas at 273\.15 K and 1 atm \(p\. 74\)\. Accuracy \+\-0\.2 % \(He \+\-0\.2 to \+\-0\.4 %\)\. shelton\_1986/dispersion\_Ar ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dispersion_Ar", "material": "Ar", "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)", "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.462 4, B (10^-8 um^4) = 0.930 8, C (10^-10 um^6) = 1.285", "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)", "state": "1atm_0C", "density": "ideal", "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A": "1.4624e-6", "B": "0.9308e-8", "C": "1.285e-10" }, "check": "Recomputed from Table I by Eq. (8): A to 1e-5, B 0.93086 (0.9308), C 1.2829 (1.285) (checked 2026-09-24); the differences in C follow from C' having three digits." } Derived from the Table I fit, not separately fitted\: use one or the other\. 'The absolute values of n\_w may be obtained from Eq\. \(8\) if the refractive index is known at a single frequency' \(p\. 74\)\. STP,i\: the density of an ideal gas at 273\.15 K and 1 atm \(p\. 74\)\. Accuracy \+\-0\.2 %\. shelton\_1986/dispersion\_H2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dispersion_H2", "material": "H2", "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)", "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.053 2, B (10^-8 um^4) = 0.680 8, C (10^-10 um^6) = 1.626", "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)", "state": "1atm_0C", "density": "ideal", "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A": "1.0532e-6", "B": "0.6808e-8", "C": "1.626e-10" }, "check": "Recomputed from Table I by Eq. (8): A exact, B 0.68097 (0.6808), C 1.6233 (1.626) (checked 2026-09-24); the differences in C follow from C' having three digits." } Derived from the Table I fit, not separately fitted\: use one or the other\. 'The absolute values of n\_w may be obtained from Eq\. \(8\) if the refractive index is known at a single frequency' \(p\. 74\)\. STP,i\: the density of an ideal gas at 273\.15 K and 1 atm \(p\. 74\)\. Accuracy \+\-0\.2 %\. shelton\_1986/dispersion\_D2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dispersion_D2", "material": "D2", "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)", "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.019 7, B (10^-8 um^4) = 0.690 8, C (10^-10 um^6) = 1.442", "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)", "state": "1atm_0C", "density": "ideal", "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A": "1.0197e-6", "B": "0.6908e-8", "C": "1.442e-10" }, "check": "Recomputed from Table I by Eq. (8): A 1.01980 (1.0197), B 0.69086 (0.6908), C 1.4400 (1.442) (checked 2026-09-24); the differences in C follow from C' having three digits." } Derived from the Table I fit, not separately fitted\: use one or the other\. 'The absolute values of n\_w may be obtained from Eq\. \(8\) if the refractive index is known at a single frequency' \(p\. 74\)\. STP,i\: the density of an ideal gas at 273\.15 K and 1 atm \(p\. 74\)\. Accuracy \+\-0\.2 %\. shelton\_1986/dispersion\_N2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dispersion_N2", "material": "N2", "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)", "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.576 5, B (10^-8 um^4) = 0.969 9, C (10^-10 um^6) = 1.320", "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)", "state": "1atm_0C", "density": "ideal", "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A": "1.5765e-6", "B": "0.9699e-8", "C": "1.320e-10" }, "check": "Recomputed from Table I by Eq. (8): A, B exact, C 1.3222 (1.320) (checked 2026-09-24); the differences in C follow from C' having three digits." } Derived from the Table I fit, not separately fitted\: use one or the other\. 'The absolute values of n\_w may be obtained from Eq\. \(8\) if the refractive index is known at a single frequency' \(p\. 74\)\. STP,i\: the density of an ideal gas at 273\.15 K and 1 atm \(p\. 74\)\. Accuracy \+\-0\.2 %\. shelton\_1986/dispersion\_O2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "dispersion_O2", "material": "O2", "equation": "Eq. (8), p. 74, with Table II (tbl_8b8e879c14ac)", "printed": "(n_w - n_0)_STP,i = A lambda^-2 + B lambda^-4 + C lambda^-6; A (10^-6 um^2) = 1.867 7, B (10^-8 um^4) = 1.503 6, C (10^-10 um^6) = 12.758", "variable": "lambda: vacuum wavelength in um; n_0 is the zero-frequency limit of n (not given: the formula fixes only the dispersion)", "state": "1atm_0C", "density": "ideal", "range": "229-700 nm (the harmonics and fundamentals of 458-700 nm); 'Extrapolation outside this interval will be less accurate'", "coefficients": { "A": "1.8677e-6", "B": "1.5036e-8", "C": "12.758e-10" }, "check": "Recomputed from Table I by Eq. (8): A 1.86778 (1.8677), B 1.50351 (1.5036), C 12.7574 (12.758) (checked 2026-09-24); the differences in C follow from C' having three digits." } Derived from the Table I fit, not separately fitted\: use one or the other\. 'The absolute values of n\_w may be obtained from Eq\. \(8\) if the refractive index is known at a single frequency' \(p\. 74\)\. STP,i\: the density of an ideal gas at 273\.15 K and 1 atm \(p\. 74\)\. Accuracy \+\-0\.2 %\. 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.