dalgarno_1960 ============= A\. Dalgarno and A\. E\. Kingston, Proc\. R\. Soc\. London A 259, 424 \(1960\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1098/rspa\.1960\.0237 `__ **Source kind:** fit **Recorded technique:** Semi\-empirical\: n^2 \- 1 expanded in the Cauchy sums S\(\-2\), S\(\-4\), S\(\-6\), \.\.\. \(Eq\. 2\); S\(\-2\) and S\(\-4\) estimated from long\-wavelength refractive indices of Cuthbertson and Cuthbertson \(1911, 1932\) and Korff and Breit \(1932\), the higher sums from the interpolation formula S\(k\) = {a \+ b/\(2\.5 \- k\) \+ c/\(2\.5 \- k\)^2}^k S\(0\) \(Eq\. 18\) with a the first excitation energy \(pp\. 424\-429\)\. **Materials in structured entries:** Ar, He, Kr, Ne, Xe **Dataset quantities:** cauchy\_moment, polarizability **Contents:** 15 datasets; 5 formulas. Curation notes -------------- Not independent data\: the polarizabilities and dispersion formulas are derived from Cuthbertson and Cuthbertson's refractivities \(cuthbertson\_1910, cuthbertson\_1932\) and Korff and Breit's \(1932\) compilation\. Energies are in rydbergs, so S\(k\) is in Ry^k\: the datasets convert with scale 2^\-k\. Eq\. \(1\) gives n^2 \- 1 'at atmospheric pressure' with the prefactor 2\.00e\-4 \(4 pi N a0^3 with Loschmidt's number, i\.e\. 0 degrees C, 1 atm, ideal gas\); the Cuthbertsons' data are for 0 degrees C, 760 mm\. Table 1's helium 'exact' S\(k\) are sums over Dalgarno and Stewart's \(1960\) oscillator strengths \(S\(\-1\) from Pekeris\), the other column Eq\. \(18\); atomic hydrogen's are not extracted \(H is not a curated species\)\. Table 3 \(expectation values of \(sum r\_i\)^2\) and the Verdet\-constant formulas \(in microminutes, from Eq\. 5\) are not extracted\: they are not dispersion data \(the Verdet formulas are consequences of the n^2 \- 1 series\)\. The Kr n^2 \- 1 formula has no element label in the library's Markdown; it is the fourth of the five \(He, Ne, Ar, Kr, Xe order\)\. Consistency \(checked 2026\-09\-24\)\: the static limits of the n^2 \- 1 formulas, converted with Lorentz\-Lorenz at Loschmidt's number and the paper's a0^3 = 1\.482e\-25 cm^3, give He 1\.384, Ar 11\.084, Kr 16\.737, Xe 27\.29 \(Table 2\: 1\.384, 11\.080, 16\.734, 27\.292\) but Ne 2\.668 against Table 2's 2\.663 \(0\.2 %\); not resolved\. The He formula gives n \- 1 = 3\.4897e\-5 at 5462 A, Cuthbertson and Cuthbertson's 3\.489e\-5\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- dalgarno\_1960/s\_exact\_0 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_exact_0", "material": "He", "file": "He/dalgarno_1960__s_exact_0.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 1, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(0\) 'exact' \(from Dalgarno and Stewart's oscillator strengths; S\(\-1\) from Pekeris\), in Ry^0; scale 2^\(0\) gives atomic units\. dalgarno\_1960/s\_eq18\_0 ~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_eq18_0", "material": "He", "file": "He/dalgarno_1960__s_eq18_0.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 1, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(0\) from the interpolation formula Eq\. \(18\), in Ry^0; scale 2^\(0\) gives atomic units\. dalgarno\_1960/s\_exact\_m1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_exact_m1", "material": "He", "file": "He/dalgarno_1960__s_exact_m1.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 2, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(\-1\) 'exact' \(from Dalgarno and Stewart's oscillator strengths; S\(\-1\) from Pekeris\), in Ry^\-1; scale 2^\(1\) gives atomic units\. dalgarno\_1960/s\_eq18\_m1 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_eq18_m1", "material": "He", "file": "He/dalgarno_1960__s_eq18_m1.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 2, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(\-1\) from the interpolation formula Eq\. \(18\), in Ry^\-1; scale 2^\(1\) gives atomic units\. dalgarno\_1960/s\_exact\_m2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_exact_m2", "material": "He", "file": "He/dalgarno_1960__s_exact_m2.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 4, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(\-2\) 'exact' \(from Dalgarno and Stewart's oscillator strengths; S\(\-1\) from Pekeris\), in Ry^\-2; scale 2^\(2\) gives atomic units\. dalgarno\_1960/s\_eq18\_m2 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_eq18_m2", "material": "He", "file": "He/dalgarno_1960__s_eq18_m2.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 4, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(\-2\) from the interpolation formula Eq\. \(18\), in Ry^\-2; scale 2^\(2\) gives atomic units\. dalgarno\_1960/s\_exact\_m4 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_exact_m4", "material": "He", "file": "He/dalgarno_1960__s_exact_m4.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 16, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(\-4\) 'exact' \(from Dalgarno and Stewart's oscillator strengths; S\(\-1\) from Pekeris\), in Ry^\-4; scale 2^\(4\) gives atomic units\. dalgarno\_1960/s\_eq18\_m4 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_eq18_m4", "material": "He", "file": "He/dalgarno_1960__s_eq18_m4.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 16, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(\-4\) from the interpolation formula Eq\. \(18\), in Ry^\-4; scale 2^\(4\) gives atomic units\. dalgarno\_1960/s\_exact\_m6 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_exact_m6", "material": "He", "file": "He/dalgarno_1960__s_exact_m6.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 64, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(\-6\) 'exact' \(from Dalgarno and Stewart's oscillator strengths; S\(\-1\) from Pekeris\), in Ry^\-6; scale 2^\(6\) gives atomic units\. dalgarno\_1960/s\_eq18\_m6 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "s_eq18_m6", "material": "He", "file": "He/dalgarno_1960__s_eq18_m6.csv", "table": "tbl_61824749ae46", "quantity": "cauchy_moment", "unit": "au", "scale": 64, "abscissa": "order", "abscissa_unit": "1", "use": "check" } Table 1, helium S\(\-6\) from the interpolation formula Eq\. \(18\), in Ry^\-6; scale 2^\(6\) gives atomic units\. dalgarno\_1960/alpha\_He ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_He", "material": "He", "file": "He/dalgarno_1960__alpha_He.csv", "table": "tbl_07f29f0464dd", "quantity": "polarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "check" } Table 2, static polarizability in a0^3 \(the paper's a0^3 = 1\.482e\-25 cm^3\), derived from the Cuthbertsons' refractivities through the formulas below; not independent of those data\. dalgarno\_1960/alpha\_Ne ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_Ne", "material": "Ne", "file": "Ne/dalgarno_1960__alpha_Ne.csv", "table": "tbl_07f29f0464dd", "quantity": "polarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "check" } Table 2, static polarizability in a0^3 \(the paper's a0^3 = 1\.482e\-25 cm^3\), derived from the Cuthbertsons' refractivities through the formulas below; not independent of those data\. dalgarno\_1960/alpha\_Ar ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_Ar", "material": "Ar", "file": "Ar/dalgarno_1960__alpha_Ar.csv", "table": "tbl_07f29f0464dd", "quantity": "polarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "check" } Table 2, static polarizability in a0^3 \(the paper's a0^3 = 1\.482e\-25 cm^3\), derived from the Cuthbertsons' refractivities through the formulas below; not independent of those data\. dalgarno\_1960/alpha\_Kr ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_Kr", "material": "Kr", "file": "Kr/dalgarno_1960__alpha_Kr.csv", "table": "tbl_07f29f0464dd", "quantity": "polarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "check" } Table 2, static polarizability in a0^3 \(the paper's a0^3 = 1\.482e\-25 cm^3\), derived from the Cuthbertsons' refractivities through the formulas below; not independent of those data\. dalgarno\_1960/alpha\_Xe ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_Xe", "material": "Xe", "file": "Xe/dalgarno_1960__alpha_Xe.csv", "table": "tbl_07f29f0464dd", "quantity": "polarizability", "unit": "au", "abscissa_unit": "angstrom", "use": "check" } Table 2, static polarizability in a0^3 \(the paper's a0^3 = 1\.482e\-25 cm^3\), derived from the Cuthbertsons' refractivities through the formulas below; not independent of those data\. Formulas -------- dalgarno\_1960/n2\_minus\_1\_He ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_minus_1_He", "material": "He", "equation": "unnumbered, p. 428", "printed": "n^2 - 1 = 6.927 x 10^-5 (1 + 2.24 x 10^5/lambda^2 + 5.94 x 10^10/lambda^4 + 1.72 x 10^16/lambda^6 + ...)", "variable": "lambda: wavelength in Angstrom (medium not stated; the fitted data mix the 1910 air and the 1932 vacuum wavelengths, a difference of 3e-4 in lambda)", "state": "1atm_0C", "density": "ideal", "range": "the Cuthbertsons' visible and near-UV data; 'values in harmony with the available experimental data at all wavelengths' (p. 428)", "coefficients": { "a": "6.927e-5", "c2": "2.24e5", "c4": "5.94e10", "c6": "1.72e16" }, "check": "Static limit 1.3845 a.u. (Table 2: 1.384); n - 1 = 3.4897e-5 at 5462 A (Cuthbertson 1932: 3.489e-5). Lorentz-Lorenz at Loschmidt's number with a0^3 = 1.482e-25 cm^3. Checked 2026-09-24." } A Cauchy series in S\(\-2k\); the paper's form of n^2 \- 1 \(Eq\. 2\), not n \- 1\. Truncated where printed \('\.\.\.'\)\. dalgarno\_1960/n2\_minus\_1\_Ne ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_minus_1_Ne", "material": "Ne", "equation": "unnumbered, p. 428", "printed": "n^2 - 1 = 1.335 x 10^-4 (1 + 2.24 x 10^5/lambda^2 + 8.09 x 10^10/lambda^4 + 3.56 x 10^16/lambda^6 + ...)", "variable": "lambda: wavelength in Angstrom (medium not stated; the fitted data mix the 1910 air and the 1932 vacuum wavelengths, a difference of 3e-4 in lambda)", "state": "1atm_0C", "density": "ideal", "range": "the Cuthbertsons' visible and near-UV data; 'values in harmony with the available experimental data at all wavelengths' (p. 428)", "coefficients": { "a": "1.335e-4", "c2": "2.24e5", "c4": "8.09e10", "c6": "3.56e16" }, "check": "Static limit 2.668 a.u. against Table 2's 2.663 (0.2 %, unexplained); the leading term equals twice the Cuthbertsons' 1932 C/n0^2 = 6.673e-5. Lorentz-Lorenz at Loschmidt's number with a0^3 = 1.482e-25 cm^3. Checked 2026-09-24." } A Cauchy series in S\(\-2k\); the paper's form of n^2 \- 1 \(Eq\. 2\), not n \- 1\. Truncated where printed \('\.\.\.'\)\. dalgarno\_1960/n2\_minus\_1\_Ar ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_minus_1_Ar", "material": "Ar", "equation": "unnumbered, p. 428", "printed": "n^2 - 1 = 5.547 x 10^-4 (1 + 5.15 x 10^5/lambda^2 + 4.19 x 10^11/lambda^4 + 4.09 x 10^17/lambda^6 + 4.32 x 10^23/lambda^8 + ...)", "variable": "lambda: wavelength in Angstrom (medium not stated; the fitted data mix the 1910 air and the 1932 vacuum wavelengths, a difference of 3e-4 in lambda)", "state": "1atm_0C", "density": "ideal", "range": "the Cuthbertsons' visible and near-UV data; 'values in harmony with the available experimental data at all wavelengths' (p. 428)", "coefficients": { "a": "5.547e-4", "c2": "5.15e5", "c4": "4.19e11", "c6": "4.09e17", "c8": "4.32e23" }, "check": "Static limit 11.084 a.u. (Table 2: 11.080). Lorentz-Lorenz at Loschmidt's number with a0^3 = 1.482e-25 cm^3. Checked 2026-09-24." } A Cauchy series in S\(\-2k\); the paper's form of n^2 \- 1 \(Eq\. 2\), not n \- 1\. Truncated where printed \('\.\.\.'\)\. dalgarno\_1960/n2\_minus\_1\_Kr ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_minus_1_Kr", "material": "Kr", "equation": "unnumbered, p. 428", "printed": "n^2 - 1 = 8.377 x 10^-4 (1 + 6.70 x 10^5/lambda^2 + 8.84 x 10^11/lambda^4 + 1.49 x 10^18/lambda^6 + 2.74 x 10^24/lambda^8 + 5.10 x 10^30/lambda^10 + ...)", "variable": "lambda: wavelength in Angstrom (medium not stated; the fitted data mix the 1910 air and the 1932 vacuum wavelengths, a difference of 3e-4 in lambda)", "state": "1atm_0C", "density": "ideal", "range": "the Cuthbertsons' visible and near-UV data; 'values in harmony with the available experimental data at all wavelengths' (p. 428)", "coefficients": { "a": "8.377e-4", "c2": "6.70e5", "c4": "8.84e11", "c6": "1.49e18", "c8": "2.74e24", "c10": "5.10e30" }, "check": "Static limit 16.737 a.u. (Table 2: 16.734). Lorentz-Lorenz at Loschmidt's number with a0^3 = 1.482e-25 cm^3. Checked 2026-09-24." } A Cauchy series in S\(\-2k\); the paper's form of n^2 \- 1 \(Eq\. 2\), not n \- 1\. Truncated where printed \('\.\.\.'\)\. dalgarno\_1960/n2\_minus\_1\_Xe ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "n2_minus_1_Xe", "material": "Xe", "equation": "unnumbered, p. 428", "printed": "n^2 - 1 = 1.366 x 10^-3 (1 + 9.02 x 10^5/lambda^2 + 1.81 x 10^12/lambda^4 + 4.89 x 10^18/lambda^6 + 1.45 x 10^25/lambda^8 + 4.34 x 10^31/lambda^10 + ...)", "variable": "lambda: wavelength in Angstrom (medium not stated; the fitted data mix the 1910 air and the 1932 vacuum wavelengths, a difference of 3e-4 in lambda)", "state": "1atm_0C", "density": "ideal", "range": "the Cuthbertsons' visible and near-UV data; 'values in harmony with the available experimental data at all wavelengths' (p. 428)", "coefficients": { "a": "1.366e-3", "c2": "9.02e5", "c4": "1.81e12", "c6": "4.89e18", "c8": "1.45e25", "c10": "4.34e31" }, "check": "Static limit 27.29 a.u. (Table 2: 27.292). Lorentz-Lorenz at Loschmidt's number with a0^3 = 1.482e-25 cm^3. Checked 2026-09-24." } A Cauchy series in S\(\-2k\); the paper's form of n^2 \- 1 \(Eq\. 2\), not n \- 1\. Truncated where printed \('\.\.\.'\)\. 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.