dawes_1968 ========== E\. L\. Dawes, Phys\. Rev\. 169, 47 \(1968\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1103/physrev\.169\.47 `__ **Source kind:** theory **Recorded technique:** One\-term \(average\-energy\) estimate of the third\-harmonic coefficient alpha\_zzzz\(3nu\) of the rare gases at 6943 A\: Armstrong et al\.'s sum over states with every excitation replaced by one average frequency nu\_0 and the sums reduced with the Thomas\-Reiche\-Kuhn rule, the effective electron number N\_0 and nu\_0 taken from Cuthbertson's fits to refractive\-index data\. **Materials in structured entries:** Ar, He, Kr, Ne, Xe **Dataset quantities:** hyperpolarizability, hyperpolarizability\_ratio **Contents:** 9 datasets; 5 formulas. Curation notes -------------- Definition \(p\. 47\)\: Eq\. \(3\), alpha\_zzzz\(3nu\) = \(e^4/h^3\) sum nu\_ig Z\_gi Z\_ij Z\_jk Z\_kg A\_ijk, 'our Eq\. \(3\) is their \[Armstrong, Bloembergen, Ducuing and Pershan, Phys\. Rev\. 127, 1918 \(1962\)\] Eq\. \(2\.22\), except for a factor of 1/4 which has been removed as per Ref\. 1' \(New and Ward 1967\)\: the same coefficient as New and Ward's, i\.e\. Ward and New's chi = gamma\_T/6 \(see ward\_1969\), in esu per atom\. Lehmeier et al\. 1985 divide these values by 4 \(their Table 2 column \(g\)\); Ward and New 1969 normalise them to He = 4\.0\. Result \(Eq\. \(9\), p\. 48\)\: alpha\_zzzz\(3nu\) ~ N\_0 e^4 / \(16 pi^4 m^2 h\) \(nu\_0^2 \+ 6 nu^2\) / \[nu\_0 \(nu\_0^2 \- 9 nu^2\)\(nu\_0^2 \- 4 nu^2\)\(nu\_0^2 \- nu^2\)\], recorded as formulas below\. Table I's N\_0 and nu\_0 are reproduced from C\. and M\. Cuthbertson, Proc\. Roy\. Soc\. \(London\) 135A, 40 \(1932\), whose fit is of the linear polarizability, Eq\. \(2\), alpha^L = N\_0 e^2 / \(4 pi^2 m \(nu\_0^2 \- nu^2\)\); not extracted as data\. Columns \(C\) and \(D\) are New and Ward's \(1967\) two sets of experimental ratios as ranges \(not extracted\)\. Table I's caption prints the unit as '10^39 esu/atom', a misprint for 10^\-39 \(abstract\: 0\.9 x 10^\-39 esu/atom for He\)\. No uncertainty is given; Ward and New 1969 find the helium value a factor 4 below Sitz and Yaris's\. Electronic, static atoms; nu \(the fundamental\) is the ruby 6943 A \(printed\)\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- dawes\_1968/alpha\_3nu\_He ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_3nu_He", "material": "He", "file": "He/dawes_1968__alpha_3nu_He.csv", "table": "tbl_f296ada000f8", "quantity": "hyperpolarizability", "unit": "esu", "scale": 1e-39, "abscissa_unit": "nm", "use": "primary", "parameters": { "process": "thg", "convention": "ward_1969", "part": "electronic" } } Table I \(p\. 48\), column \(A\)\: alpha\_zzzz\(3nu\) calculated by Eq\. \(9\) at 6943 A, 10^\-39 esu/atom, New and Ward's convention \(= gamma\_T/6\)\. A crude one\-term model\: see the source notes\. dawes\_1968/alpha\_3nu\_Ne ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_3nu_Ne", "material": "Ne", "file": "Ne/dawes_1968__alpha_3nu_Ne.csv", "table": "tbl_f296ada000f8", "quantity": "hyperpolarizability", "unit": "esu", "scale": 1e-39, "abscissa_unit": "nm", "use": "primary", "parameters": { "process": "thg", "convention": "ward_1969", "part": "electronic" } } Table I \(p\. 48\), column \(A\)\: alpha\_zzzz\(3nu\) calculated by Eq\. \(9\) at 6943 A, 10^\-39 esu/atom, New and Ward's convention \(= gamma\_T/6\)\. A crude one\-term model\: see the source notes\. dawes\_1968/alpha\_3nu\_Ar ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_3nu_Ar", "material": "Ar", "file": "Ar/dawes_1968__alpha_3nu_Ar.csv", "table": "tbl_f296ada000f8", "quantity": "hyperpolarizability", "unit": "esu", "scale": 1e-39, "abscissa_unit": "nm", "use": "primary", "parameters": { "process": "thg", "convention": "ward_1969", "part": "electronic" } } Table I \(p\. 48\), column \(A\)\: alpha\_zzzz\(3nu\) calculated by Eq\. \(9\) at 6943 A, 10^\-39 esu/atom, New and Ward's convention \(= gamma\_T/6\)\. A crude one\-term model\: see the source notes\. dawes\_1968/alpha\_3nu\_Kr ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_3nu_Kr", "material": "Kr", "file": "Kr/dawes_1968__alpha_3nu_Kr.csv", "table": "tbl_f296ada000f8", "quantity": "hyperpolarizability", "unit": "esu", "scale": 1e-39, "abscissa_unit": "nm", "use": "primary", "parameters": { "process": "thg", "convention": "ward_1969", "part": "electronic" } } Table I \(p\. 48\), column \(A\)\: alpha\_zzzz\(3nu\) calculated by Eq\. \(9\) at 6943 A, 10^\-39 esu/atom, New and Ward's convention \(= gamma\_T/6\)\. A crude one\-term model\: see the source notes\. dawes\_1968/alpha\_3nu\_Xe ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_3nu_Xe", "material": "Xe", "file": "Xe/dawes_1968__alpha_3nu_Xe.csv", "table": "tbl_f296ada000f8", "quantity": "hyperpolarizability", "unit": "esu", "scale": 1e-39, "abscissa_unit": "nm", "use": "primary", "parameters": { "process": "thg", "convention": "ward_1969", "part": "electronic" } } Table I \(p\. 48\), column \(A\)\: alpha\_zzzz\(3nu\) calculated by Eq\. \(9\) at 6943 A, 10^\-39 esu/atom, New and Ward's convention \(= gamma\_T/6\)\. A crude one\-term model\: see the source notes\. dawes\_1968/ratio\_Ne ~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ratio_Ne", "material": "Ne", "file": "Ne/dawes_1968__ratio_Ne.csv", "table": "tbl_f296ada000f8", "quantity": "hyperpolarizability_ratio", "abscissa_unit": "nm", "use": "check", "parameters": { "reference_material": "He", "process": "thg", "part": "electronic" } } Table I \(p\. 48\), column \(B\)\: column \(A\) relative to He \(1\.7/0\.9 = 1\.89, 28/0\.9 = 31\.1, 71/0\.9 = 78\.9, 222/0\.9 = 246\.7\: reproduce\)\. Not independent of alpha\_3nu\_Ne\. dawes\_1968/ratio\_Ar ~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ratio_Ar", "material": "Ar", "file": "Ar/dawes_1968__ratio_Ar.csv", "table": "tbl_f296ada000f8", "quantity": "hyperpolarizability_ratio", "abscissa_unit": "nm", "use": "check", "parameters": { "reference_material": "He", "process": "thg", "part": "electronic" } } Table I \(p\. 48\), column \(B\)\: column \(A\) relative to He \(1\.7/0\.9 = 1\.89, 28/0\.9 = 31\.1, 71/0\.9 = 78\.9, 222/0\.9 = 246\.7\: reproduce\)\. Not independent of alpha\_3nu\_Ar\. dawes\_1968/ratio\_Kr ~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ratio_Kr", "material": "Kr", "file": "Kr/dawes_1968__ratio_Kr.csv", "table": "tbl_f296ada000f8", "quantity": "hyperpolarizability_ratio", "abscissa_unit": "nm", "use": "check", "parameters": { "reference_material": "He", "process": "thg", "part": "electronic" } } Table I \(p\. 48\), column \(B\)\: column \(A\) relative to He \(1\.7/0\.9 = 1\.89, 28/0\.9 = 31\.1, 71/0\.9 = 78\.9, 222/0\.9 = 246\.7\: reproduce\)\. Not independent of alpha\_3nu\_Kr\. dawes\_1968/ratio\_Xe ~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ratio_Xe", "material": "Xe", "file": "Xe/dawes_1968__ratio_Xe.csv", "table": "tbl_f296ada000f8", "quantity": "hyperpolarizability_ratio", "abscissa_unit": "nm", "use": "check", "parameters": { "reference_material": "He", "process": "thg", "part": "electronic" } } Table I \(p\. 48\), column \(B\)\: column \(A\) relative to He \(1\.7/0\.9 = 1\.89, 28/0\.9 = 31\.1, 71/0\.9 = 78\.9, 222/0\.9 = 246\.7\: reproduce\)\. Not independent of alpha\_3nu\_Xe\. Formulas -------- dawes\_1968/eq9\_He ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq9_He", "material": "He", "equation": "Eq. (9), p. 48, with Table I's N_0 and nu_0", "printed": "alpha_zzzz(3nu) = N_0 e^4 / (16 pi^4 m^2 h) (nu_0^2 + 6 nu^2) / [nu_0 (nu_0^2 - 9 nu^2) (nu_0^2 - 4 nu^2) (nu_0^2 - nu^2)]", "variable": "nu: the fundamental frequency in Hz (cycles per second); Gaussian units (e, m, h in cgs); result in esu per atom, New and Ward's convention (gamma_T/6)", "range": "Below the first resonance nu_0/3; evaluated by the paper at 6943 A only", "coefficients": { "N_0": "1.224", "nu_0_1e15_cps": "6.19" }, "check": "With nu = c/6943 A and CODATA e, m, h (Gaussian), Eq. (9) gives 0.923 x 10^-39 esu against Table I's 0.9 (computed 2026-09-26; He-Kr within 2.5 %, Xe 1.7 % low: the printed values are rounded, and no single correction is implied)." } Process\: THG, gamma\(\-3w; w, w, w\), electronic\. N\_0 and nu\_0 are Cuthbertson's \(1932\) refractive\-index fit parameters \(reproduced in Table I\), not fitted to nonlinear data\. Dawes notes the same procedure could give the dc and Kerr coefficients \(p\. 48\)\. dawes\_1968/eq9\_Ne ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq9_Ne", "material": "Ne", "equation": "Eq. (9), p. 48, with Table I's N_0 and nu_0", "printed": "alpha_zzzz(3nu) = N_0 e^4 / (16 pi^4 m^2 h) (nu_0^2 + 6 nu^2) / [nu_0 (nu_0^2 - 9 nu^2) (nu_0^2 - 4 nu^2) (nu_0^2 - nu^2)]", "variable": "nu: the fundamental frequency in Hz (cycles per second); Gaussian units (e, m, h in cgs); result in esu per atom, New and Ward's convention (gamma_T/6)", "range": "Below the first resonance nu_0/3; evaluated by the paper at 6943 A only", "coefficients": { "N_0": "2.411", "nu_0_1e15_cps": "6.26" }, "check": "With nu = c/6943 A and CODATA e, m, h (Gaussian), Eq. (9) gives 1.715 x 10^-39 esu against Table I's 1.7 (computed 2026-09-26; He-Kr within 2.5 %, Xe 1.7 % low: the printed values are rounded, and no single correction is implied)." } Process\: THG, gamma\(\-3w; w, w, w\), electronic\. N\_0 and nu\_0 are Cuthbertson's \(1932\) refractive\-index fit parameters \(reproduced in Table I\), not fitted to nonlinear data\. Dawes notes the same procedure could give the dc and Kerr coefficients \(p\. 48\)\. dawes\_1968/eq9\_Ar ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq9_Ar", "material": "Ar", "equation": "Eq. (9), p. 48, with Table I's N_0 and nu_0", "printed": "alpha_zzzz(3nu) = N_0 e^4 / (16 pi^4 m^2 h) (nu_0^2 + 6 nu^2) / [nu_0 (nu_0^2 - 9 nu^2) (nu_0^2 - 4 nu^2) (nu_0^2 - nu^2)]", "variable": "nu: the fundamental frequency in Hz (cycles per second); Gaussian units (e, m, h in cgs); result in esu per atom, New and Ward's convention (gamma_T/6)", "range": "Below the first resonance nu_0/3; evaluated by the paper at 6943 A only", "coefficients": { "N_0": "4.352", "nu_0_1e15_cps": "4.12" }, "check": "With nu = c/6943 A and CODATA e, m, h (Gaussian), Eq. (9) gives 28.48 x 10^-39 esu against Table I's 28 (computed 2026-09-26; He-Kr within 2.5 %, Xe 1.7 % low: the printed values are rounded, and no single correction is implied)." } Process\: THG, gamma\(\-3w; w, w, w\), electronic\. N\_0 and nu\_0 are Cuthbertson's \(1932\) refractive\-index fit parameters \(reproduced in Table I\), not fitted to nonlinear data\. Dawes notes the same procedure could give the dc and Kerr coefficients \(p\. 48\)\. dawes\_1968/eq9\_Kr ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq9_Kr", "material": "Kr", "equation": "Eq. (9), p. 48, with Table I's N_0 and nu_0", "printed": "alpha_zzzz(3nu) = N_0 e^4 / (16 pi^4 m^2 h) (nu_0^2 + 6 nu^2) / [nu_0 (nu_0^2 - 9 nu^2) (nu_0^2 - 4 nu^2) (nu_0^2 - nu^2)]", "variable": "nu: the fundamental frequency in Hz (cycles per second); Gaussian units (e, m, h in cgs); result in esu per atom, New and Ward's convention (gamma_T/6)", "range": "Below the first resonance nu_0/3; evaluated by the paper at 6943 A only", "coefficients": { "N_0": "4.932", "nu_0_1e15_cps": "3.57" }, "check": "With nu = c/6943 A and CODATA e, m, h (Gaussian), Eq. (9) gives 71.33 x 10^-39 esu against Table I's 71 (computed 2026-09-26; He-Kr within 2.5 %, Xe 1.7 % low: the printed values are rounded, and no single correction is implied)." } Process\: THG, gamma\(\-3w; w, w, w\), electronic\. N\_0 and nu\_0 are Cuthbertson's \(1932\) refractive\-index fit parameters \(reproduced in Table I\), not fitted to nonlinear data\. Dawes notes the same procedure could give the dc and Kerr coefficients \(p\. 48\)\. dawes\_1968/eq9\_Xe ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "eq9_Xe", "material": "Xe", "equation": "Eq. (9), p. 48, with Table I's N_0 and nu_0", "printed": "alpha_zzzz(3nu) = N_0 e^4 / (16 pi^4 m^2 h) (nu_0^2 + 6 nu^2) / [nu_0 (nu_0^2 - 9 nu^2) (nu_0^2 - 4 nu^2) (nu_0^2 - nu^2)]", "variable": "nu: the fundamental frequency in Hz (cycles per second); Gaussian units (e, m, h in cgs); result in esu per atom, New and Ward's convention (gamma_T/6)", "range": "Below the first resonance nu_0/3; evaluated by the paper at 6943 A only", "coefficients": { "N_0": "5.548", "nu_0_1e15_cps": "3.00" }, "check": "With nu = c/6943 A and CODATA e, m, h (Gaussian), Eq. (9) gives 218.2 x 10^-39 esu against Table I's 222 (computed 2026-09-26; He-Kr within 2.5 %, Xe 1.7 % low: the printed values are rounded, and no single correction is implied)." } Process\: THG, gamma\(\-3w; w, w, w\), electronic\. N\_0 and nu\_0 are Cuthbertson's \(1932\) refractive\-index fit parameters \(reproduced in Table I\), not fitted to nonlinear data\. Dawes notes the same procedure could give the dc and Kerr coefficients \(p\. 48\)\. 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.