bhatia_1998
===========

A\. K\. Bhatia and R\. J\. Drachman, "Optical properties of helium including relativistic corrections", Phys\. Rev\. A 58, 4470 \(1998\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/bhatia_1998.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1103/PhysRevA\.58\.4470 <https://doi.org/10.1103/PhysRevA.58.4470>`__

**Source kind:** theory

**Recorded technique:** Variational \(Hylleraas\-basis pseudostate\) calculation of the helium dynamic polarizability as a series in omega^2 \(Eq\. 6\) from generalized dipole polarizabilities \(Eq\. 5\), with the Breit\-Pauli alpha^2 corrections to the first three coefficients from third\-order perturbation theory \(Eqs\. 8\-12\); converted to a series in lambda^\-2, to n \- 1 through Lorentz\-Lorenz at Loschmidt's density and to the Verdet constant \(Eqs\. 13\-18\), pp\. 4470\-4472\.

**Materials in structured entries:** He

**Dataset quantities:** polarizability, refractivity, verdet\_constant

**Contents:** 6 datasets; 5 formulas.

Curation notes
--------------

Units \(p\. 4470\)\: the Hamiltonian is in reduced Rydberg units with lengths in reduced Bohrs, K = 2 m\_e/\(M \+ m\_e\) = 2\.741 493e\-4 for 4He; the polarizabilities of Eqs\. \(5\), \(12\), \(13\) are in reduced atomic units \(a\_mu^3\) and omega in reduced Rydberg units\. Eq\. \(15\) onwards is in a0^3 with lambda in A \(vacuum wavelengths\: omega = 4 pi a0/\(alpha \(1 \- K/2\) lambda\), Eq\. 14\)\. No QED; the relativistic terms are for the 4He reduced mass \(Puchalski et al\. 2016 show that their mass dependence is not the true recoil correction\)\. No uncertainties are given for the coefficients \('it is not clear that they are all accurate', p\. 4471\)\.

Generalized polarizabilities for 4He in reduced atomic units \(p\. 4471; the first five 'calculated recently with high accuracy' in Bhatia and Drachman, Can\. J\. Phys\. 75, 11 \(1997\), the last two new here\)\: alpha\_1 = 1\.383 241 01, beta\_1 = 0\.707 521 493, gamma\_1 = 0\.385 538 368, delta\_1 = 0\.218 735 026, epsilon\_1 = 0\.127 538 649, zeta\_1 = 0\.075 827 657, eta\_1 = 0\.045 731 135 \(the sums of \<0\|Z\|p\>\<p\|Z\|0\>/\(E\_p \- E\_0\)^j, j = 1\.\.7, Z = 2\(z\_1 \+ z\_2\), energies in reduced Rydbergs\)\. They are not extracted as Cauchy moments\: the conversion to atomic units, S\(\-2k\-2\) = c\_k 4^k \(1 \+ m\_e/m\_alpha\)^\(3\+2k\) for alpha\_1, gamma\_1, epsilon\_1, eta\_1 \(checked\: it gives Puchalski's converted BD values of Table II, e\.g\. 4 gamma\_1 \(1 \+ 1/7294\.2995\)^5 = 1\.543 210 9\), is not expressible as one dataset scale; Puchalski et al\. 2016 print alpha\_0\.\.alpha\_6 so converted \(puchalski\_2016 dataset cauchy\_bhatia\_drachman\) and 1e3 to 1e5 times more accurate values\.

Unit\-conversion errors \(Puchalski et al\. 2016, Table VI notes c\-e; confirmed here\)\: Eq\. \(15\)'s lambda^\-2 and lambda^\-6 coefficients carry \(1 \+ m\_e/m\_alpha\)^\(3\+2k\) instead of \(1 \+ m\_e/m\_alpha\)^3 \(with omega already scaled by \(1 \- K/2\) in Eq\. 14\)\: 3\.204 546e5 and 2\.624 092e16 are reproduced that way from Eq\. \(13\), while the correct conversion gives 3\.2036675e5 and 2\.621934e16 \(Puchalski's BD column\); the lambda^\-4 coefficient 8\.822 907e10 is reproduced by neither \(correct\: 8\.8032341e10\), a typographical error\. Eqs\. \(17\) and \(18\) inherit the errors\: Eq\. \(17\) as printed gives 34\.88646 and 34\.81673 \(the printed values\), the corrected coefficients 34\.88638 and 34\.81667 \(x 1e\-6\), i\.e\. 2\.3 and 1\.7 ppm lower \(computed here\)\. Eq\. \(13\) and Stone and Stejskal's use of it \(stone\_2004 Eq\. 5\: the same dynamic coefficients 0\.385530216, 0\.12753895, 0\.04573114, with its static term replaced by Lach et al\.'s QED value\) are unaffected; stone\_2004's Eq\. \(8\) lambda^\-2 coefficient 1197\.5410 agrees with Puchalski's nonrelativistic \+ relativistic A\_2 converted to cm^3/mol to 4e\-7 \(checked 2026\-09\-24\)\.

Comparisons in the text \(p\. 4472\)\: Leonard's recommended \(n \- 1\) x 1e6 = 34\.895 \+\- 0\.01 at 5462\.258 A \(which is Leonard's value derived from theory, see leonard\_1974 recommended\_He\) and Achtermann et al\.'s \(1993\) 34\.887 \+\- 0\.007 at 6329\.9 A, 'differ by 10 times the standard deviation' from Eq\. \(17\); Verdet constants from Leonard's table \(Ingersoll and Liebenberg\), 0\.637 at 5000 A and 0\.246 at 8000 A\. Pendrill \(1996\), cited for the comparison, is not in the library\.

**Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets
--------

bhatia\_1998/alpha\_static
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "alpha_static",
     "material": "He",
     "table": "text",
     "quantity": "polarizability",
     "unit": "au",
     "abscissa_unit": "angstrom",
     "wavelength_medium": "vacuum",
     "use": "check"
   }

1 inline points in the source record.

Eq\. \(15\), p\. 4472\: the constant term 1\.383 729 930 a0^3, the 4He static polarizability with the alpha^2 relativistic correction \(\-8\.0029e\-5 in reduced units, Eq\. 12\), no QED; Eq\. \(13\) gives it in reduced units as 1\.383 160 981\. Puchalski et al\. \(2016\) quote 1\.383729929; their recommended total is 1\.38376077\(14\), 22 ppm higher, mostly QED\. Superseded\: a check\.

bhatia\_1998/refractivity\_theory
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "refractivity_theory",
     "material": "He",
     "table": "text",
     "quantity": "refractivity",
     "scale": 1e-06,
     "abscissa_unit": "angstrom",
     "wavelength_medium": "vacuum",
     "state": "1atm_0C",
     "density": "ideal",
     "use": "check"
   }

2 inline points in the source record.

p\. 4472\: Eq\. \(17\) evaluated, '34\.886 46 \+\- 0\.000 02' at 5462\.258 A and '34\.816 73 \+\- 0\.000 01' at 6329\.9 A, \(n \- 1\) x 1e6 at Loschmidt's density \(Eq\. 16\: 0 degrees C, 1 atm, ideal gas\); the \+\- is 'the value of the last term included', not an uncertainty, and is not stored\. Eq\. \(17\) as printed reproduces both; with the corrected coefficients they are 34\.88638 and 34\.81667 \(see the source notes\)\. A check\.

bhatia\_1998/leonard\_1974
~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "leonard_1974",
     "material": "He",
     "table": "text",
     "quantity": "refractivity",
     "scale": 1e-06,
     "abscissa_unit": "angstrom",
     "wavelength_medium": "vacuum",
     "state": {
       "pressure_atm": 1.0,
       "temperature_k": 273.16
     },
     "density": "real",
     "use": "reproduced",
     "reproduced_from": [
       "leonard_1974"
     ],
     "uncertainty": {
       "column": true,
       "level": "probable error"
     }
   }

1 inline points in the source record.

p\. 4472\: 'one absolute value of n \- 1 has been recommended by Leonard \[9\] at the common calibration wavelength lambda = 5462\.258 A\: \(n \- 1\) x 1e6 = 34\.895 \+\- 0\.01'\. Leonard's recommended value \(leonard\_1974 recommended\_He, a probable error, derived from a calculated static polarizability\); state and level as recorded there\.

bhatia\_1998/achtermann\_1993
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "achtermann_1993",
     "material": "He",
     "table": "text",
     "quantity": "refractivity",
     "scale": 1e-06,
     "abscissa_unit": "angstrom",
     "wavelength_medium": "vacuum",
     "state": "1atm_0C",
     "density": "unknown",
     "use": "reproduced",
     "reproduced_from": [
       "H. J. Achtermann, J. G. Hong, G. Magnus, R. A. Aziz and M. J. Slaman, J. Chem. Phys. 98, 2308 (1993)"
     ],
     "uncertainty": {
       "column": true,
       "level": "not stated"
     }
   }

1 inline points in the source record.

p\. 4472\: 'the single carefully analyzed point given by Achtermann et al\. \[10\] at lambda = 6329\.9 A gives the value \(n \- 1\) x 1e6 = 34\.887 \+\- 0\.007', compared with Eq\. \(17\) at Loschmidt's density; the state \(taken as 0 degrees C, 1 atm\) and the level are not stated\. The original \(J\. Chem\. Phys\. 98, 2308\) is not in the library\.

bhatia\_1998/verdet\_theory
~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "verdet_theory",
     "material": "He",
     "table": "text",
     "quantity": "verdet_constant",
     "unit": "micro-arcmin/(Oe cm)",
     "abscissa_unit": "angstrom",
     "wavelength_medium": "vacuum",
     "state": "1atm_0C",
     "density": "ideal",
     "use": "check"
   }

2 inline points in the source record.

p\. 4472\: 'At lambda = 5000 A, Eq\. \(18\) gives V = 0\.661 22 \+\- 0\.000 25 \.\.\. while at lambda = 8000 A we get V = 0\.254 816 \+\- 0\.000 015' \('all in units of u min/oer cm'\); the \+\- is presumably the size of the last term, as for n \- 1, and is not stored\. Eq\. \(18\) as printed reproduces both \(0\.6612242, 0\.2548158\)\. It inherits Eq\. \(17\)'s conversion errors\.

bhatia\_1998/verdet\_leonard\_1974
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "verdet_leonard_1974",
     "material": "He",
     "table": "text",
     "quantity": "verdet_constant",
     "unit": "micro-arcmin/(Oe cm)",
     "abscissa_unit": "angstrom",
     "wavelength_medium": "standard_air",
     "state": {
       "pressure_atm": 1.0,
       "temperature_k": 273.16
     },
     "density": "unknown",
     "use": "reproduced",
     "reproduced_from": [
       "leonard_1974"
     ]
   }

2 inline points in the source record.

p\. 4472\: 'the experimental value of 0\.637' \(5000 A\) and '0\.246' \(8000 A\) from 'a table of values of the Verdet constant' in Leonard \(1974\)\: Ingersoll and Liebenberg's smoothed values \(leonard\_1974 verdet\_ingersoll\_liebenberg\_He, 0\.637 at 5000 A\); medium and state as recorded there\.

Formulas
--------

bhatia\_1998/eq12
~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "eq12",
     "material": "He",
     "equation": "Eq. (12), p. 4471",
     "printed": "Delta alpha(omega) = -8.0029 x 10^-5 - 8.1516 x 10^-6 omega^2 + 3.006 x 10^-7 omega^4",
     "variable": "omega in reduced Rydberg units; Delta alpha in reduced atomic units (4He)",
     "state": "none (per atom)",
     "density": "none",
     "range": "visible light (omega/(E_p - E_0) <= 0.2)",
     "coefficients": {
       "C0": "-8.0029e-5",
       "C1": "-8.1516e-6",
       "C2": "3.006e-7"
     },
     "check": "alpha_1 + C0 = 1.383160981, gamma_1 + C1 = 0.3855302164 and epsilon_1 + C2 = 0.1275389496 give Eq. (13)'s coefficients (checked 2026-09-24)."
   }

The Breit\-Pauli alpha^2 correction to the dynamic polarizability \(Eqs\. 9\-11\)\.

bhatia\_1998/eq13
~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "eq13",
     "material": "He",
     "equation": "Eq. (13), p. 4471",
     "printed": "alpha_rel(omega) = 1.383 160 981 + 0.385 530 216 omega^2 + 0.127 538 95 omega^4 + 0.045 731 14 omega^6",
     "variable": "omega: photon energy in reduced Rydberg units, omega = 911.391 98/lambda with lambda in A (Eq. 14); alpha in reduced atomic units a_mu^3 (4He)",
     "state": "none (per atom)",
     "density": "none",
     "range": "visible light (omega/(E_p - E_0) <= 0.2, p. 4471)",
     "coefficients": {
       "c0": "1.383160981",
       "c2": "0.385530216",
       "c4": "0.12753895",
       "c6": "0.04573114"
     },
     "check": "Equals the generalized polarizabilities plus Eq. (12) (the omega^6 term unchanged: eta_1 = 0.045 731 135 rounded). Converted with (1 + m_e/m_alpha)^3 for a0^3 and Eq. (14) for omega it gives Puchalski's BD column 1.3837299, 3.2036675e5, 8.8032341e10, 2.621934e16 (checked 2026-09-24)."
   }

The series Stone and Stejskal \(2004\) use for their dynamic terms\. No form\: alpha\_series is in atomic, not reduced, units \(a reduced\-unit variant with the mass ratio as a parameter would take it; the equivalent atomic\-unit series is c\_k 4^k \(1 \+ m\_e/m\_alpha\)^\(3\+2k\) omega\_au^\(2k\)\)\.

bhatia\_1998/eq15
~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "eq15",
     "material": "He",
     "equation": "Eq. (15), p. 4472",
     "printed": "alpha(lambda) = 1.383 729 930 + 3.204 546 x 10^5/lambda^2 + 8.822 907 x 10^10/lambda^4 + 2.624 092 x 10^16/lambda^6 a0^3",
     "variable": "lambda: vacuum wavelength in A; alpha in a0^3",
     "state": "none (per atom)",
     "density": "none",
     "range": "visible light (as Eq. 13)",
     "coefficients": {
       "A0": "1.383729930",
       "A2": "3.204546e5",
       "A4": "8.822907e10",
       "A6": "2.624092e16"
     },
     "check": "Does not follow from Eq. (13): the correct conversion gives A2 = 3.2036675e5, A4 = 8.8032341e10, A6 = 2.621934e16 (Puchalski et al. 2016, Table VI, BD column; reproduced here). The printed A2 and A6 are Eq. (13) times (1 + m_e/m_alpha)^(3+2k) with k = 1, 3; A4 is a misprint (the same wrong conversion would give 8.80806e10). Checked 2026-09-24."
   }

Corrected coefficients \(no form takes this lambda^\-2 series in A, see puchalski\_2016 table6\_total\)\: A0 = 1\.383729930, A2 = 3\.2036675e5, A4 = 8\.8032341e10, A6 = 2\.621934e16 \(Puchalski's correctly converted values; printed evidence\: puchalski\_2016 Table VI notes c\-e\)\. Nonrelativistic \+ alpha^2 only, no QED\: superseded by puchalski\_2016\.

bhatia\_1998/eq17
~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "eq17",
     "material": "He",
     "equation": "Eqs. (16)-(17), p. 4472",
     "printed": "(n^2 - 1)/(n^2 + 2) = (4 pi/3) N_L a0^3 alpha(lambda) = 16.677 18 x 10^-6 alpha(lambda) = z; n - 1 = (3/2) z + 3z^2/8 = 34.615 27 x 10^-6 + 8.016 511/lambda^2 + 2.207 154 x 10^6/lambda^4 + 6.564 503 x 10^11/lambda^6",
     "variable": "lambda: vacuum wavelength in A; N_L Loschmidt's number",
     "state": "1atm_0C",
     "density": "ideal",
     "range": "visible light",
     "coefficients": {
       "A": "34.61527e-6",
       "B": "8.016511",
       "C": "2.207154e6",
       "D": "6.564503e11"
     },
     "check": "Reproduces the printed 34.886 46e-6 at 5462.258 A and 34.816 73e-6 at 6329.9 A (checked 2026-09-24); it is Eq. (15) with its conversion errors (the corrected coefficients give values 2.3 and 1.7 ppm lower)."
   }

No form\: n \- 1 as a polynomial in lambda^\-2 \(lambda in A\) at a stated state; a form 'dn\_lambda\_series' \(n \- 1 = sum\_k B\_k lambda^\(\-2k\)\) would take it\. Superseded by puchalski\_2016\.

bhatia\_1998/eq18
~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "eq18",
     "material": "He",
     "equation": "Eq. (18), p. 4472",
     "printed": "V = -(e/2mc^2) lambda dn/dlambda = (1.616 813 x 10^7/lambda^2) [1 + 5.506 521 x 10^5/lambda^2 + 2.456 618 x 10^11/lambda^4] u min/oer cm",
     "variable": "lambda: vacuum wavelength in A; V in microminutes of arc per oersted per cm",
     "state": "1atm_0C",
     "density": "ideal",
     "range": "visible light",
     "coefficients": {
       "V0": "1.616813e7",
       "V2": "5.506521e5",
       "V4": "2.456618e11"
     },
     "check": "Gives 0.6612242 at 5000 A and 0.2548158 at 8000 A, the printed 0.661 22 and 0.254 816; its bracket terms are 4C/2B and 6D/2B of Eq. (17) (checked 2026-09-24)."
   }

Becquerel's relation applied to Eq\. \(17\); inherits its conversion errors\.

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 </register>` for the checks and decisions.