pachucki_2019 ============= K\. Pachucki and M\. Puchalski, "Refractive index and generalized polarizability", Phys\. Rev\. A 99, 041803\(R\) \(2019\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1103/PhysRevA\.99\.041803 `__ **Source kind:** theory **Recorded technique:** Theory \(QED of a dilute medium\)\: the refractive index from the pole of the photon propagator in a gas of atoms, to first order in the density, with the forward off\-shell scattering amplitude t^ij = 4 pi \[omega^2 alpha\(omega, k\) delta^ij \+ k^2 chi\(omega, k\) \(delta^ij \- k^i k^j/k^2\)\] defining the generalized electric and magnetic polarizabilities \(Eq\. 4\); n^2 = 1 \+ 4 pi rho \[alpha\(omega, omega\) \+ chi\(omega, omega\)\] on shell \(Eqs\. 10\-12\)\. The multipole expansion of the atom\-field coupling to cubic terms in a0/lambda \(Eq\. 2\) gives the retardation correction \(electric quadrupole, electric dipole\-octupole, and the magnetic\-quadrupole and orbital terms\), Eqs\. 20\-29, for the helium atom with infinite nuclear mass and nonrelativistic wave functions; the static second\-order expectation values alpha\_0 \.\.\. alpha\_3 are computed for He \(Table I\)\. 4 pages\. **Materials in structured entries:** He **Dataset quantities:** None recorded. **Contents:** 0 datasets; 1 formula. Curation notes -------------- Page numbers are the article pages 041803\-1 to \-4 \(= pdf pages\)\. Theoretical units hbar = c = eps0 = 1 in Secs\. II\-V; atomic units in Eqs\. 28\-29 and Table I\. THE RESULT \(Eqs\. 28\-30, p\. 3\)\: for small real frequencies the retardation adds to the refractive polarizability alpha \+ chi the term \[alpha \+ chi\]\_ret = alpha\_nrel \(a0 k\)^2 kappa, with alpha\_nrel = \(2/3\) alpha\_0\(0\) the nonrelativistic static polarizability and kappa = \[alpha\_1\(0\)/20 \- 2 alpha\_2\(0\)/45 \+ alpha\_3\(0\)/9\]/alpha\_nrel, a dimensionless coefficient; kappa = \-0\.507 for He, so that delta\_ret n/\(n \- 1\) = \(2 pi a0/lambda\)^2 kappa = \-1\.40e\-7 at 633 nm \(Eq\. 30\) and \-1\.98e\-7 at 532 nm \(p\. 4\)\. In atomic units, with k = alpha\_em omega, the correction is kappa alpha\_nrel alpha\_em^2 omega^2 = \-3\.7341e\-5 omega^2 \(a0^3, omega in hartree; computed here\)\: a correction of \-24\.2 ppm to the omega^2 coefficient S\(\-4\) of the refractive polarizability\. The authors call it 'larger than the previously calculated omega^2 term in the expansion of the relativistic polarizability', delta n = \-1\.24e\-7 \(n \- 1\) at 633 nm \(puchalski\_2016's alpha\_2^\(20\) \+ alpha\_2^\(21\)\)\. Only t\_ret is new\: the relativistic Breit\-Pauli correction and the diamagnetic term \(the magnetizability chi\_0, Eq\. 19\) were in puchalski\_2016 \(p\. 3\)\. No uncertainty is stated for kappa; Table I prints eight significant figures\. TABLE I \(p\. 3, 'Expectation values in a\.u\. in the helium atom'; read on the image\: the extracted cells merge the last two rows into one garbled cell, '\-\-04\.\.54016796562945', the interleaved digits of \-4\.4176694 and \-0\.5069525\)\: alpha\_0\(0\) = 2\.0747883, alpha\_1\(0\) = 4\.0751385, alpha\_2\(0\) = 9\.3176426, alpha\_3\(0\) = \-4\.4176694, kappa = \-0\.5069525\. alpha\_0 is defined without the 1/3 of the isotropic average \(Eq\. 21\)\: \(2/3\) alpha\_0\(0\) = 1\.3831922 a0^3 is the infinite\-mass nonrelativistic static polarizability \(liu\_1997\: 1\.38319194\)\. alpha\_1 \(quadrupole\-quadrupole, Eq\. 22\), alpha\_2 \(dipole\-r^2 dipole, Eq\. 23\), alpha\_3 \(dipole\-orbital, Eq\. 24\) and alpha\_4 \(monopole r^2, Eq\. 25; not printed\: it cancels in alpha \+ chi on shell\) have no quantity in the package; they are recorded as the coefficients of the formula 'retardation', not as data\. LIMITS\: the derivation is to first order in the atomic density \('we limit ourselves only to terms which are linear in the atomic density', p\. 2\); the authors could not derive the Clausius\-Mossotti analogue\. The expansion uses the static alpha\_i\(0\), valid for photon frequencies much below the excitation energy \(p\. 1\); the omega^4 terms are not given\. For a general field \(not a plane wave\) the k dependence makes the D\-E relation nonlocal \(Eq\. 16\), which the authors suggest may matter for boundary conditions near metal walls \(p\. 2\); not relevant to propagation in a bulk gas\. Referred to by puchalski\_2020 \(Ref\. 20\: the index depends 'at the 10^\-7 level' on these terms\), rourke\_2021 \(A\_R\_He\: \-0\.140 ppm at 633 nm\) and yang\_2025 \(puchalski\_He, he\_A2\: the theoretical A\_2 = 1\.332465\(5\)e\-4 cm^3/mol includes it\)\. **Recorded bibliographic identity status:** VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- No datasets are recorded. Formulas -------- pachucki\_2019/retardation ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "retardation", "material": "He", "equation": "Eqs. (29)-(30), p. 3, with Table I (p. 3)", "printed": "[alpha + chi]_ret = a0^5 k^2 [alpha_1(0)/20 - 2 alpha_2(0)/45 + alpha_3(0)/9] = alpha_nrel (a0 k)^2 kappa; delta_ret n/(n - 1) = (2 pi a0/lambda)^2 kappa = -1.40e-7 (lambda = 633 nm); kappa = -0.5069525", "variable": "k = 2 pi/lambda (lambda the vacuum wavelength; a0 k = alpha_em omega with omega in hartree); a0 = 0.529e-10 m in Eq. (30); alpha_nrel = (2/3) alpha_0(0) in a0^3", "state": "none (per atom; the relative correction to n - 1 holds at any density to first order)", "density": "none", "range": "Photon frequencies well below the first excitation of He (58.4 nm; the paper evaluates 633 and 532 nm); first order in the density; the omega^4 terms (the frequency dependence of alpha_1 ... alpha_3) are not given", "coefficients": { "alpha_0": "2.0747883", "alpha_1": "4.0751385", "alpha_2": "9.3176426", "alpha_3": "-4.4176694", "kappa": "-0.5069525" }, "check": "From the printed alpha_i(0): alpha_nrel = 1.3831922, kappa = (0.20375693 - 0.41411745 - 0.49085216)/1.3831922 = -0.50695245 (printed -0.5069525). Eq. (30) with a0 = 0.529e-10 m: -1.3978e-7 at 633 nm (printed -1.40e-7) and -1.9789e-7 at 532 nm (printed -1.98e-7, p. 4); with CODATA a0 -1.3987e-7 and -1.9802e-7. The comparison term: puchalski_2016's (alpha_2^(20) + alpha_2^(21)) omega^2/alpha(633 nm) = -33.20976e-6 x 0.0051811/1.391812 = -1.236e-7 (printed -1.24e-7). Computed 2026-09-29." } The correction to the refractive polarizability as a coefficient of omega^2 \(atomic units, computed here\)\: kappa alpha\_nrel alpha\_em^2 = \-3\.73405e\-5 a0^3 per hartree^2, \-24\.2 ppm of S\(\-4\) = 1\.5432395\. It lowers n \- 1 by 0\.087 ppm at 800 nm, 0\.140 ppm at 633 nm and 0\.196 ppm at 532 nm; since it is all in the omega^2 term, it lowers the group\-velocity dispersion by 23\.9 ppm at 800 nm and 23\.6 ppm at 633 nm \(against fit/He, 2026\-09\-29\)\. Not a form\: the package has no quantity or correction part for it \(a polarizability\_correction part 'retardation', or a fixed omega^2 term in the fit configuration, would take it\)\. 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.