pachucki_2019#

K. Pachucki and M. Puchalski, “Refractive index and generalized polarizability”, Phys. Rev. A 99, 041803(R) (2019)

Source record · Catalogue · Review register

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#

{
  "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 review register for the checks and decisions.