The Landé gJg_J factors of the 2s22p4 3P12s^22p^4\,{}^3P_1 and 2s22p4 3P22s^22p^4\,{}^3P_2 levels of neutral atomic oxygen were measured by Rawson and Beringer (1952) and, more precisely, by Radford and Hughes (1959) — 1.5009861.500986 and 1.5009211.500921, each with an uncertainty of 2×10−62\times10^{-6} — and calculated by Kambe and Van Vleck (1954), with numerical corrections by Innes and Ufford (1958), as one of the early tests of the Schwinger anomalous magnetic moment in a many-electron atom. In the difference Δg1−Δg2=(g1−3/2)−(g2−3/2)\Delta g_1-\Delta g_2=(g_1-3/2)-(g_2-3/2) the common corrections (anomalous moment, orbit–orbit, nuclear motion) cancel exactly and the departure from Russell–Saunders coupling is isolated; here the corrected 1954 theory (64×10−664\times10^{-6}) and the 1959 measurement (65×10−665\times10^{-6}) already agree, but Radford and Hughes reported a discrepancy of 7 parts in 10610^6 between the individual calculated gJg_J values and their 1959 measurement, which they attributed to the inaccuracy of the uncorrelated Schrödinger wave functions entering the relativistic-diamagnetic corrections, and the departure-from-LSLS-coupling term itself rests on an off-diagonal spin–orbit matrix element ⟨3P2∣ζ2p∣1D2⟩\langle{}^3P_2|\zeta_{2p}|{}^1D_2\rangle that the 1954 Hartree–Fock-based calculation could only estimate by extrapolation from spectra. Later work showed the apparent 1954 agreement for the individual factors was partly fortuitous and built correlated many-body methods for exactly these corrections, but no complete correlated determination of the oxygen ground-term gJg_J corrections (including the common-mode discrepancy and the mixing term), no determination of the effective off-diagonal spin–orbit element, and no measurement beyond the 1959 precision have been published. The problem is to determine these quantities at the few-10−610^{-6} level.

Background

The Zeeman effect — the splitting of atomic levels in an external magnetic field — is quantified for a level with total angular momentum JJ by the Landé factor gJg_J, the level's magnetic moment in units of the Bohr magneton. In Russell–Saunders (LSLS) coupling, the angular-momentum coupling scheme that is a good approximation in light atoms, gJg_J is fixed purely angularly: for both fine-structure levels of the ground term 2s22p4 3P2s^22p^4\,{}^3P of neutral oxygen (O I) it equals 3/23/2 exactly, so any measured departure from 3/23/2 is a direct probe of small physical corrections. In the early 1950s this made O I a benchmark for the newly predicted quantum-electrodynamic correction to the electron's magnetic moment (the Schwinger anomalous moment, of relative size ∼10−3\sim10^{-3}): Rawson and Beringer (1952) measured gJ(3P1)=1.500971g_J({}^3P_1)=1.500971 and gJ(3P2)=1.500905g_J({}^3P_2)=1.500905 by gas-phase electron paramagnetic resonance (EPR, the resonant absorption of microwave radiation by Zeeman-split levels); Hughes and Geiger (1955) observed the two-quantum transitions of the same spectrum; and Abragam and Van Vleck (1953) and then Kambe and Van Vleck (1954) developed the full theory of the corrections of order 10−610^{-6} to 10−310^{-3}: the anomalous-moment term (+1145×10−6+1145\times10^{-6} on both levels, as evaluated then); the relativistic and diamagnetic corrections of Breit–Margenau type (the relativistic kinetic-energy correction and the field-modulated spin–orbit interaction) and the Lamb spin–other–orbit term (the interaction of each electron's spin magnetic moment with the orbital motion of the other electrons), evaluated with radial integrals from the self-consistent-field (Hartree–Fock-type) wave functions of Hartree, Hartree and Swirles (1939); the motion of the nucleus; and the departure from LSLS coupling caused by spin–orbit mixing of 3P2{}^3P_2 with the 1D2{}^1D_2 level of the same 2p42p^4 configuration. Kambe and Van Vleck obtained Δg1=974×10−6\Delta g_1=974\times10^{-6} and Δg2=913×10−6\Delta g_2=913\times10^{-6} (i.e. gg values 1.5009741.500974 and 1.5009131.500913), in good agreement with experiment for the individual factors. However, in the difference Δg1−Δg2\Delta g_1-\Delta g_2 the anomalous-moment, orbit–orbit (the magnetic interaction between the electrons' orbital motions), and nuclear-motion terms cancel exactly, leaving only the difference of the Breit–Margenau and Lamb corrections and the departure from LSLS coupling (a contribution of −21×10−6-21\times10^{-6} to Δg2\Delta g_2 only): the calculated difference was 61×10−661\times10^{-6} against the 66×10−666\times10^{-6} implied by the 1952 measurement. The LSLS-coupling term depends on the off-diagonal spin–orbit matrix element ⟨3P2∣ζ2p∣1D2⟩\langle{}^3P_2|\zeta_{2p}|{}^1D_2\rangle, where ζ2p\zeta_{2p} is the spin–orbit parameter of the 2p2p electron; since the self-consistent-field value of ζ2p\zeta_{2p} is about 10% too large and the off-diagonal element was only estimated by extrapolation from observed spectra, the 1954 authors blamed the residual difference discrepancy on this term and noted that King and Van Vleck (1939) had found diagonal and off-diagonal spin–orbit elements to differ appreciably in mercury. Two later developments sharpened the comparison. Innes and Ufford (1958) supplied numerical corrections to the theory, with which, using the then-updated value of the electron anomaly, the theoretical values become 1.5009961.500996 and 1.5009321.500932 (Δg1−Δg2=64×10−6\Delta g_1-\Delta g_2=64\times10^{-6}); and Radford and Hughes (1959) remeasured both factors by paramagnetic resonance calibrated against the proton gg factor, obtaining gJ(3P1)=1.500986g_J({}^3P_1)=1.500986 and gJ(3P2)=1.500921g_J({}^3P_2)=1.500921, each ±2×10−6\pm2\times10^{-6} — the most precise determination to date. In Δg1−Δg2\Delta g_1-\Delta g_2 the 1959 measurement (65×10−665\times10^{-6}) and the corrected theory (64×10−664\times10^{-6}) agree, but Radford and Hughes reported a discrepancy of 7 parts in 10610^6 between their measured values and the corresponding theoretical gJg_J factors used in that paper — theoretical values of unconfirmed provenance, the electron-anomaly value and radial integrals that entered the comparison not being specified — and attributed it to the inaccuracy of the Schrödinger wave functions entering the relativistic-diamagnetic corrections: a 5% error in the mean kinetic energy ⟨T⟩2p\langle T\rangle_{2p} of a 2p2p electron would suffice to close the gap. Subsequent work reshaped the theoretical picture: Veseth (1980) computed the gJg_J corrections for first-row atoms with correlated spin-extended Hartree–Fock (a generalization of Hartree–Fock that assigns different orbitals to different spin directions) wave functions and concluded that the 1954 agreement for oxygen was fortuitous, arising from a cancellation of errors; Veseth (1983, 1987, 1988) developed many-body perturbation-theory and configuration-interaction treatments of these same corrections with correlation for first- and second-row atoms; Persson (1993) treated the analogous np2np^2 elements; and modern relativistic coupled-cluster and multiconfiguration Dirac–Hartree–Fock (the relativistic, multi-configuration generalization of the self-consistent-field method) calculations of gJg_J factors are routine for ions, as in Yu, Sahoo and Suo (2020), but have not been applied to the neutral-oxygen ground term. On the experimental side, the 1959 measurement remains the most precise determination — no later measurement has been published, and the NIST Atomic Spectra Database carries no Landé-factor data for O I at all — so the common-mode discrepancy, the departure-from-LSLS-coupling contribution, and the effective off-diagonal spin–orbit element remain unsettled.

Problem Statement

Determine the corrections to the Landé gJg_J factors of the 2s22p4 3P12s^22p^4\,{}^3P_1 and 2s22p4 3P22s^22p^4\,{}^3P_2 levels of neutral atomic oxygen at the few-10−610^{-6} level of the existing measurements, and in particular settle the quantities that remain open: the difference Δg1−Δg2=(g1−3/2)−(g2−3/2)\Delta g_1-\Delta g_2=(g_1-3/2)-(g_2-3/2), the quantity in which the common corrections (anomalous magnetic moment, orbit–orbit, nuclear motion) cancel exactly and which isolates the departure from Russell–Saunders coupling; the common-mode offset between the individual calculated values and the values measured in 1959; and the effective off-diagonal spin–orbit matrix element ⟨3P2∣ζ2p∣1D2⟩\langle{}^3P_2|\zeta_{2p}|{}^1D_2\rangle. The relativistic and diamagnetic (Breit–Margenau, Lamb, orbit–orbit) corrections and the departure from LSLS coupling are to be determined with an uncertainty small enough to decide whether the common-mode discrepancy of 7 parts in 10610^6 reported in 1959 between the measured and theoretical values of both levels (attributed to the Schrödinger wave functions, through the mean kinetic energy ⟨T⟩2p\langle T\rangle_{2p} of a 2p2p electron) and any residual discrepancy in Δg1−Δg2\Delta g_1-\Delta g_2 (1954 calculation 61×10−661\times10^{-6}, or 64×10−664\times10^{-6} after the Innes–Ufford corrections, versus 66×10−666\times10^{-6} observed in 1952 and 65×10−665\times10^{-6} in 1959) are removed. The effective off-diagonal spin–orbit matrix element ⟨3P2∣ζ2p∣1D2⟩\langle{}^3P_2|\zeta_{2p}|{}^1D_2\rangle is to be determined, and it is to be settled whether it differs from the diagonal spin–orbit elements in the way found for mercury. The quantities above are to be settled at a precision better than the 1959 values (gJ(3P1)=1.500986g_J({}^3P_1)=1.500986, gJ(3P2)=1.500921g_J({}^3P_2)=1.500921, each ±2×10−6\pm2\times10^{-6}), which remain the most precise determinations to date.

References

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