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Accueil > A propos du LPP > Communication > Actualités archivées > 2022 > A simple formula to choose the energy levels of xenon and krypton most appropriate for temperature diagnostics, thanks to minimal hyperfine broadening

A simple formula to choose the energy levels of xenon and krypton most appropriate for temperature diagnostics, thanks to minimal hyperfine broadening

Summary : Spectral broadening of fluorescence lines usually stems both from Doppler broadening and hyperfine splitting. For temperature measurements, either in atomic vapours or low temperature plasmas, only the first broadening source is of interest. In order to measure it in a simple way, it appears profitable to aim at atomic levels with minimal hyperfine structure broadening or even with no hyperfine structure at all. Meanwhile, in noble gases such as xenon and krypton, which are commonly used in “two-photon absorption laser induced fluorescence” (TALIF) measurements, hyperfine structure essentially stems from a single origin. Within a very good approximation, it gets determined only by the coupling of the unexcited electron core (aka “ionic core”) with the magnetic dipole and electric quadrupole of the nucleus. Variations of the hyperfine splitting, from one to another level, thus only come from the variety of ways the angular momentum of the ionic core can couple with the orbital angular momentum and spin of the excited electron. The way the total electron angular momentum is built up appears as the key factor to make the hyperfine energy levels more or less sensitive to the nuclear orientation. This sensitivity can be quantified in an explicit function of the angular quantum numbers.

Excited states of noble gas atoms, especially Xe et Kr, are characterized by quantum numbers usually listed in the order of decreasing influence on the final energy levels : jC, the total angular momentum of the inner, unexcited electrons (the "ionic core”), n and ℓ, the principal quantum number and orbital angular momentum of the excited electron, respectively, K the angular momentum that totalizes jC and ℓ, J the total electron angular momentum including the spin s=1⁄2 of the excited electron and F the total angular momentum, including the nuclear spin I. All sums of angular momenta must remain within the limits set by the addition rules, either the classical rules depicted by the figure (below) or the more rigorous quantum rules. Hyperfine structure is the variation of the atomic energy as a function of F, when I and J have given values. The dipolar component of the hyperfine structure, ΔD, appears proportional to the scalar product of vectors I and J : ΔD=A I·J, with a proportionality constant A, or "dipolar hyperfine coefficient", that measures the sensitivity of the electron-to-nucleus coupling with respect to the relative orientation of the electron and nuclear angular momenta, J and I, respectively.

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Schéma de couplage des moments cinétiques pour un niveau simplement excité d’un gaz rare
Comme l’énergie hyperfine n’y dépend, dans une bonne approximation, que de l’orientation relative du moment cinétique total des électrons de cœur jC et du moment cinétique nucléaire I, la sensibilité de l’énergie des niveaux à l’orientation relative de I et de J, qui détermine F, dépend de la façon dont la variation de l’orientation de I (sur le cercle tireté) fait varier I·jC. La précession de jC autour de J, due au couplage entre le cœur électronique et l’électron excité, diminue d’autant plus l’amplitude de cette projection que jC et J pointent dans des directions différentes. A contrario, on s’attend à une sensibilité maximale pour les états dans lesquels jC est parallèle à J, en particulier dans les états de J maximum, J=jC+ℓ+1/2, où tous les moments cinétiques sont alignés.

Quantum calculations show that, for all states, the hyperfine coefficient A can be obtained as the product of a single hyperfine parameter, which characterizes the ionic core, with a dimensionless function of the angular momentum quantum numbers (with jC just written j for the sake of brevity) :

As we have shown in a study to be published soon [1], the above formula reproduces the measured variations of the hyperfine coefficient A quite suitably, both for Xe and Kr. It fully backs the observation that levels with jC=3/2, ℓ=3, K=5/2, J=2 – in standard notation nf[5/2]2 levels – have a particularly small hyperfine splitting [2]. Meanwhile, according to the same formula, no angular momentum combination can be found that would make hyperfine coefficient A strictly equal to zero.

A similar formula can be established for the ratio Λ expected between the quadrupolar coefficient B and the analogous quadrupolar coefficient BC of the ionic core :

This second formula also reproduces the measured variations quite satisfactorily, for the atoms that give rise to quadrupolar hyperfine coupling (i.e., for xenon, 131Xe only). Remarkably, when all angular momenta get aligned to produce a maximum J value (i.e. J=K+1/2 and K=ℓ+3/2), ratio Λ appears to be exactly 1, as suggested by the classical picture (given by the figure above) ; explicitly, when jC gets aligned with J, the projections of I on jC and on J are equal, which makes the quadrupole splitting due to core-nucleus coupling pass unaltered to the energy spectrum of the whole neutral atom. In krypton for instance, one has checked experimentally that s[3/2]2, p[5/2]3 and d[7/2]4 levels have identical B values, which are also the maximum ones [3].

These “geometrical” variation laws of the hyperfine structure of noble gases had been alluded to, more or less explicitly, in atomic spectroscopy theses defended in the 70s, but they had, to our knowledge, never been published.

[2“Sub-Doppler two-photon-excitation Rydberg spectroscopy of atomic xenon : mass- selective studies of isotopic and hyperfine structure”, M. Kono et al., J. Phys. B : At. Mol. Opt. Phys. 49 (2016) 065002

[3“Hyperfine spectra of the radioactive isotopes 81Kr and 85Kr”, B.D. Cannon, Phys. Rev. A 47 (1993) 1148

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