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Physical Review

Rare-Gas-Induced gJ Shifts in the Ground States of Alkali Atoms

Roger M. Herman

Published Nov 5, 1968• 15 citations
Publisher

Abstract

Based on the formal similarity between perturbations governing gyromagnetic ratio shifting and spin relaxation of alkali atoms in collisions with rare-gas atoms, a relatively direct semiempirical estimate of the ${g}_{j}$ shift as a function of buffer-gas density can be achieved. The expression $\frac{\ensuremath{\Delta}{g}_{J}}{{g}_{J}}=\ensuremath{-}{(\frac{4}{3\ensuremath{\pi}})}^{\frac{1}{2}}n{{\ensuremath{\sigma}}_{\mathrm{dis}}}^{\frac{1}{2}}{\ensuremath{\sigma}}_{\mathrm{kin}}$ is obtained for the fractional shift in ${g}_{J}$, where $n$ is the rare-gas number density, ${\ensuremath{\sigma}}_{\mathrm{kin}}$ the alkali-rare-gas kinetic cross section, and ${\ensuremath{\sigma}}_{\mathrm{dis}}$ the cross section for alkali spin disorientation. With this relationship, we predict a fractional ${g}_{J}$ shift for Rb by He equal to -5. 3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}11}$ ${\mathrm{Torr}}^{\ensuremath{-}1}$, as compared with a recent value, -(5\ifmmode\pm\else\textpm\fi{}3) \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}11}$ ${\mathrm{Torr}}^{\ensuremath{-}1}$, obtained through measurement by Hayne, Robinson, and their co-workers.

DOI
10.1103/physrev.175.10

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Based on the formal similarity between perturbations governing gyromagnetic ratio shifting and spin relaxation of alkali atoms in collisions with rare-gas atoms, a relatively direct semiempirical estimate of the gj{g}_{j}gj​ shift as a function of buffer-gas density can be achieved. The expression \ensuremathΔgJgJ=\ensuremath−(43\ensuremathπ)12n\ensuremathσdis12\ensuremathσkin\frac{\ensuremath{\Delta}{g}_{J}}{{g}_{J}}=\ensuremath{-}{(\frac{4}{3\ensuremath{\pi}})}^{\frac{1}{2}}n{{\ensuremath{\sigma}}_{\mathrm{dis}}}^{\frac{1}{2}}{\ensuremath{\sigma}}_{\mathrm{kin}}gJ​\ensuremathΔgJ​​=\ensuremath−(3\ensuremathπ4​)21​n\ensuremathσdis​21​\ensuremathσkin​ is obtained for the fractional shift in gJ{g}_{J}gJ​, where nnn is the rare-gas number density, \ensuremathσkin{\ensuremath{\sigma}}_{\mathrm{kin}}\ensuremathσkin​ the alkali-rare-gas kinetic cross section, and \ensuremathσdis{\ensuremath{\sigma}}_{\mathrm{dis}}\ensuremathσdis​ the cross section for alkali spin disorientation. With this relationship, we predict a fractional gJ{g}_{J}gJ​ shift for Rb by He equal to -5. 3 \ifmmode\times\else\texttimes\fi{} 10\ensuremath−11{10}^{\ensuremath{-}11}10\ensuremath−11 Torr\ensuremath−1{\mathrm{Torr}}^{\ensuremath{-}1}Torr\ensuremath−1, as compared with a recent value, -(5\ifmmode\pm\else\textpm\fi{}3) \ifmmode\times\else\texttimes\fi{} 10\ensuremath−11{10}^{\ensuremath{-}11}10\ensuremath−11 Torr\ensuremath−1{\mathrm{Torr}}^{\ensuremath{-}1}Torr\ensuremath−1, obtained through measurement by Hayne, Robinson, and their co-workers.
The spin relaxation of optically pumped rubidium vapor induced by rubidium-rare-gas atomic collisions has been investigated in an ultraclean environment. The experiment was performed under conditions such that the partial pressure of nonrare-gas impurities was less than 1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}8}$ Torr, and that the time between Rb-Rb spin exchange collisions was short compared to the spin relaxation time. The cross sections (in cm2{\mathrm{cm}}^{2}cm2) for disorientation of rubidium were found to be: He, 3.3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}25}$; Ne, 3.3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}24}$; Ar, 1.1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}22}$; Kr, 7.3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}21}$; Xe, 1.3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}19}$. The fact that these cross sections are considerably smaller than previously reported values is attributed primarily to the low degree of impurity contamination in the present work. When compared with already existing values for cesium and sodium, the disorientation cross sections of these three alkali metals are seen to have a remarkably similar dependence on buffer-gas atomic number. The new experimental values are also compared with recent theoretical calculations.
The collision cross sections for alkali-metal--rare-gas spin orbit mixing between the ${n}^{2}{P}_{3/2}\ensuremath{\rightarrow}{n}^{2}{P}_{1/2}$ levels trend strongly with the Massey parameter, or adiabaticity of the collisions. The strength of the interaction, as characterized by the C6{C}_{6}C6​ dispersion coefficient, is a secondary influence on the rates. An analytic expression for the probability of energy transfer in alkali-metal--rare-gas collisions is derived using time-dependent perturbation theory. The model agrees well with a broad literature survey of the observed temperature-dependent cross sections. A simple interaction potential successfully organizes the alkali-metal--rare-gas database. The rates become very large for high-lying states, as the collisions are quite sudden and the radius of the valence electron is large. In contrast, the highly adiabatic cesium 62P{6}^{2}P62P mixing rates are six to eight orders of magnitude smaller. The mixing rate for the Rb-He diode pumped alkali laser system varies from $0.20--1.53\phantom{\rule{4pt}{0ex}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}\mathrm{cm}{}^{3}/\mathrm{at}.\phantom{\rule{0.16em}{0ex}}\mathrm{s}$ for T=279−−893KT=279--893\phantom{\rule{0.222222em}{0ex}}\mathrm{K}T=279−−893K.
We report the first studies of magnetic decoupling of the spin relaxation of alkali-metal atoms due to binary collisions with buffer gases. When binary collisions are the dominant relaxation mechanism, the relaxation and its magnetic decoupling are well described by the SSS-damping rate ${\ensuremath{\Gamma}}_{\mathrm{SD}}$ due to the spin-rotation interaction $\ensuremath{\gamma}\mathbf{N}\ifmmode \dot{}\else \.{}\fi{}\mathbf{S}$, the spin exchange rate ${\ensuremath{\Gamma}}_{\mathrm{EX}}$ for collisions between alkali atoms, and a new ``Carver rate'' ${\ensuremath{\Gamma}}_{\mathrm{C}}$, due to the pressure-shift interaction $\ensuremath{\delta}A\mathbf{I}\ifmmode \dot{}\else \.{}\fi{}\mathbf{S}$, which can substantially broaden the magnetic decoupling curve while having no influence on the zero-field rates.
Relaxation experiments performed on optically polarized Rb atoms in a rare gas have been previously reported; their detailed interpretation is given below. It is shown that the relaxation governed by the spin-orbit interaction is strongly affected by the formation of chemically unstable Rb-Kr molecules bound by van der Waals forces. Two processes of molecule formation are analyzed: binary resonant collisions leading to metastable states and three-body collisions producing actual bound states. A relaxation model valid for any disorientation probability per single Rb-Kr interaction is developed. Aside from clear evidence for the existence of alkali-rare-gas molecules, the success of the theoretical interpretation of the relaxation experiments yields the equilibrium constant $\mathcal{K}=1.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}22}$ cm3{\mathrm{cm}}^{3}cm3/molecule for the reaction Rb+Kr \ensuremath{\rightleftharpoons} Rb-Kr at 300\ifmmode^\circ\else\textdegree\fi{}K, the average lifetime of a Rb-Kr molecule in the gas phase, $\ensuremath{\tau}=0.65\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ sec, at a krypton pressure of one Torr, and the average spin-orbit coupling constant in a Rb-Kr molecule, $\overline{\ensuremath{\gamma}}{h}^{\ensuremath{-}1}=0.63$ MHz. It is also shown that the spin-orbit potential is predominantly short-range.
The difference in the dispersion force between an alkali atom in a particular hyperfine level belonging to the groundstate and a perturbing molecule is computed. These asymptotic forces, if active alone, generally produce red shifts and suffice to account for the results obtained for the heavier buffer gases. Experimental data exhibit blue shifts for the lighter gases and therefore indicate that the net frequency shifts are the result of exchange as well as dispersion forces. The former cannot be determined theoretically. Therefore, a simple model is devised in which the difference in the forces is given a positive trend at distances of separation smaller than ddd, while beyond ddd it is given the calculated form. The experimental data for all alkali-rare gas interactions can then be fitted by values of ddd which, for the different foreign gases, are of the order of their gas-kinetic diameters.