The distribution of population in the ground-state sublevels of an optically pumped alkali-metal vapor has been found to be strongly dependent on the mechanism assumed responsible for relaxation. Two modes of optical pumping zero mixing and complete mixing in the excited state, and three modes of alkali relaxation, uniform, Zeeman, and electron randomization, are considered. Calculations of the sublevel relative populations are made in terms of experimentally measurable parameters for all six combinations of pumping and relaxation modes. Optical-pumping transient signals are then used to demonstrate that alkali relaxation due to wall collisions is of the low-frequency "Zeeman" variety, while relaxation due to alkali-rare-gas collisions takes place through randomization of the spin of the alkali valence electron. Equations for the rate of change of the electronic spin polarization are derived for all cases, and are found to be nonexponential, in contrast with the case of an alkali of zero nuclear spin.
The importance of the nuclear spins in the analysis of alkali spin-exchange experiments employing optical pumping by circularly polarized light and performed in low magnetic fields is demonstrated. Since the exchange interaction is an electronic process, the spin-exchange cross section is expected to be essentially independent of the nuclear spins. However, the hyperfine coupling is sufficient to make the expressions for the signals depend upon the nuclear spins. Failure to include the nuclear-spin effects in the analysis can lead to errors as large as several hundred percent in the deduced cross sections. The signal for general nuclear spin is found for the Franzen-type transient experiment and for the Dehmelt-type steady-state experiment in the limit of low light intensity. The results are quite sensitive to the process assumed for the relaxation of the ground-state populations. The solutions are given for a general process in which randomly oriented disorientation fields interact with the spins of the alkali atom only through the electron spin. The steady-state-signal expression includes the effects of self spin exchange and partial disorientation in the excited state. Only the diagonal elements of the density matrix are included.
We present a comprehensive theory of nuclear spin polarization of
3He and
129Xe gases by spin-exchange collisions with optically pumped alkali-metal vapors. The most important physical processes considered are (1) spin-conserving spin-exchange collisions between like or unlike alkali-metal atoms; (2) spin-destroying collisions of the alkali-metal atoms with each other and with buffer-gas atoms; (3) electron-nuclear spin-exchange collisions between alkali-metal atoms and
3He or
129Xe atoms; (4) spin interactions in van der Waals molecules consisting of a Xe atom bound to an alkali-metal atom; (5) optical pumping by laser photons; (6) spatial diffusion. The static magnetic field is assumed to be small enough that the nuclear spin of the alkali-metal atom is well coupled to the electron spin and the total spin is very nearly a good quantum number. Conditions appropriate for the production of large quantities of spin-polarized
3He or
129Xe gas are assumed, namely, atmospheres of gas pressure and nearly complete quenching of the optically excited alkali-metal atoms by collisions with
N2 or
H2 gas. Some of the more important results of this work are as follows: (1) Most of the pumping and relaxation processes are sudden with respect to the nuclear polarization. Consequently, the steady-state population distribution of alkali-metal atoms is well described by a spin temperature, whether the rate of spin-exchange collisions between alkali-metal atoms is large or small compared to the optical pumping rate or the collisional spin-relaxation rates. (2) The population distributions that characterize the response to sudden changes in the intensity of the pumping light are not described by a spin temperature, except in the limit of very rapid spin exchange. (3) Expressions given for the radio-frequency (rf) resonance linewidths and areas can be used to make reliable estimates of the local spin polarization of the alkali-metal atoms. (4) Diffusion effects for these high-pressure conditions are mainly limited to thin layers at the cell surface and at internal resonant surfaces generated by radio-frequency magnetic fields when the static magnetic field has substantial spatial inhomogeneities. The highly localized effects of diffusion at these surfaces are described with closed-form analytic functions instead of the spatial eigenmode expansions that are appropriate for lower-pressure cells.
The effect of the hyperfine interaction on collisional relaxation among Zeeman sublevels is evaluated for a
J=23 state coupled to nuclear spins of
23,
25, and
27. Standard depolarization experiments, performed on atoms with hyperfine structure, are discussed in terms of relaxation-probability matrices parametrized in terms of the relative collisional-transition probabilities for a nuclear-spin-zero atom. Experimental results involving
D2 optical pumping of
85Rb and
87Rb are shown to support the approximation of nuclear decoupling in collisional relaxation. The cross sections for electronic collisional relaxation, assuming a van der Waals model for the collisional interaction, are, in units of ${10}^{\ensuremath{-}14}$
cm2, 1.19 (
85Rb-He), 1.09 (
85Rb-Ne), 2.44 (
85Rb-Ar), 1.13 (
87Rb-He), 1.03 (
87Rb-Ne), and 2.26 (
87Rb-Ar).
The effect of excited-state mixing on the optical pumping of alkali-metal vapors is considered for the case where the energy separation between the
2P21 and
2P23 first excited states is relatively large, allowing the isolation of mixing effects within a particular
J level. Two models for the mixing process are proposed, one a random reorientation of the total electronic angular momentum J, and the other a reorientation of J subject to the selection rule $\ensuremath{\Delta}{m}_{J}=0, \ifmmode\pm\else\textpm\fi{}2$. The probabilities for mixing transitions among
J-state sublevels have been calculated for an alkali atom of nuclear spin
23, and can be used to calculate the optical-pumping transition probabilities for a vapor subject to any degree of excited-state mixing. Several possible experiments suggested by these calculations are discussed.
Detailed studies have been made of the light transmitted through rubidium-87 vapor during the optical-pumping process, both with and without buffer gas present. The observed transients are single exponentials with no buffer gas present, and double exponentials with buffer gas present, over a wide range of pumping intensities and relaxation times. These results are in excellent agreement with the predictions based on phenomenological equations in which nuclear spin is included and a single relaxation time is assumed. From a study of the amplitudes of the two exponential components of the optical-pumping transient, an effective cross section ${\ensuremath{\sigma}}_{\mathrm{eff}}$ is deduced for the disorientation of rubidium-87 within the
52P21 state as a result of its collision with a buffer-gas atom. It is shown that ${\ensuremath{\sigma}}_{\mathrm{eff}}={\ensuremath{\sigma}}_{\frac{1}{2}}+{\ensuremath{\sigma}}_{\frac{3}{2}}$, where ${\ensuremath{\sigma}}_{\frac{1}{2}}$ is the cross section for disorientation within the
52P21 level, and ${\ensuremath{\sigma}}_{\frac{3}{2}}$ is the cross section for transfer from the
52P21 level to the
52P23 level by means of collisions with the buffer gas. On the basis of recently measured values for ${\ensuremath{\sigma}}_{\frac{3}{2}}$, values are deduced for ${\ensuremath{\sigma}}_{\frac{1}{2}}$. The cross sections are: ${\ensuremath{\sigma}}_{\frac{1}{2}}(\mathrm{R}\mathrm{b}\ensuremath{-}\mathrm{H}\mathrm{e})=1.5(0.8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}$
cm2, ${\ensuremath{\sigma}}_{\frac{1}{2}}(\mathrm{R}\mathrm{b}\ensuremath{-}\mathrm{N}\mathrm{e})=4.4(2.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}$
cm2, and ${\ensuremath{\sigma}}_{\frac{1}{2}}(\mathrm{R}\mathrm{b}\ensuremath{-}\mathrm{A}\mathrm{r})=3.5(1.8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$
cm2. These cross sections are deduced from a model in which the probabilities for
Rb87 to relax from any hyperfine level to any other are all equal. This model gives a better fit with the observed transients than the assumption that the electron spin only is randomized in the
P21 state with the nuclear spin unaffected. We show that accurate relaxation-time measurements can be made by measuring the time constants associated with the double exponentials as a function of light intensity, and extrapolating them to zero light intensity. Moreover, since no shutter is required, relaxation times ${10}^{\ensuremath{-}3}$ sec can be easily observed.