Multiphoton Ionization of Hydrogen and Rare-Gas Atoms

A perturbation theory of the ionization of atoms by simultaneous absorption of several photons, each of whose energy is less than the ionization potential, is developed from the evolution-operator formalism. A precise computation is made for the hydrogen atom, giving transition rates as a function of photon energy for two- through twelve-photon photoionization. The eighth-order ionization rate (in cgs units) at the 1.78-eV ruby-laser line is found to be ∼${10}^{-244\mathrm{}}$×${\mathrm{}\left(\mathrm{photon}\mathrm{flux}\right)\mathrm{}}^8$ and should be observable using available techniques. Good agreement is obtained with Zernik's exact calculation of the two-photon ionization rate of metastable $2S$ hydrogen. Approximate calculations are made for the rare gases. Assuming "typical" experimental conditions of a gas density of ∼${10}^{20}$ atoms ${\mathrm{cm}}^{-3\mathrm{}}$ and a ruby laser focused into a volume of ∼${10}^{-8\mathrm{}}$ ${\mathrm{cm}}^3$, we find that the flux required to liberate one electron during a 10-nsec pulse is ∼${10}^{29}$ ${\mathrm{cm}}^{-2\mathrm{}}$ ${\sec }^{-1\mathrm{}}$ for Xe, Kr, and Ar and ∼5×${10}^{30}$ photons ${\mathrm{cm}}^{-2\mathrm{}}$ ${\sec }^{-1\mathrm{}}$ for Ne and He. These gases ionize with the simultaneous absorption of 7, 8, 9, 13, and 14 photons, respectively. The predicted rate for Xe is found to be in excellent agreement with the recent direct measurements of Voronov and Delone. We conclude that multiphoton ionization provides the initial electrons required for the optical breakdown of gases, though it does not account for the over-all growth of the discharge except possibly at very low pressures. Impurity atoms (particularly heavy rare gases) may be the source of "initiating" electrons in Ne and He.