Theory of the Stark effect

Theoretical photoabsorption cross sections for alkali-metal atoms in a static electric field $F$ are derived for energies near the zero-field ionization threshold ($\epsilon =0\mathrm{}$), extending a previous development for hydrogen. Spectra result from zero-field dipole matrix elements and a density-of-states matrix $D_{l^{\prime }l}^F$. The factors $D_{l^{\prime }l}^F={\left({〈{\Psi }^{\prime }|\Psi 〉}^{-1\mathrm{}}\right)}_{l^{\prime }l}$ represent a renormalization of the photoelectron's wave function ${\Psi }_l$ necessitated by the long-range Coulomb plus Stark potential. The matrix $D^F$ contains all spectral information on quasidiscrete Stark levels and continuum resonances, expressed as an algebraic function of (1) quantum defects ${\mu }_l$ embodying core effects, (2) a frame transformation between spherical and parabolic coordinates in the pure Coulomb field outside the alkali-metal core, and (3) factors $H_{l^{\prime }l}^F$ and $h_{l^{\prime }l}^F$ calculated from asymptotic amplitudes and phases of hydrogen-Stark wave functions of the parabolic dissociation channels. Hydrogenic parameters are calculated semianalytically within the WKB approximation; ${\mu }_l$ and dipole matrix elements are known independently. Predicted cross sections for photoionization of the excited $3{}_{}{}^2{}_{}{}^{}P_{\frac{3}{2}}^{}{}_{}{}^{}$ state of Na agree with experiment. The theory reproduces (a) asymmetric resonances observed at $\epsilon <0\mathrm{}$, parametrized as a Beutler-Fano profile for a simple case, and (b) the oscillations observed at $\epsilon >0\mathrm{}$, attenuated by factors $cos2\pi {\mu }_l$ from their predicted depth in H. Dependences on light polarization are sorted out for two-photon excitation.