Closed-orbit theory and the photodetachment cross section ofH−in parallel electric and magnetic fields

Abstract

In this paper we obtain a simple analytic formula for the photodetachment cross section of H 2 in parallel electric and magnetic fields. The three-dimensional semiclassical approximation predicts oscillations in the spectrum and correlates these oscillations with closed classical orbits. The cylindrical symmetry of the Hamiltonian produces some interesting effects. In particular, at boundary energies the semiclassical approximation fails as a focused cusp approaches the origin. @S1050-2947~97!03306-4# PACS number~s!: 32.80.Gc I. INTRODUCTION Experimental measurements of the photodetachment cross section of H 2 in strong static electric fields were made by Bryant et al. @1#. For energies above the threshold energy the resulting cross section was found to be a smooth background upon which was superposed sinusoidal oscillations. Theoretical discussions of the measurements have been given by a number of authors @2,3#. The oscillations arise as an interference effect because the outgoing electron can move against the electric force and then return to the atom. It was shown in Ref. @3# that the cross section could be expressed in the following way: s~ E!5s 0~ E!1C~ E!sinF~ E!, where s 0 is the cross section in the absence of any external fields andC(E) and F(E) are called the recurrence strength and recurrence phase associated with the returning orbit. Theoretical calculations @4,5# show that analogous oscillations occur for photodetachment in crossed electric and magnetic fields, and that the oscillations are again associated with closed orbits. Quantum calculations have shown that very strong oscillations are present in parallel fields @6#, but the relationship to closed orbits was not made clear. Our original motivation in the present paper was to complete this subject by calculating the closed orbits and the resulting spectral oscillations for photodetachment of an electron from H 2 in parallel electric and magnetic fields. This is a system in which the recurrences are simple, strong, relatively easy to observe, and easy to understand. We found results which go far beyond this topic, and which are connected with many other problems of current interest. We know from study of nonlinear dynamics that even when long-time motion is chaotic, short-time motion remains relatively simple and predictable. Periodic orbits play a central role: in Poincare ´ ’s words, they offer ‘‘the only opening through which we might try to penetrate the fortress ~chaos! which has the reputation of being impregnable.’’ This classical statement holds also in quantum mechanics, wherein the periodic-orbit theory of Gutzwiller, Balian and Bloch, and Berry and Tabor @7# provides a general theoretical framework for studying quantum manifestations of classical chaos. Besides producing oscillations in absorption spectra @8#, periodic orbits produce scars in wave functions @9#, oscillations in the density of states @10#, and real-time recurrences that have been observed in many atoms and molecules @11#. @Similar phenomena are observed in microwaves in cavities @12~a!#, in microjunctions @12~b!#, and they are calculated to be consequences of certain models in nuclear physics @12~c!#.# Bifurcations of periodic orbits are of particular interest. A