Semiclassical treatment of Wannier's theory when the exponent diverges

• Published in: Journal of physics. B, Atomic, molecular, and optical physics Atomic, molecular, and optical physics, 2006-09, Vol.39 (17), p.3639-3648
• Main Authors: Condren, D S, McCann, J F, Crothers, D S F
• Format: Article
• Language: English
• Subjects: Atomic and molecular physics; Calculations and mathematical techniques in atomic and molecular physics (excluding electron correlation calculations); Electronic structure of atoms, molecules and their ions: theory; Exact sciences and technology; Physics
• ISSN: 0953-4075; 1361-6455
• DOI: 10.1088/0953-4075/39/17/019

We consider a non-standard application of the Wannier model. A physical example is the single ionization of a hydrogenic beryllium ion with a fully stripped beryllium ion, where the ratio of the charge of the third particle to the charges of the escaping particles is 1/4; we investigate the single ionization by an electron of an atom comprising an electron and a nucleus of charge 1/4. An infinite exponent is obtained suggesting that this process is not tractable within the Wannier model. A modified version of Crothers' uniform semiclassical wavefunction for the outgoing particles has been adopted, since the Wannier exponents and are infinite for an effective charge of Z = 1/4. We use Bessel functions to describe the Peterkop functions u1 and u2 and derive a new turning point rho+. Since u1 is well behaved at infinity, there exists only the singularity in u2 at infinity, thus we employ a one- (rather than two-) dimensional change of dependent variable, ensuring that a uniform solution is obtained that avoids semiclassical breakdown on the Wannier ridge. The regularized final-state asymptotic wavefunction is employed, along with a continuum-distorted-wave approximation for the initial-state wavefunction to obtain total cross sections on an absolute scale.