J. Phys. 6: At. Mol. Opt. Phys. 27 (1994) 5717-5730. Printed in the UK Small-energy three-body systems: V. Threshold laws when Wannier theory fails M S Dimitrijevitt, P V GrujiEtt and N S SimonoviCt 7 Astronomical Observatory, Volgina 7, I1050 Belgrade, Yugoslavia 1 Institute of Physics, PO Box 57. I1000 Belgrade. Yugoslavia Recei\,ed 17 May 1994. in final form 21 July 1994 Abstract We investigate cases of Coulombic systems near the break-up threshold for which the Wannier model holds, but not Wannier theory. Making use of the classical trajeclory method, we derive threshold laws for a model system of fractional charge (Z = a au) nucleus and eleclrons. and a red (though perhaps impractical) system of two beryllium nuclei and an antiproton. For the first system we find the threshold law af the form exp(-A/&), where E is the t a d energy, and for the second one a number of characteristic features above the classical lhreshold have been obwined. Finally we investigate numerically a realislic case of an electron and two beryllium nuclei and discuss some general features of the ionization probability above the classical threshold. 1. Introduction We continue with investigations of three-body systems near the break-up thresholds. This paper is a sequel to the first one in the series (Simonovi6 and Cruji6 1987, hereafter referred to as I), where a general case of the near-threshold behaviour of the Coulombic systems with arbitrary masses and charges has been treated. We are interested in finding out the threshold law for the process when a charged particle impinges on a binary system itself consisting of two charged particles. As usual, one assumes the threshold law in the form a - EX E -+ +O (1) where U is the break-up cross section, E is the total energy of the system and K is the threshold law exponent to be found out. In the case of a symmetric system, when the two wing (outgoing) particles are identical, one has for the exponent (equation (38) of I) (atomic units are used throughout) where m3 (= mz) and 93 (= 42) are the masses and charges of the wing particles respectively, with particle 1 (with a charge ql opposite in sign to the outer ones) staying in the middle. If m3 < ml and q3 < -4 qt one recovers the standard Wannier result (Wannier 1953). In particular, for 91 = 1 and q3 = -1, one has K = 1.12689, a well known value for the case of ionization of hydrogen by electrons. Looking at (2) as it stands, one observes two important points: (i) formally, the exponent assumes real values only outside the region: -% 5: q1/q2 < -&; (ii) approaching the right 0953-4075/94/235717114$19.50 @ 1994 1OP Publishing Ltd 5717