Z. Phys. D 27, 173-183 (1993) ZEITSCHRIFT FORPHYSIK D © Springer-Verlag 1993 Rotational-vibrational rainbows in impulsive electron- diatomic molecule collisions: I. State-to-state transitions A. Ernesti*, I-LJ. Korsch Fachbereich Physik, Universit~it Kaiserslautern, W-6750 Kaiserslautern, Germany Received: 12 November 1992/Final version: 23 December 1992 Abstract. Collisional induced combined rotational-vibra- tional excitation of diatomic molecules is discussed in a simple quantum mechanical spectator model with appli- cations to electron-molecule collisions at intermediate collision energies (,~ 102 eV) within the rigid rotor/har- monic oscillator approximation. Quantum mechanical transition probabilities of the rotational-vibrational exci- tation, which show typical vibrational and rotational rainbow patterns, are calculated and compared with the structure of classical rainbow singularities. PACS: 34.80.Gs; 34.50.Ez 1. Introduction The theory of rotational rainbows has been extensively developed in the last decade (see, e.g., the review by Schinke and Bowman [1] based on the infinite-order sudden (IOS) approximation and a semiclassical analy- sis; the detailed analysis of classical hard-shell scattering by Beck and coworkers [2] and Bosanac [3], as well as the more recent review by the authors [4]). Quite generally such rainbow structures are observed for strong coupling between translational and rotational de- grees of freedom and yield important information on the anisotropic part of the interaction. The theory of vibrational rainbows, on the contrary, is much less ex- plored, despite of the fact that much of the early work on singularites in inelastic molecular collisions has been devoted to collinear vibrationally inelastic processes (see, e.g., the articles by Miller [5]). In many systems studied so far the vibrational coupling is weak, only small changes in the vibrational quantum numbers are impor- tant and the rotational rainbow patterns are only slightly modified by the vibration. Here we are concerned with the strong coupling regime, where new rainbow phenom- ena can be expected. Vibrational rainbows have been * Present address: Department of Chemistry, University of Durham, South Road, Durham, DH13LE, UK discussed many years ago by Toennies and coworkers [6] on the basis of a breathing sphere model, which is basically a three-dimensional extension of collinear vibrationally inelastic scattering and ignores the rota- tional degrees of freedom. In a model study for Li + -H 2 collisions vibrational rainbows appeared as broad maxi- ma in the differential cross sections for 0 ~ 1 and 0 --* 2 vibrational transitions. In experiments [7] and semiclass- ieal calculations [8] structures consistent with vibration- al rainbows have been found for high energy/small angle Ar--COz scattering. Independently, vibrational rainbow effects have been observed and analysed by Leonas and Rodionov (see [9] and references given there) for very high energy (~ 1 keV) He-Nz scattering. All these stu- dies are restricted to very low lying vibrational transi- tions, where the vibrational effects are not very pro- nounced, as discussed below. An early example of rotational-vibrational rainbow structures can be found in a series of papers by one of the authors (HJK) on the impulse approximation with application to atom-diatom collisions at about 1-10 eV collision energies [10-14] (the observed structures have not yet been termed rainbows at those times). The present article can be considered as a continuation of these early studies. Recently, there has been a renewed interest in the study of collision induced combined rotational-vibra- tional excitation of diatomic molecules under impulsive conditions stimulated by experimental measurements of differential cross sections for state-to-state transitions in electron-Na2 collisions at about 102 eV collision energy [15, 16]. The results of a simple spectator model (closely related to the impulse approximation [17]) were in excel- lent agreement with measured rotational rainbow pat- terns [-4, 18, 19]. More recently the experiments have been extended to high vibrational states and pronounced vibrational transitions have been reported [20], which were again found to agree well with the predictions of an extended spectator model [4, 20-22]. In the present paper we will study this model in more detail and analyse the predicted rotational-vibrational rainbow patterns in