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Journal of Applied Mathematics and Physics

Theory of Electronic Transitions

H. Klar

Published Jan 1, 2025• 0 citations
Publisher

Abstract

We investigate scattering of electrons from atoms in the nonrelativistic energy range. In contrast to the pioneer work by Born, we take correlation properly into account and describe electrons by waves rather than by mass points. To this end, we start from a novel parabolic partial differential equation, which resembles the inhomogeneous heat equation. Calculation of the kernels for incoming and outgoing waves allows us to formulate initial value problems. A converging Fresnel distribution is shown to control the incident electron and pulls the target electron onto an equilibrium location. The electron-electron interaction is here attractive. Finally, the two electrons are attracted by the nucleus and arrive at a triple condensation point, where they form a compound state comparable to a Cooper pair. This three-body configuration is highly unstable, described by an unstable Fresnel distribution, and leads finally to excited target states plus an escaping electron. There is no way for a one-step transition. The excitation process always goes through a ladder consisting of Fresnel states. The Wannier mode has been identified as an unstable Fresnel distribution, which is not an energy eigenstate, but it is an eigenstate of the action curvature. This unusual mode cannot be expected as a discrete line in a Rydberg spectrum, but seems to manifest itself as noise. We show that this noise is heavily suppressed near the threshold.

DOI
10.4236/jamp.2025.1311221

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We present the exact solution via Bethe's ansatz to an integrable model consisting of two parabolic bands of electrons of equal mass with a local \ensuremath{\delta}-function exchange interaction. The interaction is attractive or repulsive depending on whether the interacting particles are in a spin-singlet or spin-triplet state. The structure of the Bethe ansatz and the classification of states is similar to the one of the two-channel Kondo problem. The interest in two-band models arises from the possibility that both the 3dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 and 3dz2{\mathit{d}}_{\mathit{z}}^{2}dz2​ orbitals may play a role in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprates. Some low-temperature properties of this model are discussed as a function of the interaction strength, a magnetic field, and the (crystalline field) splitting between the bands. The attractive interaction leads to Cooper-pair-like bound states, which exist at every temperature. There is no long-range order and no condensation of the Cooper pairs. A threshold magnetic field is required to overcome the binding energy of the Cooper pairs and there is no response to fields smaller than the critical one. The low-temperature specific heat is proportional to T, except at critical points (van Hove singularities of the bands) where C\ensuremath{\propto}T1/2{\mathit{T}}^{1/2}T1/2. The spectrum of elemental excitations is approximately parabolic.