Scattering Angle Dependence of Fano Resonance Profiles in Cold Atomic Collisions Analyzed with the Complex Valued $w$ Parameter

The scattering angle dependence of Fano resonance profiles in cold atomic collisions has been theoretically studied. A complex-valued parameter $ w $ with an analytical formula describing the asymmetry of the resonance profile is proposed as a new development from previous work on electron resonance scattering from atoms [F. Koike, J. Phys. $\mathbf{B10}$, 2883 (1977)]. It serves as the general formulation which leads to the angle-dependence of Fano’s $ q $ parameter. Calculations for the case of cold elastic collisions of hydrogen atoms with krypton atoms have been accomplished. The strong angle dependence of the resonance profile asymmetry in the differential scattering cross section due to the interference with non-resonant partial waves is demonstrated. The scattering angle dependence of the resonance profile and thus the newly proposed asymmetry parameter is highly sensitive to the inter-atomic interaction potentials. This is likely to prove useful for the study of interaction potentials themselves.

💡 Research Summary

This paper presents a comprehensive theoretical study on the scattering angle dependence of Fano resonance profiles in cold atomic and molecular collisions. The primary motivation stems from recent experimental observations of angle-dependent resonance structures in systems like He* + D2, which lacked a robust theoretical framework for precise analysis. To address this, the authors generalize a concept from electron-atom scattering theory: they introduce a complex-valued profile parameter, w(θ), which provides a complete description of the resonance lineshape in the differential elastic scattering cross-section as a function of the center-of-mass scattering angle θ.

The core theoretical development builds upon the work of Koike (1977). For an isolated resonance occurring in a partial wave with angular momentum ℓ_r, the parameter w_ℓr(θ) is derived from the background phase shift of the resonant channel (δ_ℓr) and the phase shifts of all non-resonant partial waves (δ_ℓ for ℓ ≠ ℓ_r). It is expressed as a polynomial in cosθ (Eq. 6). From this complex w, an angle-dependent real Fano q-parameter is naturally defined as q(θ) = -cot