On the inversion of atomic scattering data: A new algorithm based on functional sensitivity analysis

A new iterative inversion scheme of atomic scattering data within the framework of functional sensitivity analysis is presented. The inversion scheme is based on the first order Fredholm integral equation δσ(θ) = ∫₀^∞(θ,R)δV(R)R² dR, K(θ,R) ≡ δσ(θ)/δV(R), or symbolically, δσ = KδV, which relates infinitesimal functional changes in the elastic differential cross section δσ(θ) and in the underlying interatomic potential δV(R). This equation can be written equivalently, via integration by parts, as δσ(θ) = ∫₀^∞K^[n](θ,R){R ⁿ×δV⁽ⁿ⁾(R)}R² dR, or δσ = K^[n]{R ⁿ×δV⁽ⁿ⁾}, under the a priori assumption that {R^{(2 + n)}×K^[n](θ,R) ×δV^(n-1)(R)}|_R=0^∞ =0.Here K^[n](θ,R) ≡ -R ^{-(2 + n)} ×∫ ₀^RK^{[n - 1]}(θ,R′)R′ ^{(1 + n)} dR′, δV⁽ⁿ⁾(R) ≡ (d ⁿ/dRⁿ)δV(R), with n = 0,1,2,..., and K^[0] (θ,R) ≡ K(θ,R). A choice of n corresponds to a particular level of additional stabilization in inverting the scattering data. By invoking a least squares regularization procedure and singular system analysis, the new indirect inversion scheme solves the linear relation δσ = K ^[n]{Rⁿ×δV⁽ⁿ⁾} and results in the approximate solution Rⁿ×δV⁽ⁿ⁾(R), which can in turn be integrated, n times, to yield the potential correction δV(R). The new algorithm not only makes the inversion more stable and more efficient, but also increases the sensitivity of the large angle scattering data to the repulsive part of the potential, in comparison with a previous method that directly solves the relation δσ = KδV. For illustration, the model system He-Ne is considered at both low- and high-collision energies, relative to the well depth of the potential. It is found that the indirect method based on the linear relation δσ = K^[2]{R ²×δV⁽²⁾} can more accurately determine both attractive and repulsive parts, including a large section of classically forbidden region, of the potential than its direct counterpart. Furthermore, the indirect method allows for substantial deviations of the initial reference potential from final result, not only in well depth, but in range parameter and asymptotic behavior. Finally, these observations are not significantly dependent on the magnitude of the collision energy.