Analysis of isolated resonances with weak background

Analysis of isolated resonances with weak background 

The asymptotic wavefunctions obtained from multichannel scattering calculations provide the S matrix, the eigenphases and eigenphase sum, and the cross sections. In principle, any of these quantities may be chosen for fitting the resonance formula to determine accurate values of the resonance parameters E, and T. 

In the field of photoionization, the Fano formula for the cross section has often been used for resonance fitting. Note, however, that the same resonance can sometimes stand out sharply from the background, but can also fail to manifest themselves clearly in the photoionization cross section, depending upon the initial bound state of the dipole transition [51]. Thus, the cross-section inspection might miss some resonances. The asymptotic quantities of the final continuum-state wavefunction, if available, should be much more convenient in general for the purpose of resonance search and analysis. 

In actual computations, the numerical differentiation can introduce a large error and should be avoided. A simple solution to this would be to fit spline functions, or a piecewise polynomial and overall smooth function of E, to the numerically calculated eigenphase sum 5(E), and then to differentiate the spline functions analytically [52, 53]. In the E-matrix method [44], the analytic E dependence of the R matrix and associated matrices can be taken advantage of in the direct differentiation of these quantities. This technique was found to be useful for automatic and fast analysis of the results of E-matrix method calculations [54-56]. 

It should be born in mind that our discussion now centers on extracting the resonance position Er and the total width T from the much richer scattering information that the S and Q matrices contain. Multichannel continuum wavefunctions are usually calculated for more general purposes of obtaining the scattering amplitudes and the cross sections for various state-to-state processes and of unraveling the dynamics of the whole continuum system including both resonance and nonresonance mechanisms and the intricate interference between them. 

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