Narrow Results

Within the proposed method, a set of experimental data points are fitted using a multi-channel S-matrix. Then the resonance parameters are located as its poles on an appropriate sheet of the Riemann surface of the energy. The main advantage of the method is that the S-matrix is constructed in such a way that it has proper analytic structure, i.e. for any number of two-body channels, the branching at all the channel thresholds is represented via exact analytic expressions in terms of the channel momenta. The way the S-matrix is constructed makes it possible not only to locate multi-channel resonances but also to extract their partial widths as well as to obtain the scattering cross-section in the channels for which no data are available. The efficiency of the method is demonstrated by two model examples of a single-channel and a two-channel problems, where known resonance parameters are rather accurately reproduced by fitting the pseudo-data artificially generated using the corresponding potentials.

A new parametrization of the multi-channel $S$-matrix is used to fit scattering data and then to locate the resonances as its poles. The $S$-matrix is written in terms of the corresponding “in” and “out” Jost matrices which are expanded in the Taylor series of the collision energy $E$ around an appropriately chosen energy $E_0$. In order to do this, the Jost matrices are written in a semi-analytic form where all the factors (involving the channel momenta and Sommerfeld parameters) responsible for their “bad behavior” (i.e., responsible for the multi-valuedness of the Jost matrices and for branching of the Riemann surface of the energy) are given explicitly. The remaining unknown factors in the Jost matrices are analytic and single-valued functions of the variable $E$ and are defined on a simple energy plane. The expansion is done for these analytic functions and the expansion coefficients are used as the fitting parameters. The method is tested on a two-channel model, using a set of artificially generated data points with typical error bars and a typical random noise in the positions of the points.

Partial cross-sections of the $n\alpha$ and $dt$ collisions in the quantum state $J^{\pi }={\frac{3}{2}}^{+}$ near the $dt$-threshold, taken from an available $R$-matrix analysis, are fitted using the semi-analytic multi-channel Jost matrix with proper analytic structure and some adjustable parameters. Then the spectral points are sought as zeros of the Jost matrix determinant (which correspond to the $S$-matrix poles) at complex energies. The correct analytic structure makes it possible to calculate (analytically continue) the fitted Jost matrix on any sheet of the Riemann surface whose topology involves not only the square-root but also the logarithmic branching caused by the Coulomb interaction. As a result of such an analytic continuation, it was found that the previously established ${\frac{3}{2}}^{+}$-resonance (at $\sim 47\mathrm{\text{keV}}$) and its shadow pole (at $\sim 80\mathrm{\text{keV}}$) both are split in overlapping pairs. Apart from studying the properties of a specific nuclear state, it is also proved for a general multi-channel problem that the Coulomb forces change the topology of the Riemann surface as well as destroy the so-called mirror symmetry of the $S$-matrix.

The available $R$-matrix parametrization of experimental data on the excitation functions for the elastic and inelastic $p{}^7\mathrm{\text{Be}}$ scattering at the collision energies up to 3.4MeV is used to generate the corresponding partial-wave cross-sections in the states with $J^{\pi }=0^{+}$. Thus, obtained data are fitted using the semi-analytic two-channel Jost matrix with a proper analytic structure and some adjustable parameters. Then the spectral points are sought as zeros of the Jost matrix determinant (which correspond to the $S$-matrix poles) at complex energies. The correct analytic structure makes it possible to calculate the fitted Jost matrix on any sheet of the Riemann surface whose topology involves not only the square-root but also the logarithmic branching caused by the Coulomb interaction. In this way, two overlapping $0^{+}$ resonances at the excitation energies $\sim 1.79$ and $\sim 1.96$MeV have been found.

The Jost functions, constructed by fitting available partial cross-sections for the elastic p⁷Be scattering with $J^{\pi }=3^{+},2^{-}$, are analytically continued to complex energies, where the resonances are located as their zeros. In addition to the resonance energies and widths, the residues of the $S$-matrix at the corresponding poles, as well as the Asymptotic Normalization Constants (ANC) are determined. The fitting is done using the semi-analytic representation of the Jost function with proper analytic structure, defined on the Riemann surface whose topology involves not only the square-root but also the logarithmic branching caused by the Coulomb interaction.