Role of the Pauli Principle in Collective-Model Coupled-Channel Calculations

Abstract

A multi-channel algebraic scattering theory, to find solutions of coupled-channel scattering problems with interactions determined by collective models, has been structured to ensure that the Pauli principle is not violated. By tracking the results in the zero coupling limit, a correct interpretation of the sub-threshold and resonant spectra of the compound system can be made. As an example, the neutron-12 C system is studied defining properties of 13 C to 10 MeV excitation. Accounting for the Pauli principle in collective coupled-channels models is crucial to the outcome. 25.40.Dn;25.40.Ny;28.20.Cz At energies above 25 MeV, by using optical potentials formed by full folding effective two-nucleon interactions with microscopic (nucleon based) descriptions of the target structure, the importance of treating the Pauli principle has been well established [1]. However, in the domain of low-energy nucleon scattering for which an explicit coupled-channels theory of scattering is essential, the significance of Pauli exclusion effects has not been well defined. Many coupled-channels codes are available, some of which perform phenomenological collective-model calculations searching on parameter values of the chosen function forms to find a best fit to experimental data. But while it has long been known that any such models violate the Pauli principle [2,3], quantification of that violation is lacking.To study the effects of the Pauli principle in a macroscopic (collective model) approach is not a trivial task. In a recent publication [4], the orthogonalizing pseudopotential (OPP) method [5,6] was generalized to treat this problem. That was a small though important part of the full theoretical framework of the multi-channel algebraic scattering (MCAS) theory of scattering [4]. Therein the OPP was used in finding the spectra, bound and resonance properties, of 13 C. However, implications of the role of the Pauli principle in collective model coupledchannel calculations arising from the use of the OPP was not discussed. Such is a purpose of this letter. Another is that the method could be pertinent for any study requiring coupled channel solutions of quantal systems involving fermions. As the example, we study the effects introduced by the Pauli principle in collective, geometrical-type, models for low-energy nucleon-nucleus processes that can be characterized from the spectrum of the compound nucleus. That spectrum includes the states that lie below the nucleon-nucleus threshold and in the continuum as revealed by the narrow and broad resonances that lie upon a smooth but energy dependent background of the elastic scattering cross section. This can be done in a systematic and self-consistent way since the MCAS approach facilitates such a determination of the sub-threshold bound states and resonances of the compound nucleus. This theory, with which one solves the coupled-channel Lippmann-Schwinger (LS) equations for the nucleon-nucleus system considered, is built upon sturmian expansions of an interaction matrix of potent...

Compound and quasicompound states in low-energy scattering of nucleons from C12

“…Couplings of the input channel with the ground state (ǫ 1 = 0), with the 2 + (ǫ 2 = 4.4389 MeV) and with the 0 + 2 (ǫ 3 = 7.6542 MeV) excited states of the 12 C target have been considered. Moreover, the Pauli principle has been taken into account throughout using the OPP procedure [1,3]. However, note that the OPP treatment [1] we use is an approximation.…”

“…At these energies that violation is severe. But it is possible [1,3] in the MCAS approach to account for Pauli blocking of occupied nucleon states in the target, with this or any collective model specification of the matrix of potentials. That is achieved by introducing orthogonalizing pseudo-potentials [3,4,5] (OPP) into the scheme by which the sturmians are specified.…”

“…But it is possible [1,3] in the MCAS approach to account for Pauli blocking of occupied nucleon states in the target, with this or any collective model specification of the matrix of potentials. That is achieved by introducing orthogonalizing pseudo-potentials [3,4,5] (OPP) into the scheme by which the sturmians are specified. In that way all sturmians in the (finite) set selected as the basis of expansion of the matrix of potentials contain few or no components equivalent to the external nucleon being placed in an already densely occupied orbit.…”

“…In that way all sturmians in the (finite) set selected as the basis of expansion of the matrix of potentials contain few or no components equivalent to the external nucleon being placed in an already densely occupied orbit. Treatment of the Pauli principle has a significant effect on results [1,3].…”

Recent developments in Multi-Channel Algebraic Scattering calculations

“…After truncation, the sturmian representation leads to a finite-rank expansion of the potential, and to an algebraic solution of the scattering problem. In defining the sturmians in the selected set, we consider in addition orthogonalising pseudo-potentials (OPPs) [5]. The OPP technique was developed in studies of cluster physics [6,7] as a variant of the Orthogonality Condition Model (OCM) of Saito [8].…”