Multi-channel Algebraic Scattering Research Articles

A multichannel algebraic scattering approach to astrophysical reactions

The investigation of many astrophysical processes is dependent upon an understanding of nuclear reaction rates. However, nuclear capture reactions of astrophysical interest occur at extremely low energies, taking place at the Gamow energy within the stellar environment. Hence, they are hard to study experimentally due to Coulomb repulsion. They may also involve compound resonances stemming from a delicate interplay of many quantum states in the colliding bodies. The multi-channel algebraic scattering (MCAS) method is one that addresses both of these challenges; it has a history of successfully modelling narrow compound resonance structures, incorporating as many channels as are important for a given problem, but is also proven in recreating the lowenergy, non-resonant elastic scattering cross sections needed for these astrophysics problems. We provide an overview of MCAS’ techniques of modelling elastic scattering reactions, how these may be extended to capture reactions, and current work in this area.

Coupled-channel description for mirror mass-11 nuclei compared to shell-model structures

The spectra of mass-11 nuclei are unusual, and so pose a challenge for nuclear-structure theory. Relating to nucleon emission, the set of isobars range from being well-bound (^{11}B,^{11}C) through weakly bound (^{11}Li, ^{11}Be), to being proton unstable (^{11}N,^{11}O). To add complexity, the weakly bound ^{11}Li takes the form of a two-nucleon halo nucleus. A self-consistent approach to understand this set of nuclei is especially important as the mirror pair ^{11}Be-^{11}N exhibit a parity-inverted ground state compared to their neighboring nuclei. Herein, the Multi-Channel Algebraic Scattering method (MCAS) has been used to describe the low excitation spectra of those isobars in terms of nucleon-nucleus clusters. A collective model description of the low-excitation states of the mass-10 mass-10 core nuclei has been used to form the coupled-channel interactions required in the method. For comparison, and to understand the underlying configurations, a shell model approach has been used to obtain those spectra with no-core (0+2+4)hbar omega and (0+2)hbar omega shell-model spaces for the mass 10 and mass 11 nuclei respectively. The results of the calculations suggest the need of a strong coupling in the collective coupled-channel vibrational model. In particular, the strong coupling of the collective 2^+_1 state of ^{10}Be to the valence neutron plays a decisive role in forming the positive parity ground state in ^{11}Be; an effect confirmed by the shell-model results.

Low-energy spectra of mirror mass-19 nuclei with a collective coupled-channel scattering model

The spectra of mass-19 nuclei are unusual and pose a challenge for theoretical models of their structure. Herein the Multi-Channel Algebraic Scattering (MCAS) method has been used with a collective (vibrational model) description of the low-excitation states of $$^{18}$$ O and $$^{18}$$ Ne to describe the spectra of four mass-19 nuclei, specifically the mirror pairs ( $$^{19}$$ O, $$^{19}$$ Na) and ( $$^{19}$$ F, $$^{19}$$ Ne). This coupled-channel approach allows for the effects of the Pauli principle to be included using non-local orthogonalizing pseudo-potentials. By the coupling of neutrons to the core nuclei, we obtain states in $$^{19}$$ O (a neutron coupled to $$^{18}$$ O) and $$^{19}$$ Ne (a neutron coupled to $$^{18}$$ Ne). Mirror symmetry and addition of the Coulomb interaction gives states in $$^{19}$$ F (a proton coupled to $$^{18}$$ O) and $$^{19}$$ Na (a proton coupled to $$^{18}$$ Ne). However, the same compound nuclei may be described as the coupling of more complicated nuclear clusters. We obtain the states in $$^{19}$$ F as the coupling of an $$\alpha $$ to $$^{15}$$ N, as well as the coupling of $$^3$$ H to $$^{16}$$ O. Similarly, states in $$^{19}$$ Ne have been obtained from the coupling of an $$\alpha $$ to $$^{15}$$ O and of the coupling of $${}^3$$ He to $$^{16}$$ O. Finally, as the method allows extraction of scattering matrices, it has been used to evaluate cross sections for the elastic scattering of low-energy neutrons from $$^{18}$$ O and of low-energy $$\alpha $$ particles from $$^{15}$$ O.

Modelling nuclei far from stability with a multichannel approach

Recently, new high-precision data reconfirmed the existence of narrow resonances in the p+14O cross section, first predicted by the multi-channel algebraic scattering (MCAS) theory of light-mass, low-energy scattering and later found by other experimental means. Prompted by this and a decade of extension to the method, we have updated the original MCAS study.Additionally, we use MCAS to interpret the low-energy spectrum of 19 F, which has an interesting structure that appears to stem from clusterisation, and we compare the results of several types. As only early, small-basis shell model investigations of this spectrum exist, we also perform a complete (0 + 2)ħω calculation.MCAS is well suited to these studies, having the advantage of accounting for the Pauli principle between the nucleons of the clusters, and so spurious states of the compound nucleus are removed.

Mass-15 nuclei and predicting narrow states beyond the proton drip line

In a previous letter (Phys. Rev. Lett. 96, 072502 (2006)), the multi-channel algebraic scattering (MCAS) technique was used to calculate spectral properties for proton-unstable $^{15}$F and its mirror, $^{15}$C. MCAS achieved a close match to the then-new data for $p+^{14}$O elastic scattering and predicted several unusually narrow resonances at higher energies. Subsequently, such narrow resonance states were found. New cross section data has been published characterising the shape of the $J^\pi =\frac{1}{2}^-$ resonance. Herein we update that first MCAS analysis and its predictions. We also study the spectra of the set of mass-15 isobars, ${}^{15}$C, ${}^{15}$N, ${}^{15}$O, and ${}^{15}$F, using the MCAS method and seeking a consistent Hamiltonian for clusterisation with a neutron and a proton, separately, coupled to core nuclei ${}^{14}$C and ${}^{14}$O.

A multi-channel model for an $ \alpha$ α plus 6He nucleus cluster

A multi-channel algebraic scattering (MCAS) method has been used to solve coupled sets of Lippmann-Schwinger equations for the $\alpha$+${}^6$He cluster system, so finding a model spectrum for ${}^{10}$Be to more than 10 MeV excitation. Three states of ${}^6$He are included and the resonance character of the two excited states taken into account in finding solutions. A model Hamiltonian has been found that gives very good agreement with the known bound states and with some low-lying resonances of ${}^{10}$Be. More resonance states are predicted than have as yet been observed. The method also yields $S$-matrices which we have used to evaluate low-energy ${}^6$He-$\alpha$ scattering cross sections. Reasonable reproduction of low-energy differential cross sections and of energy variation of cross sections measured at fixed scattering angles is found.

Mass-23 nuclei in astrophysics

The formation of mass-23 nuclei by radiative capture is of great interest in astrophysics. A topical problem associated with these isobars is the so-called 22Na puzzle of ONe white dwarf novae, where the abundance of 22Na observed is not as is predicted by current stellar models, indicating there is more to learn about how the distribution of elements in the universe occurred. Another concerns unexplained variations in elements abundance on the surface of aging red giant stars. One method for theoretically studying nuclear scattering is the Multi-Channel Algebraic Scattering (MCAS) formalism. Studies to date have used a simple collective-rotor prescription to model the target states which couple to projectile nucleons. While, in general, the target states considered all belong to the ground state rotor band, for some systems it is necessary to include coupling to states outside of this band. Herein we discuss an extension of MCAS to allow coupling of different strengths between such states and the ground state band. This consideration is essential when studying the scattering of neutrons from 22Ne, a necessary step in studying the mass-23 nuclei mentioned above.

Coupling to two target-state bands in the study of then+Ne22system at low energy

One theoretical method for studying nuclear scattering and resonances is via the multi-channel algebraic scattering (MCAS) formalism. Studies to date with this method have used a simple collective-rotor prescription to model target states with which a nucleon couples. While generally these target states all belong to the same rotational band, for certain systems it is necessary to include coupling to states outside of that main band. Here, we extend MCAS to allow coupling of different strengths between such states and the rotor band. This is an essential consideration in studying the example examined herein, the scattering of neutrons from 22Ne.

Comparison of multi-ħωshell-model results with MCAS

A multi-channel algebraic scattering (MCAS) method has been used to obtain spectra of a number of light-mass nuclei, which are treated as a two-cluster system, here specifically a nucleon plus nucleus. To date, collective models have been used to specify the interactions between the nucleon and low-lying states of the nucleus that form the compound. For the case of the carbon isotopes, these studies have been complemented by sufficiently complex and complete shell-model calculations. Comparisons with the multi-ħω shell-model results provide new insights into the validity of those from MCAS.

Investigating the astrophysical22Ne(p,γ)23Na and22Mg(p,γ)23Al reactions with a multi-channel scattering formalism

The reaction 22 Ne(p, ) 23 Na is key to the NeNa cycle of stellar nucleogenesis, and better understanding of the 22 Mg(p, ) 23 Al reaction is needed to understand the 22 Na puzzle in ONe white dwarf novae. We aim to study these reactions using a multi-channel algebraic scattering (MCAS) formalism for low-energy nucleon-nucleus scattering, re- cently expanded to investigate radiative capture. As a first step towards this goal, we here calculate the energy levels of the mass-23 (Ne, Mg, Na, Al) nuclei. This is not only because the resonant structure of these nuclei are related to the astrophysical -rates of interest, but also because the interaction parameters determined for describing the energy levels are an integral part of the future calculation of the astrophysical reactions when using the MCAS scheme.

Analysis of a coupled-channel continuum approach for spectra of mass-17 compound systems

By performing studies of the structure of the spectra of mass-17 nuclei we discuss specific aspects of the Multi-Channel Algebraic Scattering (MCAS) method. We devote particular attention to the comparison with results from large-scale shell-model calculations, the treatment of Pauli-forbidden or -hindered states, and to the collective/coupled nature of the states obtained in our model. By using different shell-model spaces as platform to assess the MCAS method, we question statements raised recently about the validity of this method on the basis of a small-scale shell-model approach.

Scattering and resonances on p-shell nuclei

We review recent results on low energy nucleon scattering from some light-mass nuclei and particularly consider resonances seen or expected in cross sections. We present results found using the multichannel algebraic scattering (MCAS) method which was developed to analyze such cross sections. The method gives both bound and resonant spectra of the compound nuclei. In 2006, using this method, we predicted nontrivial narrow resonances due to non-valence p-shell effects in the nucleus 15F, a nucleus that lies beyond the proton drip-line. Subsequently, in 2009, those predictions were verified experimentally. Research on other such mirror systems is underway. In particular, the structure of 17C has been used to define an appropriate MCAS nuclear interaction for the n+16C system. Adding a Coulomb field to that interaction, an MCAS evaluation has been made for the mirror system p+16Ne, to specify the low excitation spectrum of the yet unobserved, particle-unstable, 17Na.We also investigated how the scattering cross-section changes when the spectra of the colliding nuclei have low-excitation particle-emitting resonances. As a test case, we considered 8Be, being particle-unstable, and analyzed neutron scattering cross sections and the spectra of the compound 9Be system. If the 8Be excited states have fixed-energy decay widths, we find that the bound states of the compound system are affected in an unphysical way. The shapes of the target resonances must vary from the usual Lorentzian in an energy-dependent way. Resonance width functions must go smoothly to zero at the elastic threshold. Ways of achieving this condition are being explored.

COUPLED-CHANNEL CALCULATION OF BOUND AND RESONANT SPECTRA OF $^9_\Lambda{\rm Be}$ AND $^{13}_\Lambda\rm C$ HYPERNUCLEI

A multi-channel algebraic scattering (MCAS) approach has been used to analyze the spectra of two hypernuclear systems, [Formula: see text] and [Formula: see text]. The splitting of the two odd-parity excited levels ([Formula: see text] and [Formula: see text]) at 11 MeV excitation in [Formula: see text] is driven mainly by the weak Λ-nucleus spin–orbit force, but the splittings of the [Formula: see text] and [Formula: see text] levels in both [Formula: see text] and [Formula: see text] have a different origin. These cases appear to be dominated by coupling to the collective 2+ states of the core nuclei. Using simple phenomenological potentials as input to the MCAS method, the observed splitting and level ordering in [Formula: see text] is reproduced with the addition of a weak spin–spin interaction acting between the hyperon and the spin of the excited target. With no such spin–spin interaction, the level ordering in [Formula: see text] is inverted with respect to that currently observed. In both hypernuclei, our calculations suggest that there are additional low-lying resonant states in the Λ-nucleus continua.

Extending MCAS to hypernuclei and radiative-capture reactions

Using a Multi-Channel Algebraic Scattering (MCAS) approach we have analyzed the spectra of two hyper-nuclear systems, Lambda9Be and Lambda13C. We have studied the splitting of the two odd-parity excited levels (1/2- and 3/2-) at 11 MeV excitation in Lambda13C, originated by the weak Lambda-nucleus spin-orbit force. We have also considered the splittings of the 3/2+ and 5/2+ levels in both Lambda9Be and Lambda13C, finding how they originate from couplings to the collective 2+ states of the core nuclei. In both hyper-nuclei, we suggest that there could be additional low-lying resonant states in the Lambda-nucleus continua. From the MCAS approach one can extract also the full coupled-channel scattering wave-function to be used in the calculation of various transition matrix elements. As a first application, we have considered the EM-transition matrix elements for the capture reaction Alpha + 3He -> 7Be + Gamma .

Low-energy radiative captures with gauge independence and the MCAS description

We describe radiative-capture reactions at low energies within the nuclear coupled-channel approach denoted MCAS (multi-channel algebraic scattering). The amplitude of single-photon emission is constructed via a generalization of the Siegert theorem in an explicit gauge-independent way, without any multipole decomposition of the electromagnetic (e.m.) current operator. As a first test application, we calculate the 3He(α, γ)7Be reaction cross section at astrophysical energies using a simplified two-cluster model. The energy dependence of the calculated astrophysical factor reproduces the experimental situation fairly well, though the cross-section normalization turns out to be overestimated.

Constraints on the spectra of 17,19C

Diverse means are used to investigate the spectra of the radioactive, exotic ions, 17,19C. First, estimates have been made using a shell model for the systems. Information from those shell model studies were then used in evaluating cross sections of the scattering of 70 A MeV 17,19C ions from hydrogen. Complementing those studies, a multichannel algebraic scattering (MCAS) theory for n+ 16,18C coupled-channel problems has been used to identify structures of the compound systems. The results show that the shell model structure assumed for these ions is reasonable with little need of effective charges. The conditions that two excited states exist within a few hundred keV of the ground state places some restriction upon the structure models. Other positive parity states are expected in the low-lying spectra of the two nuclei.

Low-energy radiative-capture reactions within two-cluster coupled-channel description

The formalism that describes radiative-capture reactions at low energies within an extended two-cluster potential model is presented. Construction of the operator of single-photon emission is based on a generalisation of the Siegert theorem with which the amplitude of the electromagnetic process is constructed in an explicitly gauge-independent way. While the starting point for this construction is a microscopic (single-nucleon) current model, the resulting operator of low-energy photon emission by a two-cluster system is expressed in terms of macroscopic quantities for the clusters and does not depend directly on their intrinsic coordinates and momenta. The multichannel algebraic scattering (MCAS) approach has been used to construct the initial- and final-state wave functions. We present a general expression for the scattering wave function obtained from the MCAS T matrix taking into account inelastic channels and Coulomb distortion. The developed formalism has been tested on the He 3 ( α , γ ) Be 7 reaction cross section at astrophysical energies. The energy dependence of the evaluated cross section and S factor agrees well with that extracted from measurement though the calculated quantities slightly overestimate data.

Low-energy neutron-C12analyzing powers: Results from a multichannel algebraic scattering theory

Analyzing powers in low-energy neutron scattering from $^{12}\mathrm{C}$ are calculated in an algebraic momentum-space coupled-channel formalism, that is, a multichannel algebraic scattering (MCAS) theory. The results are compared with recently obtained experimental data. The channel-coupling potentials have been defined previously to reproduce the total cross section and subthreshold bound states of the compound system. Without further adjustment, good agreement with data for the analyzing powers is obtained.