Analysis of a coupled-channel continuum approach for spectra of mass-17 compound systems
Introduction
Radioactive nuclei, particularly when they are close to the driplines, have low particle-emission thresholds, and consequently their low-lying spectra are made in large part of particle-emission resonant states. Sometimes the experiments based on Radioactive-Ion Beams are still precluded from a direct observation of these low-energy resonant structures, sometimes observations are possible [1], [2] but it remains very difficult to find evidences for a direct observation of these resonances. A similar situation occurs in nuclear reactions for astrophysical interests, where the possible existence of selected resonances in the appropriate reaction channels could be crucial for the solution of fundamental cosmological problems [3], however the cross sections are typically too small to be detected with standard experimental methods.
Nuclear theory and phenomenology are essential for estimating masses and spectra in absence of direct measurements. The existence of a variety of models of nuclear structure and scattering offers an opportunity to describe the diverse facets of the physics of radioactive/weakly-bound nuclei. We take the case of mass-17 nuclei, where the ground state of 17Na is still unmeasured. Ref. [4] describes various approaches to get estimates for the mass of 17Na. They range from simple extrapolation based on data systematics in the nearby mass region, to the use of mass formulae for nuclei within isotopic multiplets and, finally, to the use of mirror symmetry to get estimates for the 17Na spectrum, starting from diverse nuclear structure calculations for the spectra of the bound 17C nucleus. The predictions from different theoretical or phenomenological approaches can change significantly from each other [4], and this could point to a possibly different selection of parameters, but also to different or complementary physics embedded in the diverse approaches. With this state of matters, it is important to judge with great attention and sagacity the comparisons amongst different theoretical and phenomenological methods.
In this article, we consider results obtained from shell-model calculations that are more extended than the simple shell-model considerations in Ref. [5]. Some of these more extended calculations were already available in the literature (see Section 5 and references therein), other calculations are new, and based on a complete shell model. They have been performed with the purpose to counter the statements raised in Ref. [5]. They lead us to conclude that a plain comparison and interpretation of the results of Ref. [6] in terms of a simple shell-model picture [5] is not justified and may lead to serious misinterpretations.
This article contains also a discussion of the Multi-Channel Algebraic Scattering (MCAS) approach, that has been used to calculate the spectra given in Ref. [6]. The approach has been used before for determining spectra, resonances and nucleon cross sections of several nuclei in the range of mass-13 [7], mass-15 [8], mass-7 [9], and mass-17/19 [10]. Essential aspects of these calculations, such as the treatment of the Pauli principle, the use of core deformation in the treatment of channel couplings, the role of the coupled-channel scattering orbits in the calculation of the resonance widths, and the connection with shell-model studies are discussed in detail.
In the next section, we give a brief overview of the MCAS methodology and then give particular emphasis on how Pauli-principle effects have been treated in cluster models of structure. Based upon this, in MCAS, we use an Orthogonalizing Pseudo-Potential approach to treat Pauli-principle effects at the level of inter-cluster dynamics. Subsequently, in Sections 3 MCAS and widths of resonances, 4 MCAS and single-particle orbits, we use new MCAS results to specify resonances in a mass-17 (simplified) model commenting on what physics underlies the widths of resonances so generated, and on what single-nucleon orbits are inherent in the channel-coupled Hamiltonian matrix. Then, in Section 5, the MCAS spectrum of 17C is compared with that found using a complete shell model; finding results that offset many of the comments made in [5]. Consideration of the role of deformation in obtaining results of the coupled-channel evaluations is then given in Section 6. Conclusions follow thereafter.
Section snippets
Synopsis of MCAS
The details of MCAS theory have been published [11] and so only salient information is given here. MCAS is a method of solving coupled-channel Lippmann–Schwinger equations. It has been applied mostly to coupled equations describing the interaction of a nucleon with a nucleus; coupling taken between a chosen set of states of that nucleus. We consider only applications to a system hereafter.
All applications of MCAS to date have used a deformed collective model to describe the
MCAS and widths of resonances
MCAS finds resonance centroids and widths from the properties of the coupled-channel S-matrix, determined by solving the coupled-channel Lippmann–Schwinger equations with a defined Hamiltonian. That Hamiltonian specifically involves the nucleon emission threshold but no other. Thus the widths of resonances found are for nucleon emission solely. When there is no other threshold in the energy range under study, the MCAS widths will be total ones. When there are one or more particle-emission
MCAS and single-particle orbits
It should be noted that MCAS is not a shell model. All MCAS solutions will be admixtures of the basic wave functions for all allowed channels in the coupled-channel problem. The stated spin-parity values for resonances ascertained using the MCAS approach are the total spins of the compound system so treated. Depending upon just how many and what target states are included, the underlying admixtures of the single-nucleon quantum sets and target spins for each (conserved) total spin in the
MCAS and shell-model results
As criticisms of our results were based upon information provided by an sd shell model of 16, 17C; and one in which only neutrons were active, we have made complete space shell-model calculations of those nuclei. The OXBASH shell-model code was used with the WBP [36] interaction. For 16C, that evaluation showed the states with which we are concerned to be purely of character. The spectrum found is quite reasonable for a shell-model calculation. In order (experimental values in
Deformation and core states
The ground state of 12C is usually assumed to have a matter distribution reminiscent of an oblate spheroid. The ground state of 14C usually is considered to have a spherical matter distribution. Deformed Hartree–Fock studies [41] expect that the ground states of 15, 16, 17C have prolate deformation. But it is not any specific deformation of the ground state matter distribution that is relevant. Rather the deformation strengths, , are measures of channel-coupling defined by using a collective
Conclusions
In conclusion, we reconsidered specific characteristics of the MCAS approach, that seem not yet widely understood. It should be noted that the MCAS approach considers (in one single coupled-channel Hamiltonian) aspects of nuclear collectivity and inter-cluster orbital dynamics, and includes effectively peculiarities of the effects due to the Pauli principle. At the present stage of development, the approach still has a certain degree of phenomenology, since the mean-field, collective, and
Acknowledgements
SK acknowledges support from the National Research Foundation of South Africa. JPS acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC). LC and PRF acknowledge funds from the Dipartimento di Fisica e Astronomia dellʼUnivesità di Padova and the PRIN research project 2009TWL3MX.
References (43)
Nucl. Phys. A
(2012)- et al.
Nucl. Phys. A
(2012) - et al.
Nucl. Phys. A
(2008) - et al.
Nucl. Phys. A
(2003) - et al.
Ann. of Phys.
(1978) Nucl. Phys. B
(1981)- et al.
Nucl. Phys. A
(1981) Phys. Lett. B
(2011)- et al.
Phys. Lett. B
(2007) - et al.
At. Data and Nucl. Data Tables
(2001)
Phys. Rev. C
Phys. Rev. C
JCAP
Europhys. Lett.
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. C
Soviet J. Nucl. Phys.
Prog. Theor. Phys.
Cited by (19)
High-J states in <sup>17</sup>C
2014, Nuclear Physics ALow-energy spectra of mirror mass-19 nuclei with a collective coupled-channel scattering model
2021, European Physical Journal AModelling nuclei far from stability with a multichannel approach
2020, Journal of Physics: Conference SeriesMass-15 nuclei and predicting narrow states beyond the proton drip line
2019, Physical Review C