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Register a new userFormal and Physical R-matrix parameters
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Carl R. Brune
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Notes from 11 October 2004 lecture presented at the Joint Institute for Nuclear Astrophysics R-Matrix School at Notre Dame University.
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Proton emission from deformed nuclei is described within the non-adiabatic weak coupling model which takes into account the coupling to $gamma$-vibrations around the axially-symmetric shape. The coupled equations are derived within the Gamow state formalism. A new method, based on the combination of the R-matrix theory and the oscillator expansion technique, is introduced that allows for a substantial increase of the number of coupled channels. As an example, we study the deformed proton emitter $^{141}$Ho.
An alternative parameterization of R-matrix theory is presented which is mathematically equivalent to the standard approach, but possesses features which simplify the fitting of experimental data. In particular there are no level shifts and no boundary-condition constants which allows the positions and partial widths of an arbitrary number levels to be easily fixed in an analysis. These alternative parameters can be converted to standard R-matrix parameters by a straightforward matrix diagonalization procedure. In addition it is possible to express the collision matrix directly in terms of the alternative parameters.
The Lagrange-mesh $R$-matrix method is generalized to inhomogeneous equations. This method is numerically stable and efficient. It can be directly used for transfer reactions with the formalism discussed by Ascuitto and Glendenning [Phys. Rev. 181,1396 (1969)] and for inclusive breakup reactions modeled by Ichimura, Austern, and Vincent [Phys. Rev. C 32, 431 (1985)]. We first present a simple example to assess the method. Then the application to the $^{93}$Nb($d$,$pX$) non-elastic breakup is discussed.
R-matrix theory was originally developed to describe nuclear reactions. The framework was further extended to describe {beta} decay to unbound states. However, at the time writing, no clear description of {gamma} decays to unbound states exist. Such a description will be presented in this note.
Background: Spectroscopic factors, overlaps, and isospin symmetry are often used in conjunction with single-particle wave functions for the phenomenological analysis of nuclear structure and reactions. Many differing prescriptions for connecting these quantities to physically relevant asymptotic normalization constants or widths are available in the literature, but their relationship and degree of validity are not always clear. Purpose: This paper derives relationships among the above quantities of interest using well-defined methodology and starting assumptions. Method: $R$-matrix theory is used as the primary tool to interoperate between the quantities of interest to this work. Particular attention is paid to effects arising from beyond the nuclear surface, where isospin symmetry is strongly violated. Results: Relationships among the quantities of interest are derived. Example applications of these methods to mirror levels in nucleon+${}^{12}{rm C}$, nucleon+${}^{16}{rm O}$, and nucleon+${}^{26}{rm Al}$ are presented. A new approach to multi-level mirror symmetry is derived and applied to the first three $2^+$ states of ${}^{18}{rm O}$ and ${}^{18}{rm Ne}$. Conclusions: The relationship between the quantities of interest is clarified and certain procedures are recommended. It is found that the asymptotic normalization constant of the second $2^+$ state in ${}^{18}{rm Ne}$ deduced from the mirror state in ${}^{18}{rm O}$ is significantly larger than found in previous work. This finding has the effect of increasing the ${}^{17}{rm F}(p,gamma){}^{18}{rm Ne}$ reaction rate in novae.
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