A new R-matrix module for multi-channel calculations with GECCCOS

. A versatile new R-matrix module for multi-channel reaction calculations was introduced into the code GECCCOS (GEneral Coupled-Channel COde System) which has been developed by the nuclear data group at TU-Wien to perform nuclear reaction calculations especially for light nuclear systems. It provides a tool for phenomenological R-matrix analyses of reaction data combined with calculations of a potential-based calculable R-matrix using the Lagrange-mesh technique. In addition it provides a platform for the development of non-standard extensions of R-matrix theory such as Reduced R-matrix analyses and the Hybrid R-matrix. A successful run of the code yields the complete S-matrix (collision matrix) as well as observables for unpolarized beams, angle-di ↵ erential cross sections, excitation functions and, if existing, angle-integrated cross sections. Recently, extensions to polarization observables for spin-1 / 2 and spin-1 particles were implemented and tested. For phenomenological R-matrix analyses a separate module assembles calculated and available experimental values, automatically performs transformations with regard to reference frame and matching radii. Furthermore it allows to switch between incident channel and compound nucleus representation and provides the necessary feedback for the chi 2 ﬁtting process.


Introduction
The knowledge of consistent nuclear reaction data is crucial for the development of novel nuclear technology such as fusion devices as well as maintaining and improving existing solutions. Nuclear data evaluation provides such information by combining experimental data with nuclear model calculations. However, the latter are quantitatively not reliable for the resonance region which is usually described by R-matrix theory [1], an elegant tool satisfying conservation rules but not accounting for the complex microscopic structure. Thus R-matrix theory provides consistent reaction data, but has no predictive power. Especially, for nuclear systems with mass numbers A . 20 R-matrix analyses play an important role because the resonance regime extends up to rather high energies and the applicability of the statistical nuclear model or mean field approaches are questionable. However, the extension of the phenomenological R-matrix analysis to high energies is not feasible at present and requires new concepts to account for the increasing number of open channels.
In order to address these challenges the GECCCOS code has been set up by the group at TU-Wien. It has been constructed from scratch as a basic tool for the development and implementation of new and extended techniques for the calculation of nuclear reactions. In particular a new R-matrix module was implemented which includes both the calculable as well as the phenomenological R-matrix together with various non-standard extensions such as the Reduced R-matrix, an hybrid approach and a capability to ⇤ e-mail: thomas.srdinko@tuwien.ac.at ⇤⇤ e-mail: helmut.leeb@tuwien.ac.at transform the matching radius while maintaining the scattering matrix. For the optimization process the DAKOTApackage [2] is used. It o↵ers a huge variety of optimization and fitting algorithms as well as taking advantage of high performance computing.

GECCCOS and the R-matrix Module
The code GECCCOS is aimed to provide a versatile platform for the development and test of novel and extended techniques for coupled-channel calculations of nuclear reactions. Important aspects are the full knowledge and control of the implementation details. In the implementation emphasis was put on easy expandability and maintainability with well-defined interfaces between modules. The code set up a full coupled-channel environment and evaluates from a known S -matrix the scattering amplitude and the associated reaction observables. The S -matrix is obtained by calculations within specific models which are numerically implemented in dedicated modules. One of these is the new R-matrix module presented in this contribution. It can be used as an elegant tool to solve the coupled-channel equations, e.g. for potential scattering and direct reactions (calculable R-matrix), or provide a simple parametrization of the R-matrix as a sum of pole terms for optimization (phenomenological R-matrix).

Concept of R-matrix theory
The key idea in R-matrix theory is the division of the configuration space into an interior and exterior region. Originally introduced by Wigner and Eisenbud [1] the formalism has been elaborated by many authors, e.g. [3,4]. It is assumed that in the exterior region no polarizing forces are acting on the particles and the external wave function u ext (c 0 )c in channel c with incident channel c 0 is known up to the S -matrix element S c(c 0 ) , (1) Here, k c is the wave number of channel c and the functions I c and O c in Eq. (1) represent the incoming and outgoing solutions of the Coulomb functions for open channels, respectively, whereas W ⌘,l+ 1 2 describes the Whittaker function for closed channels with  c = |ik c |. The remaining quantities C c and A c are normalization constants.
The wave function in the interior region is of complex n-body structure and usually not known. For simplicity it is assumed [1] to be given by a superposition of basis functions j (r), The above coe cients c c j in Eq. (2) have to ensure continuity of the logarithmic derivative of the wave function at the intersection of the regions, i.e. at the matching radius r = a. Thus the R-matrix can be defined where B c is an arbitrary constant which does not a↵ect the observables [4] and µ c and µ c 0 are the reduced masses in the corresponding channels c and c 0 , respectively. The choice of B c might be of advantage in R-matrix analyses. Apart of the choice of any constant value of B c , e.g. B c = 0, the option B c = `(`is the angular momentum quantum number in channel c) is implemented in the Rmatrix module at present.

Calculable R-matrix
The R-matrix module o↵ers the option of calculable Rmatrix calculations [4] which make use of the R-matrix concept to solve the coupled-channel Schrödinger equa- where V c,c 0 (r) is the potential including coupling between the channels c and c 0 and T c and E c are the kinetic energy and the threshold in channel c, respectively. Nonorthogonality terms of the coupled-channel wave functions are neglected in this considerations. Due to the restriction of Eq. (4) to the interior region one has to restore hermiticity of the Hamiltonian via the introduction of a Bloch operator L c [5] both sides of Eq.(4) leading to the Bloch-Schrödinger equation Here the right hand side of Eq. (5) contains only the wave function u ext c (r = a) while the left hand side contains only quantities in the internal region.
Because of the restored hermiticity of the Hamiltonian on 0  r  a in Eq. (5) it is straightforward to show (see e.g. [4]) that the R-matrix can be written in the form where E n are real-valued pole energies and nc are the reduced width amplitudes. It should be remarked that the number of pole terms is equal to the product of channel number and number of basis functions. The scattering matrix S can be easily obtained from the R-matrix via From the known S-matrix the scattering amplitude and all observables can be calculated (e.g. [4]).
The calculable R-matrix formalism was implemented in the module using the Lagrange-mesh technique with associated basis functions [4]. The method proved to be very e cient and only a relatively small set of basis functions is required to describe scattering and bound states with high numerical accuracy.
The verification of the implementation was performed by comparison of a test example with results obtained by the FRESCO code [6]. In Fig. 1 the angle di↵erential cross sections of (n+ 12 C) are shown for the elastic as well as for two inelastic channels of rotational excitations. For all included channels the results of both calculations agree within the numerical accuracy.

Phenomenological R-matrix
The simple form of the R-matrix, Eq. (7), in terms of poles and reduced widths for an hermitean Hamiltonian o↵ers a phenomenological application of the R-matrix formalism without recourse to detailed microscopic interactions. In such an application all open channels should be included in the considered energy regime. Fixing the matching radius a and the included partial waves, one can adjust an arbitrary number of poles E n and associated reduced widths nc in order to achieve agreement with available experimental data of reaction observables. The method is widely and successfully used to describe reactions with resonant behaviour, especially at low incident energies.
The R-matrix module includes an option for phenomenological R-matrix analyses which evaluates the Smatrix from a set of resonance parameters {E n , nc }, n = 1, 2, . . . , a given a and a given set of partial waves. Choosing this option the GECCCOS code allows adjustment of the resonance parameters {E n , nc }, n = 1, 2, . . . and compare them with experimental data using the DAKOTApackage [2]. GECCCOS also o↵ers options for the adjustment of the normalization of experimental data, the matching radius a and the boundary parameter B c . A separate module of GECCCOS automatically links experimental data with calculated ones, transforms units and reference frames and returns an error estimate.
The participation in an R-matrix code comparison, coordinated by the Nuclear Data Section of IAEA, showed excellent agreement with other participating R-matrix codes in a test calculation of the 7 Be-system with a specific set of parameters defined [7].

Hybrid R-matrix
The phenomenological R-matrix has been successfully applied to describe resonant reactions at low energy. However, the applicability of R-matrix analyses is actually limited in energy because the number of open channels increases with energy and thus the number of resonance parameters to be adjusted. In general statistical model calculations and/or mean field models are used to describe the reaction data at higher energies. Frequently the two energy regimes are considered independently and the transition between R-matrix regime and statistical model calculations is not guaranteed.
In order to obtain a smooth transition between Rmatrix and statistical model regime a pseudo-potential V cc 0 is introduced which generates part of the observables of the statistical nuclear model in the transition region. Using the option of the calculable R-matrix one can determine R-matrix parameters associated with the pseudo-potential V cc 0 thus determining a background R-matrix R bg cc 0 . The subsequent analysis is based on the R-matrix It is assumed that R bg cc 0 accounts for all background poles, while the parameters of the second term describe only sharp resonances in the low energy region not accounted for by the potential. If the additional poles are rather sharp an almost smooth transition for those reactions described by the pseudo potential should be achieved. This pseudopotential defines the background poles but its choice is not trivial. A more elaborated description as well as example calculations using the GECCCOS-code were given in [8].

Transformation of the matching radius
The observables do not depend on the choice of the matching radius a and the boundary parameter B c as long as the exterior region satisfies the criterion given in Sect. 2.1. However, the associated R-matrix strongly depends on these parameters. The modification of the R-matrix with changes of B c is well known [4]. The impact of changes of a on R cc 0 is more subtle, but important because phenomenological R-matrix analyses frequently use small unphysical a-values in order to facilitate adjustment procedures.
The transformation of R cc 0 with regard to changes of a can be derived from Eq. (8) keeping the S-matrix unchanged. For a transformation from a toã one can rewrite Eq. (8) where we put the corresponding matching radius into the argument. Because the R-matrix enters linearly into Z O (ã) and Z I (ã) an equation for R(ã) can be obtained Here we made use of the notations⇢ = (k c ·ã cc 0 ), O = (O c (k cã ) cc 0 ), I = (I c (k cã ) cc 0 ) and R = (R cc 0 (ã)). Using Eq. (12) allows a reconstruction of the R-matrix at a given matching radiusã from a known S-matrix at each energy.
The transformation of the R-matrix with regard to the matching radius was implemented into the R-matrix module. The impact of the transformation on the R-matrix is illustrated in a schematic example of a two-channel system for n+ 16 O. Originally using a matching radius a = 3.3 fm we evaluated the R-matrix assuming poles in three partial waves, i.e. J ⇡ = 1/2 + (E n = 4.0 MeV, n1 =1.0, n2 =1.0), J ⇡ = 3/2 (E n = 1.5 MeV, n1 =0.2, n2 =0.0) and J ⇡ = 5/2 + (E n = 7.0 MeV, n1 =0.3, n2 =0.3). The R-matrix elements of the elastic channel for the matching radius a = 3.3 fm and the transformed one at a = 6.0 fm are displayed in Fig. 2 for several partial waves. It can be seen that the original poles of the R-matrix are shifted and new poles arise. In partial waves with originally no pole terms background structures appear in the R-matrix. It must be remarked that the S-matrix remains unchanged in all channels by this transformation. This behavior confirms analytical considerations in simplified models which clearly indicate that a transformation of the R-matrix with regard to the matching radius may lead to a countably infinite number of poles due to periodicity e↵ects.

Polarization Observables in GECCCOS
Recently, the set of observables calculated by GECCCOS was extended to polarization observables for spin-1/2 and spin-1 particles which contain additional information enhancing the sensitivity to partial wave contributions. As a first step the vector analyzing power A(✓) of spin-1/2 particles was implemented which equals in this case the vector polarization P(✓) given by where represents a vector of Pauli-matrices and f (✓) = f m 0 ,m (✓) is a matrix of scattering amplitudes for each possible spin transition m ! m 0 . Based on Eq. (13) the calculation of the polarization was implemented in GECCCOS.
A first validation of the extension was the calculation of the analyzing power of elastic n+ 16 O scattering using the parameters of an R-matrix analysis. Although the polarization data were not part of the R-matrix analysis there is fair agreement between the experimental values [9] and the one obtained by GECCCOS (Fig. 3).

Summary and Outlook
Up to now GECCCOS features standard R-matrix methods and several non-standard procedures. Currently the calculable multi-channel R-matrix with Lagrange-mesh technique [4] as well as the phenomenological R-matrix approach are implemented and successfully validated [7]. GECCCOS was also successfully used to develop and implement non-standard R-matrix procedures like the hybrid R-matrix and the reduced R-matrix. The first promises a smooth transition to the statistical nuclear model regime [8] while the latter aims to account for channels which cannot explicitly be treated in R-matrix analyses, e.g. nonbinary channels. First reduced R-matrix analyses are in progress as well as the development of an R-matrix Faddeev formalism for three-body breakup channels. The recent addition of polarization observables enhances the capabilities of the code.
Apart of the extended capabilities the code is still in a prototypic state. The interface is command-line driven with a set of shell scripts. However, a graphical user interface is already in development including a prototype of a spreadsheet editor for pole terms. Tools to make the code useful for a broader user community, such as version control (git) and automated code documentation (Doxygen) are already in place. Streamlining code e ciency and the program flow for nuclear data evaluation are ongoing.