Recent Advances in R-matrix Data
Analysis
Ian J. Thompson
1 R-matrix Theory
R-matrix theory is designed to describe individual resonances in two-body scattering even when overlapping, and the non-resonant background between them. It
describes all the asymptotic properties of the relative wave function outside some
fixed radius a in terms of pole energies e p and reduced width amplitudes γ pα for
each partial-wave channel α and pole p. The γ pα can be calculated from some
structure theory, or fitted to data.
R-matrix theory is the starting point for compound-nucleus models. It is the basis
for making statistical approximations, such as the Reich–Moore approximation, and
Hauser-Feshbach models. It can be used to check the accuracy of those approximate
models, as well as models for the width-fluctuation corrections.
The foundation of R-matrix theory is summarized in the landmark paper of
Lane and Thomas [1]. In that paper is the foundational “R-matrix Theorem”: For
Hermitian H = T + V + ˆ
B with Bloch operator ˆ
B = δ(r−a)(
d
dr −
B
r ), with V = 0
only for r ∈ [0, a] and E-independent, then the exact scattering solution H ψ = Eψ
can be represented by a R-matrix at r = a with a set of pole energies e p and reduced
width amplitudes γ pα as
R αα (E) =
∞
p=1
γ pα γ pα
e p − E
.
(1)
I. J. Thompson ()
Lawrence Livermore National Laboratory, Livermore, CA, USA
e-mail: I-Thompson@llnl.gov; thompson97@llnl.gov
© This is a U.S. government work and not under copyright protection
in the U.S.; foreign copyright protection may apply 2021
J. Escher et al. (eds.), Compound-Nuclear Reactions, Springer Proceedings in
Physics 254, https://doi.org/10.1007/978-3-030-58082-7_4
37
Analysis
Ian J. Thompson
1 R-matrix Theory
R-matrix theory is designed to describe individual resonances in two-body scattering even when overlapping, and the non-resonant background between them. It
describes all the asymptotic properties of the relative wave function outside some
fixed radius a in terms of pole energies e p and reduced width amplitudes γ pα for
each partial-wave channel α and pole p. The γ pα can be calculated from some
structure theory, or fitted to data.
R-matrix theory is the starting point for compound-nucleus models. It is the basis
for making statistical approximations, such as the Reich–Moore approximation, and
Hauser-Feshbach models. It can be used to check the accuracy of those approximate
models, as well as models for the width-fluctuation corrections.
The foundation of R-matrix theory is summarized in the landmark paper of
Lane and Thomas [1]. In that paper is the foundational “R-matrix Theorem”: For
Hermitian H = T + V + ˆ
B with Bloch operator ˆ
B = δ(r−a)(
d
dr −
B
r ), with V = 0
only for r ∈ [0, a] and E-independent, then the exact scattering solution H ψ = Eψ
can be represented by a R-matrix at r = a with a set of pole energies e p and reduced
width amplitudes γ pα as
R αα (E) =
∞
p=1
γ pα γ pα
e p − E
.
(1)
I. J. Thompson ()
Lawrence Livermore National Laboratory, Livermore, CA, USA
e-mail: I-Thompson@llnl.gov; thompson97@llnl.gov
© This is a U.S. government work and not under copyright protection
in the U.S.; foreign copyright protection may apply 2021
J. Escher et al. (eds.), Compound-Nuclear Reactions, Springer Proceedings in
Physics 254, https://doi.org/10.1007/978-3-030-58082-7_4
37