Direct Boundary Matching: A Bound-State Technique for Nuclear Scattering with Lagrange-Legendre Functions

TL;DR

Direct boundary matching (DBMM) reframes nuclear scattering as a bound-state problem on a finite interval using a $L^2$-convergent $Lagrange$-$Legendre$ basis, embedding the outgoing-wave boundary condition in the last row of the matrix. This real-space approach avoids Bloch operators and two-step R-matrix matching, yielding scattering observables directly from a single solve. The formalism naturally extends to $n_c$ coupled channels via a block-matrix structure that differentiates the entrance channel through an effective source potential. Benchmarking proton+$^{12}$C scattering at $E_{lab}=30$ MeV against Numerov shows exceptional agreement for the S-matrix elements across partial waves, with $|S_\ell|$ and phase matching to within $2.5\times 10^{-5}$ and $0.01^\circ$, respectively. Implemented in SLAM.jl, DBMM offers conceptual clarity, straightforward implementation, and potential for CDCC and unified structure-reaction calculations.

Abstract

I present a direct boundary matching method (DBMM) for solving nuclear scattering problems using Lagrange-Legendre basis functions. This approach belongs to the family of bound-state techniques for the continuum, reformulating scattering problems into a localized, square-integrable ($L^2$) representation. The key feature is the direct incorporation of the outgoing wave boundary condition into the last row of the matrix equation, eliminating the need for Bloch operators and two-step matching procedures required in traditional R-matrix methods. Unlike the complex scaling method that rotates coordinates into the complex plane, DBMM operates entirely in real coordinate space. The formalism is extended to coupled-channel problems, where the wave function decomposition naturally leads to an effective source potential that distinguishes between the entrance channel and other channels. Benchmark calculations for p~+~$^{12}$C scattering demonstrate excellent agreement with the Numerov integration method.

Figures (3)

• Figure 1: S-matrix elements for p + $^{12}$C scattering at $E_{\rm lab} = 30$ MeV. (a) Modulus $|S_\ell|$ and (b) phase $\arg(S_\ell)$ as functions of orbital angular momentum $\ell$. Circles: Numerov method; squares: DBMM. The two methods show excellent agreement for all partial waves.
• Figure 2: Argand diagram showing the S-matrix trajectory in the complex plane for p + $^{12}$C at $E_{\rm lab} = 30$ MeV. Numbers indicate the orbital angular momentum $\ell$. The dashed gray circle represents the unitarity limit $|S| = 1$. Low partial waves lie inside the unit circle (absorption), while high partial waves approach unity (elastic scattering).
• Figure 3: Radial wave functions for p + $^{12}$C at $E_{\rm lab} = 30$ MeV. (a) Real part and (b) imaginary part for $\ell = 0$; (c) real part and (d) imaginary part for $\ell = 5$. Solid lines: Numerov method; circles: DBMM evaluated at Lagrange mesh points. The excellent agreement validates the Lagrange-Legendre expansion across all partial waves.