Module rmatrix_limited 

R-Matrix Limited (LRF=7) cross-section calculation.

Computes energy-dependent cross-sections (total, elastic, capture, fission) from ENDF LRF=7 resonance parameters using the full multi-channel R-matrix formalism.

Relationship to Reich-Moore (LRF=3)

Reich-Moore is a special case of R-matrix theory that eliminates the capture channel via the Wigner-Eisenbud approximation, collapsing the level matrix to a scalar per spin group. LRF=7 retains all channels (elastic, capture, fission) explicitly, requiring an NCH×NCH complex matrix inversion per spin group per energy point.

Key Formulas

(ENDF-6 Formats Manual Appendix D; SAMMY manual §3.1; SAMMY rml/mrml07.f Pgh + mrml11.f Setxqx)

R-matrix:
  R_cc'(E) = Σ_n γ_nc · γ_nc' / (E_n - E)   [complex for KRM=3, NCH×NCH, symmetric]
  γ_nc = reduced width amplitude for resonance n in channel c (eV^{1/2})

Level denominator per channel:
  L_c(E) = S_c(E) - B_c + i·P_c(E)
  P_c = penetrability,  S_c = shift factor,  B_c = boundary condition

Reduced level matrix (SAMMY "Ymat"):
  Ỹ_cc'(E) = δ_cc' / L_c(E) - R_cc'(E)   [complex, NCH×NCH]
  → Ỹinv = Ỹ⁻¹  (invert Ỹ)

Intermediate matrix (SAMMY "XXXX"):
  Ξ_cc'(E) = (√P_c / L_c) · (Ỹinv · R)_cc' · √P_c'

Collision matrix (SAMMY manual eq. III.D.4):
  W_cc' = δ_cc' + 2i·Ξ_cc'
  U_cc' = Ω_c · W_cc' · Ω_c'    where Ω_c = exp(-iφ_c)

  TRUTH SOURCE: SAMMY rml/mrml11.f lines 14-18 (W = I + 2i·XXXX)
  and lines 84-88 (elastic formula consistent with e^{-2iφ}).
  Unitarity: |U| ≤ 1 always; hard sphere (R=0) → U = exp(2iφ)·I  ✓

Cross sections per spin group (J,π), summed over entrance neutron channels c0:
  σ_total   = Σ_{c0} 2·(π/k²)·g_J·(1 - Re(U_{c0,c0}))
  σ_elastic = Σ_{c0} (π/k²)·g_J·|1 - U_{c0,c0}|²
  σ_fission = Σ_{c0} Σ_{c'∈fission} (π/k²)·g_J·|U_{c0,c'}|²
  σ_capture = σ_total - σ_elastic - σ_fission

Photon Channels

For channels involving zero-rest-mass particles (photons, MA=0), the penetrability is set to 1.0 and phase shifts to 0.0, following the standard R-matrix convention (ENDF-6 Formats Manual §2.2.1.6 Note 4).

SAMMY Reference

• rml/mrml01.f — LRF=7 reader (Scan_File_2, particle pair loop)
• rml/mrml09.f — Level matrix inversion (Yinvrs, Xspfa, Xspsl)
• rml/mrml11.f — Cross-section calculation (Sectio, Setxqx)
• SAMMY manual §3.1 (multi-channel R-matrix)

Functions

cross_sections_for_rml_range

Compute cross-section contributions from an LRF=7 energy range.