A unified analytical framework for Mössbauer synchrotron sources.

Next-generation Mössbauer spectroscopy at synchrotron and X-ray free-electron laser facilities demands rapid, accurate and polarization-aware modeling of nuclear hyperfine interactions. We present a unified analytical framework that provides exact, rotationally invariant expressions for resonance energies and transition probabilities in the presence of simultaneous magnetic dipole and electric quadrupole interactions. Unlike conventional approaches, our method avoids Hamiltonian diagonalization by expressing intensities entirely in terms of hyperfine invariants, enabling efficient global fitting and modeling of hyperfine-interaction distributions in complex materials. We further introduce a quantitative identifiability metric and demonstrate, via Monte Carlo sampling, that polarization control-particularly orthogonal linear polarizations-substantially improves hyperfine-parameter determination. This work offers a mathematically transparent and computationally efficient toolset for modern Mössbauer spectroscopy, accelerating studies of iron-based compounds and magnetic, electronic and structural order under extreme conditions and at nanoscale geometries.