Micro-environment of the Eu interstitial in $\beta$-SiAlON:Eu$^{2+}$ green phosphor

The precise atomic-scale structure around Eu$^{2+}$ activators in the $\beta$-Si$_{6-z}$Al$_z$O$_z$N$_{8-z}$:Eu$^{2+}$ commercial green phosphor remains elusive. We use the first-principles $\Delta$SCF excited-state method, embedding of the interatomic force constants for supercells up to 3501 atoms, and Huang-Rhys theory to clarify this issue. Monte Carlo exploration is used to identify representative low-energy structural models spanning different levels of Al/O concentration $z$. For the lowest-energy structure at low $z$, our computed photoluminescence spectrum reproduces the experimental vibronic peaks at 6~K with excellent agreement in peak positions and intensities, validating the Eu-N$_9$ coordination model with Al, O, and Eu confined to the same crystallographic plane. Analysis of the low-energy structures reveals that the electron-phonon coupling is weak ($S \approx 2.15$) with a robust characteristic phonon signature across different Al/O arrangements, explaining the surprising persistence of resolved phonon replicas with increasing $z$. We explain the experimentally observed red-shift of emission with increasing $z$ through systematic trends in zero-phonon line energies, modest increases in Huang-Rhys factors, and larger configurational diversity at higher compositions.