Probing the Linewidth of the 12.4-keV Solid-State $^{45}$Sc Isomeric Resonance

The $^{45}$Sc nuclear transition from the ground to the isomeric state at 12.389~keV, with a lifetime of 0.46~s, exhibits an extraordinarily narrow natural width of 1.4~feV and a quality factor $\simeq 10^{19}$ -- surpassing those of the most precise atomic clocks -- making $^{45}$Sc a compelling platform for advanced metrology and nuclear clocks. Here we investigate how closely the spectral width and quality factor of the solid-state $^{45}$Sc resonance can approach these natural limits. Using the European X-ray Free-Electron Laser, we confirm the isomer's lifetime via time-delayed incoherent $K_{\alpha,\beta}$ fluorescence and observe previously unreported elastic fluorescence, yielding a partial internal conversion coefficient of 390(60). The absence of a clear nuclear forward scattering signal beyond a 2-ms delay implies environmental broadening of at least $500~\Gamma_{0}$ under experimental conditions, placing bounds on solid-state decoherence mechanisms. These findings set new experimental benchmarks for solid-state nuclear clock development.