  Up to now, experiments involving M\"ossbauer nuclei have been restricted to
the low-excitation regime. The reason for this is the narrow spectral line
width of the nuclei. This defining feature enables M\"ossbauer spectroscopy
with remarkable resolution and convenient control and measurements in the time
domain, but at the same time implies that only a tiny part of the photons
delivered by accelerator-based x-ray sources with orders-of-magnitude larger
pulse bandwidth are resonant with the nuclei. X-ray free-electron lasers
promise a substantial enhancement of the number of nuclear-resonant photons per
pulse, such that excitations beyond the low-excitation (LER) regime come within
reach. This raises the question, how the onset of non-linear excitations could
be experimentally verified. Here, we develop and explore a method to detect an
excitation of nuclear ensembles beyond the LER for ensembles of nuclei embedded
in x-ray waveguides. It relies on the comparison of the x-rays coherently and
incoherently scattered off of the nuclei. As a key result, we show that the
ratio of the two observables is constant within the LER, essentially
independent of the details of the nuclear system and the characteristics of the
exciting x-rays. Conversely, deviations from this equivalence serve as a direct
indication of excitations beyond the LER. Building upon this observation, we
develop a variety of experimental signatures both, for near-instantaneous
impulsive and for temporally-extended non-impulsive x-ray excitation.
Correlating coherently and incoherently scattered intensities further allows
one to compare theoretical models of nonlinear excitations more rigorously to
corresponding experiments.
