  The nature of the interaction between low-excitation gas filaments at ~10^4
K, seen in optical line emission, and diffuse X-ray emitting coronal gas at
~10^7 K in the centers of galaxy clusters remains a puzzle. The presence of a
strong, empirical correlation between the two gas phases is indicative of a
fundamental relationship between them, though as yet of undetermined cause. The
cooler filaments, originally thought to have condensed from the hot gas, could
also arise from a merger or the disturbance of cool circumnuclear gas by
nuclear activity. Here, we have searched for intrinsic line emission
polarization in cool core galaxy clusters as a diagnostic of fundamental
transport processes. Drawing on developments in solar astrophysics, direct
energetic particle impact induced polarization holds the promise to
definitively determine the role of collisional processes such as thermal
conduction in the ISM physics of galaxy clusters, while providing insight into
other highly anisotropic excitation mechanisms such as shocks, intense
radiation fields and suprathermal particles. Under certain physical conditions,
theoretical calculations predict of order ten percent polarization. Our
observations of the filaments in four nearby cool core clusters place stringent
upper limits (<0.1%) on the presence of emission line polarization, requiring
that if thermal conduction is operative, the thermal gradients are not in the
saturated regime. This limit is consistent with theoretical models of the
thermal structure of filament interfaces.
