  We report the discovery of spatially-extended, non-thermal-like emission
components in Chandra X-ray spectra for five of a sample of seven massive,
merging galaxy clusters with powerful radio halos. The emission components can
be fitted by power-law models with mean photon indices in the range 1.5 < Gamma
< 2.0. A control sample of regular, dynamically relaxed clusters, without radio
halos but with comparable mean thermal temperatures and luminosities, shows no
compelling evidence for similar components. Detailed X-ray spectral mapping
reveals the complex thermodynamic states of the radio halo clusters. Our
deepest observations, of the Bullet Cluster 1E 0657-56, demonstrate a spatial
correlation between the strongest power-law X-ray emission, highest thermal
pressure, and brightest 1.34GHz radio halo emission in this cluster. We confirm
the presence of a shock front in the 1E 0657-56 and report the discovery of a
new, large-scale shock front in Abell 2219. We explore possible origins for the
power-law X-ray components. These include inverse Compton scattering of cosmic
microwave background photons by relativistic electrons in the clusters;
bremsstrahlung from supra-thermal electrons energized by Coulomb collisions
with an energetic, nonthermal proton population; and synchrotron emission
associated with ultra-relativistic electrons. Interestingly, we show that the
power-law signatures may also be due to complex temperature and/or metallicity
structure in clusters particularly in the presence of metallicity gradients. In
this case, an important distinguishing characteristic between the radio halo
clusters and control sample of predominantly cool-core clusters is the
relatively low central X-ray surface brightness of the former.
