Clusters of galaxies are among the best candidates for particle acceleration sources in the universe, a signature of which is non-thermal hard X-ray emission from the accelerated relativistic particles. We present early results on Suzaku observations of non-thermal emission from Abell 3376, which is a nearby on-going merger cluster. Suzaku observed the cluster twice, focusing on the cluster center containing the diffuse radio emission to the east, and cluster peripheral region to the west. For both observations, we detect no excess hard X-ray emission above the thermal cluster emission. An upper limit on the non-thermal X-ray flux of $2.1times10^{-11}$ erg cm$^{-2}$ s$^{-1}$ (15--50 keV) at the 3$sigma$ level from a $34times34$ arcmin$^2$ region, derived with the Hard X-ray Detector (HXD), is similar to that obtained with the BeppoSAX/PDS. Using the X-ray Imaging Spectrometer (XIS) data, the upper limit on the non-thermal emission from the West Relic is independently constrained to be $<1.1times10^{-12}$ erg s$^{-1}$ cm$^{-2}$ (4$-$8 keV) at the 3$sigma$ level from a 122 arcmin$^2$ region. Assuming Compton scattering between relativistic particles and the cosmic microwave background (CMB) photons, the intracluster magnetic field $B$ is limited to be $>0.03mu$G (HXD) and $>0.10mu$G (XIS).
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