Cool cores of some galaxy clusters exhibit faint radio minihalos. Their origin is unclear; their study has been limited by their small number. We undertook a systematic search for minihalos in a large sample of X-ray luminous clusters with high-quality radio data. In this paper, we report four new minihalos (A 478, ZwCl 3146, RXJ 1532.9+3021 and A 2204), and five candidates, found in the reanalyzed archival Very Large Array observations. The radio luminosities of our minihalos and candidates are in the range $10^{23-25}$ W Hz$^{-1}$ at 1.4 GHz, consistent with this type of radio sources. Their sizes (40-160 kpc in radius) are somewhat smaller than those of the previously known minihalos. We combine our new detections with previously known minihalos, obtaining a total sample of 21 objects, and briefly compare the cluster radio properties to the average X-ray temperature and the total masses estimated from Planck. We find that nearly all clusters hosting minihalos are hot and massive. Beyond that, there is no clear correlation between the minihalo radio power and cluster temperature or mass (in contrast with the giant radio halos found in cluster mergers, whose radio luminosity correlates with the cluster mass). Chandra X-ray images indicate gas sloshing in the cool cores of most of our clusters, with minihalos contained within the sloshing regions in many of them. This supports the hypothesis that radio-emitting electrons are reaccelerated by sloshing. Advection of relativistic electrons by the sloshing gas may also play a role in the formation of the less-extended minihalos.
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