LOFAR detection of extended emission around a mini halo in the galaxy cluster Abell 1413

Context. The relation between giant radio halos and mini halos in galaxy clusters is not understood. The former are usually associated with merging clusters, while the latter are found in relaxed systems. In recent years, the advent of low-frequency radio observations has challenged this dichotomy by finding intermediate objects with a hybrid radio morphology. Aims. We aim to investigate the presence of diffuse radio emission in the cluster Abell 1413 and determine its dynamical status to explore the relation between mini halos and giant radio halos. Methods. We used LOFAR observations centred at 144 MHz to study the diffuse radio emission. To investigate the dynamical state of the system, we used newly analysed XMM-Newton archival data. Abell 1413 shows features that are typically present in both relaxed (e.g., peaked X-ray surface brightness distribution and some large-scale inhomogeneities) and disturbed (e.g., flatter temperature and metallicity profiles) clusters. Results. This suggests that Abell 1413 is neither disturbed nor fully relaxed, and we argue that it is an intermediate-phase cluster. At 144 MHz, we discover a wider diffuse component surrounding the previously known mini halo at the cluster center. By fitting the radio surface-brightness profile with a double-exponential model, we can disentangle the two components. We find an inner mini halo with an e-folding radius, r(e, 1) = 28 +/- 5 kpc, and an extended component with r(e, 2) = 290 +/- 60 kpc. We also evaluated the point-to-point correlation between the radio and X-ray surface brightness, finding a sublinear relation for the outer emission and a superlinear relation for the mini halo. The mini halo and the diffuse emission extend over different scales and show different features, confirming the double nature of the radio emission and suggesting that the mechanisms responsible for the re-acceleration of the radio-emitting particle might be different.