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The relation between giant radio halos and mini-halos in galaxy clusters isnot understood. The former are usually associated with merging clusters, thelatter are found in relaxed systems. In the last years, the advent oflow-frequency radio observations has challenged this dichotomy, findingintermediate objects with a hybrid radio morphology. We aim to investigate thepresence of diffuse radio emission in the cluster Abell 1413 and determine itsdynamical status. We used LOFAR HBA observations centred at 144 MHz to studythe diffuse emission hosted by this cluster.To investigate the dynamical stateof the system, we complete our study with newly analysed XMM-Newton archivaldata. A1413 shows features that are typically present in both relaxed (e.g.,peaked x-ray surface brightness distribution and little large-scaleinhomogeneities) and disturbed (e.g., flatter temperature and metallicityprofiles) clusters.This evidence supports the scenario that A1413 is neither adisturbed nor fully relaxed object. We argue that it is an intermediate-phasecluster.Using radio observations at 144 MHz, we discover the presence of awider diffuse component surrounding the previously reported mini-halo at thecluster centre. By fitting the radio surface brightness profile with adouble-exponential model, we can disentangle the two components. We find aninner mini-halo with an e-folding radius r_e1=28 kpc and the extended componentwith r_e2 = 290 kpc. We also performed point-to-point correlations betweenradio and X-ray surface brightness, finding a sub-linear relation for the outeremission and a super-linear relation for the mini-halo.The mini-halo and thediffuse emission extend over different scales and show different features,confirming the double nature of the radio emission and suggesting that themechanisms responsible for the re-acceleration of the radio-emitting particlemight be different.

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