The ultra-steep diffuse radio emission observed in the cool-core cluster
  RX J1720.1+2638 with LOFAR at 54 MHz
  Diffuse radio emission at the centre of galaxy clusters has been observed
both in merging clusters on scales of Mpc, called giant radio haloes, and in
relaxed systems with a cool-core on smaller scales, named mini haloes. Giant
radio haloes and mini haloes are thought to be distinct classes of sources.
However, recent observations have revealed the presence of diffuse radio
emission on Mpc scales in clusters that do not show strong dynamical activity.
RX J1720.1+2638 is a cool-core cluster, presenting both a bright central mini
halo and a fainter diffuse, steep-spectrum emission extending beyond the
cluster core that resembles giant radio halo emission. In this paper, we
present new observations performed with the LOFAR Low Band Antennas (LBA) at 54
MHz. These observations, combined with data at higher frequencies, allow us to
constrain the spectral properties of the radio emission. The large-scale
emission presents an ultra-steep spectrum with $\alpha_{54}^{144}\sim3.2$. The
radio emission inside and outside the cluster core have strictly different
properties, as there is a net change in spectral index and they follow
different radio-X-ray surface brightness correlations. We argue that the
large-scale diffuse emission is generated by particles re-acceleration after a
minor merger. While for the central mini halo we suggest that it could be
generated by secondary electrons and positrons from hadronic interactions of
relativistic nuclei with the dense cool-core gas, as an alternative to
re-acceleration models.
