We will unravel the physics of particle acceleration and cosmic rays in dilute cluster plasmas using the LOFAR Low-Band Antennas (LBA). With LOFAR’s sensitivity, survey speed, and resolution – combined with the latest developments in correcting for Earth’s distorting ionosphere – we can now take the first step in ultra-low frequency studies of galaxy clusters. Galaxy clusters are the Universe’s largest gravitationally bound structures. Clusters grow by a sequence of mergers and Mpc-size diffuse radio sources trace plasma shock waves and turbulence, generated when clusters collide and merge. In these magnetized plasmas, cosmic rays are accelerated to extreme energies emitting radio synchrotron radiation. It is still unclear how these particles are accelerated and what the role is of relativistic plasma deposited in the intracluster plasma by active galactic nuclei. We will develop techniques that correct for ionospheric blurring, building upon recent advances in interferometric ionospheric calibration. Developing these techniques will also be crucially important to exploit the full sensitivity of the planned Square Kilometre Array (SKA). Given its similarities, LOFAR is the ideal pathfinder instrument to advance and test low-frequency ionospheric calibration for the SKA. Enabled by the ionospheric calibration, we will carry out the first systematic study of galaxy clusters at frequencies below 100 MHz with the LBA. With our observations, we will characterize ultra-steep spectrum radio sources that only radiate at MHz frequencies. These observations will directly test models that predict that turbulence and old “fossil” plasma from active galactic nuclei play an important role in the acceleration of cosmic rays. These results will be essential for our understanding of galaxy cluster formation, its associated energy budget, and the origin of cosmic rays in dilute cluster plasmas.