  We use deep archival Chandra and XMM-Newton observations of three of the
brightest cluster-centre radio galaxies in the sky, Cygnus A, Hercules A and
Hydra A, to search for inverse-Compton emission from the population of
electrons responsible for the low-frequency radio emission. Using simulated
observations, we derive robust estimates for the uncertainties on the
normalization of an inverse-Compton component in the presence of the variations
in background thermal temperature actually seen in our target objects. Using
these, together with the pressures external to the lobes, we are able to place
interesting upper limits on the fraction of the energy density in the lobes of
Hydra A and Her A that can be provided by a population of relativistic
electrons with standard properties, assuming that the magnetic field is not
dominant; these limits are consistent with the long-standing idea that the
energy density in these lobes is dominated by a non-radiating particle
population. In Cygnus A, we find evidence in the spectra for an additional hard
component over and above the expected thermal emission, which is plausibly a
detection of inverse-Compton emission; even in this case, though, some
additional non-radiating particles and/or a departure from our standard
assumptions on the electron spectrum are necessary to allow pressure balance at
the mid-point of the lobes. As this is not the case in other Fanaroff-Riley
class II radio galaxies, we suggest that the rich environment of Cygnus A may
have some effect on its lobe particle content.
