  Some galaxy clusters show diffuse radio emission in the form of giant halos
(GHs) on Mpc scales or minihalos (MHs) on smaller scales. Comparing VLA and XMM
radial profiles of several such clusters, we find a universal linear
correlation between radio and X-ray surface brightness, valid in both types of
halos. It implies a halo central emissivity
nu*j_nu=10^{-31.4+-0.2}*(n/10^-2cm^-3)^2*(T/T_0)^{0.2+-0.5} erg/s/cm^3, where T
and T_0 are the local and central temperatures, and n is the electron number
density. We argue that the tight correlation and the scaling of j_nu, combined
with morphological and spectral evidence, indicate that both GHs and MHs arise
from secondary electrons and positrons, produced in cosmic-ray ion (CRI)
collisions with a strongly magnetized, B >3 muG intracluster gas. When the
magnetic energy density drops below that of the microwave background, the radio
emission weakens considerably, producing halos with a clumpy morphology (e.g.
RXCJ2003.5-2323 and A2255) or a distinct radial break. We thus measure a
magnetic field B=3 muG at a radius r~110 kpc in A2029 and r~50 kpc in Perseus.
The spectrum of secondaries, produced from hadronic collisions of ~20 GeV CRIs,
reflects the energy dependence of the collision cross section. We use the
observed spectra of halos, in particular where they steepen with increasing
radius or frequency, to (i) measure B~10(nu/700 MHz) muG, with nu the spectral
break frequency; (ii) identify a correlation between the average spectrum and
the central magnetic field; and (iii) infer a CRI spectral index s <-2.7 and
energy fraction xi_p~10^{-3.6+-0.2} at particle energies above 10 GeV. Our
results favor a model where CRIs diffuse away from their sources (which are
probably supernovae, according to a preliminary correlation with star
formation), whereas the magnetic fields are generated by mergers in GHs and by
core sloshing in MHs.
