4.1. Radio halos
Cluster radio halos are the most spectacular expression of clusters non-thermal emission. They permeate the cluster centers with size of more than a Mpc, showing low surface brightnesses ( 10-6Jy arcsec-2 at 21 cm) and steep spectra ( > ~ 1). A typical example is Coma C, the halo source in the Coma cluster, which was first shown to be diffuse by Willson  and mapped later at various radio wavelengths by several authors [28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. In Fig. 3 we show a radio image at 90 cm of the Coma cluster , obtained with the Westerbork Synthesis Radio Telescope (WSRT). The integrated spectrum of Coma C is 1.3, with a steepening at high frequencies . The spectral index distribution shows a radial decrease from ~ 0.8 at the cluster center, to ~ 1.8 at about 15' from the center .
Figure 3. WSRT radio image  of the Coma cluster at 90 cm with a resolution (FWHM) of 55" × 125" (RA × DEC). The cluster center is approximately located at the position RA1950 = 12h 57m 24s, DEC1950 = 28°15'00". The radio halo Coma C is at the cluster center, the radio relic 1253+275 is at the cluster periphery. The gray-scale range display total intensity emission from 2 to 30 mJy/beam whereas contour levels are at 3, 5, 10, 30, 50 mJy/beam. The Coma cluster is at a redshift of 0.023, such that 1" is 0.46 kpc.
Studies of several radio halos and of their hosting clusters have been recently performed, thus improving the knowledge of the characteristics and physical properties of this class of radio sources. Radio halos have been studied in X-ray luminous clusters such as A2163  and 1E0657-57 , and in distant clusters, such as A2744  (z = 0.308), and CL0016+16  (z = 0.5545). The latter is the most distant cluster with a radio halo known so far. Halos of small size, i.e. ~ 500-600 kpc have also been detected in some cases (e.g. in A2218  and in A3562 ).
No polarized flux has been detected so far in radio halos. In the Coma cluster, the upper limit to the fractional polarization of Coma C is of ~ 10% at 1.4 GHz . Upper limits of ~ 6.5% and of ~ 4% have been obtained for the two powerful radio halos in A2219  and A2163 , respectively. Also, no significant polarization is reported for 1E0657-57 . The interpretation of these low polarization levels is that the thermal gas has become mixed with the relativistic plasma, thus internal depolarization occurs within the radio emitting plasma. In addition, the magnetic field may be disordered on scales smaller than the observing beam, thus producing significant beam depolarization.
Due to their low surface brightness, radio halos have been studied so far with low spatial resolution. This prevents a detailed investigation of the small-scale magnetic field geometry and intensity. Using minimum energy assumptions (see Sec. 3.2), it is possible to estimate an equipartition magnetic field strength averaged over the entire halo volume, i.e. on scales as large as ~ 1 Mpc. The derived minimum energy densities in halos are of the order of 10-14 - 10-13 erg cm-3, i.e. much lower than the energy density in the thermal gas. These calculations typically assume equal energy in relativistic protons and electrons (k = 1), a volume filling factor = 1, a low frequency cut-off of 10 MHz, and a high frequency cut-off of 10 GHz. The corresponding equipartition magnetic field strengths range from 0.1 to 1 µG. In the Coma cluster, a minimum energy density of 1.9 × 10-14 erg cm-3 is derived from the radio data in Coma C. The corresponding  equipartition magnetic field is 0.45 µG. In the equipartition approximation, a homogeneous cluster magnetic field is assumed, but this is probably a too simple picture. Important clues about a radial decrease of the magnetic field strength in clusters of galaxies are given in Sec. 8.
Several models for the origin of the relativistic radiating electrons in the ICM have been proposed (see, e.g., recent reviews [44, 45, 46, 47] and references therein). These can be basically divided in two different scenarios:
Primary electron models [48, 49] in which relativistic electrons are injected in the ICM from AGN activity (quasar, radio galaxies, etc.) and/or from star formation in galaxies (supernovae, galactic winds, etc.). The radiative lifetime of the relativistic electrons is relatively short (~ 107-8 yrs). Therefore models involving a primary origin of the relativistic electrons require continuous injection processes and/or reacceleration processes in order to explain the presence of diffuse non-thermal emission out to Mpc scales. Electrons are likely reaccelerated in the gas turbulence [31, 50, 51, 52] or in shocks [53, 54, 55], although the efficiency of the latter process is debated [56, 57].
Secondary electron models [58, 59, 60] in which cosmic ray electrons result as secondary products of hadronic collisions between relativistic protons and ICM thermal protons. The relativistic protons in the ICM have lifetimes of the order of the Hubble time. Thus they are able to travel a large distance from their source before they release their energy. In this way, electrons are produced through the whole cluster volume and do not need to be reaccelerated. The production of relativistic electrons by secondary models predict large gamma-ray fluxes from neutral pion decay which could be tested by future gamma-ray missions.
On the observational side, it is possible to draw some of the general characteristics of radio halos and derive correlations with other cluster properties:
i) Halos are typically found in clusters with significant
substructure and deviation from spherical symmetry in the X-ray morphology
This is confirmed by the high resolution X-ray data
obtained with Chandra and XMM
In addition to the
distorted X-ray morphology, all the clusters with halos exhibit
strong gas temperature gradients. Some clusters show a spatial
correlation between the radio halo brightness and the hot gas regions,
although this is not a general feature
ii) In a number of well-resolved clusters, a point-to-point spatial correlation is observed between the radio brightness of the halo and the X-ray brightness as detected by ROSAT . This correlation is visible e.g. in A2744 also in the Chandra high resolution data .
iii) Halos are present in rich clusters, characterized by high X-ray luminosities and temperatures . The percentage of clusters with halos in a complete X-ray flux-limited sample (that includes systems with LX > 5 × 1044 h50-2 erg s-1 in the 0.1 - 2.4 keV band) is 5%. The halo fraction increases with the X-ray luminosity, to 33% for clusters with LX > 1045 h50-2 erg s-1.
iv) The radio power of a halo strongly correlates with the cluster luminosity [39, 43, 72] the gas temperature [39, 73], and the total mass .
Therefore the available data suggest that radio halos seem to be strictly related to the X-ray properties of the host clusters and to the presence of cluster merger processes, which can provide the energy for the electron reacceleration and magnetic field amplification on large scales. From energetic grounds, mergers can indeed supply enough kinetic energy for the maintenance of a radio halo, as first suggested by Harris et al. .
The observed link between radio halos and cluster mergers is in favor of primary electron models. These are also supported by the high frequency steepening of the integrated radio spectra (e.g. in Coma C ) and by the radial steepening of the two-frequency spectra in Coma C , A665  and A2163 . These spectral behaviors can be easily reproduced by models invoking reacceleration of particles. On the contrary, they are difficult to explain by models considering secondary electron populations.