  We developed a magneto-turbulent model for the cosmic ray (CR) electrons seen
in the radio halo clusters of galaxies. Steady state momentum distribution
functions of the CR electrons are calculated for given spectra of the turbulent
Alfven waves. The radio spectrum produced by the obtained CR electron
distribution is compared to the observed radio spectrum of the Coma radio halo.
We find that the observed radio spectrum of the Coma cluster is well reproduced
when the spectral index of the turbulent Alfven waves is 2.8. The obtained
energy spectrum of the turbulent Alfven waves is steeper than that expected
from the turbulence theory, suggesting back reaction of the particle
acceleration. The fitting procedure constraints the amplitude of the turbulent
Alfven waves as well as the spectral index. Then we estimate the dissipation of
the turbulent Alfven waves, which is found to be less than the cooling rate by
X-ray radiation. We suggest that the turbulence which is sufficient for
particle acceleration is developed in the clusters containing the radio halo.
It is most likely that cluster mergers create the turbulence and seed
relativistic electrons.
