  The sign, magnitude, and range of the exchange couplings between pairs of Mn
ions is determined for (Ga,Mn)N and (Ga,Mn)N:Si with x < 3%. The samples have
been grown by metalorganic vapor phase epitaxy and characterized by
secondary-ion mass spectroscopy; high-resolution transmission electron
microscopy with capabilities allowing for chemical analysis, including the
annular dark-field mode and electron energy loss spectroscopy; high-resolution
and synchrotron x-ray diffraction; synchrotron extended x-ray absorption
fine-structure; synchrotron x-ray absorption near-edge structure; infra-red
optics and electron spin resonance. The results of high resolution magnetic
measurements and their quantitative interpretation have allowed to verify a
series of ab initio predictions on the possibility of ferromagnetism in dilute
magnetic insulators and to demonstrate that the interaction changes from
ferromagnetic to antiferromagnetic when the charge state of the Mn ions is
reduced from 3+ to 2+.
