A method for the calculations of the Gilbert damping parameter $alpha$ is presented, which based on the linear response formalism, has been implemented within the fully relativistic Korringa-Kohn-Rostoker band structure method in combination with the
coherent potential approximation alloy theory. To account for thermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. This allows the determination of $alpha$ for various types of materials, such as elemental magnetic systems and ordered magnetic compounds at finite temperature, as well as for disordered magnetic alloys at $T = 0$ K and above. The effects of spin-orbit coupling, chemical and temperature induced structural disorder are analyzed. Calculations have been performed for the 3$d$ transition-metals bcc Fe, hcp Co, and fcc Ni, their binary alloys bcc Fe$_{1-x}$Co$_{x}$, fcc Ni$_{1-x}$Fe$_x$, fcc Ni$_{1-x}$Co$_x$ and bcc Fe$_{1-x}$V$_{x}$, and for 5d impurities in transition-metal alloys. All results are in satisfying agreement with experiment.
Doping NiFe by heavy rare earth atoms alters the magnetic relaxation properties of this material drastically. We show that this effect can be well explained by the slow relaxing impurity mechanism. This process is a consequence of the anisotropy of t
he on site exchange interaction between the 4f magnetic moments and the conduction band. As expected from this model the magnitude of the damping effect scales with the anisotropy of the exchange interaction and increases by an order of magnitude at low temperatures. In addition our measurements allow us to determine the relaxation time of the 4f electrons as a function of temperature.
