  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.
