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A method for the calculations of the Gilbert damping parameter $\alpha$ ispresented, which based on the linear response formalism, has been implementedwithin the fully relativistic Korringa-Kohn-Rostoker band structure method incombination with the coherent potential approximation alloy theory. To accountfor thermal displacements of atoms as a scattering mechanism, an alloy-analogymodel is introduced. This allows the determination of $\alpha$ for varioustypes of materials, such as elemental magnetic systems and ordered magneticcompounds at finite temperature, as well as for disordered magnetic alloys at$T = 0$ K and above. The effects of spin-orbit coupling, chemical andtemperature induced structural disorder are analyzed. Calculations have beenperformed for the 3$d$ transition-metals bcc Fe, hcp Co, and fcc Ni, theirbinary alloys bcc Fe$_{1-x}$Co$_{x}$, fcc Ni$_{1-x}$Fe$_x$, fccNi$_{1-x}$Co$_x$ and bcc Fe$_{1-x}$V$_{x}$, and for 5d impurities intransition-metal alloys. All results are in satisfying agreement withexperiment.

physical review. b, condensed matter and materials physics

First-principles calculation of the Gilbert damping parameter via the  linear response formalism with application to magnetic transition-metals and  alloys