Local and nonlocal atomic contributions to unit-cell damping in near-adiabatic collinear magnetization dynamics

Recently a unified theory for the near-adiabatic magnetization dynamics in collinear and noncollinear systems has been presented [D. Steiauf et al., Phys. Rev. B 78, 020410(R) (2008)]. There, an equation of motion for atomic magnetic moments is derived which includes anisotropic and nonlocal atomic damping matrices. In the present paper the connection between damping for atomic and unit-cell magnetic moments in collinear systems is investigated. Using the unified theory, numerical values for the local and nonlocal atomic contributions to the unit-cell damping in collinear hcp-Co, bcc-Fe, and a Co layer are calculated. In addition, the damping matrix for the whole unit cell is calculated by a formerly established method, the torque-operator method. The results of the unified theory and of the torque-operator method which are based on different approximations agree very well.