Wavenumber dependent Gilbert damping in metallic ferromagnets*

New terms to the dynamical equation of magnetization motion, associated with spin transport, have been reported over the past several years. Each newly identified term is thought to possess both a real and an imaginary effective field leading to fieldlike and dampinglike torques on magnetization. Here we show that three metallic ferromagnets possess an imaginary effective-field term which mirrors the well-known real effective-field term associated with exchange in spin waves. Using perpendicular standing spin wave resonance between 2-26 GHz, we evaluate the magnitude of the finite-wavenumber ($k$) dependent Gilbert damping of the uniform mode ($\alpha_u$) and the first spin wave mode ($\alpha_s$) in three typical ferromagnets, Ni$_{79}$Fe$_{21}$, Co, and Co$_{40}$Fe$_{40}$B$_{20}$. By taking the difference of $\alpha _s$ and $\alpha _u$ and excluding the eddy current damping $\alpha_E$ ($\Delta\alpha_k=\alpha_s-\alpha_u+\alpha_E$), we find the presence of a $k^2$ term, as $\Delta\alpha_k=\Delta\alpha_0+A_{k}\cdot k^2$ in all three metals. We interpret the new term as the continuum analog of spin pumping, predicted recently, and show that its magnitude, $A_{k}$=0.07-0.1 nm$^2$, is consistent with transverse spin relaxation lengths (1-3 nm) as measured by conventional spin pumping.