Spin Dynamics with Inertia in Metallic Ferromagnets

Non-adiabatic contribution of environmental degrees of freedom yields effective inertia of spin in effective spin dynamics. In this paper, we study several aspects of the inertia of spin in metallic ferromagnets. (i) a concrete expression of the spin inertia $m_s$: $m_s=\hbar S_c/(2g_{\rm sd})$, where $S_c$ is the spin polarization of conduction electrons and $g_{\rm sd}$ is the $sd$ coupling constant. (ii) dynamical behavior of spin with inertia, discussed from viewpoints of a spinning top and of a particle on a sphere. (iii) behavior of spin waves and domain walls in the presence of inertia, and behavior of spin with inertia in the case of a time-dependent magnetic field.