content_image/363798.jpg content_image/363797.jpg
where E$_{‖}$ ≡ n · E and ∇ ‖ ≡ n · ∇ . 

In addition to the change of the magnitude, Eq. (45), there is a torque, which is per- pendicular to n . Such torque arises when the magnetization is not homogeneous, and plays important roles in current-induced magnetization dynamics. We have carried out the calcu- lation of the current-induced torque done in Ref. [10] on the same footing as the derivation of Eq. (5). As a result, we found that the β term becomes asymmetric as (see Appendix D for details of the calculation) 

<img src='content_image/363798.jpg'>

where j$_{‖}$ ≡ n · j is the current along the local magnetization and δβ/β = − 1 / 5 in the present model. The spin transfer torque due to the spin-orbit interaction is thus different from that due to the spin-flip scattering. The expression of the total torque allowing for the spatially varying current density and the magnetization is therefore obtained as 

<img src='content_image/363797.jpg'>

This expression clearly demonstrates that the spin relaxation torque requires some inhomo- geneity either of the applied current or the spin structure, in addition to the spin-orbit (or spin flip) interaction. Totally homogeneous system does not relax. The last term in Eq. (47) gives useful information for measuring the spin accumulation induced by the spin current. 

## VII. CONCLUSION 

We have carried out a microscopic calculation of the spin relaxation torque and the spin current induced in disordered ferromagnetic metals by the applied electric field. The spin-orbit interaction arising from the random impurities is included as a source of spin relaxation, and inhomogeneity of the applied electric field is taken into account. We found that the spin relaxation torque in the uniform magnetization case is written as a divergence of the electric field plus an anisotropic term. The spin current was shown to be made up of field-driven (local) and diffusive (nonlocal) contributions, the latter written as a gradient of a spin chemical potential. We have derived a general linear response expression for the 

