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spin chemical potential. The spin injection effect was briefly discussed based on our results. When the analysis is applied to the inhomogeneous magnetization case, we argued that the β torque in the current-induced magnetization dynamics can be anisotropic. 

Before finishing, we emphasize that the expression for the spin current and µ$_{s}$ are mean- ingless without specifying the physical observable to be measured. In the inverse spin Hall effect, which was originally proposed as [3, 23, 24] j$_{µ}$ ∝ ϵ$_{µνρ}$j ρ s $_{,ν}$, it has recently been demon- strated that the charge current is not directly proportional to the spin current [7, 25]. Solving for the spin current only does not therefore provide physical information. 

## Acknowledgment 

The authors thank E. Saitoh, J. Shibata, H. Kohno, S. Murakami for valuable discus- sions. This work was supported by a Grant-in-Aid for Scientific Research in Priority Areas, ”Creation and control of spin current” (1948027), the Kurata Memorial Hitachi Science and Technology Foundation and the Sumitomo Foundation. 

## Appendix A: Derivation of the spin continuity equation 

We calculate the commutators which appears in Eq. (13) with the total Hamiltonian H ≡ H$_{0}$ + H$_{so}$ 0 + H$_{so}$ A + H$_{em}$ + H$_{imp}$ . As for the free electron part, H$_{0}$ , the commutator reads 

The spin-orbit contributions read 

and the electromagnetic field contribution is 

<img src='content_image/327857.jpg'>

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<img src='content_image/327855.jpg'>

