where ℓ$_{s}$ in the spin diffusion length in the normal metal ( ℓ$_{s}$ is a long ( ∼ µ m) length scale and thus does not depend on the spin). We therefore obtain from Eq. (30) the chemical potential as where q$_{s}$ ≡ ( ∑ σ σD$_{σ}$ν$_{σ}$ ) eV$_{FN}$ ) A$_{FN}$/d , is the spin accumulation rate at the interface (per unit time), and A$_{FN}$ is the area of the junction. This result of µ$_{s}$ is consistent with intuitive and phenomenological results of the spin injection in the perpendicular structure shown in Fig. 4(a). In contrast, when the voltage is applied parallel to the ideal interface as shown in Fig. 4(b), spin injection does not occur since ∇ · E = 0 at the interaface and thus µ (N) s = 0. The continuity equation (1), indicates that the spin polarization (magnetization) changes due to the spin relaxation. (Change of the magnetization magnitude is a feature of the itinerant magnetism.) By use of Eqs. (5)(33)(37), we see that Eq. (1) results in where γ$_{τ}$ ≡ γ − δσ$_{s}$ and δγ$_{τ}$ ≡ δγ + δσ$_{s}$ . General case with uniform magnetization along any unit vector n is given by ( s ≡ s n ) ˙ s z = γ$_{τ}$ ∇ · E + δγ$_{τ}$ ∇ $_{z}$E$_{z}$ + ∑ ± ( ± ) µ$_{±}$ ( ℓ$_{s}$$_{,}$$_{±}$ ) $_{2}$, (44) ˙ s = n ( γ$_{τ}$ ∇ · E + δγ$_{τ}$ ∇ $_{‖}$E$_{‖}$ + ∑ ± ( ± ) µ$_{±}$ ( ℓ$_{s}$$_{,}$$_{±}$ ) 2 ) , (45) V F N $^{∇·}$E =0 (b) V F N V FN d j s (a) FIG. 4: (a) Creation of diffusive spin current (spin injection) by applying the electric voltage perpendicular to the F-N interface. (b) When the voltage is applied parallel to an ideal interface, no spin current is induced since ∇ · E = 0. 13 VI. TOTAL TORQUE AND ASYMMETRIC β TERM µ (N) s ( x ) = q$_{s}$ 4 π | x | e −| x | $^{/ℓ$_{s}$}$, (43)