The factor Γ$_{σσ}$ ′ ( q , Ω) contains all the vertex corrections due to the normal impurities and the spin-orbit interaction shown in Fig. 3. The equation of motion for Γ$_{σσ}$ ′ is derived in the same manner as in Ref. [21] carried out in the context of quantum correction (the diffusion without the spin-orbit interaction was considered in Ref. [22]). The equation is obtained as where where D$_{σ}$ ≡ ( k$_{F σ}$ ) 2 3 m 2 τ$_{σ}$ is the diffusion constant. Neglecting quantities of order of ( κ$_{z}$, κ$_{⊥}$ ) $^{2}$, Eq. (25) is solved as Using Eq. (26), we obtain Γ$_{σσ}$ ′ as (assuming the rotational symmetry for the wave vectors when averaging over the spin-orbit potential) By use of Eq. (28) and summing over the wave vectors, the normal spin current, Eq. (24), reads where σ 0 s ≡ e ∑ $_{±}$( ± ) D$_{±}$ν$_{±}$ is the bare spin conductivity divided by e . The first term of Eq. (29) is the field-driven contribution. The second gradient term is a diffusive contribution (vertex corrections), arising from the spin accumulation. The effective potential describing the spin accumulation, µ$_{s}$ , reads µ$_{s}$ ≡ ∫ d $^{3}$x $^{′}$χ ( x − x $^{′}$)( ∇ · E )( x $^{′}$) , (30) j (n) ,z s ,i = σ 0 $_{s}$E$_{i}$ − ∇ $_{i}$µ$_{s}$ , (29) Γ$_{σσ}$ = 1 D$_{σ}$q $^{2}$τ$_{σ}$ + κ$_{⊥}$γ$_{σ}$ Γ$_{σ,}$$_{−}$$_{σ}$ = κ$_{⊥}$ [ D$_{+}$q $^{2}$τ$_{+}$ + κ$_{⊥}$γ$_{+}$ ][ D$_{−}$q $^{2}$τ$_{−}$ + κ$_{⊥}$γ$_{−}$ ] . (28) Γ$_{σσ}$ = 1 + κ$_{z}$ − (1 + 2 κ$_{z}$ )Π$_{−}$$_{σ}$ [1 − (1 + κ$_{z}$ )Π$_{+}$][1 − (1 + κ$_{z}$ )Π$_{−}$] Γ$_{σ,}$$_{−}$$_{σ}$ = κ$_{⊥}$ [1 − (1 + κ$_{z}$ )Π$_{+}$][1 − (1 + κ$_{z}$ )Π$_{−}$] . (27) Π$_{σ}$( q , Ω) ≡ n$_{i}$v 2 i ∑ k g r k − q $_{$_{2}$,σ}$g a k + q $_{2}$,σ ≃ [1 − ( D$_{σ}$q $^{2}$τ$_{σ}$ + κ$_{z}$ + κ$_{⊥}$γ$_{σ}$ )] , (26) Γ$_{σσ}$ = (1 + κ$_{z}$ )(1 + Π$_{σ}$Γ$_{σσ}$) + κ$_{⊥}$ Π$_{−}$$_{σ}$Γ$_{−}$$_{σ,σ}$ Γ$_{σ,}$$_{−}$$_{σ}$ = κ$_{⊥}$ (1 + Π$_{−}$$_{σ}$Γ$_{−}$$_{σ,}$$_{−}$$_{σ}$) + (1 + κ$_{z}$ )Π$_{σ}$Γ$_{σ,}$$_{−}$$_{σ}$ , (25) 10