is a sudden jump in $\theta'(x)$ at $x=\pm w$. There is a corresponding jump in the direction of ${\bf B}_{\rm eff}(x)$ as defined by Eq. (\[eqn:gradientfield\]). Spins propagating along the the $x$-axis cannot follow this abrupt jump and thus precess around the post-jump field direction with an amplitude determined by the sine of the angle between the before-and-after field directions. The latter is proportional to the jump in $\theta'(x)$, which is $\pi/2w$ for the wall Eq. (\[WVwall\]).

### Zhang & Li

In spin spirals and long domain walls, we find that the non-equilibrium spin current is adiabatic, [*i.e.*]{}, ${\bf Q}(x)$ is aligned with ${\bf M}(x)$ \[or ${\bf B}_{\rm ex}(x)$\]. At the same time, we find in both cases that the non-equilibrium spin density ${\bf s}(x)$ is [*not*]{} aligned with the magnetization; there is component of ${\bf s}(x)$ transverse to ${\bf M}(x)$. The corresponding transverse component of the spin current density cancels between pairs of electrons moving in opposite directions. Zhang and Li[@Zhang:2004] found exactly the same form of non-equilibrium spin accumulation (called $\delta {\bf m}(x)$ by them) using a phenomenological theory. They proposed that this non-equilibrium spin density relaxes by spin-flip scattering toward alignment with the magnetization. Such a relaxation 