perfectly adiabatic. Wessely [*et al.*]{}[@Nordstrom:2005] found consistent results in their density functional calculation of the steady-state spin current density associated with the helical spin density wave in erbium metal. We emphasize that Eq. (\[eqn:spiralcd\]) is independent of pitch for an infinite spin spiral. As we discuss below, a similar independence does [*not*]{} hold for domain walls. In that case, wide walls are adiabatic, but narrow ones are not.

![Electrons (wave vector $k_x$) and holes (wave vector $-k_x$) move in an effective field that is the sum of the exchange field ${\bf B}_{\rm ex}(x) \parallel \hat{\bf z}'$ and a fictitious, velocity-dependent “gradient” field induced by the spatial dependence of the exchange field. The spins align to the total effective field in an infinite spin spiral. The $x$ axis lies in $x'$-$z'$ plane. []{data-label="fig:curvature"}](fig4.eps){width="8cm"}

The semi-classical formula Eq. (\[eqn:scscd\]) provides an appealing way to understand the adiabaticity of the spin current density in the spin spiral defined by Eq. (\[thetas\]). The key point is that the angle $\alpha$ in Eq. (\[alpha\]) which fixes the direction of ${\bf
s}_\pm(x,k_x)$ in Eq. (\[eqn:spirald\]) is positive when $k_x$ is positive and negative when $k_x$ is negative (Fig. \[fig:curvature\]). Moreover, for every $k_x$ electron that contributes 