is indicated in Eq. (\[eqn:spirald\]) and in Fig. \[fig:curvature\]. More generally, the expected motion of the magnetic moment is precession around ${\bf B}_{\rm
eff}(x)$.Nevertheless, as indicated above, the total spin current density for the spin spiral aligns with ${\bf B}_{\rm
ex}(x)$ (which is the conventional definition of adiabaticity for this quantity) when the net effect of all conduction electrons is taken into account.

Domain Walls {#domainwalls}
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Our main interest is the spin-transfer torque associated with domain walls that connect two regions of uniform and antiparallel magnetization. A realistic wall of this kind can be described by Eq. (\[eqn:magnetization\]) with[@Hubert:1998] $$\label{eqn:asine}
\theta(x) = \pi/2 - \arcsin{\left[ \tanh{\left( x/w \right)} \right]} ,$$ The wall is Néel-type if $\phi(x)=0$ and Bloch-type if $\phi(x)=\pi/2$. We will speak of the domain wall width $w$ as “long” or “short” depending on whether $w$ is large or small compared to the characteristic length $$\label{eqn:length}
L={E_F\over E_{\rm ex}}{1\over k_F}={k_F\over k_B^2}.$$ Intuitively, the adiabatic approximation should be valid when $w \gg
L$. When applied to Eq. (\[eqn:spinandcurrent\]), the predicted adiabatic spin-transfer torque for our model is $$\label{eqn:adit}
{\bf N}_{\rm ad}(x) = -{\hbar\over 2}\eta{neE\tau\over m}\partial_x\hat{\bf M}(x),$$ where $n$ is the electron density, and $\eta$ is the polarization of the 