of non-adiabaticity of the spin-transfer torque is $$\label{eqn:nan}
{\cal Q}= {{\rm max}\,|\,b(x)\,| \over {\rm max}\,|\,N_{\rm ad}(x)\,|}.$$

![Non-adiabaticity in Eq. \[eqn:nan\] versus wall width scaled by the characteristic length in Eq. (\[eqn:length\]). Note the logarithmic scale.[]{data-label="fig:nonadiabaticity"}](fig7.eps){width="8cm"}

Fig. \[fig:nonadiabaticity\] plots this quantity as a function of scaled domain wall width $w/L$ on a log scale. The observed exponential decrease of the non-adiabatic torque as the wall width increases can be understood from the work of Dugaev [*et al.*]{}[@Dugaev:2002] These authors treat the gradient field in Eq. (\[eqn:gradientfield\]) as a perturbation and calculate the probability for an electron in a $(k_x \uparrow)$ state to scatter into a $(k'_x \downarrow)$ state in the Born approximation. If we choose $k_x$ and $k'_x$ as $k_F^+$ and $k_F^-$, respectively, their results imply that the probability ${\cal P}$ that a majority electron retains its spin and becomes a minority electron as it passes through a domain wall is $$\label{eqn:Dugaev}
{\cal P} \propto \exp(-\gamma w/L),$$ where $\gamma$ is a constant of order unity. This rationalizes the result plotted in Fig. \[fig:nonadiabaticity\] because the magnitude of the minority spin component determines the amplitude of the spin precession around ${\bf B}_{\rm eff}(x)$ and thus the magnitude of the non-adiabatic component of 