${\bf s}$ and ${\bf Q}$ in Eq. (\[eqn:scscd\]). In fact, $N_{\rm ad} \propto 1/w$, so it is the case that

$$\label{over}
 {\rm max}\,|\,b(x)\,|\propto {1\over w} \exp(-\gamma w/L).$$

The slope of the straight line in Fig. \[fig:nonadiabaticity\], [*i.e.*]{}, the value of the constant $\gamma$ in Eq. (\[eqn:Dugaev\]) depends on the sharpness of the domain wall. Using Eq. (\[eqn:asine\]) and other simple domain wall profile functions, it is not difficult to convince oneself that a suitable measure of domain wall sharpness is the maximum value of the second derivative $\theta''(x)$ for walls with the same width. The numerical results shown in Fig. \[fig:slope\] confirm this to be true. The sharper the domain wall, the less rapidly the non-adiabatic torque disappears with increasing domain wall width.

![Dependence of $\gamma$ in Eq. (\[eqn:Dugaev\]) on domain wall sharpness: Squares are calculated points. Straight line is a guide to the eye that passes through the origin.[]{data-label="fig:slope"}](fig8.eps){width="8cm"}

Relation to Other Work {#other}
======================

### Waintal & Viret

Waintal and Viret[@Waintal:2004] (WV) used a free-electron Stoner model and the Landauer-Büttiker formalism to calculate the spin transfer torque associated with a Néel wall with magnetization Eq. (\[eqn:magnetization\]) and $$\begin{aligned}
\label{WVwall}
  \quad
   \theta(x) &=& \left\{ \begin{array}{lr}
   0, & x < 