such component due to pairwise cancellation between forward and backward moving spins of the same type (majority or minority). Moreover, since the cancellation occurs within each band, the final result is insensitive to the details of intraband scattering.

It remains only to understand the origin of the misalignment angle $\alpha$. Why does each spin not simply align itself with ${\bf
B}_{\rm ex}$? Berger[@Berger:1978] was the first to notice this fact and the physics was made particularly clear by Aharonov and Stern.[@Aharonov:1992] These authors studied the adiabatic approximation for a classical magnetic moment that moves in a slowly varying field ${\bf B}(x)$. Not obviously, the moment behaves as if were subjected to a effective magnetic field ${\bf B}_{\rm eff}(x)$ that is the sum of ${\bf B}_{\rm ex}(x)$ and a fictitious, velocity-dependent, “gradient” field ${\bf B}_{\rm g}(x)$ that points in the direction $\nabla
\hat{\bf B}(x)\times \hat{\bf B}(x)$. For our problem, $$\label{eqn:gradientfield}
    {\bf B}_{\rm eff} 
    = {\bf B}_{\rm ex}+{\hbar^2k_x\over 2m\mu}{d\hat{\bf B}_{\rm ex}\over dx}\times\hat{\bf B}_{\rm ex}.$$ The presence of this gradient field is apparent from Eq. (\[eqn:spiralenergy\]) where the square root is proportional to $\vert {\bf B}_{\rm eff}\vert$. [*The adiabatic solution corresponds to perfect alignment of the moment with ${\bf B}_{\rm
eff}(x)$*]{}. This alignment 