discusses the effects of scattering. We summarize our results and offer some conclusions in Section \[sumcon\]. Two appendices provides some technical details omitted in the main body of the paper.

Model & Methods {#models}
===============

The free electron Stoner model provides a first approximation to the electronic structure of an itinerant ferromagnet. The Hamiltonian is $$\label{eqn:Hamiltonian}
H=-{\hbar^2\over 2m}\nabla^2 - \mu \bm{\sigma} \cdot {\bf B}_{\rm ex}(x),$$ where $\bm{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ is a vector composed of the three Pauli matrices and $\mu=g\hbar e/2mc$. The magnetic field ${\bf B}_{\rm ex}(x)$ is everywhere parallel to ${\bf M}(x)$ but has a constant magnitude.[@MagSign] That magnitude is chosen so the Zeeman splitting between the majority and minority spins bands reproduces the quantum mechanical exchange energy in the limit of uniform magnetization: $$\label{eqn:exchange}
E_{\rm ex}=2\mu |{\bf B}_{\rm ex}|=\hbar^2 k_B^2/m.$$ If $E_F=\hbar^2 k_F^2/2m$ is the Fermi energy, the constant $k_B$ in Eq. (\[eqn:exchange\]) fixes the Fermi wave vectors for up and down spins, $k_F^+$ and $k_F^-$, from $$\label{eqn:kpm}
k_F^\pm = \sqrt{k_F^2\pm k_B^2}.$$

Given ${\bf B}_{\rm ex}(x)$, we use both quantum mechanics and a semi-classical approximation to calculate the spin accumulation, spin current density, and spin-transfer torque. The building blocks are the single-particle spin density ${\bf s}_\pm(x,k_x)$ and the single-particle spin 