  We consider a model of an electron in a crystal moving under the influence of
an external electric field: Schroedinger's equation in one spatial dimension
with a potential which is the sum of a periodic function $V$ and a smooth
function $W$. We assume that the period of $V$ is much shorter than the scale
of variation of $W$ and denote the ratio of these scales by $\epsilon$. We
consider the dynamics of $\textit{semiclassical wavepacket}$ asymptotic (in the
limit $\epsilon \downarrow 0$) solutions which are spectrally localized near to
a $\textit{crossing}$ of two Bloch band dispersion functions of the periodic
operator $- \frac{1}{2} \partial_z^2 + V(z)$. We show that the dynamics is
qualitatively different from the case where bands are well-separated: at the
time the wavepacket is incident on the band crossing, a second wavepacket is
`excited' which has $\textit{opposite}$ group velocity to the incident
wavepacket. We then show that our result is consistent with the solution of a
`Landau-Zener'-type model.
