= Solution
For an electron wavepacket in one isolated band under a uniform <electric field>,
$$
\hbar\dot k=-eE,\qquad v(k)=\frac1\hbar\frac{d\epsilon}{dk}.
$$
Because $\epsilon(k+2\pi/a)=\epsilon(k)$, acceleration through successive <Brillouin zones> makes the velocity periodic rather than increasing indefinitely. For $\epsilon=\epsilon_0-2J\cos(ka)$,
$$
\boxed{\omega_B=\frac{e|E|a}{\hbar},\qquad T_B=\frac{h}{e|E|a},\qquad v(t)=\frac{2Ja}{\hbar}\sin(k_0a-\omega_Bt)}
$$
for a positive field along the chosen axis. Integration gives
$$
x(t)-x(0)=\frac{2J}{eE}[\cos(k_0a-\omega_Bt)-\cos(k_0a)].
$$
The <excursion amplitude of a Bloch-oscillating wavepacket> is $2|J|/(e|E|)$ about its mean and peak-to-peak extent $4|J|/(e|E|)$. The turning of velocity corresponds semiclassically to Bragg reflection near a zone boundary.
Coherent <Bloch oscillations> require scattering/<dephasing> times long compared with $T_B$ and a field weak enough to avoid substantial interband tunnelling. Ordinary metals usually scatter too rapidly; <semiconductor> superlattices and cold atoms in periodic optical potentials provide cleaner realizations. A completely filled band has canceling velocities, so its net current does not oscillate simply because individual wavepackets can.<Solid State Physics by David Tong>[Chapter 3: semiclassical band dynamics and Bloch oscillations]{ref}
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2015/paper-1-bloch-oscillations.png]
{title=Constant force sweeps crystal momentum linearly, but a periodic band produces oscillating velocity and bounded displacement}
{source=paper-1-bloch-oscillations.py}
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