= Solution
At large inverse field, $B$ and the <cyclotron frequency> are small, so the orbital spacing $\hbar\omega_c$ becomes comparable with both disorder broadening and the Fermi occupation's thermal width. Scattering of <quantum lifetime> $\tau_q$ gives a typical half-width $\Gamma=\hbar/(2\tau_q)$; overlapping levels lose their density-of-states contrast. The fundamental <Dingle factor> and <thermal damping of quantum oscillations> are
$$
\boxed{R_D=e^{-\pi/(\omega_c\tau_q)},\qquad
R_T=\frac{X}{\sinh X},\qquad X=\frac{2\pi^2k_BT}{\hbar\omega_c}.}
$$
Both shrink rapidly as $B$ falls. Well-resolved oscillations require a sufficiently large $\omega_c\tau_q$ and a temperature small compared with the orbital spacing. The scattering time in this broadening argument is not automatically the transport time inferred from mobility; small-angle events affect the two differently.
The transverse Hall resistance is approximately $|R_{xy}|=B/(n_se)$ in the classical weak-field regime. At stronger fields the <integer quantum Hall effect> produces plateaus
$$
\boxed{|R_{xy}|=\frac{h}{\nu e^2},\qquad \nu\in\mathbb N,}
$$
accompanied by nearly vanishing longitudinal resistance. Localized bulk states between extended Landau states allow plateau intervals, while current can be carried by <quantum Hall edge states>. At still larger fields and sufficiently low temperature, the cleanest samples show the <fractional quantum Hall effect>, with interaction-generated incompressible states at fractional fillings such as $\nu=1/3$. The sign of $R_{xy}$ depends on the contact and magnetic-field orientation; the sketch shows its magnitude.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2019-q5-hall-plateaus.png]
{title=Classical Hall slope and schematic integer quantum Hall plateaus}
{description=Original schematic magnitude of transverse Hall resistance. The plateau transitions are chosen for illustration, not fitted to the supplied longitudinal-resistance trace. The vertical scale is $h/e^2$ and the field scale is $B_1=n_sh/e$, so the classical line has unit slope.}
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