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The Dingle factor describes the damping of quantum oscillations by Landau-level broadening. For a Lorentzian half-width $\hslash /(2{\tau }_q)$, the fundamental harmonic has $R_D=\exp [-\pi /({\omega }_c{\tau }_q)]$. Since ${\omega }_c$ decreases with field, the amplitude decays at large inverse field. The lifetime here is the quantum lifetime, not automatically the mobility-derived transport time.

At large inverse field, $B$ and the cyclotron frequency are small, so the orbital spacing $\hslash {\omega }_c$ becomes comparable with both disorder broadening and the Fermi occupation's thermal width. Scattering of quantum lifetime ${\tau }_q$ gives a typical half-width $\Gamma =\hslash /(2{\tau }_q)$; overlapping levels lose their density-of-states contrast. The fundamental Dingle factor and thermal damping of quantum oscillations areBoth 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 $|R_{xy}|=B/(n_{s}e)$ in the classical weak-field regime. At stronger fields the integer quantum Hall effect produces plateausaccompanied 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.