= Solution
As <magnetic field> grows from zero, a clean <two-dimensional electron gas> first has an ordinary finite <longitudinal resistivity>, then develops Shubnikov–de Haas oscillations as broadened <Landau levels> pass the <chemical potential>. Thermal and disorder broadening obscure the low-field oscillations; their visibility grows when the level spacing exceeds those broadening scales. In a well-developed integer quantum Hall regime, $\rho_{xx}$ has minima close to zero on Hall plateaux and maxima during transitions between them, when extended states cross the <chemical potential>. Detailed peak heights and envelopes depend on disorder and scattering and are not fixed by quantization alone.
At fixed density the nominal filling is $\nu(B)=N_Sh/(eB)$, so level-filling features are approximately periodic in $1/B$, with period $e/(N_Sh)$ for resolved <spin> levels or $2e/(N_Sh)$ for an unresolved <spin> pair. Increasing $B$ lowers the filling; <spin> splitting can create additional odd-index features. The original schematic below shows these relations without pretending to reconstruct the missing data.
\b[TODO — missing source graph:] the actual $0$–$12\,\mathrm T$ positions and integer plateau labels require $N_S$ or surviving plateau fields from part (b). Neither is provided independently. Thus a uniquely labelled field-axis sketch for this sample cannot be completed. The following DOS diagram distinguishes extended centres and localized tails generically; its chemical-potential markers are illustrative and do not stand for the requested 8 T or 10 T assignments.
\Image[/past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2018-landau-level-schematic.png]
{title=Generic longitudinal oscillations and broadened Landau levels without inventing the missing Hall data}
{description=Original qualitative illustration. The horizontal filling scale is dimensionless and is not a reconstruction of the missing 0–12 T graph. Chemical-potential positions in the DOS panel illustrate two regimes, with no assignment to 8 T or 10 T.}
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