 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2022/paper-2/a9/a/ii/solution

Write

t_{s} = ∣ β_{s} ∣

and

t_{p} = ∣ β_{p} ∣

, so

E_{s} = α_{s} - 2 t_{s} cos (ka), E_{p} = α_{p} + 2 t_{p} cos (ka) .

The labelled key points of the two dispersion relations are

k 0 π / (2 a) π / a E_{s} α_{s} - 2 t_{s} α_{s} α_{s} + 2 t_{s} E_{p} α_{p} + 2 t_{p} α_{p} α_{p} - 2 t_{p}

The

s

band rises from centre to boundary and the axial

p

band falls. The separated-band assumption requires

α_{p} - α_{s} > 2 (t_{s} + t_{p})

. The upper-left panel shows a representative pair, with the other panels supplying the requested comparisons in part (b).

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