Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / Section D / 43 / b / iii / 2 / Solution

The eight body-centred cubic neighbour vectors are

(a / 3) (\pm 1, \pm 1, \pm 1)

with independent signs. Their diagonal sums are

8 a^{2} /3

and the sign-paired off-diagonal sums cancel. Hence

σ_{bcc} = \frac{8}{3} K a^{2} I = \frac{4}{3} \frac{n Q ^{2} F a ^{2}}{k _{B} T} I .

The conventional cubic cell edge is

2 a / 3

in this comparison; using that edge as the hopping distance would spoil the stipulated equal-distance comparison.

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