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ID: past-exam-of-the-materials-science-course-of-the-university-of-cambridge/2018/paper-1/1/c/solution

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Past exam of the materials science course of the University of Cambridge / 2018 / Paper 1 / 1 / c / Solution by

Codex  0 2026-10-20

The metallic conventional cell has four tin atoms, whereas the conventional diamond cubic structure cell has eight. Therefore the volume per atom is

v_{β} = \frac{( 0.5831 ) ^{2} ( 0.3182 )}{4} = 0.0270474 n m^{3}, v_{α} = \frac{( 0.6489 ) ^{3}}{8} = 0.0341541 n m^{3} .

(1)

The fractional change for metallic to nonmetallic tin is

\frac{v _{α} - v _{β}}{v _{β}} ≃ 0.2627, about 26.3% expansion .

(2)

The atom count is unchanged, so this is also the unconstrained bulk volume change for complete transformation under the model. If it were macroscopically isotropic, the corresponding free linear expansion would be

(1.2627)^{1/3} - 1 ≃ 8.1%

Such a large transformation strain creates stresses where transformed and untransformed regions constrain one another. Cracking, loss of mechanical integrity and disintegration can follow: the phenomenon is tin pest. The calculation establishes the volume mismatch, not the transformation rate; temperature, nucleation and impurities affect when a real specimen changes.

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