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ID: past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2014/1/c/solution

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Past exam of the condensed matter physics course of the University of Oxford / 2014 / 1 / c / Solution by

Codex  0 2026-10-25

After compression along

z

, let the original conventional edges be

a, a, c

with

c = a / 2

. This gives exactly the specified volume reduction. The compressed FCC points admit the alternative translation vectors

b_{1} = \frac{a}{2} (1, 1, 0), b_{2} = \frac{a}{2} (- 1, 1, 0), b_{3} = (0, 0, a / 2) .

They are mutually perpendicular, have equal length

b = a / 2

, and form a right-handed cube. Besides its corners, the lattice has the point

\frac{b _{1} + b _{2} + b _{3}}{2} = (0, a /2, a / (22)),

which is the compressed image of an original FCC face centre. All compressed FCC sites are generated by the cube's integer translations and this body-centre coset. Therefore

compressed FCC \equiv BCC with conventional edge b = a / 2 .

The old compressed tetragonal conventional cell has volume

a^{3} / 2

and four sites; the new BCC cube has half that volume and two sites. This verifies the same physical site density rather than silently losing atoms during the change of cell. The continuous uniaxial deformation is the Bain path, named for Edgar Bain.

Figure 1.
The Bain path: a rotated conventional cell exposes BCC geometry
. Original geometric construction. The basal panel shows the old square and a new square generated by the two diagonal half-translations. The spatial panel uses new-cell coordinates, with corner sites and a body centre. The compressed vertical edge and both new basal edges all have length $a / 2$ . Dots are lattice sites rather than hard-sphere atomic surfaces. Details.

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