Equilibrium Crystal Shapes

An equilibrium crystal simulated by Gerard Barkema and Mark Holzer, here at Cornell (unpublished).
A perfect crystal, sitting in a large container, in perfect equilibrium, will have a shape which is determined by the different energies needed to make a crystal/vapor boundary in different directions with respect to the crystalline lattice. The faces aligned along certain crystalline directions have lower energy, and so one gets faceted crystal shapes (crystals with flat walls along low-energy directions) at low temperatures.
This is not the reason diamond rings and salt grains have facets! The salt crystals on your table have facets because they break that way: diamonds have facets because jewellers grind them with great patience. It takes careful experiments to see these shapes: they've been seen in salt (NaCl), gold, lead, and in the blue phases of chiral nematic liquid crystals.
A salt crystal (NaCl), at 710 C. The six faces along the crystal lattice directions are flat, or faceted. (Heyraud and Métois, J. Cryst. Growth 84, 503 (1987)). A gold crystal at about 1000 C, (Heyraud and Métois, J. Cryst. Growth 50, 571 (1980); Acta Metal. 28, 1789 (1980)). A lead crystal at about 300 C. Prepared by Heyraud and Métois (Rottman et al., Phys. Rev. Lett. 52, 1009 (1984).

Rock crystals like quartz aren't showing equilibrium facets either: they develop facets because of the way they grow (rather than because it's the shape they like the best).

A simulated copper cluster.
Around (but below) their melting temperatures, crystals tend to have shapes which are pretty round: not a complete sphere, but with no regions which are flat (faceted). This is because at high temperature the atoms on the surface jiggle and wiggle more: they don't care so much which places are easier to sit because they have so much energy to spare. The facets appear at lower temperatures, as the crystal is cooled: the first temperature at which a facet occurs is called the roughening temperature. The facets start out small and close to the special directions; as the temperature goes down, they become larger.

Heating up to the roughening transition.

(Perversely, it's called the roughening temperature because the nice, flat face becomes all bumpy and rugged above that temperature. Of course, the microscopic bumpy, rugged face with the wiggly and jiggly atoms makes the nice smooth, round shapes you see in the experiments: the ``rough'' phase has round faces and the ``smooth'' phase has facets.)

A salt (NaCl) crystal in equilibrium at a cooler temperature of 620 C. (Heyraud and Métois, J. Cryst. Growth 84, 503 (1987)).
At even lower temperatures, for some crystals, the flat facets begin to merge, leading to a shape with sharp edges (at the edge rounding temperature). At lower temperatures still (below the corner rounding transition), the equilibrium crystal shape is predicted to show sharp corners. (It's easy to have sharp edges and corners if you cut or cleave the crystal that way. It's easy to grow crystals with edges and corners. Think, though, of how weird it would be to sand off the corners of your diamond ring, wait for a while, and see them re-form by themselves!)

It's been really hard for experimentalists to see these corners. They tell us that the crystals get really slow and sluggish just around where the corners would form. Joel Shore studied this problem in our group, when we were trying to understand why glasses get sluggish just before they freeze. The way the crystal finds the right shape is a lot like the way coarsening occurs when a material (like a metal) with lots of small crystalline regions is annealed to let the big regions grow at the expense of small ones. Joel showed that coarsening looks completely different if the regions have sharp corners!

More Information

Much of the information here is derived from Michael Wortis, in ``Chemistry and Physics of Solid Surfaces VII,'' Springer Series in Surface Science 10, ed. R. Vanselow and R. Howe, pp. 367 (1988).

Last modified: February 11, 1995

Jim Sethna, sethna@lassp.cornell.edu

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).