Exploring Crystal Lattices

2: Enter the Model Universe

Although we're restricted by the discrete sets of strut lengths and connection angles available in Zometool, we've seen that some fun stuff can be built, and some crystallographic lessons can be extracted from it.

Let's become a little more adventurous, and go beyond the isometric crystal system. Now we quickly hit another obstacle: Zometool has been designed around icosahedral symmetry. It is well known that periodic crystal lattices in three dimensions cannot have fivefold rotational symmetries, so at first glance we seem to have little use for all those 5-sided R struts. (Which is why I didn't hesitate to "waste" dozens of R1 struts in my little stands for node-painting.) Conversely, icosahedra admit twofold and threefold and fivefold rotations, but no fourfold or sixfold ones, nor reflections in the diagonal {110} planes of their circumscribed cubes.

Sadly, there's no apatite and no beryl in the Zometool universe!

And anything with a 4-fold rotation or screw or rotation-inversion axis is iffy at best. Not even our fluorite model from the previous chapter really admitted 4-fold rotations, once you look at the cross sections of the struts: That model really only achieved pyrite symmetry.

And there are no 180° rotations around a 3-fold (Y) axis, nor reflections in planes at right angles to such axes; and therefore no spinel-law twins and no benitoite.

So the Zometool universe is not a subset of the crystallographic universe, nor vice versa... but they overlap. I'll be referring to the common ground as the model universe.

Fortunately, the model universe does contain a lot of interesting stuff... more than I had anticipated!

A few compromises need to be made. Sometimes, the overall axis ratio can only be approximated; sometimes, there are local distortions. And sometimes both.

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Chalcopyrite

An easy way to get from an isometric lattice to an example with tetragonal symmetry is to play with site occupants. Positions which are equivalent under the symmetry (space group) operations of a given lattice may be occupied by atoms/ions of more than one kind, and these may be ordered in ways that reduce the original symmetry whilst (usually) preserving some part of it.

Recall that sphalerite (no pictures, sorry: I had built one, but scavenged its parts for the following...) has a diamond-type lattice with zinc cations and sulphur anions strictly alternating. Each Zn is surrounded by a tetrahedral arrangement of four S, and vice versa. The Zn sites considered on their own form one face-centred cubic (fcc) lattice, alias cubic close-packing (ccp), and the S sites form another, shifted a quarter of the way along a fundamental cell's body diagonal in relation to the first.

Now there are ways to "halve" an fcc lattice again, partitioning its sites into two subsets of equal density - but certainly not in strict alternation, since nearest neighbours within an fcc lattice form lots of triangles.



One way of partitioning produces an arrangement which preserves one of the 4-fold rotation-inversion axes, and thus takes us into the tetragonal system. Looking sideways through the lattice along a 2-fold rotation axis, cations of the same kind (black or red in the picture) are aligned in uniformly coloured rows extending along the line of sight. (Ok, so this preliminary model is too small and we see only one such row of two black nodes, right of centre. Let your imagination fill in the others.) Pairs of such rows are stacked above each other (in the top and middle layers of the picture) to form zigzag ribbons of each colour. The colours then swap places for the next (bottom and beyond) pair of layers. Rotating the whole affair 90° around the vertical axis, the view would be similar, but we would then see the rows in the middle and bottom layers paired into ribbons along the new line of sight, with colours swapping places for the top row and beyond. Each cation is on the top edge of one zigzag ribbon and at the bottom edge of another zigzag ribbon orthogonal to the former.

If we declare our two cation species to be Cu and Fe, and use nodes painted to match, with yellow nodes for S^2- anions, we get chalcopyrite:

(To make the copper nodes stand out better, I used white Y0 struts for all their bonds, instead of the basic yellow ones.)

This model still fits into a cubic framework. All the tetrahedra we see have all their edges of the same length. But in the real crystal, there no longer exist any symmetry elements that would force all the real-world distances and angles to stay as they were... and thus they really don't stay unchanged! We should imagine everything stretched along the new c direction by about 1.5%. This isn't much, and would hardly be visible in the model. (The fact that the new a1,a2 axes can be chosen at 45° to the old horizontal cube axes then throws another square-root-of-2 factor into the axis ratio; but that's an artifact of our human conventions, not a property of the lattice as such.)

Yet this stretching makes a big difference from the point of view of crystal chemistry. Because the Fe/Cu ordering in one small corner of the crystal causes the lattice there to stretch a little in one direction, the cations in the next cell will preferentially occupy sites which are again compatible with a stretching in the same direction. This back-and-forth coupling between the geometry and the bond chemistry is what allows the local ordering of a small handful of atoms to extend into a long-range order extending over many millions of cells, resulting in a macroscopically tetragonal crystal. (And twins may form when there are mis-orderings.)

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A rough rutile

G struts give us, inside the model universe, all the possible face-diagonal directions of each cube. As long as we use these and no other more-or-less-diagonal struts (in particular, no R struts), we can construct some genuinely tetragonal lattice models, departing from isometric symmetry in ways other than mere site ordering.

Using white nodes (which we possess in abundance) for oxygen (which exists in abundance in the lithosphere), and for example black nodes for the metal sites, and B1 and G0 struts for the bonds, we can build an approximation to the rutile lattice. It consists of ribbons (formed by the G0s) parallel to c, with metal sites aligned above each other and a pair of side-by-side oxygens between each pair of cations; adjacent ribbons are rotated 90° relative to each other around c and shifted vertically by half the cell height. The B1 struts bond the metals of one ribbon to oxygens at the same height of two adjacent ribbons.

The long horizontal G2 struts outline several fundamental cells and lend stability to the model. (Note that the a1,a2 axes run in G directions of the model universe, not in B directions.)

Cassiterite shares the rutile structure, as does the ultra-high pressure modification of SiO₂, stishovite.

While this model correctly captures the space group and the nearest-neighbour relationships, it is seriously distorted overall: The c unit is a whopping 20% too short, and thus the angles of the pyramid-forming planes are quite wrong. The ribbons should not consist of successive squares; they should be a bit narrower and taller. In reality, the two oxygens facing each other at opposite edges of a ribbon "touch" each other: they're as close as the oxygen ion radii allow. The oxygens lined up above each other along a vertical ribbon edge, however, should be a little farther apart.

We'll return to this topic later. For now, let me promise that this is the most distorted axis ratio you're going to see in these pages.

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- Gerhard Niklasch ©2017