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Exploring Crystal Lattices - 6: Holes for Giants

Last Updated: 9th Feb 2017

By Gerhard Niklasch

Exploring Crystal Lattices

6: Holes for Giants

A giant in its hole

When your toolkit consists of just a hammer, everything looks like a nail. We'll now add cardboard and scissors and glue and a little bit of string to our kit in order to illustrate something that's hard to show with Zometool parts alone.

All our nodes are the same size, but ion sizes vary considerably. Many common anions are larger than many cations (and therefore often dictate the arrangements, which is one reason why we classify minerals by anion types: oxides, carbonates, silicates,...). But among the cations, the alkali ions are especially large. (The unionized alkali atoms would be much larger yet.) Caesium, in particular, takes up an enormous amount of space, and few crystal lattices have that much room to offer.

Paradoxically, one such lattice can be constructed from building blocks borrowed from the rather rigid spinel structure, replacing the MgO4 tetrahedra with somewhat-too-small BeO4 ones, and then stitching adjacent blocks together with similarly small BO4 tetrahedra.

In spinel, there were numerous groups of four AlO6 octahedra sharing edges in pairs and oxygens in threes (and with adjacent such groups overlapping in a common octahedron). We'll now start over with just one such Al4O16 group, represented by blue and white nodes as before. In order to have a flat surface at the bottom of our model, we'll cut this group in half. Then we'll attach BeO4 tetrahedra to the inner oxygens (those shared among three octahedra). One oxygen per tetrahedron was already there, so the entire group (including the bottom half we've omitted) now accounts for Be4Al4O28. For the beryllium sites, I chose nodes painted a light shade of mint green:

Starting from octahedrally coordinated Al...
...and tetrahedrally coordinated Be...
We're fudging a little here: These tetrahedra should be somewhat smaller, or the Al-O bonds somewhat longer in comparison (but not as long as in the exaggerated picture in chapter 4). The model will be idealized in this regard.

Next, we attach four nodes painted fern green and representing boron to the six white oxygen nodes at the top:

...we add boron...
A view along a future mirror plane
The top is just one of six faces of the fundamental cell, so doing this on all sides would require twenty-four borons - but they sit on the walls of the cubic fundamental cell we're building, and thus each B is shared between two cells, bringing the grand total per cell to Be4B12Al4O28.

Completing the BO4 tetrahedra with the adjacent oxygens from the next cell, we get another "domino-tile six" of oxygen attachment points for the next Be4Al4O28 group at the heart of this next cell. (The BO4 tetrahedra should be tilted a bit to account for the Al-O bond lengths; we're idealizing this tilt away.)

Ready to attach the next cell (stereo).
One cell nucleus and a half, along {110}.
One cell nucleus and a half, along {100}, stereo.
One cell nucleus and a half, along {111}, stereo.
Note that all of the little tetrahedra are pointing the same way. The structure as a whole will have tetrahedral symmetry, lacking 4-fold rotations (or screws) and lacking reflections in {100} planes. There are {110} mirror planes, though.

And now we can continue in the same way, in every direction: left and right, and front and back, and up (and down where the table isn't in the way).

The oxygens (white nodes) occupy the sites of an fcc/ccp lattice... but they occupy only 28 of the 32 sites in each cell.

At the corners where eight cells meet, this leaves a large hole - nay, a GIANT hole -, shaped like a truncated tetrahedron.

Rhodizite and londonite

And this is where there's plenty of room left to fit a large K+ ion, or a huge Rb+ one, or a GIANT Cs+ one (though a modest-sized Na+ wouldn't be a problem either). I made a truncated tetrahedron from red cardboard, tried at first to suspend it on an array of short auxiliary struts, and then decided to suspend it from two short loops of thin black thread instead:

The alkali site and its surroundings
In fact, I think of this red polyhedron as representing rubidium - caesium would be even larger, and there's still room to put something larger in this place. I'll try that once I've found some nice light-blue cardboard!

Before I reached for ruler and scissors and glue, I had used four red nodes clustered in a small volume to represent a single (potassium, say) cation. This contraption has survived in another corner of the model:
Another alkali site and its surroundings
If you have kept careful count, you'll have noticed that having enough room for another cation isn't all - we also need to balance our charges, and (K,Rb,Cs)Be4B12Al4O28 would be off by one!

The current IMA formulas KBe5B11Al4O28 - CsBe5B11Al4O28 achieve charge balance at the expense of another idealization. In natural crystals, charge balance is maintained by a combination of three factors, in varying proportions: A fraction (15-20%) of the alkali sites remains vacant; some excess beryllium (usually rather less than one atom per formula unit) does substitute for boron; and some of the boron sites remain vacant - each of the latter vacancies removes three positive charges from the picture.

Also, in real crystals Be and B are disordered to some extent across all the small tetrahedral sites.

Londonite-rhodizite, stereo

Tetrahedral faces

Macroscopically, most rhodizite(/londonite) crystals only show the faces of the rhombic dodecahedron {110}, which looks the same regardless whether the lattice inside it has full cubic/octahedreal or merely tetrahedral or pyrite-type symmetry. But occasionally one encounters a specimen where an equilateral triangle of the positive tetrahedron {111} truncates a corner where three obtuse rhomb angles meet. Here's one where the triangle, although not quite complete, reaches an impressive size:

Rhodizite, Antsongombato gem mine
Rhombic dodecahedron truncated by the positive tetrahedron
Rhodizite, Antsongombato gem mine
Rhombic dodecahedron truncated by the positive tetrahedron
When you do spot such a crystal, it's worth looking at the opposite obtuse corners under magnification. You may well find a tiny face of the negative tetrahedron {111} as well - 0.4mm wide on this same specimen:

Rhodizite, Antsongombato gem mine, stereo
Rhombic dodecahedron truncated by positive and negative tetrahedra
Conceivably this tiny face might be a cleavage. The larger positive triangle clearly has grown that way.

Back to: Introduction, Contents, and Glossary

- Gerhard Niklasch ©2017

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