Exploring Crystal Lattices

7: The Mother of all Silicates

...meaning quartz, of course. And my hat is off to Amir C. Akhavan whose Quartz Page is a fabulous resource. (Mindat's own quartz page has been augmented with several excerpts from it.)

Quartz has its own unique lattice (unique up to a reflection!), and a very distinctive repertoire of shapes. Many macroscopic quartz crystals are immediately recognizable - it's almost impossible to mistake a quartz termination with well-developed faces for anything else.

That our model universe contains a fairly good approximation to the quartz lattice may not be immediately obvious. It is easier to see that, up to a reflection, it can contain only one - and then, as a second step, we can try out the only possibility, and find that it works. The starting point is that the Si atoms lie on 2-fold rotation axes, and these axes run in three different directions.

There are ten ways to inscribe a tetrahedron into a fixed dodecahedron, and therefore ten ways to stick four Y struts into a fixed node in a tetrahedral arrangement: five "left" tetrahedra and five "right" ones. For the moment, I'll use white Y0 struts as bonds for the left and yellow ones for the right tetrahedra.

If we keep the node fixed with a 3-fold axis vertical, one of each will have a strut going straight down:

(Mnemonic: Stick one Y strut into a node. Hold the combination as if it were a lollipop, thumb on a flat side at the upper end of the strut. To build a left tetrahedron, stick the second Y strut into the triangular hole to the left of the B (rectangular) hole in front of your nose. Rotate the affair through 120° for the third and fourth Y struts. To build a right tetrahedron, use the hole to the right of this B hole instead.)

One of the five left tetrahedra and one of the five right ones have a Y strut going straight up; and three of each will have a horizontal 2-fold rotation axis, bringing the total to ten. Rotations through 120° around the vertical axis will permute the members of each set of three:

From now on, we'll stick with left tetrahedra, all with horizontal symmetry axes; and this will result in a model of left(-handed) quartz. The grey nodes will stand for the Si atoms, and the white ones for O.

In any given horizontal layer, all the tetrahedra and all their 2-fold axes are parallel:

The next layer is rotated 120° against the first, along a right-hand screw motion. By a fortuitous arithmetic coincidence, the B2 struts representing our 2-fold axes are just the right length for the Y0 bond struts to reach the common white oxygen nodes:

In the same way, the tetrahedra in the third layer are attached, after another such screw motion, to those in the second layer:

And so we continue, for as long as our parts suffice. The black and yellow parts are scaffolding, but they also outline one fundamental cell (placed at a nonstandard origin). Near the top, I've added the contours of three of the tetrahedra (using six G0 struts for each).

Unfortunately, the 3-fold screw axes along the narrow helices are out of reach - they would need custom parts, one third or two thirds the design length of the regular ones. (A lone teal HG1 strut with a broken end points horizontally from a grey node into one narrow helix, roughly to where it would meet its screw axis.)

Likewise, the screw axes of the wide helices cannot be reached by standard parts, but in two places I have used 3D-printed additions to place a blue node at the intersection of such a screw axis with a horizontal 2-fold axis. The blue B2 is subdivided in the ratio √5:2 (very close to the correct value). The longer segment consists of a B0 and a bob (one node diameter long), which just touches the blue node. (It's not attached, so the contraption sags a little under its weight. There's an imaginary zero-length "B minus 2" connection missing.) The shorter segment is made of two "B minus 1" struts and a bob in the middle.

There are way more 2-fold axes than the model shows explicitly. For example, one set of 2-fold rotation axes passes through the auxiliary yellow nodes in the preceding stereo view, parallel to the line of sight. And any pair of adjacent parallel 2-fold rotation axes in the same horizontal plane has a 2-fold screw axis halfway between them.

Note that the two narrow helices outlined by the green and red B1 struts are not mirror images of each other - both are right-handed, and related by "upside-down" rotations around the 2-fold horizontal axes passing through the Si nodes they have in common. In the green helix, the connecting oxygens are below the green struts; in the red one, they're above the red struts.

Note also that all the horizontal blue struts are tilted sideways by a small amount. The model universe does not contain the high-temperature hexagonal modification (called β-quartz or quartz-β to distinguish it from ordinary α-quartz, although some older literature had the greek letters the other way round); but in our imagination we could visualize it as the outcome when we seize all the blue struts and forcefully twist them until their rectangular cross sections are horizontal. The top and bottom edges of all the tetrahedra would then also be horizontal. If we then let go again (allowing our imagined high-temperature quartz to cool), they might either snap back, or snap to the opposite inclination, ending up with the Dauphiné twin partner of our original lattice (but staying a left quartz all the time). No bonds are broken or reconnected during this process!

It would be truly a miracle if nature was in exact agreement with all the constrained angles of the model universe, and indeed it isn't quite: Our Si-O-Si angles should be slightly more obtuse, our tetrahedra slightly less tilted out of the hexagonally symmetric position (the model shows a somewhat exaggerated low quartz), and the height of the unit cell about 5% larger than it is in the model. This means that the top node of the model should be higher up by about one node diameter. Not dramatic, but if you look carefully, the rhombohedral planes should be visibly a little bit steeper.

Which face is which?

Now that we have a model which we know to be of untwinned, left-handed quartz, wouldn't it be nice to point at it and say: Here is an r face, and here is a z face, and an s face would form here? This plane belongs to the positive rhombohedron, and that one to the negative rhombohedron? Can't we just use the Miller indices?

Unfortunately, it's not that obvious. One part of the issue is that we generally insist on using right-handed coordinate systems even when we want to describe chiral objects (like quartz crystals), which differ from their mirror images. Also, there's more than one way to choose the axes for a trigonal crystal. The coordinate axes x1,x2,x3 (NB sometimes called a1,a2,a3 but I prefer to reserve that notation for the cell edge lengths along the axes) will obviously be oriented parallel to the 2-fold rotation axes, but we still need to choose where the positive coordinate directions shall point, and there are two possible choices.

And more than one convention, unfortunately, is in common and widespread use for quartz.

The convention chosen by the 19th century crystallographers is the one which assigns Miller indices (1011) to an upper r face, regardless whether we're dealing with a left or a right quartz.

An upper s face of a left quartz, if present, will then get indices (2111) - but (2111) when it's a right quartz. (For the faces of the lower half of the crystal, apply a 2-fold rotation, which swaps two of the three x coordinates whilst changing the sign of the z coordinate.)

With perfect hindsight, it would have been wiser to keep the s face indices the same for left and right quartz, and allow the r face indices to change. Maybe this would have happened in an alternate universe where the piezoelectric properties of quartz had been known four decades earlier. As it was, Jacques and Pierre Curie discovered piezoelectricity in 1880 - the year William Miller died.

When a 20th (or 21st) century crystallographer reports the site coordinates derived from X-ray diffraction measurements, the convention for quartz is to put the origin on the intersection point between a 2-fold axis and a 3-fold screw axis passing through a wide helix, and have the x1 axis point at the nearest Si atom, less than half of the cell width away.

We can readily see how this fits into our model (using one of our blue nodes as the origin), but how does it relate to the older macroscopic convention? Single-crystal XRD often starts from a small slice cut at a known angle to the z axis (verifiable by optical means), but at an unknown angle to any r faces, from an untwinned, defect-free, preferably lab-grown crystal. (Powder XRD preserves no information about the macroscopic crystal orientation at all.)

So couldn't one just put a large, pretty quartz from the Swiss Alps with well-developed s and x and other faces...

...into an X-ray diffractometer to settle the matter...?

Well, no: Firstly, a large one won't even fit. Second, it would be a shame to cut it up. Third, large naturally grown crystals tend to have lots of growth defects, resulting in noisy measurements. Fourth, and worst,...

...they are often twinned (even when this isn't immediately obvious); and any Dauphiné-law twinning would ruin the exercise!

Piezoelectricity comes to our rescue. It turns out that the two conventions agree for left quartz, but are opposites for right quartz.

When an (untwinned) macroscopic quartz crystal is compressed along a horizontal axis, the prism edge between two s faces becomes negatively charged, and the opposite edge becomes positively charged.

When the crystal lattice is compressed, we can readily see from first principles, model in hand (the Quartz Page has some simplified diagrams), that the positive direction of the x1 axis in the X-ray convention will point towards the negatively charged edge. The reason is that the Si-O bond lengths are quite rigid and won't change much under pressure, while the bond angles will flex a bit. Those tetrahedra which have 2-fold rotational symmetry around an axis parallel to the x1 coordinate axis won't contribute to the effect: Each of their four bonds forms the same angle with the line of force; the tetrahedron will be distorted a little but the Si atom and its positive charge will remain at the centroid of the four oxygens. But all the other tetrahedra in the layers above and below will contribute: Each of their Si atoms has one oxygen on its +x1 side, almost along the line of force, and will thus get pushed off-centre towards the -x1 direction; the other three bonds are at large angles to the line of force and will bend, displacing these three O ligands towards +x1.

Thus the +x1,x2,x3 directions as defined by the X-ray convention will always point towards those prism edges which connect two s faces - those where the negative charge will appear under compression.

And the rhombohedral face above our green and red helices is a z face. In the chapter title picture, repeated here for convenience,
several parallel r planes are seen from the side (in-plane) going from bottom left to top right, and z planes from top left to bottom right.

Till we have faces

Our first model spans well more than one fundamental cell, and is big enough to show the essential features of the interior of the quartz lattice: narrow and wide helices, 3-fold screw axes in the z coordinate direction, 2-fold rotation and screw axis in the three x coordinate directions. However, it is not big enough to see how the various faces really "work".

To that end, I built a second model, using many more parts, showing just the outer layer of a quartz termination, with (slightly off-white) nodes on yellow Y0 struts representing the (not entirely hypothetical) OH groups sticking out into the surrounding medium.

First, a couple of side views in stereo. In both, we're looking in-plane along an r face (almost flat) at top right, and along a z face (wrinkled) at top left - the opposite of the chapter title picture.

After rotating this model through 120°, its front side features an r face above an m (prism) face with green helices on the left, and a z face above (the other end of) an m face with red helices on the right.

Here is the termination, seen from a very steep angle, with an r face below it. It so happens that the one lone topmost tetrahedron continues the pattern of each of the six rhombohedral faces: they all meet in a perfect point!

Next, looking straight at an r face, we can see it consists of chevron-shaped tiles surrounded by rings of eight Si and eight O, each tile bisected by a 2-fold rotation axis. Looking straight at a z face, we see C-shaped tiles, similarly bisected. Of course, the topology (what connects to what) of both kinds of rhombohedral face is the same - remember that we could turn one into the other by heating and cooling without rearranging any bonds.

The r, z, and m faces are formed from tetrahedra which are either connected to four other tetrahedra inside the body of the lattice, or to three, with one OH group sticking out. The same is true of the edges between an r face and a z face (except for the tetrahedron at the very top). Tetrahedra with two OH appear around the edges of the m faces. (They would also appear where two r faces or two z faces meet.)

In one corner of this model, to the lower left of an r face as it must be in a left quartz, I constructed an s face along what looks to be the least reactive plane in the correct orientation. Looking diagonally downward in the plane of this face, we see a shallow groove; the face presents a more or less flat contour when seen from the side. Note that, unlike the r and z and m faces, this one contains Si atoms with only two attachments to adjacent tetrahedra and thus two OH groups sticking out, explaining in part why an s face is less inert than r and z and m and therefore less likely to become prominent in naturally grown quartz crystals. (The largest faces are those which had been the slowest to accumulate new layers during growth.)

Looking straight at our s face, we see square-ish (but wrinkled) tiles surrounded by rings of ten Si and ten O, plus there's a lone OH sticking out of a shallow depression near the middle of the square. The little tick marks point to five Si sites in one common (2111) lattice plane which are equivalent under translations; each is the topmost Si of one such tile.

In stereo - this one is a bit difficult to blend; you may find it easiest if you first focus on the node at bottom centre:

Unlike the rhombohedral faces, an s face one would not stay the same when turned inside out. (In real quartz crystals, an s face would have a corner diametrally opposite to it, rather than another "anti-s" face.)

It would take an even larger model to exhibit a recognizable x face...

Faces up close

Quartz, widely known for centuries and widely used in modern technology, continues to be an active research topic. Let me mention one 21st century example: Michel Schlegel et al. used atomic force microscopy (AFM) to study prism and rhombohedron faces (of natural crystals from Herkimer Co.) and their interaction with water, right down to the scale of the first several individual layers of atoms. (It is somewhat difficult to find such publications... Just about every paper about AFM mentions quartz, not as a topic of investigation, but because a quartz tuning fork or a quartz microbalance is part of the measurement apparatus!) Note that their lattice diagrams (e.g. Fig. 6 and 9) show right quartz, but they're using the 19-th century coordinate system, not the X-ray one: Their (1011) is indeed an r face.

Reference: Schlegel, M.L., Nagy, K.L., Fenter, P., Sturchio, N.C. (2002) Structures of quartz (1010)- and (1011)-water interfaces determined by X-ray reflectivity and atomic force microscopy of natural growth surfaces. Geochimica et Cosmochimica Acta: 66(17): 3037-3054.

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- Gerhard Niklasch ©2017