What are crystals really? — a physicist’s perspective

Quartz crystal (from Wikimedia Commons)

If I asked you to imagine a crystal, what would come to your mind?

Most people would probably think of some kind of quartz crystal as in the picture above. Of some color — maybe white, red, or green. Transparent or translucent. With flat and elegant facets. With well-defined edges and corners. The ones that can be found in mines and are commonly sold in souvenir shops throughout (the touristic parts of) the mountains. In fact, as a kid I was very proud of my own little collection of minerals, which I gathered during my family’s summer vacations in the Alps and which also contained some quartz crystals. They are among the most abundant minerals on earth and come in different varieties. Because of their beautiful appearance they have also been used as gemstones for jewelry throughout history. But what does this have to do with physics?

Quartz is indeed a prototypical crystal. But crystals are so much more than minerals and gemstones, especially from the perspective of a physicist. In my last blog post, where I introduced the field of condensed matter physics — check it out, if you haven’t already — I told you that atoms typically arrange themselves into certain patterns to form the physical matter that surrounds us. Some of these patterns are characterized as crystals. Keeping this in mind, let’s have a closer look at quartz… a much closer look down to the atomic scale. This is actually possible with modern electron microscopes, which can resolve even individual atoms. Quartz consists of silicon and oxygen atoms. More precisely, the oxygen atoms sit at the corners of tetrahedra that form a particular kind of network (see the image below). The silicon atoms are positioned at the centers of these tetrahedra. Most importantly, the pattern of these tetrahedra is not random. The pattern repeats periodically in certain directions. Have a look yourself in the image below: can you find the periodic directions?

Pictorial structure of the atoms in a quartz crystal: the oxygen atoms (red spheres) sit at the corners of tetrahedra (blue facets), while the silicon atoms (blue spheres) are located inside the tetrahedra. The black lines in the center indicate the periodic directions along which the pattern of the atoms repeats. Taken together, these lines form the unit cell of the quartz crystal (image source)

The atoms form a periodic pattern, or lattice, which is the defining property of a crystal.

The volume spanned by going only one period along each of these periodic directions defines a so-called unit cell. The entire crystal can be thought of as stacking millions of these unit cells together — just like LEGO bricks. This kind of spatial periodicity is an important concept in condensed matter and solid state physics. We could literally move — or “translate” — the entire crystal by some number of unit cells in any of the periodic directions and the crystal would look locally identical. We couldn’t tell the difference only from looking at a small fraction of the crystal. Such an invariance of the spatial pattern is called translational symmetry.

I hope you’re still with me.

Let’s look at a simpler example to get a better feeling for the concept of translational symmetry. Recall again my last blog post, particularly the part about pencil lead, which isn’t actual lead but graphite. Graphite consists of stacks of carbon sheets. A single one of these sheets is called graphene. It’s a single layer of carbon atoms which is only one atom thick — for real. Remarkably, in 2004, the physicists Andrei Geim and Konstantin Novoselov managed to analyze pure graphene by isolating it from graphite using common Scotch tape, which got them the Nobel prize in Physics only a few years later. In graphene, the carbon atoms form a two-dimensional lattice in the shape of a honeycomb — a so-called honeycomb lattice (see image below). Its unit cell is spanned by two periodic directions forming a parallelogram with a small angle of 60°. The unit cell contains two carbon atoms. We could build the entire graphene crystal from scratch by repeating these unit cells and by seamlessly covering the entire plane with them — just as with quartz or any other crystal. And again: shifting the lattice along the periodic directions doesn’t change the pattern. That’s the essence of translational symmetry!

Pictorial structure of the atoms in graphene: the carbon atoms (blue spheres) form a two-dimensional lattice in the shape of a honeycomb. The green lines indicate the periodic directions. They form the unit cell of graphene.

While we’re at it: translational symmetry is not the only symmetry a crystal can have. Take a closer look at the graphene lattice above. You might notice that we can rotate the lattice by 60° around the center of any hexagon and the resulting lattice would look identical. This is called rotational symmetry. We could also put up a mirror going through the center of a hexagon right in between two carbon atoms. The picture you would see in the mirror would be identical to what the lattice looks like behind the mirror. This is called — you might already guess it at this point — mirror symmetry.

That’s all you need to know about crystals and symmetries to follow me on my journey. In my next blog post, I will shed light on the question: “What are electrons and why do they like to form bands?”

As always, thanks for reading!

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This popular-science blog is about a fascinating link between the physics of condensed matter and a mathematical discipline called topology. I write about the basic concepts and about my own research revolving around this topic in an accessible language.

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Alexander Lau

Alexander Lau

Theoretical physicist and science enthusiast

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