# Of Temples and Table Salt

Have you ever seen a diagram of a substance’s atomic structure (like the one below, for table salt) and wondered where it came from? If you haven’t, I’ll try to quickly explain why the existence of such a picture might be a mystery:

• Typical atomic sizes are between 30 and 300 picometers. A picometer is one thousandth of a nanometer, which is one millionth of a millimeter. Atoms are really small!
• Typical interatomic spacings in solids are bigger, but not by much; they are usually between 0.1 and 1 nanometers. So at the very least, we need to be able to detect things that are about 1 nanometer in size to determine the atomic structure of a molecule or material.
• The physical size of the average pupil prevents human eyes from seeing anything much smaller than a hair, which is about 0.1 millimeters wide– far larger than a nanometer! (Go here for an explanation.)
• In fact, the large wavelength of visible light prevents any standard optical device (including eyes and microscopes) from seeing anything smaller than a few hundred nanometers.
• X-rays have wavelengths comparable to interatomic spacings, but it is nearly impossible to build lenses for x-rays. Thus, x-ray microscopes don’t really exist.

So how do we know what anything looks like on such a small scale? Well, people have actually invented a whole bunch of clever methods for seeing very tiny things. I would love to talk about all of them, but for now I am just going to focus on one of the oldest and most widely-used techniques: x-ray diffraction. X-ray diffraction is a tool for determining the structure of a crystal, which is a solid material that has a repeating atomic structure. It’s true that not every material has a repeating structure, but many do. What’s even better is that many molecules can be tricked into growing in a crystalline form; for example, proteins can be stacked into periodic arrays. This allows x-ray diffraction to determine their molecular structures. This information is invaluable to physicists, chemists, biologists, doctors, and pharmaceutical companies. (Here is a big ol’ pile of protein structures.)

# Of Temples

Before I explain exactly how x-ray diffraction works, I’d like to talk about a similar (although slightly contrived) situation that will provide some great insight. Imagine, for a second, that you are an anthropologist studying hypostyle halls in ancient temples, like the one shown below. The roofs of these beautiful halls are supported by huge arrays of pillars arranged into repeating patterns. You are hoping to show that different cultures each use different pillar patterns. You have just arrived at your first temple; you plan to go in to the hypostyle hall and write down the location of each pillar so you can compare it to other temples. However, a temple guard informs you that uninitiated guests are forbidden from entering the sacred hall! You can observe it from the outside but cannot enter. Is your quest doomed, or is there some other way for you to determine how the pillars are arranged?

Of course there’s a way! Although you can’t walk into the hall, you can still walk around it.

In order to explain how this could be useful, I have built a model hypostyle hall out of some nails and cardboard and “walked” around it with a camera. The picture below shows the temple as it appears from a few selected points along the walk. (You can click on the picture to see a larger version if it is too small.) The walk begins at a random point, which I have characterized by an angle $\theta$. The pillars look like a confusing jumble from this point. As the walk continues, the pillars eventually appear to arrange themselves into lines. This first occurs at $\theta + 15^{\circ}$, and then it happens again at $\theta + 60^{\circ}$ and $\theta + 105^{\circ}$. Although I don’t show it in this diagram, the pillars also line up at $\theta + 150^{\circ}$, $\theta + 195^{\circ}$, $\theta + 240^{\circ}$, $\theta + 285^{\circ}$, and $\theta + 335^{\circ}$. Furthermore, the lines of pillars appear farther apart at some angles (e.g., $\theta + 15^{\circ}$) and closer together at other angles (e.g., $\theta + 60^{\circ}$). The ratio between the long and short inter-row distances seems to be about 1.46 to 1.

These results are definitely not random. The pillars line up every $45^{\circ}$, and the length ratio is awfully close to the square root of two ($\sqrt{2} \approx 1.414$). Our task is now to find out the significance of these observations. Let us first ask ourselves what we would expect to see if the pillars were all arranged in a square lattice (like graph paper). The picture below shows such a situation, where the pillars are shown as black dots. Red and blue dotted lines show two different ways for the pillars to line up. It can be shown that the angle between the red and blue lines is $45^{\circ}$, and that the ratio between $d$ and $d'$ is $\sqrt{2}$ to 1. This exactly matches our observations from the temple! It seems that this particular temple has a square lattice of pillars.

We shouldn’t jump to conclusions quite yet though; we should also check to make sure that different pillar patterns yield different angles and distance ratios. Another one is shown in the picture below. The angles between “pillar-lining-ups” for this pattern will be either $26.5^{\circ}$ or $63.5^{\circ}$. The ratio between $d$ and $d'$ is 2 to 1, and the ratio between $d'$ and $d''$ is $\sqrt{5}/4$ to 1. Clearly this is quite different than the square lattice!

We can now be confident that our observed temple had a square lattice of pillars. The aerial photograph shown below confirms our conclusion. (It is pretty much impossible to get all the nails to point perfectly straight upward; they form a perfect grid at their bases, but their tops aren’t so perfect…)

So now let’s put it all together. It seems that each pillar pattern produces a unique set of “lining-up-angles” and distance ratios. If we can measure the “lining-up-angles” for a particular temple, then we can figure out how the pillars are arranged! Once we know what type of pattern we have, we can then figure out how each observed inter-row distance relates to the distance between nearest-neighbor pillars. Thus, we can use each “lining-up-angle” to make an independent distance measurement! Even if our measurements have some uncertainty, we can average them together for a good estimate of the true value.

To put it together even more succinctly: if we measure the “lining-up-angles” and inter-row distances for a particular temple, then we can determine both the shape and size of its pillar pattern.

# Of Table Salt

Now let’s get back to physics and think about crystals. A crystal is actually a lot like a forbidden hypostyle hall. We know it is made of a repeating pattern of atoms, but we are unable to measure each individual atom’s position. Our experience with temples suggests that we might be able to determine the crystal’s structure by observing it from different angles. However, we can’t just look with our eyes to see when all the atoms line up in certain directions. We’ll have to be a little more creative! We’ll have to send in a probe that will be able to notify us of when the atoms have lined up.

For example, we could use a ball as a probe for the hypostyle halls. If we throw the ball forward and it bounces back, we know the pillars haven’t lined up; if it rolls between the pillars and out the other side, then we know the pillars have lined up. The ball obviously needs to be smaller than the distances between the pillars for this to work.

Similarly, we can use x-rays to probe a crystal. They’ll work because their wavelengths are smaller than interatomic separations. The only catch is that instead of looking for angles where they pass through the crystal, we’ll actually be looking for angles where they get reflected. For those who care, here’s why. (I think the rest of this post will make sense without this explanation, but it’s pretty interesting, so you can read it if you want to.)

Let’s consider two parallel x-rays traveling towards two parallel planes of atoms, as shown in the picture below. (Please recall that x-rays are electromagnetic waves.) The two waves start with their peaks and troughs lined up, but each wave travels a slightly different distance before being reflected back out of the crystal. In the top part of the picture, the angle of the incoming waves is such that the outgoing waves now have their peaks and troughs anti-lined up. This is called destructive interference. The two outgoing waves will cancel each other out and there will be no reflected x-ray. In the bottom part of the picture, a different incident angle results in the outgoing waves still having their peaks and troughs lined up. This is called constructive interference and will result in a reflected x-ray. For angles between the two that I’ve shown, the outgoing waves will have their peaks and troughs slightly shifted, but not so much as to cause complete destructive interference.

In a real crystal, there will be millions of planes stacked on top of each other. Therefore, if the angle of the incoming x-rays is such that the outgoing ones don’t line up perfectly, they will completely destructively interfere and there will be no reflection. (They might not destructively interfere with their immediate neighbors, but they will with the reflected waves from some other plane.) So for most angles, there will be no reflection.

The special angles that allow reflections will be the keys to unlocking the crystal’s structure. For hypostyle halls, each specific pillar pattern had its own unique “lining-up-angles” that allowed us to identify it. In the same way, each specific atomic structure has its own set of special angles where x-ray reflection occurs. For example, here is some x-ray diffraction data of aluminum that I took in one of my classes. The vertical axis shows the intensity of the reflected x-rays, which has sharp peaks at a few special angles.

Ratios of $\sin^2{\theta}$ for each special angle allowed me to determine that aluminum has a face-centered-cubic (FCC) lattice, as shown below. Actual values of $\theta$ (along with the wavelength of the incoming x-rays) allowed me to conclude that the sides of the cube are each 0.4050 nanometers long. I won’t go into the details of these calculations here because I think this post is long enough already. But please note that this is just like the hypostyle hall situation, in which relationships between angles determined structure and actual distance values determined size.

So, to summarize: we can determine the atomic structure of a crystal by shining x-rays at it from different angles, since the x-rays will only reflect off the crystal at certain special angles. This process shares a lot with the hypostyle hall situation. In both cases:

1. We know the object (crystal or hypostyle hall) has a periodic structure
2. We are unable to directly measure the positions of its constituent parts (atoms or pillars)
3. We are able to observe the object from different angles (using x-rays or our eyes)
4. There are certain special angles where something different happens to our observations (x-rays get reflected or pillars line up)
5. These special angles are unique to specific periodic structures (atomic structures or pillar patterns)

The main idea is that in both cases, we have found something that rarely occurs (either pillars line up or x-rays reflect) and use it to our advantage to learn about the internal structure of the object we’re observing. This is a common idea in physics, and occurs in other measurement techniques like nuclear magnetic resonance, infrared absorption spectroscopy, and many others.

# A Few Final Thoughts

Although I pretty thoroughly covered the main ideas behind x-ray diffraction, I have really only shown you a small part of what the technique is capable of. An in-depth analysis can show how to use the intensities of the peaks and background in a diffraction spectrum to measure the size of atomic lattice vibrations, the distribution of electrons in a crystal’s constituent atoms, the strength of the binding between neighboring atoms, and a whole variety of other interesting things. And diffraction doesn’t end with x-rays; electrons and neutrons are small enough to use for diffraction as well. I made a big effort to not use any formulas in this explanation, but reading some more detailed treatments of x-ray diffraction can really enrich your understanding (once you’re familiar with the basic ideas). In particular, it’s pretty interesting to learn why x-rays reflect off of parallel planes of atoms; the answer involves applying scattering theory to interactions between x-rays and electrons. This scattering process is the heart of an in-depth treatment of x-ray diffraction.