Are Miller Indices used for anything, or are they just a convenient notation?

October 17, 2009 9:52 AM

I am taking an intro to materials science class, and in which we have learned about Miller indices, which are a way of denoting directions and planes in a unit cell. So far we have done nothing with this other than draw directions and planes. I was wondering, is this just a notational convention or is there some reason why they are the way they are, as in does it make some more advanced math easier or something?
posted by selenized to science & nature (6 answers total)

Well it has been a while, but the indices describe the periodic structure of a crystal. This means that the electron number density exhibits a periodicity in the crystal described by the miller indices. This periodicity can be exploited through Fourier analysis and properties such as xray diffraction and really everything interesting about crystals is related to this Fourier analysis.
posted by wigner3j at 11:05 AM on October 17, 2009

I'm not really sure what you're asking, but the gist is that they tie into the physical nature of the lattice properties of the material. Since you need three numbers to denote any direction in three dimensions and the natural directions of a lattice are the three lattice vectors, the Miller indices give you an efficient way of describing directions within a lattice based on these fundamental directions. It also turns out that much of what goes on in a lattice has certain constraints about being the same for any unit cell, so directions that are similar in all cells turn out to be preferred for a lot of behaviors. These directions correspond to the possible (100) (010) (111), etc combinations with only 0s and 1s.

In general, the indices are good at bridging the notion of direction with the notion of the physical nature of the lattice. For example, when a material is cut along a particular plane, the indices tell you which sides of the interface are bound and which are free in a way that directly reflects the lattice. You can basically read off the free energy of the interface from the indices, for instance, and use that to figure out the shape that the interface will tend towards.

Perhaps you can get more useful answers if you mention what seems unnatural about them to you?
posted by Schismatic at 11:06 AM on October 17, 2009

They're used for computations. Your book probably has formulas for things like the distance between families of planes and the angle between intersecting planes. This last one is used all the time in anisotropic etching of silicon; for example, the angle between the (100) (a common surface orientation for silicon wafers) and (111) (the slowest etching plane, which emerges when a concave mask pattern is employed) is 54.7 degrees. This is technologically important for bulk micromachined devices like pressure transducers and accelerometers.
posted by Wet Spot at 11:12 AM on October 17, 2009

miller indices are an easy and visual way to get you thinking more intuitively about the notion of reciprocal space and to ease you into thinking reciprocally about crystal planes. parallel sets of planes in real space correspond to points in reciprocal space; understanding this and how to go back and forth between the two lets you turn diffraction data into meaningful information about the physical location of things. really understanding this stuff is what you need to fully get what a brillouin zone is, what the ewald sphere is, how the mathematical description of diffraction works, and so on. basically all this is what wigner3j said, though more eloquently.

(it is, by the way, a fantastically useful skill on its own, to be able to visualize any particular plane for a given crystal structure. what you are learning now is one of those things that in five years you'll realize is actually incredibly useful if you are ever doing anything like crystallography, surface microscopy, TEM, etc etc etc. it's nice to be able to look at an STM image and know you're looking at the 110 surface of an FCC structure, f'rinstance. take my word for it.)
posted by sergeant sandwich at 12:58 PM on October 17, 2009

Schismatic: to clarify: It's not that I find the Miller index system unnatural, it just seemed like an arbitrary notational system. Basically my question is: given a set of indices for something (plane, direction, or family of) what can I do with that? other than just draw them on a unit cell and say "ah"

sergeant sandwich: so, presuming you had some x-ray data and wanted to figure out the unit cell from that, would various parameters be functions of the miller index for things. Or would you be exploiting a more fundamental symmetry of the crystal? (or have I missed the boat entirely)
posted by selenized at 3:41 PM on October 17, 2009

It is arbitrary in the same way we the x,y,z basis set for Cartesian space. There are some many ways to describe 3D space, why choose the one we do? A lot of science is purely arbitrary in this way.

Miller indices are used in X-ray crystallography all the time. Basically, the whole of mineral science is based on it. Also, a lot of material behaviour and properties can be described using Miller indices, such as slip planes, micro structures, defects, etc. What about looking at anisotropic metallurgy? You need it to describe planes in metals. Not to mention the whole concept of reciprocal space is made a whole lot easier (this concept is used in solid state physics to explain Bravais lattices, which leads to semiconductors...). In fact, it is even used in chemistry, catalytic surface side chemistry is a great example.

So it short, it is damned useful. Although it's pretty boring if you learn it by itself. I know the pain, did 2 years of materials science and the X-ray stuff was by far the most boring.
posted by dragontail at 6:04 AM on October 18, 2009