# What, physically, are the Miller planes of a crystal?

1. Apr 9, 2012

### Roo2

I'm trying to learn crystallography and I've had trouble with this concept since the very beginning of the course. It's been so long since it's been introduced that I'd be embarrassed to ask the prof. Right now, I seem to understand the principles of diffraction based on the Miller model; that is, based on the phase changes of radiation scattered from Miller planes at non-n*λ spacings. However, I don't understand what these planes actually are in real life. For a diffraction grating, I can physically see the ridges on the grating, which "validates" the model for me. However, for something as complicated and non-plane-like as a protein, I completely fail to see where the planes originate.

Could someone please give me some help?

2. Apr 9, 2012

### dllahr

Well, I always remember that the Miller indices indicate the vector that is perpendicular to the miller plane. The vector is defined in terms of lattice vectors. So in the simplest case, a lattice is 3 orthogonal vectors of the same length. For any arbitrary miller indices, you now have a vector you can draw in 3D. Now using your vector math, you can define the plane that is perpendicular to that vector. The atoms that lie in that plane are the diffraction grating.

3. Apr 10, 2012

### mcodesmart

They are like real space crystallographic planes but in reciprocal space

4. Apr 14, 2012

### Roo2

But aren't the crystalographic planes arbitrarily defined? How can mathematical constructs produce physical effects?

5. Apr 15, 2012

### sam_bell

Imagine the argument made by Miller applied not to the planes, but simply to two point particles (proteins in your case). If the phase change is zero, the reflected waves add constructively. However, since this is just for two particles, it might not amount to much. Now consider that the incoming and reflected waves define an imaginary plane running through the particles. If this plane is one of the Miller planes, then it "hits" every particle in the lattice. Which means the argument can be repeated for every particle, and ALL the reflected waves add constructively. An effect THAT large, you WILL see.

6. Jul 5, 2012

### apung

An easy way that I found to remember / find which Miller Indices belong to which plane, is to to count how many atoms laterally, and how many atoms vertically make up the plane. That is, [1 3] would be a plane that, if you start at *any* given atom in a crystal, the plane will be defined by drawing a line from that point, to the atom that is 1 row up, and 3 atoms backwards. That is. this plane would be defined by the outermost points (X) of:

X X X X

. X X X X

7. Jul 5, 2012

### Staff: Mentor

Have you ever held a 3-D ball-and-rod model of a simple crystal (say a cubic one like NaCl) in your hands and looked at it while rotating it? You can see the planes vividly at certain orientations.

For a two-dimensional version, draw a rectangular array of dots on a sheet of paper, and then look at it at a grazing angle from different directions.

Last edited: Jul 5, 2012
8. Jul 6, 2012

### nasu

The planes are not associated with one protein molecule but with the lattice made from many protein molecules. I suppose you are talking about a crystallized protein.

9. Jul 23, 2012

### jikim

it's just convention. don't be bother too much.