# Miller indices. What determines the appropriate axis?

So, first of all, what's up with this "plane" I see two planes!

The basis of my question is that to determine miller indices the plane cannot lie on your origin, so you have to make a translation. If I move my origin to the right then I get the miller indices that are indicated in the figure. But if I move backwards then I don't get the miller indices in the figure.

So my question then is: what determines the appropriate translation?

• You want integer numbers, preferably possible smallest (except he 0,0,0 :). The physicists have a more proper definition for Miller indexes with reciprocal spaces etc.
– Greg
Oct 9, 2014 at 7:38

## 1 Answer

There is no "appropriate" translation. You must remember that everything includes translational symmetry, including the Miller planes.

The Wikipedia page on crystal structures and Miller indices go into much more detail, but some key points:

• The unit cells possess translational symmetry, so there are a set of Miller planes that also differ by translational symmetry.
• The Miller indices (hkl) (which yes should be expressed in smallest possible integers) define the normal of the Miller plane.

That is, (hkℓ) simply indicates a normal to the planes in the basis of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction.

So in your case, you have to remember if you "move the origin", you have to also consider the symmetrically identical unit cells around the one in your picture. As I said above, there are an infinite set of identical Miller planes that simply differ by the translational symmetry of the lattice.