# Miller's indices

Hello, there. I'm having a small problem with Miller's indices.

1) Imagine that the plane (2 1 1) is given in the fcc lattice. How can I determine Miller's indices of that plane in the sc and in the bcc?

2) And after that, how can I find the density of lattice's points?

1) So far I took the vectors of the reciprocal space:

a*, b* and c* and then I tried to compute the vector G=n1a* + n2b* + n3c*

But then what?

2) The only thing that I know is that the density of lattice's points is proportional of the quantity 1/G

Any help?

How can I find the density of lattice's points?

Consider that there is one lattice point per unit cell. So there is one lattice point per volume of a unit cell.

Then what is the volume per lattice point? This is the density of lattice points.

Consider that there is one lattice point per unit cell. So there is one lattice point per volume of a unit cell.

Then what is the volume per lattice point? This is the density of lattice points.

Yes, but here we have planes. What should I suppose? Is density 1/(area of plane)?

The question is ambiguous.

It can mean what is the "density of lattice points" (units cm^-3)or "area density of lattice points on the [211] planes" (units cm^-2).

The question is ambiguous.

It can mean what is the "density of lattice points" (units cm^-3)or "area density of lattice points on the [211] planes" (units cm^-2).

The second explanation seems better. So if we have a specific plane, suppose in the fcc, how should I compute density? Should I count the points "contained" in the specific plane and then divide by the area of plane?

How do you define the plane? There is a family of planes that are parallel to each other. I can draw a [001] plane in a SC lattice that contains no lattice points.

How do you define the plane? There is a family of planes that are parallel to each other. I can draw a [001] plane in a SC lattice that contains no lattice points.

I see, but there must be an answer. Something goes wrong. Is there any definition about the density of lattice's points?

The second explanation seems better. So if we have a specific plane, suppose in the fcc, how should I compute density? Should I count the points "contained" in the specific plane and then divide by the area of plane?
Think of the plane as a two-dimensional lattice. The atoms in the plane will form a periodic lattice of parallelograms (or squares or rectangles). Since there is one atom per unit cell in this 2-D lattice, the density will be the reciprocal of the area of a parallelogram. (The area is equal to the magnitude of the cross product of the vectors for two adjacent sides of a parallelogram)
How do you define the plane? There is a family of planes that are parallel to each other. I can draw a [001] plane in a SC lattice that contains no lattice points.

The only planes that are of any interest whatsoever are the ones containing atoms. These planes are separated by a distance of (area density within plane)/(volume density).