Miller's Indices: Find in FCC & BCC, Calculate Density of Lattice Points

[prochatz]

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?

 

[bpsbps]

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.

 

[prochatz]

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)?

 

[bpsbps]

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).

 

[prochatz]

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?

 

[bpsbps]

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.

 

[prochatz]

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?

 

[PhaseShifter]

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).