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

  • Thread starter prochatz
  • Start date
  • Tags
    Indices
In summary, the question is about determining Miller's indices for a specific plane in different lattice types and finding the density of lattice points on that plane. To calculate the density, we can consider the plane as a two-dimensional lattice and take the reciprocal of the area of a parallelogram in that lattice.
  • #1
prochatz
42
0
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?
 
Physics news on Phys.org
  • #2
prochatz said:
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.
 
  • #3
bpsbps said:
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)?
 
  • #4
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).
 
  • #5
bpsbps said:
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?
 
  • #6
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.
 
  • #7
bpsbps said:
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?
 
  • #8
prochatz said:
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)
bpsbps said:
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).
 

1. What are Miller's Indices?

Miller's indices are a method used to describe the orientation of crystal planes in a crystal lattice. They are represented by three numbers, (hkl), which correspond to the reciprocals of the intercepts of the plane with the unit cell axes.

2. How do you find Miller's Indices in a FCC lattice?

In a face-centered cubic (FCC) lattice, the Miller's Indices of a plane can be found by taking the reciprocals of the intercepts of the plane with the unit cell axes and then reducing them to the lowest possible integers. For example, a plane that intercepts the x-axis at 1/2, the y-axis at 1/3, and the z-axis at 1/4 would have Miller's Indices of (213).

3. How do you find Miller's Indices in a BCC lattice?

In a body-centered cubic (BCC) lattice, the Miller's Indices of a plane can be found by taking the reciprocals of the intercepts of the plane with the unit cell axes, multiplying them by 2, and then reducing them to the lowest possible integers. For example, a plane that intercepts the x-axis at 1/2, the y-axis at 1/3, and the z-axis at 1/4 would have Miller's Indices of (431).

4. How do you calculate the density of lattice points using Miller's Indices?

The density of lattice points in a crystal lattice can be calculated using the formula D = N/V, where N is the number of lattice points within a unit cell and V is the volume of the unit cell. To find the number of lattice points, you can use Miller's Indices to determine the number of planes that intersect the unit cell in each direction. The volume of the unit cell can be calculated by taking the product of the lengths of the unit cell edges.

5. What is the significance of calculating the density of lattice points?

The density of lattice points is an important factor in understanding the physical properties of a material. It can provide information about the strength and stability of a crystal lattice, as well as its thermal and electrical conductivity. Additionally, the density of lattice points can be used to determine the arrangement of atoms within a crystal, which is crucial in understanding the structure and behavior of materials.

Similar threads

  • Atomic and Condensed Matter
Replies
1
Views
4K
  • Atomic and Condensed Matter
Replies
1
Views
1K
  • Atomic and Condensed Matter
Replies
1
Views
29K
  • Atomic and Condensed Matter
Replies
2
Views
2K
  • Differential Geometry
Replies
0
Views
582
  • Atomic and Condensed Matter
Replies
8
Views
3K
Replies
1
Views
1K
Replies
3
Views
3K
  • Atomic and Condensed Matter
Replies
2
Views
1K
  • Atomic and Condensed Matter
Replies
1
Views
1K
Back
Top