Rhombohedral primitive cell on FCC

[georgeh]

How do you determine a plane or a point on a crystal lattice when translated to a rhomboidal primitive cell.
For example, rhombohedral primitive cell for an FCC is defined as:
a_1 = 1/2 a*(x+y);
a_2=1/2*a(y+z);
a_3=1/2*a(z+x);

If we have a plane, for example the 111 plane on an FCC, how do we describe this for a rhombodial primitive cell?
I have no clue how to do this, i tried finding a constant to multiple a_1,a_2 and a_3 for example to the normal vector on the 111 plane [1,1,1], but i got something i don't think is right.
Could someone point me in the right direction, or a link to an explanation?
thanks,

 

[BigJDubs]

To determine a plane or point on a crystal lattice when translated to a rhombohedral primitive cell, you need to use the Miller indices of the plane or vector in question. For example, the 111 plane on an FCC has Miller indices (1,1,1). To translate this to a rhombohedral primitive cell, you need to take the reciprocal lattice vectors of the primitive cell, which are (1/a_1, 1/a_2, 1/a_3), and multiply each component of the Miller indices by them. In this example, the vector for the 111 plane in the rhombohedral primitive cell would be (1/2*(x+y), 1/2*(y+z), 1/2*(z+x)).

 

[charng]

I would approach this question by first understanding the concept of crystal lattices and primitive cells. A crystal lattice is a repeating arrangement of atoms, ions, or molecules in a solid material. This arrangement can be described by a unit cell, which is the smallest repeating unit of the lattice. A primitive cell is a unit cell that contains only one lattice point and represents the smallest repeating unit of the lattice.

In the case of the rhombohedral primitive cell on an FCC lattice, the unit cell is defined by the three vectors a_1, a_2, and a_3, which are all half of the lattice parameter a and are oriented along the diagonal directions of the FCC lattice. This means that the rhombohedral primitive cell contains only one lattice point, located at the center of the cell.

To determine a plane or a point on the crystal lattice when translated to a rhomboidal primitive cell, we can use a concept called Miller indices. Miller indices are a method of describing planes and directions in a crystal lattice. They are represented by three numbers, (hkl), where h, k, and l are integers representing the intercepts of the plane with the crystallographic axes.

To describe the 111 plane in terms of the rhombohedral primitive cell, we can use the Miller indices (111). This means that the plane intersects the a_1, a_2, and a_3 vectors at the points (1,1,1). We can then use these points to construct a rhomboidal primitive cell that contains the 111 plane.

In terms of the vectors a_1, a_2, and a_3, we can describe the 111 plane by taking the a_1 vector and multiplying it by 1, the a_2 vector and multiplying it by 1, and the a_3 vector and multiplying it by 1. This gives us the equation for the 111 plane in terms of the rhombohedral primitive cell:

1/2 a*(x+y) + 1/2 a*(y+z) + 1/2 a*(z+x) = 1/2 a*(x+y+z)

This equation represents the 111 plane in the rhombohedral primitive cell. To find the normal vector to this plane, we can take the cross product of any two of the vectors used in the equation. For example, taking the cross product of a