Rhombohedral primitive cell on FCC

• georgeh
In summary: This method can be used for any plane or point on a crystal lattice. In summary, to determine a plane or point on a crystal lattice when translated to a rhombohedral primitive cell, you need to use the Miller indices and the reciprocal lattice vectors of the primitive cell.

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,

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)).

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

1. What is a rhombohedral primitive cell?

A rhombohedral primitive cell is a unit cell in a crystal lattice that has a rhombohedral shape, meaning its unit cell edges are all equal in length and its angles are all equal. It is the simplest repeating unit in a crystal lattice and can be described as a parallelepiped with six parallelogram faces.

2. What is the relationship between a rhombohedral primitive cell and an FCC structure?

A rhombohedral primitive cell is a type of unit cell that can be found in an FCC (face-centered cubic) crystal structure. In an FCC structure, the atoms are arranged in a cubic lattice with additional atoms at the center of each face. This creates a rhombohedral shape when the unit cell is repeated in three dimensions.

3. How can I determine the volume of a rhombohedral primitive cell?

The volume of a rhombohedral primitive cell can be calculated using the formula V = a^3 * sin(alpha), where "a" is the length of the unit cell edge and "alpha" is the angle between the unit cell edges. This formula assumes that all three angles of the rhombohedral primitive cell are equal.

4. What are some common examples of materials with a rhombohedral primitive cell?

Some common materials with a rhombohedral primitive cell include calcium carbonate (in the form of calcite), quartz, and bismuth. These materials have a rhombohedral crystal structure due to the arrangement of their atoms in a repeating unit cell.

5. How does the rhombohedral primitive cell affect the properties of a material?

The rhombohedral primitive cell, along with other factors such as the type of atoms present and their arrangement, can greatly influence the physical, chemical, and mechanical properties of a material. For example, the orientation of the rhombohedral primitive cell can affect the material's strength, hardness, and thermal conductivity.