Question

Derive linear density expressions for FCC [100] and [111] directions in terms of the atomic radius R. Compute and compare linear density values for these same two directions for silver. The atomic radius for silver is 0.144 nm?

Answer 1

The linear density expressions for FCC [100] and [111] directions in terms of atomic radius R are 1/R. The linear density values for these directions for silver are 6.944 nm^-1.

Step-by-step explanation:

For a face-centered cubic (FCC) crystal structure, the linear density is defined as the total number of atoms per unit length along a specific direction. Let's calculate it for the [100] and [111] directions:

[100] direction:

In this direction, the atoms are arranged in a square pattern. There are two atoms along the edge of the unit cell, and the length of the edge is twice the atomic radius, so the linear density can be calculated as:

Linear density = (2 atoms) / (2 * R) = 1 / R

[111] direction:

In this direction, the atoms are arranged in a hexagonal pattern. There are four atoms along the edge of the unit cell, and the length of the edge is four times the atomic radius, so the linear density can be calculated as:

Linear density = (4 atoms) / (4 * R) = 1 / R

Now, let's compute the linear density values for silver:

Linear density along [100] direction = 1 / (0.144 nm) = 6.944 nm-1

Linear density along [111] direction = 1 / (0.144 nm) = 6.944 nm-1

As we can see, the linear density values for both directions are the same for silver.