Final answer:
To derive the linear density expressions for the FCC [100] and [111] directions, we count the number of atoms along these directions. The [100] direction has a linear density of 5 atoms per unit length, while the [111] direction has a linear density of 8 atoms per unit length.
Step-by-step explanation:
Linear density refers to the number of lattice points per unit length along a specific direction in a crystal lattice. To derive the linear density expressions for the Face-Centered Cubic (FCC) [100] and [111] directions, we need to determine the number of atoms present along these directions.
For the FCC [100] direction, the atoms are aligned along the cube edges. Since there are four atoms at the corners and one in the center of each face of the unit cell, the total number of atoms per unit length is 4 + 1 = 5 atoms (considering only half of the atoms at the faces).
For the FCC [111] direction, the atoms are aligned along the diagonals of a face of the unit cell. Each face has four diagonals, and each diagonal contains one atom at the corner and one in the center of each face. So, the total number of atoms per unit length is 4 (number of diagonals) * 2 (number of atoms per diagonal) = 8 atoms.