Final answer:
The linear density for FCC [100] direction is √2 / (4R) and for [111] direction is 1½ / (4R√3). For silver with its given edge length, we calculate the atomic radius and then substitute this radius into the linear density expressions to find the values for the [100] and [111] directions.
Step-by-step explanation:
Linear Density for FCC [100] and [111] Directions
To derive the linear density expressions for face-centered cubic (FCC) [100] and [111] directions in terms of the atomic radius R, we consider the number of atoms per unit length along the specified direction.
FCC [100] direction: The [100] direction is along the edge of the cubic unit cell. Since each corner atom is shared among eight unit cells, a single edge includes ½ atom from one corner plus ½ atom from the other corner. Thus, there is 1 atom per edge length, a, in the [100] direction. The edge length can be expressed as 4R/√2. Therefore, the linear density (LD) is given by:
LD[100] = (½ + ½) atoms / a = 1 atom / (4R/√2) = √2 / (4R)
FCC [111] direction: The [111] direction passes through the diagonal of the cube, which contains atoms from three different layers. In this pathway, there are three touch points which represent ½ atom each, contributing to 1½ fully counted atoms. The length of this diagonal is √3 times the edge length, a, which is 4R. Therefore, we have:
LD[111] = 1½ atoms / (4R√3) = 1½ / (4R√3)
For silver, with an FCC structure and a given edge length of the unit cell as 409 pm, we calculate the atomic radius R using the relation between the edge length and radius in FCC, a=4R/√2. Substituting the value in the above expressions, we get:
Atomic radius R of silver = 409 pm / (4/√2) ≈ 144.5 pm
Linear density for [100] in silver = √2 / (4 * 144.5 pm)
Linear density for [111] in silver = 1½ / (4 * 144.5 pm * √3)