(a) Derive linear density expressions for FCC [100] and [111] directions in terms of the atomic radius \(R\) (b) Compute and compare linear density val use for these same two planes for copper.

A face-centered cubic (FCC) unit cell consists of atoms present at the corners and the centers of all the faces. Every corner atom is shared by 8 adjacent unit cells, and every face-centered atom is shared by 2 adjacent unit cells. The lattice parameter of an FCC unit cell is \(a = 2\sqrt{2} R\), where \(R\) is the atomic radius of the atom present in the cell.

The [100] direction is the direction parallel to the x-axis in Cartesian coordinates, passing through the centers of atoms in the FCC unit cell. The atoms involved in this direction are face-centered atoms and corner atoms. In the [111] direction, we move from one corner of the unit cell along the body diagonal to an opposite corner. This direction passes through the corners, body-center, and face-center of the cell.

Linear density, \(\rho_{L}\), is defined as the number of atoms per unit length along a specific direction in the lattice structure. For the [100] direction: There is 1/4 corner atom and 1 full face-centered atom along the [100] direction in the distance equal to the lattice parameter \(a\): \(\rho_{L[100]} = \frac{1.5\ \text{atoms}}{a\ \text{length}}\) Since \(a = 2\sqrt{2} R\), we get: \(\rho_{L[100]} = \frac{1.5\ \text{atoms}}{2\sqrt{2} R\ \text{length}}\) For the [111] direction: There are 3 face-centered atoms along the body diagonal of the unit cell, and the length of the body diagonal is \(l = \sqrt{3}a\): \(\rho_{L[111]} = \frac{3\ \text{atoms}}{l\ \text{length}}\) Since \(l = \sqrt{3}a = \sqrt{3}(2\sqrt{2}R)\), we get: \(\rho_{L[111]} = \frac{3\ \text{atoms}}{\sqrt{3}(2\sqrt{2}R)\ \text{length}}\)

For copper, the atomic radius is \(R_{Cu} = 1.278 \times 10^{-10}\ \text{m}\). Now, we can compute the linear densities for the [100] and [111] directions using the derived expressions from step 3: \(\rho_{L[100]}_{Cu}=\frac{1.5\ \text{atoms}}{2\sqrt{2}(1.278 \times 10^{-10}\ \text{m})\ \text{length}}\) \(\rho_{L[111]}_{Cu}=\frac{3\ \text{atoms}}{\sqrt{3}(2\sqrt{2}(1.278 \times 10^{-10}\ \text{m}))\ \text{length}}\)

By calculating the numeric values for \(\rho_{L[100]}_{Cu}\) and \(\rho_{L[111]}_{Cu}\), we can compare them to evaluate the relative linear density in both directions for copper: \(\rho_{L[100]}_{Cu} \approx 4.18 \times 10^9\ \text{atoms/m}\) \(\rho_{L[111]}_{Cu} \approx 3.62 \times 10^9\ \text{atoms/m}\) Comparing both values, linear density is higher along the [100] direction as compared to the [111] direction in the face-centered cubic structure of copper.