(a) Derive linear density expressions for BCC [110] and [111] directions in terms of the atomic radius \(R\). (b) Compute and compare linear density values for these same two direction for tungsten.

Recall that for a BCC lattice, there is one additional lattice point located at the center of the cubic unit cell (occupying an eighth of a sphere at each corner of the unit cell). To derive the linear density expressions, we first need to determine the side length (a) and diagonal lengths of the cubic unit cell. Let R be the atomic radius of a sphere in the unit cell. There are four spheres along the [110] and [111] directions that touch each other in these directions. In the [110] direction, the spheres are positioned along an edge of the unit cell. In the [111] direction, the spheres are positioned along the body diagonal of the unit cell. Since the spheres touch each other in the respective directions, we can use geometry to relate the radii of the spheres and the lengths of the edges and diagonals. Consider two spheres connected by an edge (in the [100] direction). The edge length is equal to twice the sphere radius: a = 2R Now, let's find the length of the face diagonal (in the [110] direction) and the length of the body diagonal (in the [111] direction).

We can use the Pythagorean theorem to find the lengths of the diagonals. For the [110] direction (along the face diagonal), we have: \(Length_{110} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}\) Since a = 2R, we find the length of the face diagonal in terms of R: \(Length_{110} = 2R\sqrt{2}\) For the [111] direction (along the body diagonal), we have: \(Length_{111} = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3}\) With a = 2R, we find the length of the body diagonal in terms of R: \(Length_{111} = 2R\sqrt{3}\) Now, the linear density expression for each direction is given by the number of lattice points along the given direction divided by the length of that direction. There are 2 lattice points (atoms) in the BCC lattice along the [110] direction, and 4 atoms along the [111] direction (including the corner atoms and the additional atom at the center). Therefore, the linear density expressions for the [110] and [111] directions are: \(Linear Density_{110} = \frac{2}{Length_{110}} = \frac{2}{2R\sqrt{2}} = \frac{1}{R\sqrt{2}}\) \(Linear Density_{111} = \frac{4}{Length_{111}} = \frac{4}{2R\sqrt{3}} = \frac{2}{R\sqrt{3}}\)

For tungsten, the atomic radius is approximately R = 137 pm. We can plug in this value into the linear density expressions to obtain the linear density values for the [110] and [111] directions. \(Linear Density_{110} = \frac{1}{137\sqrt{2}} \approx 5.15 \times 10^6 \, m^{-1}\) \(Linear Density_{111} = \frac{2}{137\sqrt{3}} \approx 5.44 \times 10^6 \, m^{-1}\) Therefore, the linear density values for tungsten in the [110] and [111] directions are approximately 5.15 × 10^6 m^-1 and 5.44 × 10^6 m^-1, respectively. Comparing these values, we find that the linear density along the [111] direction is slightly higher than that in the [110] direction in tungsten.