The highly-symmetric object on the left of the figure below is the Ist Brillouin zone in reciprocal space (k-space), for a Face Centred Cubic (FCC) crystal lattice. The shape is created in a similar way as the real-space 20 Wigner-Seitz cell described in the lecture notes. The faces originate from intersecting planes perpendicular to the reciprocal lattice vectors which contain the point half way between nearest neighbour reciprocal lattice points. Like its real-space partner. this shape is also a unit cell. It fits together filling the whole volume in &-space with no gaps (as shown on the right of the figure). The points labelled (I, L. U. X. W. A') in figure 1(a) are special, because they are points of high symmetry. I'(0. 0,0) is located at the centre of the Brillouin zone. The point _X lies at the centre of the square face, with vector coordinates (0, 1, 0). The point & lies at the centre of the hexagonal face, with vector coordinates # (1, 1, 1). These vectors are also normal to the faces. There are equivalent X' and [ points on the other faces. Akz K Ky (a) (b) Figure 1: (a) 1st Brillouin zone for a FCC crystal lattice. (b) How neighbouring Brillouin zones fit together to fil k-space. 1. Write down vector equations for the three shaded planes in figure 1taj. [3] 2. The three shaded planes intersect at the point W, which is one of the vertices of the 30 shape. Solve the vector equations of the three planes to find the coordinates of W. [2] 3. Find the vector coordinates of the remaining points K and U. [4] 4. What are the coordinates of the equivalent I' points of the neighbouring Brillouin zones in figure 1(b). (2] 5. Calculate the area of a square face. [1] 6. Calculate the area of a hexagonal face. [3] 7. Calculate the volume of the 30 shape (1st Brillouin zone) in figure 1(a). [5] 8. Starting with the primitive real-space FCC lattice vectors, calculate the corresponding reciprocal lattice vectors, and show that the volume of this (parallelepiped) primitive cell in &-space equals the volume of the 1st Brillouin zone that you've just calculated. [5]