Hexagonal space lattice. The primitive translation vectors of the hexagonal space lattice may be taken as a = (3a/2)& + (a/2)ŷ ; az = -(3Pal2)& + (a/2)§ ; az = cz . %3D (a) Show that the volume of the primitive cell is (32/2)a°c. (b) Show that the primitive translations of the reciprocal lattice are b, = (27/3a)& + (2mla)ŷ ; b2 = -(27/32a) + (27/a)ŷ ; b, = (2m/c)2, %3D %3D so that the lattice is its own reciprocal, but with a rotation of axes. (c) Describe and sketch the first Brillouin zone of the hexagonal space lattice.

6. Within a cubic unit cell, sketch the following directions: ( :)) (a) [110] (e) [771] (b) [121] (f) [722] (c) [072] (d) [133] (g) [123] (h) [103]

Determine the miller indices for the plane displayed +z 3/4 1/2 +y +X

1. Many substances crystallize in a cubic structure. The unit cell for such crystals is a cube having an edge with a length equal to do. -Face diagonal a. What is the length, in terms of do, of the face diagonal, which runs diagonally across one face of the cube? (Hint: Use the Pythagorean theorem.) b. What is the length, again in terms of do. of the cube diagonal, which runs from one corner, through the center of the cube, to the opposite corner? (Hint: Make a right triangle having a face diagonal and an edge of the cube as its sides, with the hypotenuse equal to the cube diagonal, then use the Pythagorean theorem again.) 2. In an FCC structure, the centers of the atoms are found on the corners of the cubic unit cell and at the center of each face. The unit cell has an edge whose length is the distance from the center of one corner atom to the center of another corner atom on the same edge. The atoms on the diagonal of any face are touching. One of the faces of the unit cell is shown…