1. Diamond crystallises in a face-centered cubic structure as shown in the figure. (a) Write expressions for the real-space lattice vectors a for the conventional unit cell and for a primitive unit cell of your choice and find the corresponding reciprocal lattice vectors b for both cases. (b) Specify the basis for the diamond latticeª 0 for both unit cell choices in terms of your lattice vec- tors in the form [n 1,n 2, n 3] when the position of an atom in the basis is given by r = n₁a₁ +n2a2+n3a3. How many atoms are in the unit cell for the primi- tive and the conventional unit cell, respectively? (c) Based on your answer above, how many acoustic and optical phonon branches does diamond have? "Please note that we use the following definition for the basis: A basis describes the arrangement of atoms associated with each lattice point. In the Simon textbook the definition is slightly unclear: "The description of objects in the unit cell with respect to the reference lattice point in the unit cell is known as a basis." With our definition, the basis contains the same number of atoms independent on the choice of the unit cell, in contrast to Simon's definition. There, the basis of conventional unit cell can contain more atoms than for the primitive one. Fig.1: Diamond crystal structure, conventional fcc unit cell.

1. Diamond crystallises in a face-centered cubic structure as shown in the figure. (a) Write expressions for the real-space lattice vectors a for the conventional unit cell and for a primitive unit cell of your choice and find the corresponding reciprocal lattice vectors b for both cases. (b) Specify the basis for the diamond latticeª 0 for both unit cell choices in terms of your lattice vec- tors in the form [n 1,n 2, n 3] when the position of an atom in the basis is given by r = n₁a₁ +n2a2+n3a3. How many atoms are in the unit cell for the primi- tive and the conventional unit cell, respectively? (c) Based on your answer above, how many acoustic and optical phonon branches does diamond have? "Please note that we use the following definition for the basis: A basis describes the arrangement of atoms associated with each lattice point. In the Simon textbook the definition is slightly unclear: "The description of objects in the unit cell with respect to the reference lattice point in the unit cell is known as a basis." With our definition, the basis contains the same number of atoms independent on the choice of the unit cell, in contrast to Simon's definition. There, the basis of conventional unit cell can contain more atoms than for the primitive one. Fig.1: Diamond crystal structure, conventional fcc unit cell.