Question

7. (10 points) Show that the angle between any two of the bonds joining a carbon atom to its neighbors in the diamond lattice is cos 1/3) 109°28'. (Hint: Use the vector dot product between wisely picked vectors). (20 points) Compute the Fermi energy and the Fermi temperature for silver. Show that for T 300 K about 0.1 percent of the free electrons in metallic silver have an energy greater 8. than EF.

Answer 1

We know that in diamond lattice with a basis of two identical carbon atoms associated with each lattice points one displaced from the other by a translation of (1/4, 1/4 1/4) along the body diagonal position vector of atom A is r^rightarrow_1 = a/4 (i + j + u) position vector of atom B is r^rightarrow_2 = a/2 (j + k) position vector of atom c is r^rightarrow_3 = a/2 (i + k) All there positions vectors are written considering the corner atom o as the origin (0, 0, 0). so, Now using triangle law of vector addition position vector of B with respect to A is r^rightarrow_AB = r^rightarrow_2 - r^rightarrow_1 = (a/2 j + a/2 k) - (a/4 i + a/4 j + a/4 k) Implies r^rightarrow_A B = a/4 j + a/4 k - a/4 i similarly, position vector of c with respect to A is given by r^rightarrow_A C = (r^rightarrow_3 - r^rightarrow_1) = (a/2 i + a/2 k) - (a/4 i + a/4 j + a/4 k)