Name Matrikelnr.
Problem 1
(a) All homonuclear diatomic molecules (H2, N2, O2, ...) can be described with the same set of molecular orbitals. Why is this?
(b) How many molecular orbitals are occupied in the ground state of N2? (Nitrogen has 7 protons and 7 neutrons).
(c) The bond between two atoms of N2 can can be described by a Morse potential,
\begin{equation} U(r)= U_0\left(e^{-2(r-r_0)/a}-2e^{-(r-r_0)/a}\right). \end{equation}Explain how you could estimate the vibrational frequencies of the molecule from the bond potential.
(d) How could the H-N-H bond angle be calculated for NH3?
Problem 2
(a) Calcium Fluoride, CaF2, has an fcc Bravais lattice and a basis with Ca at 000 and F's at fractional coordinantes ¼ ¼ ¼ and ¾ ¾ ¾ of the conventional (cubic) unit cell. Sketch the arrangement of the atoms in the $(1\overline{1}0)$ plane.
(b) The lattice constant of CaF2 is $a = $ 5.451 Å. This is the length of a side of the conventional unit cell. What is the volume of the primitive unit cell?
(c) How may phonon normal modes are there in 1 cm³ of CaF2?
(d) For CaF2, what is the length of the $\vec{G}_{111}$ reciprocal lattice vector? The reciprocal lattice vectors are indexed using the conventional unit cell.
Problem 3
(a) What is the electron dispersion relation in the free electron model?
(b) What experiment can be performed to show that a metal is well described by the free electron model?
(c) The electron density of states of a metal decreases as a function of energy at the Fermi energy. What consequence does this have for the temperature dependence of the chemical potential?
(d) The electron density of states of a metal decreases as a function of energy at the Fermi energy. What does this tell you about the charge carriers? Explain if they are electrons or holes.
Problem 4
(a) In one-dimensional crystals, a material is a metal if there are an odd number of electrons in the basis and it is a semiconductor or an insulator if there are an even number of electrons in the basis. Explain why this is.
(b) A one-dimensional material is a semiconductor. The lattice constant is increased while the number of electrons in the basis stays constant. Explain what consequences this will have for the effective mass of the holes.
(c) Show that a Bloch wave function is an eigen function of the translation operator.
(d) A semiconductor has a small band gap, $E_g < 3k_BT$. How can you determine the chemical potential in this case?