(a) How would you calculate the form of a bond potential for a CO molecule?

(b) What information from the calculated bond potential would you use to calculate the rotational energy levels of CO? How many rotational degrees of freedom are there?

(c) What information from the calculated bond potential would you use to calculate the vibrational states of CO? How many vibrational modes are there?

(d) What would be the first excited electronic state of CO be? Label the molecular orbitals MO1, MO2, $\cdots$.

(e) If the CO molecule was in an electronic excited state, how would that change the calculation of the bond potential?

Problem 2

α-Lanthanum forms a double hexagonal close-packed crystal structure. The Bravais lattice is hexagonal. There are two La atoms in the asymmetric unit and 4 atoms in the basis.

(a) How could you determine the conventional unit cell from the asymmetric unit? How many atoms are there in a conventional unit cell?

(b) How many nearest neighbors does every La atom have?

(c) What is the smallest angle $2\theta$ that would be observed in a powder diffraction experiment of lanthanum?

(d) Sketch the phonon dispersion for La.

(e) The electron configuration of La is [Xe] 5d¹ 6s². It is in the first column of the 5d transition metals. How could you estimate the electronic contribution to the specific heat from the information given in this problem?

Problem 3

GaAs is a semiconductor with a zincblend crystal structure. The bandgap is 1.43 eV.

(a) What is the Bravais lattice of GaAs?

(b) GaAs is a direct bandgap semiconductor where the effective mass of the electrons is $0.067m_e$, the effective mass of the light holes is $0.082m_e$ and the effective mass of the heavy holes is $0.45m_e$. Here $m_e$ is the mass of a free electron. Draw the electronic band structure and the electronic density of states for GaAs.

(c) If a light emitting diode is made from GaAs, what wavelength light will it emit? $h = 6.626\times 10^{-34}\text{ J s}$, $e = 1.6022\times 10^{-19}\text{ C}$, $c = 2.998\times 10^{8}\text{ m/s}$.

(d) Describe how the electrical conductivity of GaAs changes with temperature. How does the electrical conductivity of GaAs compare to silicon? Silicon is an indirect bandgap semiconductor with a bandgap of 1.1 eV.

(e) What rank tensor is the electrical conductivity? What form will this tensor have for GaAs?

Problem 4

The electronic band structure of a material with 3 atoms in the basis is shown below. Each atom has one valence electron.

(a) Is this material a metal, an insulator, a direct bandgap semiconductor, or an indirect bandgap semiconductor? Explain your reasoning.

(b) How many $\vec{k}$-vectors does this crystal have per primitive unit cell?

(c) The wave function of electrons in a band have Bloch form. Explain what Bloch form is.

(d) What form does the wavefunction of an electron state in a band gap have?