Problem 1
(a) The calculation of a bond potential for a molecule like N2 is computationally intensive. Describe how this calculation would be done.
(b) Once a bond potential has been calculated, it is often fit to a Morse potential of the form,
\[ \begin{equation} U(r) =U_0\left( \exp \left(-2a(r-r_0)\right) -2 \exp \left(-a(r-r_0)\right)\right). \end{equation} \] For N2, the results are: $U_0 = 1.76$ eV; $r_0=1.09$ Å; $a=2.73$Å-1;It is possible to calculate the vibrational and rotational frequencies of a diatomic molecule like N2 from the Morse potential. What are the vibrational and rotational frequencies of N2?
Problem 2
An orthorhombic crystal has primitive lattice vectors,
Here $\alpha$ is a constant. What are the reciprocal lattice vectors?
What x-ray reflections would be observed if the wavenumber of the x-rays is $k = \frac{5\pi}{12\alpha}$?
Problem 3
Consider constructing the electronic band structure of silicon using the tight binding model. The Bravais lattice of silicon is fcc and there are two atoms in the basis.
(a) In tight binding we solve for the energy of a single electron moving in a periodic potential. What wavefunction would you use to describe this electron? There should be some unknown coefficients in the expression for the wavefunction.
(b) Describe how such a trial wavefucntion can be used to calculate the $E$ vs. $\vec{k}$ dispersion relation.
(c) Sketch the electronic bandstructure for silicon. It is an indirect bandgap semiconductor with conduction band minima in the [100] directions. There is a light-hole band and a heavy-hole band that are degenerate at the top of the valence band.
(d) If the electronic band structure of silicon is known, how would you calculate the chemical potential?
Problem 4
The electrical conductivity of a metal depends on the electron density and the mobility of the electrons.
(a) What is the expression for the electrical conductivity of a metal in terms of the electron density and the mobility?
(b) How could the electron density and the mobility be determined experimentally?
(c) How is the electrical conductivity of a metal related to its thermal conductivity? What metals are poor thermal conductors?