Problem 1

Lithium has three protons. Consider a Li₂ molecule.

(a) Using a linear combination of atomic orbitals, what trial wavefunction would you use to solve the molecular orbital Hamiltonian for Li₂?

(b) How will the molecular orbitals found in (a) be occupied in the ground state of Li₂?

(c) In what ways are the molecular orbitals of Li₂ similar to the molecular orbitals of H₂?

(d) What kind of bond holds the two Li atoms together? How could you calculate the bond potential?

(e) How would you determine which vibrational modes and which rotational modes of Li₂ would be occupied at room temperature?

Problem 2

(a) Silicon and germanium both form the diamond crystal structure. Which has a higher Debye frequency? Why?

(b) Germanium has a larger lattice constant than silicon and germanium atoms are heavier than silicon atoms. Which material has a higher specific heat at high temperature? Why?

(c) Copper and gold both form an fcc crystal structure. Gold has a larger lattice constant than copper and gold atoms are heavier than copper atoms. Compare the specific heats of copper and gold at low temperature.

Problem 3

(a) If the electron dispersion relation of $E(\vec{k})$ of a crystal is known, how could the chemical potential be calculated?

(b) What is the Sommerfeld expansion?

(c) What is the Boltzmann approximation? How do you calculate the chemical potential within the Boltzmann approximation?

(d) In an intrinsic semiconductor, what determines the concentration of electrons and the concentration of holes?

Problem 4

An orthorhomic crystal with one atom in the basis has lattice constants of $a=\pi$ Å, $b=2\pi$ Å, and $c=3\pi$ Å.

(a) What is the longest wavelength that can be diffracted by this crystal?

(b) The intensities of the diffraction peaks are proportional to the square of the structure factors. If the atomic form factor is assumed to be a constant $f$, what is the structure factor for $G_{111}$?

(c) X-rays with a wavelength of 0.3 Å are used to measure this material in a powder diffraction experiment. At which angle $\theta$ will the reflection appear for $G_{201}$?