Problem 1 Vanadium is to the left of chrome in the periodic table. Both metals form an bcc crystal structure and it is possible to make VxCr1-x alloys.
(a) How could you use the empty lattice approximation to predict how the electronic contribution to the specific heat and the electrical conductivity shift as $x$ is varied from 0 to 1? Would the shifts be correlated? Why? (b) How will the phonon density of states change in VxCr1-x alloys as $x$ is varied from 0 to 1? (c) A thin film of VxCr1-x is oxidized. How could you measure the thickness of the oxide that is formed? How could you measure the chemical composition of the oxide? (d) How would the surface plasmons at a VxCr1-x surface change as the oxide thickness increases? How could you measure this? Problem 2 (a) What can you deduce about the metal from these oscillations? (b) At high temperatures the oscillations are not visible. What are the experimental conditions needed to observe de Haas - van Alphen oscillations? (c) The magnetization oscillates. What does this say about the magnetic susceptibility? How do you tell if the metal is diamagnetic or paramagnetic? (d) What will happen if you try to measure the de Haas - van Alphen oscillations of a ferromagnet? Problem 3
(a) An ellipsometery experiment is performed on some material used for optical lenses and the dielectric function is measured in the range 400 nm - 1000 nm. The imaginary part is nearly zero throughout this range as you would expect for a lens material. Can you say something about the imaginary part of the dielectric function at higher or lower frequencies or is that impossible from this measurement? (b) You could change the index of refraction of a material by straining it. Explain how you could start from the band structure and calculate the change in index of refraction with respect to strain. Problem 4 Very high dielectric constants at low frequencies are typically related to a structural phase transistion. (a) Explain why this is true. (b) How will the dielectric constant decrease as the temperature moves away from the critical temperature $T_c$? (c) Superelastic materials are also related to a structural phase transistion. How would the resistivity of a superelastic wire change as it is bent?
The de Haas - van Alphen oscillations describe how the magnetization of a metal changes as a function of the applied magnetic field.