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Solid state physics 2019 Final Exam. April 16th, 2019. Good luck! • You may not use textbooks, notes, or calculators. • When plotting, label the axes and ...
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(a) (5 points) Write down the equations of motion for the atoms in a unit cell of this chain. Formulate the running wave ansatz for computing the dispersion relation.
..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... (b) (10 points) Find the dispersion relation by solving the equations of motion. ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... .....................................................................................................
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(a) (5 points) Give a set of primitive lattice vectors and the corresponding basis.
..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... (b) (10 points) Calculate the filling factor of this crystal, assuming that the atoms have equal radii rA = rB. For what ratio of rB /rA do you expect the filling fraction to be maximal? ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... .....................................................................................................
(c) (10 points) Calculate the reciprocal lattice vectors corresponding to the lattice vectors you formulated in (a). Calculate the volume of the primitive unit cell of the reciprocal lattice.
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(d) (10 points) Compute the structure factor S corresponding to the lattice vectors found in (a) (so the primitive lattice vectors). Use fA and fB as form factors for atoms A and B. Which diffraction peaks vanish when fA = fB?
..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... .....................................................................................................
(c) (5 points) Write down an expression for the electron heat capacity of this semiconductor as an integral over energies.
..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... (d) (5 points) Discuss whether the electron heat capacity becomes larger than the phonon heat capacity as temperature is lowered towards 0K. ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... ..................................................................................................... .....................................................................................................
