Physics 5337 - Homework #3

  1. Graphite consists of layers of atoms arranged in a honeycomb lattice. A single layer of carbon atoms in this configuration is called graphene. The binding is much stronger within a layer than between different layers.
    1. Is the hexagonal mesh shown in the link a Bravais lattice? Explain.
    2. Draw the Wigner-Seitz cell on top of the mesh.
    3. Draw a different primitive cell on top of the mesh.
    4. How many carbon atoms are in either primitive cell?
    5. If the spacing between bound carbon atoms is a, what is the area of a primitive cell?

  2. Show that < cos(kp x) | cos(km x) > = δpm = < sin(kp x) | sin(km x) >
    and that < cos(kp x) | sin(km x) > = 0 for p,m integers greater than 1.

  3. Suppose that n(x) is a periodic function that has value b for the first half of the distance a between one-dimensional lattice sites, and has the value zero for the second half of the distance. Find the first four (that is, the four most important lowest frequency) non-zero terms in a Fourier expansion of n(x). [A good way to check that you have done this correctly is to plot both the original function and your Fourier approximation on top of each other, but this is not required for the assignment.]