Physics 5337 - Homework #6

  1. Consider longitudinal vibrations in a crystal with 3 atoms in the basis. The masses are m1, m2 and m3 arranged as shown below. Consider only nearest neighbor interactions and imagine all the adjacent masses connected by Hooke's law springs (----) of force constant C.

    ----m1----m2----m3----m1----m2----m3----m1----m2----m3----m1----m2----m3

    1. Write the coupled difference equations that express Newton's second law. These are functions of space and time (a*s, which is x; and t).
    2. Write the same equations after Fourier space and time transformations. These are functions of frequency and wavenumber (ω and k).
    3. Write the Fourier transformed equations in matrix form.
    4. Write the condition that the matrix must satisfy if there are to be non-trivial solutions. (The trivial solution is that none of the masses are displaced from their equilibrium positions.)
    5. Find the equation above which is cubic in &omega2.
    6. Plot (by computer, not by hand!) the three solutions for &omega versus k on the same graph. You will need to choose reasonable numerical values for the constants: k, a, m1, m2, m3, c, etc.