The harmonic crystal with alternating spring constants

Most crystals contain more than one kind of atom. This leads to both quantitative and qualitative differences in the vibration spectrum from the results derived in class. These differences occur whenever the periodicity is altered, whether by changing the masses of the atoms or changing the spring constants. As an example of the kinds of differences that can arise when the periodicity changes, consider the problem of a one-dimensional lattice with alternating spring constants K₁ and K₂. For simplicity, you can assume that all the masses are equal to m.

1. If the distance between atoms is a, the periodicity is 2a. What is the size of the Brillouin Zone?

2. Calculate the vibration spectrum of the lattice. You might find it more convenient to write down equations of motion for the two sublattices.

3. Sketch the vibration spectrum in the Brillouin Zone based on your calculations.

4. Sometimes the vibration spectrum of a crystal with more than one kind of atom is approximated by a combination of an Einstein model and a Debye model. Explain why this might make sense and, give appropriate parameters for such a description in the present case. In answering this question, consider the highand low-temperature behavior of the specific heat of this model.