Homework #4

Phonon spectrum of selected Bravais lattices

The objective of this assignment is to understand the phonon spectrum of selected Bravais lattices, where each atom interacts only with its nearest neighbors. Assume that the interaction between a pair of atoms is described by a central harmonic potential

Φ(r) = Φ(|r|) = 1 2 γ(|r|-r0)2,

where r₀ represents the equilibrium nearest neighbor distance.

1. (4 pt.) Write down the expression for the dynamical matrix D_μν (k) of a general Bravais lattice interacting with nearest neighbor pairwise harmonic potentials only.

2. (5 pt.) Write down the analytic expression for all the elements of the dynamical matrix D_μν(k) of a simple cubic lattice, interacting only by central harmonic nearest neighbor interactions.

3. (5 pt.) Write down the analytic expression for the elements D_xx(k), D_yy(k), and D_xy(k) of the dynamical matrix of a face-centered cubic lattice, interacting only by central harmonic nearest neighbor interactions. Simplify your final expressions so that they contain only simple trigonometric functions like sin(k_μa/2) and cos(k_μa/2), where μ=x,y,z and a=√2 r₀.

4. (2 pt.) How do the dispersion relations ω(k) in the sc lattice depend on the lattice constant a, the force constant γ, and the mass m? Discuss briefly your results.

5. (2 pt.) A young Physicist performed vibrational band structure calculations for different lattices using the assumptions in this homework and obtained the following results:


Classify the modes as longitudinal or transverse, acoustic or optical, and specify the degeneracy of all branches. Information about high-symmetry points in the different lattices is listed HERE.