Physics 871: Problem set 2

1. What is the reciprocal lattice of a simple hexagonal lattice?

2. M&A problem 6.3.

3. A certain polycrystalline metal film generates an xray diffraction pattern that has lines at the following Bragg angles (not 2θ) 22.3°, 26.0°, 38.3°, 46.6°, 49.4°, and 61.2°. It was a CuK_α source. What element(s) could it be? It is known to be a either a face-centered cubic cell or body-centered cubic cell.

4. Calculate and sketch the branch(es) for the phonon dispersion in a one-dimensional lattice with a basis of two identical ions that have alternating interactions. Namely the springs between the ions alternate between two values K₁ > K₂. Discuss the results as the two spring constants approach the same value. How would the results above change if K₁ < K₂?

5. A one-dimensional diatomic lattice has equally spaced masses m₁ and m₂ (alternating) that are coupled by identical springs so the nearest-neighbor interaction constant is K₁. Calculate the phonon dispersion. If a next-nearest-neightbor interaction constant K₂ is added, what are the governing equations for phonon modes? Discuss differences between the new modes and the modes with only nearest neighbor coupling.

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