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1. In this exercise, we will consider vibrations in a 1D-lattice with two atoms with mass M and m (figure 1). The equilibrium spacing of the atoms are a, and when vibrating the atoms are displaced a distance u and v from the equilibrium position. Mass 1(M) Spring constant (C) Mass, 2 (m) Equilibrium position mõõ - o o o o o o o o o o o o o o o o oooo Deformed position Vn-1 Un Figure 1: one dimensional chain of atoms with two different masses (a) Derive the dispersion relation w(k) for a wave propagating in this crystal. (b) Sketch the dispersion relation graphically, and give a physical interpretation of the two dispersion branches.? a rooooooooo
2. Derivation of the phonon density of states (DOS) in a 1D lattice containing containing N atoms (a) Introduce periodic boundary conditions, derive DOS in the k-space, and plot DOS(k) (b) Use the 1D dispersion equation as obtained in the course for 1D-lattices with one type of atom only, i.e., w = wo sin(ka/2), to find DOS as a function of w and plot DOS(w). (c) Pay attention to the relative "simplicity" of DOS(k) and significantly bigger "complexity" of DOS(w), providing an argument for calculating DOS in k-space.
3. Evaluate the progress/limitations of the Dulong-Petit, Einstein, and Debye models for explaining the temperature dependence of the lattice heat capacity, Cv(T). (Suggestion: instead of deriving the formalism for each model in full, consider making a qualitative comparison of the "oscillator models" assumed to be responsible for the corresponding Cv (T) dependencies. 4. A two-dimensional (2D) finite hexagonal lattice has a spacing of a = 3 Å. Assuming the sound velocity in this material to be c = 10³ m/s, what is the Debye frequency wp? 5. Provide a qualitative explanation for the T³ Debye law, comparing the fraction of phonon modes occupied at a given temperature T versus all modes within the Debye cut-off wavevector kp. Estimate kp and D for a Na crystal. 6. The thermal conductivity coefficient is given by K = CvA, where Cv is the heat capacity and A is the phonon mean free path. Consider the temperature dependencies for Cv (account for three- dimensional lattice related heat capacity only), A and at low/high temperature limits, and fill the table below. Make a plot illustrating the temperature dependence of K and elaborate on "Normal (N)" versus "Umklapp (U)" scattering processes. Cv A K Low T High T