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Unformatted text preview: Homework # 5 (due Thursday, 17 February)
1. Using the dispersion relation for the monoatomic linear lattice of N atoms with nearest neighbor interactions, show that the density of vibrational modes is given by 2N 1 , where m is the maximum frequency. D ( ) = 2 m - 2 2. Consider a dielectric crystal made up of layers of atoms with rigid coupling between layers so that the motion of atoms is restricted to the plane of the layer (i.e. 2D solid). Using the Debye approximation obtain the expression for the thermal energy and show that the phonon heat capacity in the low temperature limit is proportional to T2. 3. In the Debye approximation, show that the mean square displacement of an atom at 2 3 D 2 absolute zero is R = 2 3 , where v is velocity of sound. Estimate this value for Cu 4 v ( D = D / k B = 343K , = 8920 kg/m3, v = 3570 m/s). Hint: Use the result of Problem 1 4.1 for the time-average total energy of a longitudinal wave in 1D lattice E = MA2 2 . 2 By equating this energy to the quantum-mechanical energy of a harmonic oscillator of frequency in the ground state, obtain the expression for the vibration amplitude A2 . Then, average this expression over phonon frequencies within the Debye approximation for the density of states. ...
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- Spring '08