Consider the normal modes of a linear chain in which the force constants between nearest-neighbor atoms are alternately C and 10C in a 1-dimensional chain form. Let the masses (m) be equal, and let the nearest-neighbor separation be a/2. (35 points) Find the dispersion relation: (15 points) Sketch the dispersion relation focusing on K-0 and the first zone boundary: (5 points) Obtain the sound velocity of this system as a function of C. You may use appropriate approximation. (5 points) Apply reasonable values to the parameters in the real world and obtain the sound wave of this system: What is the Debye frequency in this case? (0 points)
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Dispersion relation: To find the dispersion relation, we can start by writing the equation of motion for the atoms in the chain. Let's denote the displacement of the nth atom from its equilibrium position as u_n. Then, the equation of motion for the nth atom can Show more…
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