Singularity in density of states (a) From the dispersion relation derived for a monatomic linear lattice of N atoms with nearest neighbor interactions, show the density of modes is D(w) 2N 1 (w 2 m w 2 ) 1 2 , where w m is the maximum frequency (b) Suppose that an optical phonon branch has the form w(K) w 0 AK 2 , near K 0 in three dimensions Show that D(w) (L 2) 3 (2 A 3 2 ) (w 0 w) 1 2 for w w 0 and D(w) 0 for w w 0 Here the density of modes is discontinuous

Question: Singularity in density of states (a) From the dispersion relation derived for a monatomic linear lattice of N atoms with nearest-neighbor interactions, show the density

Singularity in density of states

(a) From the dispersion relation derived for a monatomic linear lattice of N atoms with nearest-neighbor interactions, show the density of modes is D(w) = 2N/π 1 /(w²m – w²)^1/2, where w_m is the maximum frequency.

(b) Suppose that an optical phonon branch has the form w(K) = w₀ – AK², near K = 0 in three dimensions. Show that D(w) = (L/2π)³ (2π/A^3/2) (w₀ – w)^1/2 for w < w₀ and D(w) = 0 for w > w₀. Here the density of modes is discontinuous.

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