Singularity in density of states. (a) From the dispersion derived in Chapter 4 for a monatomic linear lattice of N atoms with nearest neighbor interactions, show that the density of modes is D(w) = 2N 1 π (@²-0²) ¹/2" m where is the maximum frequency. (b) Suppose that an optical phonon branch has the form near K=0 in three dimension. Show that @(K)= @ - AK², @(K)= @- AK², L D(@)= (27) (@-@)/² for @<@ and D(@)=0 for @>@. Here the 2π A³/2 density of modes is discontinuous.
Singularity in density of states. (a) From the dispersion derived in Chapter 4 for a monatomic linear lattice of N atoms with nearest neighbor interactions, show that the density of modes is D(w) = 2N 1 π (@²-0²) ¹/2" m where is the maximum frequency. (b) Suppose that an optical phonon branch has the form near K=0 in three dimension. Show that @(K)= @ - AK², @(K)= @- AK², L D(@)= (27) (@-@)/² for @<@ and D(@)=0 for @>@. Here the 2π A³/2 density of modes is discontinuous.
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Transcribed Image Text:1. Singularity in density of states. (a) From the dispersion derived in Chapter 4 for a
monatomic linear lattice of N atoms with nearest neighbor interactions, show that the
density of modes is
D(w) =
m
where is the maximum frequency. (b) Suppose that an optical phonon branch has
the form @(K) = @- AK², near K = 0 in three dimension. Show that
for @<w, and D()=0 for @>@. Here the
2N
1
π (@²-0²) 1/2"
m
L
2π
- (2-) - 250 (0
(@-@)¹/²
2π
D(@) =
density of modes is discontinuous.
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