3a.3. Consider a longitudinal wave us propagating in a linear monoatomic chain of mass M and with lattice spacing a and the nearest neighbor interaction C. us = u cos(wt - ska) a) Show that the total energy (summed for all particles) is given by, 2 Etotal = K+U = [ {/M (dus) - Σ } M (44)² + [ ² C(U₂ - 15+ 2)² Σ{ccus-us+12 S b) Show that the dispersion relation is given by dispersion relation c) time-averaged total energy per atom is 1 Eavg ==Mw²u²+ W= 4C M S Ka |sin (42)| 1 C(1 - cos Ka)u² == Mw² u²
3a.3. Consider a longitudinal wave us propagating in a linear monoatomic chain of mass M and with lattice spacing a and the nearest neighbor interaction C. us = u cos(wt - ska) a) Show that the total energy (summed for all particles) is given by, 2 Etotal = K+U = [ {/M (dus) - Σ } M (44)² + [ ² C(U₂ - 15+ 2)² Σ{ccus-us+12 S b) Show that the dispersion relation is given by dispersion relation c) time-averaged total energy per atom is 1 Eavg ==Mw²u²+ W= 4C M S Ka |sin (42)| 1 C(1 - cos Ka)u² == Mw² u²
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Transcribed Image Text:3a.3. Consider a longitudinal wave us propagating in a linear monoatomic chain of mass M and with
lattice spacing a and the nearest neighbor interaction C.
us = u cos(wt - ska)
a) Show that the total energy (summed for all particles) is given by,
2
Etotal = K+U = [ {/M (dus)
= [ { M (²) ² + Σ ² C(U₂₁ - 1₂+2)³²
Σ{ccus-us+12
S
b) Show that the dispersion relation is given by
dispersion
relation
Ka
= |sin (42)|
W=
S
4C
M
c) time-averaged total energy per atom is
1
1
Eaug = Ma²u² + C(1-cosKa)u² = Mw²u²
Cavgi =
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