Phonons driven at high frequency

[jurybrown]

Homework Statement

For one-dimensional monatomic crystals, the dispersion plot for phonons has a maximum frequency ω_max. If a driving force oscillates the crystal beyond this frequency, the phonon will no longer propagate and will instead decay as it gets further from the external source of oscillation. Obtain an expression for k (the wave number) using a decaying function rather than an oscillating one.

Homework Equations

Original "oscillatory" solution of form Ae^i(ksa-ωt) where s indexes the atoms of the crystal and a is the spacing between crystals. The equation of motion is given by d²u_s/dt² = C (u_s+1+u_s-1-2u_s) where u_s is the displacement of the sth atom from equillbrium and C is the constant of interaction between atoms.

The Attempt at a Solution

I can tack on an e^-γsa to the oscillatory solution but the result of substituting that in the equation of motion above does not look like it can be solved for 'k'. Subsequent questions suggest that a closed-form solution is what is being asked for. I also considered a solution that simply decayed (vs decaying a sinusoidal function) just of the form e^-γsa but I am not sure how k would fit into such a solution.

 

[jurybrown]

bump.