Oscillation Frequency
Frequency = 
and wave vector Assume time harmonic solution.
Then
A&B constants
Substituting 


2 equations 2 unknowns,

or 

For nontrivial A & B


Dispersion relation between 
Solving for 


where, 

Two values of for any wave number q.
Solving for constants.
When The ratio of amplitude
for lower branch.
Vibrations of neighboring atoms are in phase Similar to those when an acoustic wave propagation in the crystal, optical mode lower frequency.
3-D crystal: 2 distinct modes of transverse vibration also along with optical or acoustic longitudinal mode
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