Question: the vibrational dispersion relation for 1d atomic chain with atoms...
The vibrational dispersion relation for 1D atomic chain with atoms of alternating masses of and and alternating spring constants and can be written:
a) Sketch the behavior of the phonon branches. Identify the branches and assign the period for the branches.
b) What is the behavior of the branches as ? Solve the speed of sound.
c) What is the frequency of the branches at the Brillioun zone boundary ? Discuss how the frequencies depend on relative size of the masses and force constants.
d) Simplify the results from b) and c) for the case . Rationalize the simplified result on the characteristics of the phonon modes. How do the modes look like at and at for the simplified case?