# Question: the vibrational dispersion relation for 1d atomic chain with atoms...

###### Question details

The vibrational dispersion relation for 1D atomic chain with atoms of alternating masses of and and alternating spring constants and can be written:

,

where

.

**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?