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

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The vibrational dispersion relation for 1D atomic chain with atoms of alternating masses of M_1 and M2 and alternating spring constants K_1 and A2 can be written:

Mi + M2 sin M1M2 M1M2 K2,

where

K1 K2.

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 k o 0 ? Solve the speed of sound.

c) What is the frequency of the branches at the Brillioun zone boundary k = pm rac{pi}{a} ? 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 M_1=M_2=M. Rationalize the simplified result on the characteristics of the phonon modes. How do the modes look like at k=0 and at k = pm rac{pi}{a} for the simplified case?

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