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phonons and lattice vibrations

What is meant by the terms: (i) normal mode and (ii) phonon. Explain why phonons obey Planck-Bose/Einstein statistics. What is the difference between an acoustic mode, and optic mode?
Quantized lattice vibrations are called phonons. When a phonon propagetes to a crystal lattice the atomic oscillators excited and vibrate as per the propagation of phons in the lattice. there is a relation called dispersion relation , connecting teh frequency and veleocity of propagation of teh phonons. the phonons creates two modes of vibrations, optic and accoustic. In between these two modes of vibrations there is a region wher no sound waves are propagated, and such a lattice acts as a bandpass mechanical filter. This problem is illustrated.

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Dear student

I explained it as easy as possible. without complicating things with Mathematics. The Acoustic, Optic Mode, I just wrote the simplified final equation.

Normal Mode of Vibration
Consider a simple solid containing atoms. Now, atoms in solids cannot translate, but are free to vibrate about their equilibrium positions. Such vibrations are called lattice vibrations, and can be thought of as sound waves propagating through the crystal lattice. Each atom is specified by three independent position coordinates, y,z, and three conjugate momentum coordinates Px Py, Pz .
Px = mVx , Py=mVy and Pz = mVz, vx, vy vz are the velocity along x,y and z axes, and m is the mass of the atom
Let us only consider small amplitude vibrations. In effect, we can find 3N independent modes of oscillation of the solid. ( N atoms in x, y and z directions). Each mode has its own particular oscillation frequency, and its own particular pattern of atomic displacements. In this case, we can expand the potential energy of interaction between the atoms to give an expression which is quadratic in the atomic displacements along different axis, from their ...

Solution Summary

Quantized lattice vibrations are called phonons. When a phonon propagetes to a crystal lattice the atomic oscillators excited and vibrate as per the propagation of phons in the lattice. This creates two modes of vibrations, optic and accoustic. In between these two modes of vibrations there is a region wher no sound waves are propagated, and such a lattice acts as a bandpass mechanical filter. This problem is illustrated.

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