513.001 Molecular and Solid State Physics

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Physical
Constants

Periodic System
of Elements

      

Continuum wave equation

The equations for the displacement of atoms along a linear chain are,

Here M is the mass of the atoms, and C is the force constant for nearest neighbor atoms. The spacing between nearest neighbor atoms is a. Show that for long wavelengths these equations can be approximated by the wave equation,

where v is the velocity of sound. What is v in terms of M, C, and a?

When the equations are quantized, each normal mode solution of the wave equation can have one of the energies given by the equation

Here n is the number of phonons in that normal mode. The average value of n is given by the Bose-Einstein factor. How many phonons are there in the normal mode with λ = L/100 at temperature T?