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 The quantum mechanics of harmonic crystals,
 Including thermal properties

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 `Elastic scattering of xray photon

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 For most practical purposes phonon acts as if it had momentum
 hK

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 There is no thermal expansion if one considers only the harmonic
crystal.
 It is possible to understand thermal expansion, however, by taking into
account the effect of anharmonic terms in the potential energy on the
mean separation of a pair of atoms at a temperature T

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 Consider as the potential energy of the atoms at a displacement x from their equilibrium separation

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 We use statistical physics to calculate the average displacement
(thermal expansion). The
probability of finding a given energy U(x) is just the Boltzmann
distribution function, so

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 With this approximation we can evaluate both the numerator and
denominator:

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