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The dispersion relation for phonons in a one dimensional monoatomic Bravais lattice
with lattice spacing a and consisting of ions
of masses
M is given by
2c
o(k) =
1- cos(ka), where o is the frequency of oscillation, k is the wavevector
M
and Cis the spring constant. For the long wavelength modes (2 >> a), the ratio of the
phase velocity to the group velocity is

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