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    The dispersion relation

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    The dispersion relation for the longitudinal oscillations of a one-dimensional chain of N identical masses m connected by springs with elastic constant C is given by:

    w(k) = 2(C/m)^1/2|sin(ka/2)|

    where a is the equilibrium separation of the masses.

    (a) Show that the mode with wavevector k + 2pi/a has the same pattern of mass displacments as the mode with wavevector k, and hence that the dispersion relation is periodic in reciprocal space.
    [Hint: When the masses are oscillating in the normal mode with wavevector k the displacement from equilibrium of the nth mass is given by u_n(t) = Aexp[i(kna-wt)|.]
    (a) Derive expressions for the phase and group velocities, and sketch them as a function of k.
    (c) Find the expression for g(w), the density of modes per unit angular frequency. Sketch g(w).

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    This solution provide step by step calculations for a problem regarding a dispersion relation.