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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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