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3.13 Compute the acoustic two-phonon density-of-states, p2(o), given that the one- phonon density-of-states function, p.(w), is the Debye function (see Eq. 3.140). The two-phonon density-of-states is computed by convolution as given by p2(o) = pi(w) * Pi(w).

3 N= 340 3 -K3 V03 612v 2л where the volume V = L?. The vibrational density-of-states in the Debye approxima- tion (considering only acoustic modes) is obtained by differentiating the above result with frequency. The normalized result is given by 302 0 < @max p(w) = { (3.140) w > (max where max is the maximum acoustic frequency the lattice will support. This is, in gen- eral, a crude approximation, good only for small o (e.g., acoustic modes), but useful for wany applications. An example of a real density-of-states function is shown Fig. 3.25 vacantation of the fo