The low temperature heat capacity was experimentally shown to go as T^3 for low temeperatures (and as T for even lower temperatures for metals). This result came out of Debye's model after he modified Einstein's model, relaxing some of his assumptions. I have a few questions on Debye's model that I wish to better understand. (Ref. Ashcroft and Mermin P.458) He assumes that the dispersion relation for all branches can be approximated by a linear disperson ##\omega = ck##. Now, this would make sense to me at low k values, however this approximation is used to obtain the specific heat capacity for any ##T##. There is also the Debye temperature introduced and a Debye cut-off wave vector introduced so that we obtain the correct number of modes. I didn't really understand why we needed this. I see that we approximate the 1st Brillioun zone by a sphere with a radius such that it contains the same number of allowed modes (with radius equal to magnitude of Debye cut off wave vector). This sphere extends outside the 1st B.Z. Thanks.