- #1
CAF123
Gold Member
- 2,948
- 88
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.
(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.