When deriving an expression for the shortest possible wavelength in the Debye model it in terms of the interatomic distance a, do you take into account the fact that you have a total of 3N k-points in reciprocal space? For some reason we turn out to have a different formula for the Debye wavelength as the solutions.
You do need to take into account that there should be 3N phonon modes, but also remember that each k-point has 3 possible polarizations. Does this help?
Thank you for your response! Is it not so that the number of allowed k-points in reciprocal space is independent of polarization? I was thinking about a sphere in k-space with radius, the maximum value for k corresponding to the Debye frequency, and then dividing the total volume of this sphere by the volume of a single phonon mode, and setting this equal to the total possible points in k-space being 3N? This however does not give me the correct value. The easier way is just using the formula for the Debye frequency and deriving the minimum wavelength from here, by using the dispersion relation and the relation between wavelength and k-vector, where I do obtain the ‘correct’ result, but without taking actively into acount that k has 3 possible polarizations. I hope you could clarify some of the things where I go wrong in my reasoning.
Thank you in advance
The allowed k-values don’t depend on polarization. On the other hand, the number of independent modes existing at the same k does depend on the polarization.