What is the relationship between heat capacity and temperature?

[MMS]

Hello,
I'm reposting this 'cause I think it's more relevant here than where it was posted before. Not sure how to delete the other one...

I'm trying to plot the heat capacity as a function of the temperature from some small temperature to Debye's using numerical summation over the energy (shown in the photo below).
However, I'm struggling to determine the boundary of k (actually, of all three since there are 3 sums).

I'm using Maple for those wondering.

I'd be more than happy if someone could help me out with that.

Thank you in advance.

 

[PietKuip]

In a periodic crystal the largest k of lattice vibrations is pi/a. (I am not sure I understand the question.)

 

[MMS]

Yes, so I believe. I summed over the FBZ (-pi/a to pi/a) for each of k's products.

I'm yet to be able to work this out on Maple though.

Do you have any experience with it? Or if anyone reading this knows some Maple, how will I be able to insert the expression for C(T) above (I can upload what I've tried if needed) and actually have it work (I'm not getting anything)? :P

 

[PietKuip]

It is sufficient to sum over one irreducible wedge of the Brillouin zone, that saves some computing time.

 

[MMS]

Hi,

I tried plotting it first in 1D (hence sin(k*a/2) where k has only one product) and i receive the expected graph of C(T) vs T.

However, when I plot it in 2D and 3D it becomes this deadly noise-like signal (example shown below for what I got in 3D).

Any idea where I could be mistaken? I literally just plugged the expression in Maple..

 

[PietKuip]

So that is basically a constant large value. With numerical noise that increases linearly with temperature.
And I assume that you know that it should start out as T^3, approaching the Dulong-Petit value asymptotically for height T.

Sorry, I do not know anything about Maple.