What is the relationship between heat capacity and temperature?

Click For Summary
SUMMARY

The discussion focuses on plotting heat capacity as a function of temperature using Maple software, specifically addressing challenges in determining the boundaries of k-values in numerical summation. The user is attempting to visualize heat capacity (C(T)) from small temperatures up to Debye's temperature, but encounters issues with noise in 2D and 3D plots. Key insights include the importance of summing over one irreducible wedge of the Brillouin zone to optimize computation and the expected behavior of C(T) starting as T^3 and approaching the Dulong-Petit value at high temperatures.

PREREQUISITES
  • Understanding of heat capacity and its relationship with temperature.
  • Familiarity with numerical summation techniques in computational physics.
  • Knowledge of the Brillouin zone and lattice vibrations in periodic crystals.
  • Proficiency in using Maple software for mathematical modeling and plotting.
NEXT STEPS
  • Learn how to implement numerical summation in Maple for heat capacity calculations.
  • Explore the concept of the Brillouin zone and its significance in solid-state physics.
  • Investigate methods to reduce numerical noise in 2D and 3D plots in Maple.
  • Study the theoretical foundations of heat capacity, including the Debye model and Dulong-Petit law.
USEFUL FOR

Physicists, materials scientists, and computational researchers interested in thermodynamic properties of materials, particularly those using Maple for numerical simulations and plotting.

MMS
Messages
146
Reaction score
4
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).
M7dVt2S.png


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.
 
Physics news on Phys.org
In a periodic crystal the largest k of lattice vibrations is pi/a. (I am not sure I understand the question.)
 
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
 
It is sufficient to sum over one irreducible wedge of the Brillouin zone, that saves some computing time.
 
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).
hoHEj05.png

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

Similar threads

Undergrad Help understanding "Heat capacity"
  • · Replies 28 ·
Replies
28
Views
3K
Undergrad How to understand experiment to measure heat capacity of solid?
  • · Replies 1 ·
Replies
1
Views
2K
Adiabatic expansion with temperature dependent heat capacity.
  • · Replies 6 ·
Replies
6
Views
4K
Specific heat capacity varies with temperature.
  • · Replies 1 ·
Replies
1
Views
4K
Heat capacity using Debye dispersion relation
  • · Replies 2 ·
Replies
2
Views
3K
Thermal Conductivity of Steel experiment
  • · Replies 13 ·
Replies
13
Views
5K
Work Check: Heat transfer between reservoir and small system
  • · Replies 1 ·
Replies
1
Views
2K
Debye internal energy for heat capacity
  • · Replies 1 ·
Replies
1
Views
3K
Specific heat capacity and concentration
  • · Replies 1 ·
Replies
1
Views
10K
Undergrad Extremely strange consequence of the speed of heat value
  • · Replies 19 ·
Replies
19
Views
2K