Debye Model and Specific Heat

In summary, the conversation discusses calculating the specific heat of a linear array of N similar atoms and a 2-D square array using the Debye approximation. At low temperatures, the specific heat is expected to be proportional to T for the linear array and T^2 for the 2-D square array. The equations for specific heat and energy are given, but the method for calculating them in 1D and 2D is not clear. The conversation also mentions using different forms of g(\omega) for the 2D case and suggests a source for further information.
  • #1
TFM
1,026
0

Homework Statement



Consider a linear array of N similar atoms, the separation between nearest neighbours being a. Discuss the specific heat of the system on the basis of the Debye approximation and show that at low temperatures, the specific heat would be expected to be proportional to T.

Do the same thing for a 2-D square array and show that the expected low temperature dependence is now [tex]T^2[/tex].

You should note that

[tex] \int^{\theta_D/T}_0 \frac{x}{e^x - 1}dx \rightarrow [/tex] constant as [tex]\frac{\theta_D}{T} \rightarrow \infty[/tex]

and

[tex] \int^{\theta_D/T}_0 \frac{x^2}{e^x - 1}dx \rightarrow [/tex] constant as [tex]\frac{\theta_D}{T} \rightarrow \infty[/tex]

Homework Equations



[tex] U = \Sigma_{\omega} E_{\omega} -> \int^{\omega}_{0} \overline{E}(\omega)g(\omega)d\omega [/tex]

[tex] \overline{E}(\omega) = 1/2 \hbar \omega + \frac{1}{exp\left(\left(\frac{\hbar\omega}{kT}\right) - 1\right)} [/tex]

The Attempt at a Solution



So I know that the specific heat is:

[tex] C_V = \left(\frac{\partial U}{\partial T}\right)_V [/tex]

And U is given by the above relevant equations.

However I am not sure how to do this for the 1D and 2D.

My notes have the 3D

[tex] g(\omega) = V \frac{1}{8\pi^3}4\pi \frac{\omega^2}{C_s^3} [/tex]

And I also have

[tex] g(\omega) = g(k) \frac{1}{\frac{d\omega}{dk}} [/tex]

where

[tex] \frac{d\omega}{dk} = C_s [/tex]
 
Physics news on Phys.org
  • #2
In 2D all that changes is your g([tex]\omega[/tex]).

Then you can do the calculations for the specific heat.

The g you have is assuming that you have a sphere in k space. For 2D assume that you have a circle in k space.

Chapter 2 (around page 44) from Zieman is a great source for this type of stuff. Good luck!
 

1. What is the Debye model and how is it used to explain specific heat?

The Debye model is a theoretical model used to describe the behavior of solids at low temperatures. It is based on the idea that the atoms in a solid vibrate at specific frequencies, and these vibrations contribute to the overall heat energy of the solid. The Debye model helps explain specific heat by taking into account the number of atoms in a solid and the frequencies at which they vibrate, which ultimately determines the amount of heat energy needed to raise the temperature of the solid.

2. How does the Debye model differ from the classical model of specific heat?

The classical model of specific heat assumes that the atoms in a solid vibrate at all possible frequencies, while the Debye model takes into account the quantization of energy levels and only considers certain frequencies of vibration. This results in a more accurate prediction of the heat capacity of solids at low temperatures.

3. What is the Debye temperature and how is it related to specific heat?

The Debye temperature is a characteristic temperature of a solid that represents the maximum frequency of atomic vibrations in the solid. It is directly related to specific heat, as the specific heat of a solid starts to decrease significantly once the temperature drops below the Debye temperature.

4. Can the Debye model be applied to all types of solids?

No, the Debye model is most accurate for solids with simple crystal structures, such as metals and insulators. It is less accurate for more complex structures, such as polymers and glasses.

5. How does the Debye model account for the observed deviation in specific heat at low temperatures?

The Debye model takes into account the effects of quantum mechanics, specifically the quantization of energy levels, which can lead to deviations in specific heat at low temperatures. This is in contrast to the classical model, which does not account for these effects and therefore cannot explain the deviations.

Similar threads

Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
4
Views
881
  • Advanced Physics Homework Help
Replies
9
Views
808
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
24
Views
630
  • Advanced Physics Homework Help
Replies
1
Views
865
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
11
Views
974
Back
Top