Debye model and dispersion relation

In summary, the conversation discusses the 3D Debye model and its application to heat capacity at constant volume for temperatures much lower than the Debye temperature. It also explores the relationship between the dispersion relation and heat capacity for bosons. There is a question about how the dispersion relation affects the density of modes and the integration of energy in the calculation of heat capacity. The conversation ends with a question about the general definitions of energy and momentum in terms of the dispersion relation.
  • #1
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Homework Statement



I have seen case studies of the 3D Debye model where the vibrational modes of a solid is taken to be harmonic with dispersion relation [itex]\omega = c_sk[/itex]. It is said that for temperatures much less than the Debye temperature, the heat capacity at constant volume [tex]C_V\sim T^3[/tex].

Now I want to show that for bosons with dispersion relation [itex]\omega\sim A\sqrt k[/itex] has heat capacity [tex]C_V\sim T^4[/tex] for [itex]T\ll T_{Debye}[/itex].

In the case studies I have read, I can't find where the dispersion relation comes into play. I have no idea how to see this. Please help!

Homework Equations



Debye Temperature is given by [tex]T_{Debye}k_B=\hbar \omega_{max}[/tex]



The Attempt at a Solution



Generally, I know I need to get the density of modes -- I have a suspicion that here is where the dispersion relation kicks in, but I don't know how.

After that, I should find the ultraviolet cutoff frequency [itex]\omega_{max}[/itex].

Then I should find the energy $$E=\int_0^\omega{max}d\omega {E(\omega)g(\omega)\over \exp(\beta(E-\mu))-1}$$ But what form does $E$ in the integrand take? I know that for photons with dispersion relation [tex]\omega = c_s k[/tex] we have [tex]E=\hbar \omega[/tex].

After that, it's just a matter of taking limits and [tex]C_V=\left({\partial E\over \partial T}\right)_V[/tex] (should) give the required result...
 
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  • #2
Please, somebody?

OK, so I know that the density of state depends on the dispersion relation. What are the general definitions of [tex]E, p[/tex] in terms of [tex]\omega, k[/tex]? E.g. for the first case [tex]E=\hbar \omega[/tex] and [tex]p=\hbar k[/tex]. So the question is: what are the respective values for [tex]E,p[/tex] in general?
 

1. What is the Debye model and how does it explain thermal properties of solids?

The Debye model is a theoretical model used to describe the thermal properties of solids, specifically their heat capacity and thermal conductivity. It is based on the assumption that the atoms in a solid vibrate at different frequencies, and these vibrations can be treated as a collection of harmonic oscillators. This model explains the observed temperature dependence of heat capacity and thermal conductivity in solids.

2. How does the Debye model account for the dispersion relation in solids?

The dispersion relation in solids refers to the relationship between the frequency and wavelength of a wave traveling through the solid. The Debye model takes into account the different vibrational frequencies of atoms in a solid and how they affect the dispersion of waves. This allows for a more accurate description of the behavior of waves in solids compared to other models.

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

The Debye frequency is the highest vibrational frequency in a solid, and it is directly related to the Debye temperature. The Debye temperature is a characteristic temperature of a solid, and it represents the temperature at which the highest energy vibrational mode is excited. The Debye frequency can be calculated using the Debye temperature and other material-specific constants.

4. How does the Debye model explain the behavior of phonons in solids?

Phonons are quantized units of lattice vibrations in solids, and they play a crucial role in the thermal and mechanical properties of solids. The Debye model describes phonons as a continuum of vibrational modes with different frequencies, and it can be used to calculate the density of phonon states and their contribution to the heat capacity and thermal conductivity of a solid.

5. What are the limitations of the Debye model?

While the Debye model is a useful tool for describing the thermal properties of solids, it has some limitations. It assumes that all atoms in a solid vibrate at the same frequency, which is not always the case in real materials. Additionally, it does not take into account anharmonic effects, which become more significant at higher temperatures. Therefore, the Debye model is most accurate for describing the behavior of solids at low temperatures.

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