# Debye Model Q&A: Interpreting Expression & Link to Einstein's

• A
• guy bar
In summary, the Debye model is an expression for the averaged energy of a quantum harmonic oscillator in 1D, similar to Einstein's expression. However, Debye's model introduces a correction for different frequencies of oscillations. The interpretation and derivation of Debye's model can be found in Zimann's book "Principles of Theory of Solid" and pages 5-8 of "Phonons II - Thermal Properties" by SMU Physics.
guy bar
Hi all, I have trouble understanding some ideas relating to the Debye model.

In my text (Oxford Solid State Basics by Steven Simon, page 11), it was stated that Debye wrote the following expression
⟨E⟩=3∑→kℏω(→k) [nB(βℏω(→k))+12]
What was not stated was the meaning of this expression. The only mention was that it was completely analogous to Einstein's expression for the averaged energy of a quantum harmonic oscillator in 1D.
⟨E⟩=∑kℏω [nB(βℏω)+12]
However, I can't seem to draw the link between the 2 expressions. Could someone explain to me
1) the interpretation of Debye's expression
2) how Debye's expression arises from a partition function (and how the partition function comes about),
3) and also the link between the 2 equations?

The derivation of both model is nicely carried out in Zimann's book "principles of theory of solid" (I think in chapter 2). Although I don't really recognize the formulae you wrote...

This should answer question 1 and 2. About your question 3, if I remember correctly Einstein model was derived assuming all ions vibrate at the same frequency. You can think of the Debye model as a "correction" of the Einstein model which introduces a "wight" for different frequencies of oscillations (phonons).

Lord Jestocost
Pls. see the attached paper.
Best regards

#### Attachments

• Volume 63 issue 1 1981 [doi 10.1002_pssa.2210630159] A. A. Bahgat -- Correlation between Einst...pdf
115.3 KB · Views: 260

## 1. What is the Debye model and how does it relate to Einstein's model?

The Debye model is a theoretical model used to describe the specific heat of a solid at low temperatures. It is an improvement upon Einstein's model, which only works for high temperatures. The Debye model takes into account the vibrations of atoms in a solid and predicts a more accurate specific heat at low temperatures.

## 2. How is the Debye model expressed mathematically?

The Debye model is expressed as C = 9NkB(T/θ)30θ/T x4ex/(ex-1)2dx, where C is the specific heat, N is the number of atoms, kB is the Boltzmann constant, T is the temperature, and θ is the Debye temperature.

## 3. What is the significance of the Debye temperature in the Debye model?

The Debye temperature, θ, is a characteristic temperature of a solid that represents the maximum frequency of atomic vibrations. It is used in the Debye model to calculate the specific heat at low temperatures and is a measure of the rigidity of the solid's lattice structure.

## 4. How does the Debye model account for the different types of vibrations in a solid?

The Debye model takes into account both acoustic and optical modes of vibrations in a solid. Acoustic modes involve the entire lattice vibrating together, while optical modes involve only certain atoms vibrating in a specific pattern. The Debye model incorporates both types of vibrations in its calculations.

## 5. What are the limitations of the Debye model?

The Debye model is only accurate at low temperatures and does not account for the effects of quantum mechanics. It also assumes that all atoms in a solid are identical and that there are no interactions between them. Additionally, it does not take into account the anharmonicity of vibrations, which can affect the specific heat at higher temperatures.

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