# Question about the Debye model of solids

• I
• WWCY
In summary, the conversation discusses the Debye model and an expression given by Debye that is similar to Einstein's expression for the averaged energy of a quantum harmonic oscillator. The meaning of Debye's expression is not explicitly stated, but it is said to be analogous to Einstein's expression. The conversation also asks for an explanation of Debye's expression, its connection to the partition function, and the relationship between Debye's expression and Einstein's expression. A resource is also referenced for further understanding.

#### WWCY

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
$$\langle E \rangle = 3\sum_{\vec{k}} \hbar \omega (\vec{k}) \ [ n_B (\beta \hbar \omega (\vec{k})) + \frac{1}{2}]$$
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.
$$\langle E \rangle = \sum_k \hbar \omega \ [ n_B (\beta \hbar \omega ) + \frac{1}{2}]$$
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?

That was very helpful, many thanks!

## 1. What is the Debye model of solids?

The Debye model is a theoretical model that describes the behavior of a solid at low temperatures. It was proposed by Peter Debye in 1912 and is based on the assumption that atoms in a solid vibrate at specific frequencies, known as phonons.

## 2. How does the Debye model differ from other models of solids?

The Debye model is different from other models because it considers the solid as a whole rather than individual atoms. It also takes into account the effects of temperature on the vibrations of atoms, which is known as thermal expansion.

## 3. What factors does the Debye model take into account?

The Debye model takes into account the crystal structure, density, and atomic weights of the solid, as well as the temperature and heat capacity. It also considers the range of frequencies at which atoms vibrate.

## 4. Can the Debye model accurately predict the behavior of all solids?

No, the Debye model is not applicable to all solids. It works best for solids with low atomic weight and a simple crystal structure. For more complex solids, other models may be necessary.

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

The Debye model does not take into account the effects of quantum mechanics, such as zero-point energy, on the vibrations of atoms. It also does not consider the anharmonicity of atomic vibrations, which becomes significant at higher temperatures. Additionally, it only applies to solids at low temperatures.