Calculating Phonon Gas Density of States and Unit Cell Shape

[Karlisbad]

Let's suppose we have a Phonon gas in 1-D then:

- density of states [tex] g(k)=A/ \frac{ d\omega (k)}{dk} [/tex] (i don't remember the value of constant A sorry.. )

- The Schroedinguer equation (NO interaction) would be:

[tex] H_TOTAL =\Sum_{i}\frac{P^{2} _{i}}{2M}+ \sum_{i}B\omega ^{2}(k) (x_{i})^{2} [/tex]

B is another constant..since the SE is separable we can find the exact solution in terms of Hermite Polynomials..my question is How i could get the density of states for this gas?

Ah..sorry another question if you know the "exact" partition function of a system can you determine the exact (by numercal or other methods) "shape" of the unit cell (i'm referring to get the sides of the unit cell, if this can be a cube or a parallepipede or other ..)

 

[eljose79]

Hello, thank you for your interesting questions. Let me try to answer them one by one.

First, to calculate the density of states for a 1-D Phonon gas, we can use the formula g(k)=A/(dω(k)/dk), where A is a constant that depends on the specific system and ω(k) is the dispersion relation of the Phonon gas. The dispersion relation can be obtained from the Schrodinger equation you mentioned, H_TOTAL = \Sum_{i}\frac{P^{2} _{i}}{2M}+ \sum_{i}B\omega ^{2}(k) (x_{i})^{2}. By solving this equation, we can find the energy levels of the system, which are related to the dispersion relation through ω(k)=E/h, where h is Planck's constant. Then, we can take the derivative of ω(k) with respect to k, and plug it into the formula for g(k). This will give us the density of states for the Phonon gas in 1-D.

As for your second question, if we know the exact partition function of a system, we can use it to calculate thermodynamic quantities such as the free energy, entropy, and specific heat. However, I'm not sure how this would help us determine the shape of the unit cell. The partition function does not contain information about the physical dimensions of the system, such as the size of the unit cell. To determine the shape of the unit cell, we would need to know the specific lattice structure and the atomic positions within the unit cell. This information is not contained in the partition function. We can determine the shape of the unit cell experimentally using techniques such as X-ray diffraction or neutron scattering.

 

[charng]

I would first like to clarify that the equations and concepts presented in the content are related to the field of solid state physics, specifically the study of phonons, which are quantized lattice vibrations in a solid material. Now, to address the questions asked:

1. Calculating Density of States (DOS) for a Phonon Gas in 1-D:

The density of states (DOS) is a measure of the number of available states per unit energy interval at a given energy. In the case of a phonon gas, the DOS is given by the equation g(k) = A/ (dω(k)/dk), where A is a constant and ω(k) is the angular frequency of the phonon with wavevector k. This equation can be derived using statistical mechanics and quantum mechanics principles, and it is specific to a 1-dimensional phonon gas.

2. Solving the Schrodinger Equation for a Phonon Gas:

The Schrodinger equation presented in the content is for a non-interacting phonon gas, meaning that the phonons do not interact with each other. This simplifies the equation and allows for the use of Hermite polynomials to find the exact solution. However, in reality, phonons do interact with each other and this interaction can affect the shape and behavior of the unit cell. Therefore, the exact solution obtained from this equation may not accurately describe the behavior of a real phonon gas.

3. Determining the Shape of the Unit Cell from the Exact Partition Function:

The partition function of a system is a mathematical tool used to calculate the thermodynamic properties of a system. It is a sum over all possible energy states of the system. In theory, if we have the exact partition function of a system, we can determine the shape of the unit cell by analyzing the energy states and their corresponding wavefunctions. However, in practice, it is very difficult to obtain the exact partition function of a system, especially for complex systems like a phonon gas. Therefore, other methods, such as experimental techniques or numerical simulations, are often used to determine the shape of the unit cell.