Understanding Density of States in 3d Crystals: A Comprehensive Explanation

In summary, the book discusses the problem of crystal vibrations and mentions that there is one allowed value of K per volume for each polarization and branch. However, there are more combinations of Kx, Ky, and Kz within the confined volume. The book also talks about how applying periodic boundary conditions and fixed ends yield identical results, despite slightly different meanings when read without pausing. The term "branch" refers to different states with different energies for the same combination of kx, ky, and kz, as seen in the example of the acoustic and optic branch in a 1D chain with two types of atoms.
  • #1
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My book gives a treatment of this problem for crystal vibrations, but I don't really understand it. It says: There is one allowed value of K per volume (2[itex]\pi[/itex]/L)3. But at the same time it has just shown that Kx,Ky,Kz can take values ±2[itex]\pi[/itex]/L which would certainly lead to more combinations of Kx,Ky,Kz within the volume confined by (2[itex]\pi[/itex]/L)3. What am I misunderstanding.
Also: applying periodic boundary conditions yields the condition that Kx,Ky,Kz=±n2[itex]\pi[/itex]/L, while fixed ends yielded K=n[itex]\pi[/itex]/L, but my book says the two approaches yield identical results. How is that??
 

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  • #2
if you read the sentence without pausing, the meaning is slightly different.
"There is one allowed value of K per volume , for each polarization and for each branch."
so there are multiple K values allowed per volume
 
  • #3
Well in this context, what does the word branch refer to? Different combinations of Kx,Ky,Kz?
 
  • #4
No, same combination of kx,ky,kz may correspond to different states with different energies.
Look at the acoustic and optic branch in a 1D chain with two types of atoms. This is the simplest example of "branches".
Here for each k there are two energies (or frequencies).
 

1. What is the definition of density of states in 3d?

The density of states in 3d refers to the number of energy states per unit volume available to particles in a three-dimensional system. It is a measure of the number of possible energy states that particles can occupy at a given energy level.

2. How is the density of states in 3d calculated?

The density of states in 3d can be calculated using the following formula: D(E) = (1/8π^2) (2m/h^2)^3/2 √E, where D(E) is the density of states, m is the mass of the particles, h is the Planck's constant, and E is the energy level.

3. What is the significance of the density of states in 3d in materials science?

The density of states in 3d is an important quantity in materials science as it provides information about the electronic and optical properties of materials. It helps in understanding the behavior of electrons in a material and their contributions to its physical properties.

4. How does the density of states in 3d vary with temperature?

The density of states in 3d is directly proportional to the temperature. As the temperature increases, the number of available energy states also increases, leading to an increase in the density of states. This relationship is described by the Fermi-Dirac distribution.

5. How does the density of states in 3d differ from that in 2d and 1d systems?

The density of states in 3d is different from 2d and 1d systems as it takes into account the three-dimensional nature of the system. In 2d and 1d systems, the density of states is dependent on the dimensionality of the system, and it decreases as the dimensionality decreases.

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