Question about Spectral Density at the Fermi level

[VietBac]

Hi everybody,

I have the following homework assignment:

1. Derive the expression of density of state D(E) in free electron system.
$$D(E) = \frac{m\sqrt{2m}}{{\hbar}^3\pi^2}V\sqrt{E},$$ (1)

where $$m$$ and $$V$$ are electron mass and volume of the system, respectively.

2. Derive the expression of spectral density $$\nu_{TF}$$ at the Fermi level in the Thomas-Fermi approximation.
$$\nu_{TF}= 2\frac{V}{4\pi^{2}} (\frac{2m}{{\hbar}^2})^{3/2} \sqrt{\varepsilon_{F}},$$ (2)

where $$\varepsilon_{F}$$ is the Fermi energy. We can write the spectral density as $$V\bar{\nu}_{TF}$$, where $$\bar{\nu}_{TF} = \nu_{TF}/V$$.

It is convenient to use $$\bar{\nu}_{TF}$$ for making a theory.

About Eq. (1), I could do quicky as the way of DOS calculation shown in Kittel book, so no problem. I am confusing about "Spectral Density at the Fermi level" in Eq. (2), what does it mean? What is the difference between Eq. (1) and Eq. (2)??

Could anybody explain the difference between Thomas-Fermi approximation and free electron gas model?

I appreciate any help,

Thanks in advance.

 

[malawi_glenn]

http://en.wikipedia.org/wiki/Bose_gas

you replace the sum in the partition function with an integral.

 

[charng]

Hello there,

I am happy to help with your homework assignment. Let's start by discussing the concept of spectral density at the Fermi level. The Fermi level is the energy level at which the probability of finding an electron is 50%. In other words, it is the highest occupied energy level in a material at absolute zero temperature. The spectral density at the Fermi level, denoted as \nu_{TF}, is the number of states per unit energy at the Fermi level. It is an important quantity in solid state physics as it provides information about the electronic structure of a material.

Now, let's look at the difference between Eq. (1) and Eq. (2). Eq. (1) represents the density of states (DOS) in a free electron system. It gives the number of quantum states per unit energy per unit volume. On the other hand, Eq. (2) represents the spectral density at the Fermi level in the Thomas-Fermi approximation. This is a simplified model used to describe the electronic structure of metals. In this model, the electron density is assumed to be continuous and the potential is described by the Thomas-Fermi equation. Eq. (2) takes into account the Fermi energy, which is not considered in Eq. (1).

In summary, the main difference between the Thomas-Fermi approximation and the free electron gas model is that the former takes into account the Fermi energy while the latter does not. I hope this helps to clarify your confusion. Let me know if you have any further questions. Good luck with your assignment!