# Question about Spectral Density at the Fermi level

1. Aug 3, 2008

### 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,