Density of States Per Unit Energy

Most of the properties of the electron states are governed by energy, and sometimes, direction of propagation. So it is convenient to write

$$\int d^3k \longrightarrow \int d\Omega \int k^2dk = \int \frac{d\Omega}{4\pi} \cdot 4\pi \int k^2dk$$

If the property we are considering does not depend on angle, $\int \frac{\textstyle d\Omega }{\textstyle 4\pi} = 1$ and we are left with $4\pi \int k^2dk$.

Often it is more convenient to integrate over energy. Since $\varepsilon = \varepsilon (\vert k\vert)$ depends only on the magnitude of $\vec{k}$, $\Delta\varepsilon = (\frac{\textstyle d\varepsilon }{\textstyle dk})\Delta k$.

$$\int d^3k \longrightarrow 4\pi\int k^2 dk = 4\pi \int k^2 \frac{dk}{d\varepsilon }d\varepsilon$$

To summarize, for quantities which do not depend on angle or spin, we have

$$\sum\limits_{\vec{k}\sigma} \longrightarrow \frac{2V}{(2\pi )^3} ... ...2\frac{dk}{d\varepsilon } d\varepsilon \equiv \int g(\varepsilon ) d\varepsilon$$

where $g(\varepsilon ) = \frac{\textstyle 2V}{\textstyle (2\pi )^3}\; 4\pi k^2\;\frac... ... dk}{\textstyle d\varepsilon } = \frac{\textstyle dn}{\textstyle d\varepsilon }$ is the number of states in the interval $\Delta \varepsilon$. It is called the density of states. This formula is valid in 3D.

For a free electron gas, $\varepsilon (k) = \frac{\textstyle \hbar^2k^2}{\textstyle 2m}$, $\frac{\textstyle d\varepsilon }{\textstyle dk} = \frac{\textstyle \hbar^2k}{\textstyle m}$

For $\varepsilon < 0$, $g(\varepsilon )=0$. Note $g(\varepsilon )\propto V$.

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Fermi-Dirac Statistics

Electrons are fermions, hence the total wavefunction is antisymmetric. There can be no more than one electron per state $\vert k\sigma\rangle$. At $T = 0$, the ground state is obtained by filling up the lowest possible energy states (assuming a fixed number of electrons). In this way, we fill up a sphere in $k$-space. The radius of the sphere is fixed by the condition

$$N = \sum\limits_{k\sigma} n_{k\sigma} =$$

Call the radius of the sphere $k_F$. Then

$$N = \frac{4}{3} \pi k^3_F\cdot \frac{2V}{(2\pi )^3} = \frac{V k^3_F}{3\pi^2}$$

Thus the radius of the filled sphere is given by

$k_F$ is called the Fermi wavevector. $(k_F = \frac{\textstyle 1.92}{\textstyle r_s})$ The filled sphere is called the Fermi sphere. Fermi momentum = $p_F = \hbar k_F$. The Fermi energy

$$\varepsilon _F = \frac{p^2_F}{2m} = \frac{\hbar^2}{2m} \left( 3\pi^2 \frac{N}{V}\right)^{2/3}$$

Fermi velocity $= v_F = \frac{\textstyle p_F}{\textstyle m}$. Note that all these depend only on the density $N/V$.

Typical values: $k_F \sim 10^8$cm$^{-1}$ ( $\sim 1\AA^{-1}$), $v_F \sim 10^8$ cm/sec $( < 0.01$c), $\varepsilon _F \sim$ a few eV $\sim$ atomic energies which is no coincidence. We can also define a Fermi temperature $T_F = \frac{\textstyle \varepsilon _F}{\textstyle k_B}$. $T_F \sim 10^4 - 10^5 K$. (Note $T \ll T_F$ for all temperatures where substance is a solid or a liquid.)