CND-08

3-D free-electron DOS

g(E) = (V / 2π²) (2m/ħ²)^{3/2} √E. Spin-degenerate 3-D DOS.

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Electrons in solidsDensity of states

Governing equation

g(E)=\dfrac{V}{2\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}

$E \in \left[0.05, 20\right]$

where

V: Volume (Å³)
E: Energy (eV)
g(E): DOS (1/eV)

The figure, part by part

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①

Governing law — red

g(E)=\dfrac{V}{2\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}

$E \in \left[0.05, 20\right]$

$E = 0.05$ $g(E) = 0.03$

Along the curve

②

Domain of the variable — blue

E_{\min} = 0.05,\quad E_{\max} = 20

$E \in \left[0.05, 20\right]$

$E = 0.05$ $E = 0.05$

Along the curve

③

Response curve — green

g(E)=\dfrac{V}{2\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}

$E \in \left[0.05, 20\right]$

$E = 0.05$ $g(E) = 0.03$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}g(E)}{\mathrm{d}E}

$E \in \left[0.05, 20\right]$

$E = 0.175$ $g(E)^{\prime} = 0.176$

Along the curve

Lecture brief

Historical brief

Drude electrons, Bloch waves, BCS pairing (1957) and Wiedemann–Franz heat are the first solids-and-metals laws. The lab is conductivity, gap and phonon heat in closed form. This sheet (CND-08 — 3-D free-electron DOS) is the form associated with Density of states. Working symbols: $V$ , $E$ $\rightarrow$ $g(E)$ . At E_F this sets the Pauli susceptibility and the Sommerfeld γ.

Purpose

Purpose: compute

g(E)

from

V

E

in Condensed matter via

g(E)=\dfrac{V}{2\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}

g(E) = (V / 2π²) (2m/ħ²)^{3/2} √E. Spin-degenerate 3-D DOS. Use it when a real condensed matter question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

V = 20.000\,\mathrm{Å^{3}}

E = 5.000\,\mathrm{eV}

, the governing relation

g(E)=\dfrac{V}{2\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}

yields

g(E) = 0.305\,\mathrm{1/eV}

. A √E curve of states. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Volume V20 Å³Energy E5 eV

Outputs

DOS g(E)0.305 1/eV

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Narration of this film

A √E curve of states.

At E_F this sets the Pauli susceptibility and the Sommerfeld γ.

Reading speed

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