CND-11

Plasma frequency

ω_p = √(n e² / ε₀ m). The metal becomes transparent above ω_p.

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Electrons in solidsPlasma frequency

Governing equation

\omega_p=\sqrt{ne^2/\varepsilon_0 m}

$n \in \left[0.01, 20\right]$

where

n: Density (10²⁸/m³)
\omega_p: Angular plasma (PHz)
f_p: Plasma frequency (PHz)

The figure, part by part

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①

Governing law — red

\omega_p=\sqrt{ne^2/\varepsilon_0 m}

$n \in \left[0.01, 20\right]$

$n = 0.01$ $\omega_p = 0.09$

Along the curve

②

Domain of the variable — blue

n_{\min} = 0.01,\quad n_{\max} = 20

$n \in \left[0.01, 20\right]$

$n = 0.01$ $n = 0.01$

Along the curve

③

Response curve — green

\omega_p=\sqrt{ne^2/\varepsilon_0 m}

$n \in \left[0.01, 20\right]$

$n = 0.01$ $\omega_p = 0.09$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\omega_p}{\mathrm{d}n}

$n \in \left[0.01, 20\right]$

$n = 0.136$ $\omega_p^{\prime} = 1.469$

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-11 — Plasma frequency) is the form associated with Plasma frequency. Working symbols: $n$ $\rightarrow$ $\omega_p$ , $f_p$ . UV transparency of the alkali metals. In SI, f_p = ω_p / 2π sits in the PHz for n ~ 10²⁸.

Purpose

Purpose: compute

\omega_p

f_p

from

n

in Condensed matter via

\omega_p=\sqrt{ne^2/\varepsilon_0 m}

ω_p = √(n e² / ε₀ m). The metal becomes transparent above ω_p. 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

n = 8.500\,\mathrm{10^{2}⁸/m^{3}}

, the governing relation

\omega_p=\sqrt{ne^2/\varepsilon_0 m}

yields

\omega_p = 16.45\,\mathrm{PHz}

f_p = 2.62\,\mathrm{PHz}

. A jellium, a collective oscillation. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Density n8.5 10²⁸/m³

Outputs

Angular plasma \omega_p16.45 PHz
Plasma frequency f_p2.62 PHz

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Narration of this film

A jellium, a collective oscillation.

UV transparency of the alkali metals. In SI, f_p = ω_p / 2π sits in the PHz for n ~ 10²⁸.

Reading speed

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