Why electrons in a plasma ring like a bell

Shift every electron in a thin slab a little to the right and the slab swings back, overshoots and swings again. The frequency of that swing depends only on how many electrons share each cubic metre. It is the plasma frequency, and every time in this course is measured against it.

Sandbox Physics · 2026-10-29

Derivation

A displaced slab pulls itself back

Take a plasma with electron number density $n$ per cubic metre and an equal positive charge density from ions that stay still. Shift a slab of electrons by a small distance $\xi$ to the right. A thin layer of uncovered ions appears on the left and an extra layer of electrons on the right. Each layer carries a surface charge of magnitude $e n \xi$ .

Between two oppositely charged sheets the field is $E = e n \xi / \varepsilon_0$ , and it points back toward the original position. The force on each electron is $-eE$ , so Newton’s law gives an oscillator with one angular frequency for every size of shift.

m\ddot\xi=-\frac{e^2 n}{\varepsilon_0}\,\xi\qquad\Longrightarrow\qquad \omega_p=\sqrt{\frac{n e^2}{\varepsilon_0 m}}

Heat adds a second length and a speed

Real electrons also move randomly. Their spread in velocity is measured by the thermal speed $v_{\mathrm{th}} = \sqrt{k_B T/m}$ . In one plasma period an average electron travels about one Debye length, $\lambda_D = v_{\mathrm{th}}/\omega_p$ . On scales much longer than $\lambda_D$ the electrons move together as a fluid; on scales comparable to it, individual speeds matter.

This course uses three units throughout: time in $1/\omega_p$ , speed in $v_{\mathrm{th}}$ and length in $\lambda_D$ . A wave number written $k$ therefore means $k\lambda_D$ , and $k = 0.5$ is a wave about $12.6$ Debye lengths long.

\lambda_D=\frac{v_{\mathrm{th}}}{\omega_p}=\sqrt{\frac{\varepsilon_0 k_B T}{n e^2}}

Worked example

Worked example: the ionosphere at night

Suppose a layer of the ionosphere has $n = 10^{12}\,\mathrm{m^{-3}}$ and electron temperature $k_B T = 0.1\,\mathrm{eV}$ . With the constants inserted, $f_p = \omega_p/2\pi \approx 8.98\sqrt{n}\,\mathrm{Hz}$ gives $f_p \approx 8.98\,\mathrm{MHz}$ . A radio wave below this frequency cannot travel through the layer and is reflected, which is how an ionosonde measures the density.

The thermal speed is $\sqrt{0.1 \times 1.602\times10^{-19}/9.109\times10^{-31}} \approx 1.33\times10^{5}\,\mathrm{m/s}$ . Dividing by $\omega_p = 2\pi f_p \approx 5.64\times10^{7}\,\mathrm{s^{-1}}$ gives $\lambda_D \approx 2.35\,\mathrm{mm}$ . A metre-scale wave in that layer is hundreds of Debye lengths long and behaves almost like the cold slab above.

Short waves feel the electron speeds

When the wavelength shrinks toward a few Debye lengths, electrons can cross a crest in one period. Pressure then adds to the restoring force, and the frequency rises. For long waves this gives the Bohm–Gross correction, $\omega^2 \approx 1 + 3k^2$ in our units.

The same crossing electrons do something stranger: some of them travel with the wave and exchange energy with it. A fluid picture cannot describe that exchange. The next chapters follow the electrons in position and velocity together, which is what kinetic theory means.

\omega^2\approx 1+3k^2\qquad(\text{time in }1/\omega_p,\ \text{length in }\lambda_D)

An experiment to think through

Predict, change, explain

Before opening the Lab, predict whether a shorter ripple (larger wave number) oscillates faster or slower than a long one.
In the phase-space Lab, compare the presets for a long wave and a short wave and watch the field strip under the phase-space picture.
Use the thermal speed and the wavelength to explain why the short ripple oscillates faster and also fades sooner.

See the plasma frequency reflect radio waves Watch a ripple fold into velocity stripes

The three phase-space Labs solve a one-dimensional electron model in plasma units. Each Lab exports its parameters, model version and field history.

Try explaining it yourself

1. The electron density of a layer increases by a factor of four. What happens to the plasma frequency and the Debye length at fixed temperature?

Read an explained answer

The plasma frequency doubles because it scales as the square root of density. The Debye length halves, since it scales as one over the square root of density.

2. Using the Bohm–Gross formula, estimate the frequency of a $k = 0.3$ wave in plasma units.

Read an explained answer

The formula gives $\omega \approx \sqrt{1 + 3(0.09)} \approx 1.127$ . The full kinetic calculation in chapter 4 gives $1.160$ ; the difference grows with $k$ because the formula keeps only the first thermal correction.

3. Why does a heavier ion background that stays still not change the plasma frequency of the electrons?

Read an explained answer

The ions only supply the uncovered positive charge. The restoring force comes from that charge and the inertia comes from the electron mass, so the electron plasma frequency contains the electron mass alone.

Sources to read alongside

R. Fitzpatrick, Plasma Physics lecture notes, University of Texas at Austin: Landau damping, resonant particles and the plasma dispersion function
Three linked pages derive the linear response of a Maxwellian plasma, the role of electrons near the wave speed, and the plasma dispersion function used for our dielectric roots.
L. Tonks and I. Langmuir, Oscillations in Ionized Gases, Physical Review 33, 195 (1929)
The early paper on electron oscillations in an ionized gas, cited for the history of the plasma frequency. Only the bibliographic record was checked; nothing in chapter 1 depends on its text.
D. Bohm and E. P. Gross, Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior, Physical Review 75, 1851 (1949)
The thermal correction to the plasma frequency is usually named after this paper. Only the bibliographic record was checked; the estimate used in chapter 1 is compared with our own dielectric roots.

A displaced slab pulls itself back

Heat adds a second length and a speed

Worked example: the ionosphere at night

Short waves feel the electron speeds

Predict, change, explain

Try explaining it yourself

1. The electron density of a layer increases by a factor of four. What happens to the plasma frequency and the Debye length at fixed temperature?

2. Using the Bohm–Gross formula, estimate the frequency of a k=0.3k = 0.3k=0.3 wave in plasma units.

3. Why does a heavier ion background that stays still not change the plasma frequency of the electrons?

Sources to read alongside

2. Using the Bohm–Gross formula, estimate the frequency of a $k = 0.3$ wave in plasma units.