Illustrated Physics
physics, drawn
Concept

Undulator — where it appears

A row of magnets of alternating polarity that makes a fast electron beam wiggle and radiate. The light's wavelength is the magnet period divided by about twice the square of the Lorentz factor; synchrotron light sources and free-electron lasers are built on them.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Two factors of γ between a magnet and an X-ray. An electron of 6 GeV (γ = 11742) crossing an undulator, a row of magnets alternating every λᵤ = 3 cm. Top: the laboratory, where the electron wiggles once per period. Middle: the electron's own frame, where the row is contracted by γ to 2.55 μm and rushes past at nearly the speed of light; to the electron it is a wave of that wavelength, and the electron radiates at it, in the infrared. Bottom: that light, emitted forward and seen back in the laboratory, is Doppler-shortened by a further factor of about 2γ, to λᵤ/2γ² = 1.088 Å; the wiggle's own slowing of the electron adds a factor (1 + K²/2), giving 1.632 Å for a deflection parameter K = 1 — hard X-rays from centimetre magnets. The drawing is schematic: the three periods differ by factors of about twelve thousand and a hundred million, and cannot be drawn to one scale.

The magnet period that comes out as an X-ray

A row of permanent magnets alternating every three centimetres, with a beam of six-gigaelectronvolt electrons sent down the middle, shines X-rays with a wavelength of an ångström and a half — two hundred million times shorter than the spacing of the magnets that made them. The factor is γ squared, and the two γ's come from two different frames. In the electron's frame the row of magnets is contracted by γ and rushes past it as a wave; the electron shakes in step and radiates; and that light, sent forward, comes back to the laboratory shortened by another factor of 2γ. Length contraction is half of every X-ray a modern synchrotron makes.

relativity · Length contraction The photon a fast electron throws back. The highest energy of a laser photon scattered straight back by an electron it meets head-on, against the electron's energy, for a carbon-dioxide laser, an infrared and a green one, on logarithmic axes: dashed, 4γ² times the laser photon's energy, the double Doppler shift of a mirror moving at the electron's speed; solid, the exact edge with the electron's recoil, E·x/(1 + x) with x = 4γε/mc²; dotted, the electron's own energy, which no scattered photon reaches. A 1 GeV electron turns 1.17-eV infrared photons into gamma rays of 17.6 MeV, within 1.8 per cent of the mirror's figure. At 46.6 GeV and 2.35 eV the mirror's figure is 78.2 GeV, more than the electron has to give; the photon actually comes back with 29.2 GeV.

The photon a fast electron throws back

Shine an infrared laser at a beam of gigaelectronvolt electrons and gamma rays come back out of the collision, each carrying seventeen million times the energy of the laser photon it started as. The electron acts as a mirror moving at nearly the speed of light, and a moving mirror multiplies the energy of light by two Doppler factors. Push the electron's energy far enough and the mirror arithmetic promises photons more energetic than the electron itself — and the photon's own momentum, which the mirror ignored, steps in to stop it.

relativity · Length contraction Every colour at its own angle, from one charge. The wavelength radiated by a charge passing over a grating, as a multiple of the grating's period d, against the angle from the charge's direction, for speeds of 0.3, 0.776, 0.99 of light's, in the first order: λ = d(1/β − cos θ). The shaded band is visible light for a grating of period 1.67 micrometres, the one Smith and Purcell used in 1953. At 0.776c — 300 keV electrons, their speed — visible light comes out between 0° and 30° from the beam, blue-green straight ahead and red at the wider angles; straight ahead the wavelength is 0.29d, 482 nm, and straight back 2.29d. A slow charge at 0.3c radiates only at wavelengths longer than 2.33d. A fast one at 0.99c sends 1.0 per cent of a period straight ahead, the d/2γ² of an undulator.

The rainbow a passing electron leaves on a grating

An electron moving steadily in a straight line through empty space does not radiate, at any speed. Send it skimming over a ruled metal grating, without touching it, and it lights up — every colour at its own angle, blue-green ahead and red further out, from an electron that is neither turning nor slowing. The grating hands the electron's field the one thing it lacked: a wavenumber that lets part of it escape. It is the same mechanism as the blue glow in a reactor pool, with a ruled surface standing in for the slow light in the water.

astrophysics · Radiating charge

Named alongside it

The objects these essays reach for when they reach for this one.

Doppler effectThe Lorentz factorCherenkov radiationCoherent emissionCompton scatteringDiffraction gratingEvanescent waveField transformationFree electron laserInverse compton scatteringKlein nishinaLength contraction

All concepts