BREMSSTRAHLUNG

The German word is compact and exact: Bremse means brake; Strahlung means radiation. Bremsstrahlung is "braking radiation," a term coined in the 1920s at the Göttingen X-ray lab and never improved on since. The physics is one sentence: when a charged particle decelerates, it radiates. That is simply Larmor's 1897 formula — P ∝ a² — applied to the one kind of acceleration we haven't yet looked at. The dipole radiates by oscillating. The synchrotron radiates by bending. Bremsstrahlung radiates by braking.

The canonical setting is an X-ray tube. You have seen one in every dental clinic and every airport scanner. An electron gun — a hot tungsten filament — fires a thin stream of electrons across a vacuum gap, accelerated by a potential difference U of typically 40 to 150 kilovolts. At the far end of the gap they slam into a heavy-metal anode — tungsten, rhenium, or molybdenum — and decelerate from several tenths of the speed of light to rest over a distance comparable to an atomic radius. The braking is catastrophic. Each electron loses its entire kinetic energy eU in the space of about a femtometer, which means its acceleration is peak-ludicrous (of order 10²⁴ m/s²). Larmor's formula says it radiates, and it radiates hard.

What comes out is not a single wavelength. It is a continuous spectrum — a smooth broadband of photons from nearly zero frequency all the way up to a sharp upper cutoff. No line structure, just a shape. That continuum is the bremsstrahlung signature; the sharp spikes that sometimes ride on top of it (the Kα and Kβ lines at 59 and 67 keV for tungsten) are a different mechanism — an inner-shell electron being knocked out and replaced — and are not bremsstrahlung.

Zoom in on a single electron hitting a single tungsten nucleus. The electron is traveling fast; the nucleus is heavy, positively charged, and effectively stationary. As the electron passes through the nuclear Coulomb field, it is deflected — a classical Rutherford scattering hyperbola, the same curve Rutherford fitted to gold-foil alphas in 1911. The deflection is sharpest at the point of closest approach, where the transverse acceleration peaks. At that instant the electron is effectively "braked" perpendicular to its velocity, and Larmor tells us it radiates a photon.

The parameter that controls the event is the impact parameter b — the perpendicular distance between the electron's incoming asymptote and the nucleus. A head-on approach (b → 0) means a violent, high-acceleration kink, and the emitted photon can carry away a large fraction of the electron's kinetic energy. A glancing pass (b large) means a gentle deflection, a small transverse acceleration, and a soft photon. Integrate over the distribution of impact parameters that a random incoming electron actually sees — most passes are glancing, head-on events are rare — and you get a probability distribution over photon energies that is heavily weighted toward the soft end. That is why the bremsstrahlung spectrum rises sharply at low photon energies.

Draw the spectrum of a 50-kV tube on a horizontal axis of photon energy and you see something remarkable: the continuum climbs, peaks, falls, and then stops — absolutely stops — at exactly 50 keV. No photons above. This is the Duane-Hunt limit, discovered experimentally by William Duane and Franklin Hunt at Harvard in 1915. The rule is startlingly simple:

The maximum photon energy a tube can produce equals the kinetic energy one electron was given by the accelerating voltage. A 50-kV tube emits nothing above 50 keV. A 150-kV tube reaches 150 keV and not a single photon more.

Classically this is mysterious. Why should there be any sharp cutoff at all? An electron with kinetic energy E_k could, classically, radiate a mix of photons in any proportion, and nothing forbids a photon of energy greater than E_k — energy conservation just says the total radiated energy cannot exceed E_k. The Duane-Hunt cutoff is energy conservation at the level of one photon at a time. One electron cannot emit a photon more energetic than its own kinetic energy, because photons are quantised and a half-photon does not exist. The clean cutoff was a direct, early confirmation of Einstein's 1905 relation E = h f — it told you Planck's constant was real at the level of a single emission event.

Equivalently, the cutoff sets a minimum wavelength:

At 50 kV, λ_min ≈ 24.8 picometers — firmly in the hard-X-ray range, short enough to probe bond lengths in a crystal lattice. Medical diagnostic tubes run at 80 to 150 kV (chest X-rays need to push photons through twenty centimeters of tissue); industrial inspection tubes reach 450 kV; computed-tomography scanners sit near 120 kV. Dental bitewings use smaller 60-to-70 kV tubes — lower energy, but shorter shots at short range.

Between zero and the cutoff, what does the spectrum look like? In 1923 Hendrik Kramers (a Dutch theoretician, student of Niels Bohr, later a major figure in early quantum mechanics) worked out the shape for a thick target — a target thick enough that each electron slows to rest inside it, radiating many bremsstrahlung photons along the way. Integrate over all the deflection events an electron experiences as it cascades to a stop, and the outgoing photon-number distribution simplifies beautifully:

That is Kramers' formula. It is the shape every X-ray engineer calibrates against. Read it: at E → 0 the 1/E factor diverges (soft photons dominate); at E = E_max the (E_max − E) factor kills the spectrum linearly to zero; in between, a smooth monotonic fall. The soft-photon divergence is real in the emission but not in the usable spectrum — the tube's glass window, the aluminum filter every clinical machine carries, and the target itself all absorb soft X-rays heavily, so the observed spectrum peaks around 30 to 40 percent of E_max and the low-E side is chewed back to a rounded edge. The Duane-Hunt edge stays sharp because nothing physical lives above it.

The optional Kα/Kβ overlay shows what a real tungsten-anode medical tube actually emits above 69 kV: the continuous bremsstrahlung backdrop plus two extremely sharp spikes where inner-shell vacancies are being refilled by electrons from higher shells. The spikes are not bremsstrahlung — they are characteristic X-ray fluorescence, a line spectrum specific to tungsten. Below the K-shell ionisation threshold (69 keV for tungsten) the spikes vanish and only the smooth continuum remains.

The Kramers shape is a thick-target answer. A thin target — a foil thin enough that the electron crosses it without losing much kinetic energy — gives a different curve. The classical calculation was refined by Hans Bethe and Walter Heitler in 1934 into the first fully quantum-mechanical bremsstrahlung cross-section, and the thin-target spectrum shape comes out as

also singular at E → 0, but only logarithmically — a much gentler shape. The physical difference is clean: in a thick target the electron passes through a range of kinetic energies as it slows down, so every photon energy below E_max is sampled; in a thin target every deflection samples the full initial E_max, and the logarithm comes from the ln(b_max/b_min) impact-parameter integral.

Outside medicine, bremsstrahlung shows up in every corner of physics where a fast electron meets a strong field. In particle detectors, the characteristic length over which a fast electron radiates away half its energy is the radiation length X_0 — about 0.56 cm for lead, 47 m for air. An electromagnetic calorimeter is a block of matter thick enough in X_0 to soak up the bremsstrahlung and the pair-production cascade that follows.

And in the sky: any hot plasma of free electrons and ions radiates thermal bremsstrahlung. The intracluster medium of a galaxy cluster sits at 10⁷ to 10⁸ K — fully ionised — and its electrons scatter on the ion-field Coulomb potential exactly the way tube electrons scatter on tungsten nuclei. The emitted spectrum is the Kramers shape with the cutoff replaced by the Maxwell-Boltzmann temperature — I(E) ∝ e^{-E / k_B T} above a few keV. Fit that exponential to the X-ray continuum of Perseus or Coma and you read off the gas temperature directly. No absorption line needed.

It is all the same physics as the dental X-ray. A charge gets in the way of itself; light comes out.