Bremsstrahlung

The physics

Maxwell's equations require that any accelerating charge radiates. The classical Larmor formula gives the radiated power:

P = q² a² / (6 π ε₀ c³)

An electron passing near a nucleus is deflected by the Coulomb field 1/(4πε₀) Ze²/r². That deflection is an acceleration; the acceleration radiates. The total energy radiated in one encounter integrates over the trajectory; summed over many encounters with random impact parameters, you get a continuous EM spectrum.

The peak power is emitted when the electron is closest to the nucleus — in a small fraction of an attosecond — producing photons up to energies comparable to T_e. The radiation is mostly forward-peaked at relativistic energies (sin²θ/(1−βcosθ)⁵ angular pattern).

The continuous spectrum

The differential cross-section per unit photon energy (Bethe-Heitler formula, leading log) is:

dσ/dE_γ ≈ (4/3) α r_e² Z(Z+1) (1/E_γ) (1 − 2/3 (E_γ/T) + (E_γ/T)²) ln(...)

Roughly 1/E_γ weighting times a quadratic shape factor times Z(Z+1) scaling. The spectrum is flat in dσ/dE_γ near low energies (soft photons), then drops to zero at E_γ = T (the Duane-Hunt limit).

A "thick target" bremsstrahlung spectrum (real X-ray tube) is the convolution of the elementary spectrum with the electron's slowing-down spectrum in the anode. Result: linear-falling spectrum from E_max down to zero, plus characteristic line emission (Kα, Kβ) from the anode atoms.

Worked example: a 100 keV X-ray tube

Tube voltage V = 100 kV; electron kinetic energy on impact = 100 keV. Tungsten anode (Z = 74). Beam current I = 100 mA.

Quantity	Value
E_max (Duane-Hunt)	100 keV
λ_min	0.0124 nm
Efficiency η	9.2 × 10⁻⁴ · Z · V_kV = 0.7%
X-ray power output	0.007 × 100 kV × 0.1 A = 70 W
Heat dissipated in anode	~9930 W (99.3% of input)
Photons emitted per second	~ 10¹⁵ (broadband)
Kα characteristic lines	59.3 keV (Kα₁), 57.6 keV (Kα₂)

X-ray tubes are about 1% efficient at 100 kV; the rest is heat. That's why diagnostic tubes have rotating anodes and water/oil cooling.

Bremsstrahlung in different materials

Material	Z (eff)	E_c (MeV)	X_0 (cm)
Hydrogen	1	~ 350	~ 870
Air	7.4	~ 87	~ 36 m at sea level
Water	7.4	~ 92	9.0 cm (10 m at sea level air)
Aluminum	13	40	8.9 cm
Iron	26	22	1.76 cm
Lead	82	9.5	0.56 cm
Tungsten	74	10	0.35 cm

(Radiation length X₀ is the thickness over which a high-energy electron loses energy by a factor of e.) Lead is the textbook X-ray shield because of its short X₀.

JavaScript — bremsstrahlung calculations

// Duane-Hunt limit: maximum photon energy = electron kinetic energy
function duaneHuntLimit_keV(V_kV) {
  return V_kV;  // when accelerated through V volts, KE = eV joules
}
function minWavelength_nm(V_kV) {
  // lambda_min (nm) = h c / (e V) = 1240 nm·V / V_volts
  return 1240 / (V_kV * 1000);
}
console.log('100 kV tube: E_max =', duaneHuntLimit_keV(100), 'keV');
console.log('100 kV tube: lambda_min =', minWavelength_nm(100), 'nm');  // 0.0124 nm

// X-ray tube efficiency (Kramers' empirical rule)
function tubeEfficiency(Z, V_kV) {
  // η ≈ K * Z * V_kV, K = 1.1e-9 per V * Z = 1.1e-6 per kV * Z
  // In percent: ~0.0009 * Z * V_kV → for W Z=74 at 100 kV → ~0.7%
  return 1.1e-6 * Z * V_kV * 1000;  // dimensionless fraction
}
console.log('Tungsten @ 100 kV efficiency:', (tubeEfficiency(74, 100) * 100).toFixed(2) + '%');
// ~0.81% in this model — real-world ~ 1%

// Critical energy E_c above which bremsstrahlung dominates over ionization
function criticalEnergy_MeV(Z) {
  return 800 / (Z + 1.2);  // approximate, valid Z ≳ 1
}
console.log('Lead E_c =', criticalEnergy_MeV(82).toFixed(1), 'MeV');     // ~9.6
console.log('Water E_c =', criticalEnergy_MeV(7.4).toFixed(1), 'MeV');   // ~93

// Radiation length (per gram) approximate
function radiationLength_gcm2(Z, A) {
  // X_0 ≈ 716.4 A / [Z(Z+1) ln(287/√Z)] g/cm²
  return 716.4 * A / (Z * (Z + 1) * Math.log(287 / Math.sqrt(Z)));
}
function radiationLength_cm(Z, A, rho_gcm3) {
  return radiationLength_gcm2(Z, A) / rho_gcm3;
}
console.log('X_0 lead =', radiationLength_cm(82, 207, 11.35).toFixed(2), 'cm');  // ~0.55
console.log('X_0 water =', radiationLength_cm(7.4, 18, 1.0).toFixed(2), 'cm');   // ~36 -- careful, water is a compound

// Average photons / electron for a thick-target spectrum (rough)
function photonsPerElectron(V_kV, Z) {
  // Order-of-magnitude: ~ 1e-2 * Z * V_kV / 30
  return 1e-2 * Z * V_kV / 30;
}
console.log('Photons/electron (W, 100 kV):', photonsPerElectron(100, 74).toFixed(1));
// ~25 photons per incoming electron (rough; depends on band of integration)

Where bremsstrahlung matters

• Medical X-ray. Diagnostic tubes (~30-150 kV) and therapeutic LINACs (~6-25 MV) both rely on bremsstrahlung.
• Industrial X-ray. Non-destructive testing, baggage scanning, food inspection — all bremsstrahlung sources.
• Particle physics calorimetry. Electromagnetic calorimeters (CMS, ATLAS) measure energy via bremsstrahlung-photon-pair-production cascades; radiation length sets their geometry.
• Cosmic-ray air showers. High-energy electrons in atmosphere bremsstrahlung photons that pair-produce more electrons -- an avalanche, detected by Cherenkov telescopes (HESS, MAGIC) and surface arrays (Pierre Auger).
• Astrophysical hot plasmas. Thermal bremsstrahlung (free-free emission) dominates X-ray spectra of galaxy clusters (~ 10⁷ K), accretion-disk coronas, hot stellar winds.
• Synchrotron radiation. Bremsstrahlung's cousin -- emission from a charge accelerated by a magnetic field, not an electric field. Same Larmor mechanism; different geometry.
• Radiation shielding. X_0 of lead (0.35 cm) sets the thickness of shielding for any electron source.

Common mistakes

• Confusing bremsstrahlung with characteristic X-rays. Bremsstrahlung is continuous (any energy up to E_max). Characteristic X-rays are sharp lines at element-specific transition energies. Both appear in a typical X-ray tube spectrum.
• Forgetting 1/m² suppression for protons. Protons radiate 3.4 million times less efficiently than electrons at the same KE. Don't expect a proton beam to make X-rays the same way.
• Treating the Duane-Hunt limit as soft. The cutoff at E_γ = T is sharp — zero photons above. Tube voltage strictly bounds photon energy.
• Mixing radiation length and mean free path. X₀ is the bremsstrahlung scale length for high-energy electrons; the photon mean free path (Compton, pair production) is a different quantity.
• Assuming high Z is always best. High Z gives more bremsstrahlung but also more self-absorption inside the anode. Tube design balances production against escape.
• Ignoring the 99% heat. A clinical X-ray tube emits about 1% of input power as X-rays; the other 99% melts the anode. Rotating anodes and heat sinks are not optional.