The physics of diagnostic X-rays
X-rays are electromagnetic waves, like light, but with a much shorter wavelength (about 10⁻¹¹ to 10⁻⁸ m). Short wavelength means each photon carries a lot of energy.
The X-ray tube
- A filament (cathode) is heated. It gives out electrons. This is called thermionic emission.
- A high voltage (about 30–150 kV) speeds the electrons up towards a tungsten anode.
- When an electron stops suddenly in the target, it gives out an X-ray photon. This is called braking radiation (bremsstrahlung). It gives a smooth spread of energies.
- Some electrons knock out inner electrons of tungsten atoms. Outer electrons drop in and give characteristic X-rays: sharp lines at fixed energies.
Only about 1% of the electron energy becomes X-rays. The rest becomes heat. So the anode spins and is cooled with oil.
The biggest photon energy equals the energy of one electron: Emax = eV. So at 100 kV the top photon energy is 100 keV.
Filters (thin aluminium sheets) remove low-energy X-rays. These would only be absorbed by the skin and add dose without helping the picture.
Absorption of X-rays
As X-rays go through matter, the intensity drops smoothly. Each extra centimetre removes the same fraction. So the curve is exponential:
I = I₀ e−μx
- I₀ = intensity going in, I = intensity coming out, x = thickness.
- μ = linear attenuation coefficient (unit m⁻¹ or cm⁻¹). Big μ means the material stops X-rays well.
Half-value thickness
The thickness that cuts the intensity to half is the half-value thickness, x½ = ln 2 / μ.
Mass attenuation coefficient
μm = μ / ρ, where ρ is density. It lets us compare materials fairly.
At diagnostic energies, absorption depends strongly on atomic number Z (roughly Z³). Bone (calcium, Z = 20) stops much more than soft tissue (mostly H, C, O). That gives the contrast.
Image detection and enhancement
- Photographic film: darkens where X-rays hit. Bone blocks X-rays, so bone stays white.
- Intensifying screens: fluorescent layers next to the film. One X-ray photon makes many light photons. So a smaller dose is enough.
- Flat-panel digital detectors: turn X-rays into electric signals. The image appears on a screen at once and can be zoomed or brightened.
- Image intensifier: makes a bright, live image for watching moving organs (fluoroscopy).
- Contrast media: soft organs look alike. A patient swallows barium sulfate (barium meal) or gets an iodine injection. These have big Z, so the gut or blood vessels show up clearly.
The CT scanner
A normal X-ray squashes the whole body into one flat shadow. A CT (computed tomography) scanner fixes this.
- The tube and a ring of detectors rotate round the patient, taking many thin fan-shaped beams.
- The patient bed slides through the ring slowly.
- A computer works out μ for each tiny cube (voxel) and builds slices. These can be stacked into a 3D model.
Good: shows soft tissue differences, 3D view. Bad: much bigger dose than a single X-ray, costly machine, patient must stay still.
Safety: X-rays are ionising. Staff stand behind lead screens. Doses are kept as low as reasonably possible.
Key formulas and definitions
- I = I₀ e^(−μx)
- x½ = ln 2 / μ
- μm = μ / ρ
- Emax = eV (photon energy limit from tube voltage)
- λmin = hc / (eV)
Worked examples
1. A tube works at 80 kV. Find the highest photon energy in joules.
E = eV = 1.6×10⁻¹⁹ × 80 000 = 1.28×10⁻¹⁴ J (80 keV).
2. μ for a material is 0.5 cm⁻¹. What fraction passes through 2 cm?
I/I₀ = e^(−0.5×2) = e^(−1) = 0.37, so about 37%.
3. Find the half-value thickness when μ = 0.35 cm⁻¹.
x½ = 0.693 / 0.35 = 1.98 cm ≈ 2.0 cm.
4. Aluminium has x½ = 1.2 cm for a beam. How much passes through 3.6 cm?
3.6 / 1.2 = 3 half-thicknesses. (½)³ = 1/8 = 12.5%.
5. Find the shortest wavelength from a 100 kV tube.
λ = hc/(eV) = (6.63×10⁻³⁴ × 3×10⁸)/(1.6×10⁻¹⁹ × 10⁵) = 1.24×10⁻¹¹ m.
6. Lead has μ = 50 cm⁻¹. What thickness cuts a beam to 1%?
0.01 = e^(−50x) ⇒ x = ln 100 / 50 = 4.61/50 = 0.092 cm ≈ 0.9 mm.
Common mistakes
- Thinking intensity falls in a straight line. It falls exponentially: the same fraction per centimetre.
- Mixing cm and m: μ in cm⁻¹ needs x in cm.
- Saying bone shows white because it "reflects" X-rays. It absorbs them, so fewer reach the detector.
- Thinking a CT scan uses less radiation than a plain X-ray. It usually uses much more.