The physics of diagnostic X-rays

Making X-rays

An X-ray tube is an electron gun aimed at a metal target. Electrons boil off a heated cathode, accelerate through a pd of tens of kilovolts, and decelerate violently in the target metal. Rapid deceleration of charge radiates, and at these energies the radiation is X-rays. Around 99 per cent of the incident electrons' kinetic energy arrives as heat in the target rather than as X-rays, and that single fact explains the design. The anode is tungsten, for its melting point, and it spins so no one spot stays under the beam.

The photons come out with a spread of energies, and there is a hard ceiling on that spread. No photon can carry away more than one electron brought in. An electron accelerated through pd V arrives with energy eV, so

and the shortest wavelength in the beam follows from E = hc/λ.

Two spectra, one beam

Sort the photons leaving the window by energy and the beam turns out to be two spectra laid on top of each other. Both are asked for by name, and each has its own mechanism.

The broad background is the continuous spectrum, or bremsstrahlung, which is German for braking radiation. An electron flying past a tungsten nucleus is pulled off course by its charge and slowed, and decelerating charge radiates. How much energy any one electron loses in any one such encounter depends on how close it passed, and most electrons are braked in many stages on the way to rest, most of which only heat the target, so the photons that are produced emerge with every energy from almost nothing upwards. That is why the background is a smooth hump and not a set of lines.

The hump ends abruptly at the short-wavelength end, and that edge is the minimum wavelength. It belongs to the rare electron that gives up the whole of its kinetic energy in a single encounter, producing one photon that carries all of it. Then hf = eV, so the edge sits at λ_min = hceV$\lambda_{min} = \frac{hc}{eV}$λ_min = hc/eV with V the tube pd. Nothing about the target appears in that expression. Change the metal, the current or the exposure time and the edge does not move; change the accelerating pd and it does, which is the standard two-mark discrimination.

Sitting on the hump are a few sharp characteristic lines, and they come from the target's electrons rather than from the beam's. An arriving electron with enough energy can knock an inner-shell electron clean out of a tungsten atom. The vacancy does not last: an electron from a higher shell drops into it, and the photon emitted carries exactly the difference between the two levels. Those differences belong to tungsten alone, so the lines sit at fixed photon energies whatever the tube pd, which is where the name comes from.

The lines therefore behave quite unlike the hump. Below the pd needed to eject an inner electron they are missing altogether, and above it they only grow brighter while staying where they are. That splits the two dials on the machine cleanly. Raising the tube voltage hardens the beam, pushing the continuous ceiling up and the minimum wavelength down. Raising the tube current sends more electrons per second, so every part of the spectrum grows in proportion while the edge and the lines stay exactly where they were.

Describe the drawing

Two X-ray spectra plotted against wavelength on shared axes. Each is a smooth hump of bremsstrahlung that falls to nothing at its own minimum wavelength and stops there, the 140 kilovolt hump taller and cut off at 8.9 picometres and the 100 kilovolt one lower and cut off at 12.4 picometres. Two narrow spikes, the characteristic lines of the tungsten target, stand on both humps at the same pair of wavelengths, taller on the higher-voltage curve but not shifted.

Attenuation and contrast

Passing through matter, X-rays are absorbed and scattered, and the surviving intensity falls exponentially with thickness:

where μ is the attenuation coefficient of the material, measured per metre. The same mathematics as capacitor discharge and radioactive decay, now applied to X-rays in tissue. One caveat applies: μ is defined for one photon energy, so the clean exponential describes a narrow monochromatic beam, which is how AQA treats it; a real tube's filtered spectrum spans many energies and clinical work quotes effective values. The booklet also gives you the mass attenuation coefficient, which divides out the density so that different materials can be compared fairly:

Its unit is m² kg⁻¹, and a question that gives you μ_m and a density requires you to multiply back to μ before touching the exponential.

Four named mechanisms do the attenuating. It is OCR that expects each one named with a rough energy against it; AQA's specification asks for the differential absorption of tissues while excluding the details of the absorption processes, so on that board the list below is background.

Simple scatter matters at low photon energies, about 1–20 keV. The photoelectric effect dominates below about 100 keV, which covers most of the diagnostic range, and it is the source of bone contrast, because its strength climbs steeply with atomic number. Compton scattering dominates at higher energies, about 0.5–5 MeV, above the diagnostic range, though it still removes some photons from a diagnostic beam. Pair production cannot happen at all below 1.02 MeV, twice the electron rest energy, so it plays no part in diagnosis.

Image quality depends on contrast. Bone absorbs far more strongly than soft tissue, so bones throw crisp shadows. Two soft tissues with similar μ are nearly indistinguishable, and that is the problem a contrast medium solves. A patient drinks a barium compound before a gut X-ray, or is given iodine for blood vessels. Both have high atomic numbers, so both attenuate strongly and paint an outline into the image where the anatomy alone offered none.