Radiography gives a fast, inexpensive view of internal structures, helping clinicians find fractures, examine the lungs, plan dental work, and check implants or tubes. Bone stands out in radiographs because it is denser than soft tissue and contains elements such as calcium that interact more strongly with X-rays. Radiography uses ionizing radiation, so each examination must balance the information needed against dose. Besides anatomy, a radiograph often shows a radiopaque marker, sometimes text burned into the image by the software, or straight field borders where the collimator limited the beam.
Three radiographs, each with a side marker. Left: PA chest radiograph; the lead "L" marker at the patient's left shoulder was placed by the radiographer, and the straight edges are the collimated field (Stillwaterising, Wikimedia Commons, CC0). Middle: hand radiograph; the lead marker reads "Sin" (sinister, left) and appears mirrored because of how it was laid on the detector (Mikael Häggström, Wikimedia Commons, CC0). Right: dental panoramic radiograph; the "R" and the vendor name at the left edge were burned in by the software (Temehetmebmk, Wikimedia Commons, CC BY-SA 4.0).
X-rays are electromagnetic waves like visible light, with energies thousands of times higher. They are made in an X-ray tube where electrons accelerated by a high voltage strike a metal target, and the resulting photons leave as a diverging cone. The patient is placed between the tube and a flat detector. Photons enter the body, some pass straight through, some are absorbed, and some are scattered. The detector records how many photons arrive at each position, a two-dimensional projection of the three-dimensional body, or a radiograph.
Visible light is stopped entirely by the hand and casts a silhouette. X-rays are only partly stopped, so the image records how much each path through the hand attenuated them.
Inside tissue, three things can happen to a photon. It can pass through without interacting, it can be absorbed, or it can be scattered. In Compton scattering the photon knocks an electron loose, keeps part of its energy, and continues in a new direction. Both absorption and scattering have the same first effect, fewer photons arrive at the expected position on the detector, but a scattered photon can still hit the detector somewhere else. Absorption depends strongly on which atoms the tissue contains and dominates at low energies. Compton scattering dominates at higher energies.
Three identical photons enter the tissue. One passes straight through and is counted at the appropriate pixel. One is absorbed and nothing arrives. One is scattered, loses some energy, and is detected in the wrong pixel.
The x-ray tube does not emit one photon energy but a whole spectrum. The tube voltage sets the maximum: at 100 kV no photon carries more than 100 keV, and the mean is around 50 to 60 keV. Typical settings are 50 to 60 kV for a hand, 100 to 120 kV for a chest, and 25 to 35 kV in mammography. Low-energy photons are much more likely to be absorbed than high-energy ones, so the beam that leaves the patient has a higher mean energy than the beam that entered. This is called beam hardening.
Spectrum of a 100 kV chest exposure. No photon has more than 100 keV, most have far less, and the two spikes are characteristic lines of the tungsten target. After 10 cm of tissue the low-energy photons are gone and the mean energy has risen: beam hardening. Schematic, not to scale.
How strongly a material stops X-rays is summarized by the linear attenuation coefficient $\mu$ (unit $\mathrm{cm}^{-1}$). Consider photons of one energy and count only those that continue along the ray without scattering. Send a beam of $I_0$ photons per second into a uniform slab of thickness $d$, and write $I$ for the number still traveling at a point along the path. Every photon crossing a thin slice $dx$ has the same small chance $\mu\,dx$ of being stopped, so the number lost in a slice is proportional to the number still present: a full beam loses more per slice than a depleted one. Across one slice,
$$ dI=-\mu\,I\,dx\tag{2.2.1} $$
Along one ray the exponent is the sum of μ over the elements it crosses.
Each slice removes the same fraction of whatever is left: if 1 cm lets half the photons through, 2 cm let through a quarter and 3 cm an eighth. A quantity multiplied by the same factor per unit length is an exponential, which gives the Beer-Lambert law:
$$ I=I_0\,e^{-\mu d}\tag{2.2.2} $$
In practice, X-rays cross several materials in turn: skin, muscle, bone, muscle, skin. Each layer multiplies the beam by its own factor $e^{-\mu_i d_i}$, and a product of exponentials is the exponential of a sum, so the exponent becomes $\mu_1 d_1 + \mu_2 d_2 + \dots$ . If the tissue varies continuously, let the layers become infinitely thin: $s$ is the distance along the ray, $\mu(s)$ the coefficient at that point, and the sum turns into an integral:
$$ I=I_0\exp\!\left(-\int_{\text{ray}}\mu(s)\,ds\right)\tag{2.2.3} $$
The integral is the total attenuation along the ray, a dimensionless number. It says nothing about where along the ray the attenuation happened. Reordering the layers has no effect.