Photons are removed from a beam by photoelectric absorption, Compton scattering and, above 1.022 MeV, pair production. For a narrow beam, intensity falls exponentially, I = I₀·e^(−μx). The half-value layer (HVL = 0.693/μ) halves the intensity; ten-fold reduction takes about 3.3 HVLs. Absorption edges make every element’s attenuation energy-dependent.
Reviewed 2026-09-24
Exponential attenuation
Exponential attenuation
- In a narrow beam, transmission falls exponentially with thickness.
- Each half-value layer (HVL) halves what remains: 50%, 25%, 12.5%…
- About 3.32 HVLs (one tenth-value layer) leave 10%.
- The HVL itself depends on photon energy and material.
Each additional half-value layer halves what is left: 1 HVL leaves 50%, 2 leave 25%, 3 leave 12.5%. The tenth-value layer (TVL = 2.303/μ ≈ 3.32 HVL) leaves 10%. μ depends on photon energy and material; mass attenuation coefficients μ/ρ for every element are tabulated in the NIST XCOM database.
Three interactions
| Interaction | What happens | Where it dominates |
|---|---|---|
| Photoelectric absorption | The photon is absorbed and ejects an inner electron | Low energies and high-Z materials; probability rises steeply with atomic number |
| Compton scattering | The photon loses part of its energy and changes direction | Intermediate energies; the main source of scattered radiation |
| Pair production | The photon becomes an electron–positron pair | Only above 1.022 MeV; not relevant to diagnostic X-rays |
Absorption edges
Photoelectric absorption jumps sharply when photon energy just exceeds an electron-shell binding energy. K-edges of common shielding elements are listed below. Because each element absorbs most strongly just above its edge, combining elements lets a material cover different parts of an X-ray spectrum.
| Element | Atomic number Z | K-edge (keV) |
|---|---|---|
| Tin (Sn) | 50 | 29.20 |
| Antimony (Sb) | 51 | 30.49 |
| Barium (Ba) | 56 | 37.44 |
| Gadolinium (Gd) | 64 | 50.24 |
| Tungsten (W) | 74 | 69.53 |
| Lead (Pb) | 82 | 88.00 |
| Bismuth (Bi) | 83 | 90.53 |
K-shell binding energies from the X-Ray Data Booklet (LBNL).
Narrow beam versus broad beam
The exponential law describes a narrow beam. In a broad beam, scattered photons also reach the detector (“build-up”), so measured transmission is higher. That is why lead-equivalence values must state their geometry.
Narrow beam and broad beam
- Narrow beam: collimators restrict the beam before and after the sample.
- Most radiation scattered by the sample never reaches the small detector.
- Broad beam: a wide field and a large detector close to the sample.
- Part of the scatter is counted, so broad-beam results differ from narrow-beam results. Do not mix them when comparing materials.
References
- NIST XCOM: Photon Cross Sections Database
- X-Ray Data Booklet, Lawrence Berkeley National Laboratory (electron binding energies)
- Bushberg et al., The Essential Physics of Medical Imaging, 3rd ed., 2012
- IAEA — Radiation, People and the Environment (2004)
This encyclopedia is for general and engineering reference and is not medical advice. Consult radiation protection or medical professionals for individual exposure assessment.