understand that computed tomography (CT) scanning produces a 3D image of an internal structure by first combining multiple X-ray images taken in the same section from different angles to obtain a 2D image of the section, then repeating this process a

Cambridge A‑Level Physics 9702 – Production and Use of X‑rays (Syllabus 24.2)

Learning objectives (linked to syllabus codes)

1. Explain how X‑rays are produced in an X‑ray tube (24.2.1).
2. Describe the main interactions of X‑rays with matter and how they affect image contrast (24.2.2 & 24.2.3).
3. Understand the principle of computed tomography (CT) – how many X‑ray projections are combined to give a 2‑D slice and how successive slices give a 3‑D volume (24.2.4).
4. Identify the main advantages, limitations and safety considerations of conventional X‑ray imaging and CT.

1. Production of X‑rays (24.2.1)

1.1 Basic components of an X‑ray tube

• Cathode (focus cup) – heated filament that emits electrons by thermionic emission; the cup focuses the electron beam onto a small spot on the anode.
• Anode – high‑Z target (normally tungsten). Modern tubes use a rotating anode to spread the heat over a larger area and allow higher tube currents.
• High‑voltage supply – typically 30–150 kV applied between cathode and anode; the voltage determines the maximum photon energy (beam quality).
• Filtration – thin sheets of aluminium (or other low‑Z material) placed in the beam to remove low‑energy photons that would increase patient dose without improving image quality.
• Anti‑scatter grid (optional) – placed between patient and detector to absorb scattered photons, improving image contrast at the cost of increased dose.

1.2 Electron acceleration and kinetic energy

The electrons gain kinetic energy

\(E_{\text{kin}} = eV\)

where e is the elementary charge and V the tube voltage (kV). This energy is released as X‑rays when the electrons are abruptly decelerated on striking the anode.

1.3 Bremsstrahlung (braking radiation)

• Continuous spectrum of X‑ray photons.
• Maximum photon energy \(E_{\max}=eV\) → minimum wavelength

\(\displaystyle \lambda_{\min}= \frac{hc}{eV}\)

• Dominates the spectrum for most clinical tube voltages.

1.4 Characteristic radiation (24.2.1 – required)

• When a high‑energy electron ejects an inner‑shell electron from the anode atom, an outer‑shell electron fills the vacancy.
• The transition releases a photon of a discrete energy characteristic of the target material (e.g., K‑α, K‑β lines of tungsten).
• These lines appear as sharp peaks superimposed on the bremsstrahlung background.

1.5 Extended production details (enhancement of 24.2.1)

• Focusing cup – electrostatic lens that narrows the electron beam, increasing the intensity of X‑rays from a small focal spot.
• Rotating anode – spreads the heat generated by electron impact, permitting higher tube currents (mA) and shorter exposure times.
• Tube filtration – removes low‑energy photons (< 30 keV) that would be absorbed in the patient’s skin, reducing dose and hardening the beam.
• Beam quality (kVp) – higher kVp produces higher‑energy photons, increasing penetration but reducing contrast; lower kVp improves contrast but increases patient dose.
• Anti‑scatter grid – improves contrast by absorbing photons that have changed direction in the patient; must be used with higher mAs to compensate for the loss of signal.

Example calculation

For a tube voltage of 120 kV:

\(\displaystyle \lambda_{\min}= \frac{6.626\times10^{-34}\,\text{J·s}\; \times\; 3.00\times10^{8}\,\text{m·s}^{-1}}{1.60\times10^{-19}\,\text{C}\; \times\; 1.20\times10^{5}\,\text{V}} \approx 1.03\times10^{-11}\,\text{m}\) (0.010 nm).

2. Interaction of X‑rays with Matter (24.2.2)

2.1 Principal interaction mechanisms

• Photoelectric absorption – photon is completely absorbed, ejecting an inner‑shell electron. Dominant at low photon energies and in high‑Z materials (e.g., bone, contrast agents).
• Compton scattering – photon is scattered with reduced energy, transferring part of its energy to an outer‑shell electron. Dominant at intermediate energies (≈ 30–150 keV) for soft tissue.
• Pair production – photon converts into an electron‑positron pair in the nuclear field; becomes significant only for \(E_{\gamma}>1.022\) MeV (outside the range of diagnostic imaging).

2.2 Dependence on atomic number (Z) and photon energy

• Photoelectric cross‑section \(\propto Z^{3}/E^{3}\) → strong increase with Z and decrease with photon energy.
• Compton cross‑section \(\propto Z\) and varies slowly with energy.
• These dependencies explain why bone (high Z) appears white, while soft tissue (low Z) appears grey.

2.3 Image contrast and the role of contrast agents

• Contrast arises from differences in the linear attenuation coefficient \(\mu\) of adjacent structures.
• Intravenous contrast agents (iodine‑based, high‑Z) increase \(\mu\) of blood vessels and organs, enhancing their visibility, especially at lower kVp where the photoelectric effect is stronger.

2.4 Dose–quality trade‑off

• Increasing tube current (mA) or exposure time (s) raises the number of photons → better signal‑to‑noise (image quality) but higher patient dose.
• Choosing an appropriate kVp balances penetration (dose) against contrast (image quality).
• Use of filtration and grids improves contrast without proportionally increasing dose.

3. Attenuation law and image formation (24.2.3)

The intensity of an X‑ray beam after passing through a material of thickness \(x\) is described by the exponential attenuation law:

\(I = I_{0}\,e^{-\mu x}\)

where \(\mu\) is the linear attenuation coefficient (units m\(^{-1}\)). Often the mass attenuation coefficient \(\mu/\rho\) (units m\(^2\) kg\(^{-1}\)) is used to compare different materials.

Example

For soft tissue (\(\mu = 0.20\ \text{cm}^{-1}\)) and a thickness of 10 cm:

\(I = I{0}e^{-0.20\times10}=I{0}e^{-2}=0.135\,I_{0}\) – only 13.5 % of the original beam emerges.

4. Computed Tomography (CT) – principle and practice (24.2.4)

4.1 Acquisition of 2‑D projections

• The X‑ray source and a linear detector array rotate together (typically 0° → 360°) around the patient.
• For each rotation a large number of projections are recorded; each projection records the line integral of \(\mu\) along many X‑ray paths through the same thin slice.
• Modern scanners use a fan‑beam or cone‑beam geometry; the latter acquires multiple slices simultaneously.

4.2 Reconstruction of a 2‑D slice

• Mathematically the set of projections is the Radon transform of the slice.
• Reconstruction algorithms:

  • Filtered back‑projection (FBP) – fast, widely used; each projection is “back‑projected” and then filtered to remove blurring.
  • Iterative reconstruction – models noise and system geometry; reduces dose for a given image quality.

• The output is a matrix of attenuation coefficients \(\mu(x,y)\) displayed as a greyscale image.

4.3 Slice thickness, spatial resolution and voxels

• Slice thickness (z‑dimension) is set by the collimation of the X‑ray beam; typical values: 0.5 mm – 5 mm.
• Spatial resolution in‑plane is limited by detector element size (pixel) and focal‑spot size; expressed as the modulation transfer function (MTF).
• Each voxel (3‑D pixel) has dimensions \(\Delta x \times \Delta y \times \Delta z\); smaller voxels give higher resolution but increase noise and dose.

4.4 Dose metrics for CT

These quantities allow comparison of patient dose between different protocols and scanners.

4.5 Hounsfield units (HU)

Attenuation values are expressed relative to water:

\(\displaystyle \text{HU}=1000\;\frac{\mu{\text{pixel}}-\mu{\text{water}}}{\mu_{\text{water}}}\)

• Water = 0 HU, air = –1000 HU, cortical bone ≈ +1000 HU.
• HU provide a quantitative basis for tissue classification and for measuring contrast‑agent enhancement.

4.6 Common CT artefacts (syllabus‑relevant)

• Partial‑volume effect – averaging of different tissues within a thick slice → blurring of small structures.
• Beam hardening – low‑energy photons are preferentially absorbed, causing cupping artefacts; mitigated by filtration and software correction.
• Motion artefact – patient movement during acquisition leads to streaks; reduced by faster rotation or breath‑hold techniques.
• Metal artefact – high‑Z implants cause severe photon starvation and streaking; corrected with metal‑artifact reduction algorithms.

4.7 Building the 3‑D volume

• The patient table translates a fixed distance (slice thickness) after each rotation.
• Contiguous slices are stacked to form a volumetric data set.
• Software can display:

  • Axial, coronal or sagittal cross‑sections.
  • Multiplanar re‑formations (MPR).
  • Surface or volume renderings for 3‑D visualisation.

5. Advantages, limitations and safety (ALARA)

Safety measures embodying ALARA

• Optimise kVp and mAs – use the lowest settings that still give diagnostic quality.
• Automatic exposure control (AEC) – modulates tube current in real time to match patient attenuation.
• Collimation and limited scan length – restrict the X‑ray beam to the region of interest.
• Lead shielding – protect staff and adjacent patients.
• Use of iterative reconstruction – allows dose reduction while maintaining image quality.

6. Key equations

• Maximum photon energy: \(E_{\max}=eV\)
• Minimum wavelength: \(\displaystyle \lambda_{\min}= \frac{hc}{eV}\)
• Exponential attenuation: \(I = I_{0}\,e^{-\mu x}\)
• Hounsfield unit definition: \(\displaystyle \text{HU}=1000\;\frac{\mu{\text{pixel}}-\mu{\text{water}}}{\mu_{\text{water}}}\)
• CT dose metrics:

  • CTDI_vol (mGy)
  • DLP = CTDI_vol × scan length (mGy·cm)

7. Suggested illustrative diagrams (for classroom use)

8. Practice questions (aligned with syllabus 24.2)

1. Calculate the minimum wavelength of X‑rays produced when the tube voltage is 120 kV. (Use \(h c = 1.986\times10^{-25}\,\text{J·m}\)).
2. Explain how the three interaction mechanisms (photoelectric, Compton, pair production) influence the contrast between bone and soft tissue in a conventional radiograph.
3. A CT scanner uses a slice thickness of 1 mm and a pitch of 1.2. If the scan covers 180 mm in the longitudinal direction, how many slices are acquired? (Assume one rotation per slice.)
4. Discuss how changing the tube voltage (kVp) affects both image contrast and patient dose in conventional X‑ray imaging and in CT.
5. Identify two common CT artefacts and describe a practical method to reduce each.
6. Define CTDI_vol and DLP, and explain how they are used to compare patient dose between different CT protocols.
7. Two safety measures that illustrate the ALARA principle are (i) and (ii). Explain why each is important.